Com piling Modelica
A bout t he Separ ate T r anslation of Models from Modelica to
OCaml and its Im pact on V ariable-S tructure Modeling
v org elegt v on
Dipl. Inf orm.
Chris toph Hög er
v on der F akultät IV - Elektro technik und Inf ormatik
der T echnischen U niv ersität Ber lin
zur Er langung des akademischen Grades
Doktor der N atur wissenschaf ten
- Dr . rer . nat. -
g enehmigte Dissertation
Promo tionsausschuss:
V orsitzender : Prof. Dr . V olker Mar kl
Gutachter : Prof. Dr . P eter P epper
Prof. Da vid Broman, PhD
Prof. Dr . Stef an Jähnichen
T ag der wissenschaftlichen A usspr ache: 07.05.2018
Ber lin, 2019

zus ammenf assun g
Die gleichungsbasierte, objektorientierte Modellierungssprache Modelica
is t standar disiert und hat viele v erschiedene An w endungsg ebiete und Im-
plementierungen. Die meis ten dieser Im plementierungen f olgen einem
g emeinsamen Schema: Modelle w erden ers t k ombiniert, dann zu einem
Gleichungssys tem reduziert, anal ysiert und schließlich in effizienten Si-
mulationscode übersetzt. Anal yse und Übersetzung sind nicht T eil der
Spr achspezifikation. Die Details ihrer U msetzung sind der Im plementie-
rung über lassen. Dieser A blauf hat einen Entscheidenden Einfluss auf die
Beziehung zwischen Modellen und ihrer Simulation. N ach der Erzeugung
des Simulationscodes kann ein Modell seine S truktur , d.h. seine Gleichun-
g en, nicht mehr ändern.
Diese Dissertation untersucht einen alternativ en Ansatz, in dem je-
des Modell g etrennt betr achtet wird und so eine intrinsische, k ontext-
unabhängig e Bedeutung hat. A us dieser P erspektiv e ist die Simulation
kein v on der Modellierungssprache unabhängig er Schritt, sonder n wir d in
dieser selbs t ausg edrückt. So kann jedes Modell separat in Maschinencode
übersetzt w erden. Dar aus f olgt die Möglichkeit, den Simulationsv er lauf aus
den Modellen heraus zu s teuer n und eine mächtige V ariante strukturdyna-
mischer Sys teme zu im plementieren.
Die Übersetzung v on Modellen ber uht auf einer rigorosen Definition
einzelner Aspekte der Semantik v on Modelica in F orm einer Kernsprache,
Mcl. Diese K er nspr ache k onzentriert sich auf die w esentlichsten Aspekte
v on Modelica: Die hier archische Org anisation v on Modellen und ihre R e-
duktion zu Gleichungssys temen. Die F or malisier ung der R eduktion erfolgt
mit Continuations und ermöglicht so die Implementierung v on Struktur -
dynamik. Der so im plementierte Com piler kann mit vielen Sprachmittel
v on Modelica umgehen. Darüber hinaus unterstützt er eine unk om plizierte
F or m zur Modellierung strukturdynamischer Sys teme mit vielen Modi.
W eil Modelle in Mcl Objekte erster Klasse sind, kann ihre symbolische
und numerische V erarbeitung in der Spr ache selbst im plementiert w erden.
Eine K ernel Bibliothek stellt die Mittel zur Er zeugung, T r ansformation und
Simulation v on kontinuier lichen Gleichung en diskreten Signalen bereit.
Dieser Ansatz reduziert die K om plexität der Sprachsemantik er heblich und
er leichtert die Änderung oder Er w eiter ung der Algorit hmen die Modelle
v erarbeiten.

abs tra ct
The equation-based object-oriented modeling languag e Modelica is an
openl y accessible standar d wit h man y im plementations and applications.
Mos t, if not all, t ools t hat ex ecute a Modelica simulation follo w a common
scheme: Models are loaded, composed, anal yzed, and transf ormed into
a sys tem of equations. This system is then furt her simplified, tr anslated
into efficient code, and simulated. The latter steps are ex ecuted b y t he
tool at hand and no t specified b y t he modeling language’ s semantics. This
architecture has prof ound im pact on t he relationship betw een models and
t heir simulation. Af ter t he simulation code has been g ener ated, a model
canno t chang e its s tr ucture, i.e. its equations, during simulation.
This t hesis in v estig ates an alternativ e approach: Here, each model is
considered separ atel y and has an intrinsic context-independent meaning.
In t his view , simulation is not a t ool-specific step, disconnected from the
languag e semantics but expressed in the modeling language itself and each
model can be translated separ atel y into ex ecutable code. Consequently , a
model can t hen control the course of t he simulation, enabling a pow erful
v ariant of v ariable-structure systems.
The tr anslation of models builds upon a partial but rigorous definition
of t he modeling languag e’s semantics in t he f orm of a core language, Mcl.
The core languag e f ocuses on t he mos t im portant aspects of Modelica: t he
org anization of models in hierarchies and their reduction to sys tems of
equations. The formalization of the reduction in a continuation-passing
s ty le permits t he definition of v ariable-structure models. The com piler ,
alt hough onl y a prot otype, is capable of tr anslating sev er al im portant fea-
tures of Modelica. As an extension to the language, it has a s traightf or w ard
support f or v ariable-structure systems with man y modes.
W it h t he help of first-class models it becomes possible t o implement
t he symbolical and numerical treatment of models in Mcl itself. A kernel
libr ar y pro vides t he means for the description of equations, discrete sig-
nals, symbolic manipulation, and simulation. This architecture reduces t he
com plexity of the modeling language t o a manag eable magnitude and also
makes it r at her simple t o extend or change the features of that library .

a ckn o wledgement
It took me man y y ears to dev elop t his dissertation, and during t his time,
I could rel y on t he support and advice of sev eral people. W ithout t hem, I
certainl y w ould not ha v e finished t his w or k.
F irs t, I w ant to t hank m y super visor , Prof. Dr . P eter P epper for his
expert advice and unconditional support and especiall y f or t he friendl y
and familiar atmosphere that he created in our g roup. P eter encourag ed
me to visit man y conferences, w or k shops, and other con v entions all around
t he w or ld. I am deepl y g rateful f or ev eryt hing I lear ned t here and ev er y
contact I made at t hese meetings.
During t he y ears I w or ked with man y colleagues from man y different
circles, far too man y to name them all. I am especiall y t hankful for the
cor dial recep tion into the Modelica Association, t he stimulating academic
discussions in t he community around the EOOL T w or k shop series, and t he
fascinating inter disciplinar y w or k in the AMSUN project. Ev er y one who
discussed argued or collabor ated wit h me on t hese occasions: Thank y ou!
There w ere, of course, some colleagues closer t han others, literall y and
figur ativ el y . Dr . Judit h R ohloff and Dr . Florian Lorenzen w or ked in t he
office next door . Bo th alw a ys had an open ear when I s tumbled upon t he
minor differences betw een our academic surrounding and t he of ten more
pr agmatic w or ld of Modelica, and I vividl y recall our lengt h y and liv ely
discussions. I cannot begin t o enumer ate t he topics where I learned from
t he tw o, and I am deepl y g r ateful for the time w e collaborated.
W riting a dissertation is no t just a prof ession but easil y influences ones
priv ate life as w ell. Here, I could res t on t he un w a v ering support of m y
whole famil y , especiall y m y parents W era and Bernhard. They ne v er com-
plained when I came to a w eekend’ s visit just t o disappear behind a screen
or concentr ate on some sheet of paper full of nonsensical symbols.
There is t he one person that I ow e the most. My wife Mareik e has no t
onl y endured me t hroughout all t he ups and dep t hs, taken the load off me
when I needed it the most, and helped me to clear m y mind when I w as
s tuck again with some proof. She also w as alw a ys ready to lis ten to the
tiresome technical details, the tedious problems of specifying a prog ram-
ming languag e, and m y con v oluted attemp ts at a solution. W or ds cannot
describe ho w lucky I am to ha v e found such a perf ect partner . I lo v e y ou!

F or Lena & Johanna

c ontents
1 Introduction 13
1.1 Differential Alg ebr aic Equations 14
1.2 Modelica 24
1.3 Detailed R esearch Ques tions 33
1.4 Contributions and Outline 35
1.5 R esearch Method 36
1.6 N otational Con v entions 37
2 The Model Com position Languag e 39
2.1 Base Languag e Definition 39
2.2 Modeling R outines 45
2.3 Classes and Object-Oriented Modeling 52
2.4 V ariable-S tructure Modeling 66
2.5 T op-Le v el Entities: P ackages and Modules 71
3 Modeling & Simulation as a Library 75
3.1 Outline 75
3.2 Fundamental Data S tructures 78
3.3 Quasi-Primitiv e Operations 79
3.4 Embedding Equations 82
3.5 Elaboration Interf ace 98
3.6 Exam ple 102
4 F rom Modelica via Mcl to OCaml 105
4.1 F ront End 105
4.2 The Com pilation Pipeline 120
4.3 Desugaring Model R outines 132
4.4 Separ ating T erms and Hierarch y 140
4.5 Ev aluating Hierarchical T erms 146
4.6 Resol ving R eferences 151
4.7 Correctness of the Second Phase 157
4.8 Back End 159

12
5 Exam ple Applications 165
5.1 Sim ple Models 166
5.2 Im plementing a Libr ar y 172
5.3 P arameter Op timization 180
5.4 Reusing Models b y Inheritance 183
6 Conclusion 187
6.1 R ecapitulation 187
6.2 Related W or k 189
6.3 F uture W or k 196
6.4 Discussion 200
Appendices 216
A N otations 217
B Com plete Syntax of Mcl 219
C Extensiv e Definitions 221
D Sketches of Selected Proofs 233
E Obtaining t he Im plementation 243
F K er nel Library 247
G Example Sources 271

1 intr oduction
Modeling and simulation ha v e become indispensable tools f or moder n
engineering. Modeling languages con v ey methods, tools, and techniques
from t he domain of softw are dev elopment into the w or ld of modeling
and engineering and especiall y into y oung industries that cannot rel y on
centuries of experience.
In 2010, an engineer who w as w or king on t he modeling & simulation
of offshore wind farms asked the aut hor of t his t hesis the seemingly inno-
cent ques tion, whet her it could be possible to com pile Modelica libraries
separ atel y on a file-b y -file basis. Separate com pilation is a common feature
of com pilers f or prog ramming languag es like C or Ja v a. Its motiv ation is
quite pr agmatic: Successiv e compilation onl y considers chang ed in put and
minimizes t he time necessary to translate and tes t a change. This, in turn,
increases t he productivity of the prog rammer and enables the distribution
and deplo yment of pre-com piled libraries or packag es. The demand for
such a feature w as natural, as the modeling of wind-turbines and t heir en-
vironmental conditions had lead to the dev elopment of rather larg e model
libr aries at t hat point [S trobel et al., 2011]. Separate com pilation of Model-
ica models w as not possible at that time, t hough and at least one author
addressed t he ques tion as a topic of research [Zimmer, 2009]. The reason
f or t his apparent difficulty lies in a particularity of Modelica ’ s semantic
domain: Systems of differential equations need to be pre-processed bef ore
t he y can be com piled into an efficient, lo w -lev el language (see Section 1.1 f or
a discussion of t his process). These transf or mations are t he sine-qua-non of
an y practicall y relev ant implementation, but they require a holis tic repre-
sentation of t he simulated model. Separate Com pilation, on t he other hand,
is achiev ed b y splitting t he input of the source language along a certain
axis of com position (e.g. files or classes) and transf or ming t he fr agments
into independent units (e.g. object files, libr aries or modules) of the targ et
languag e.
When asked this question, the aut hor of t his t hesis had just begun an
in v estig ation of Mosilab, a research implementation and dialect of Modelica
[Nytsch-Geusen and Er ns t, 2005]. Mosilab supports a limited form of
s tructural v ariability , a f eature t hat allo ws a model to chang e its under l ying
set of equations, called t he mode of t he model, at simulation-time. The
exis ting im plementation of Mosilab had g ro wn old and did not support

14 1 Introduction
moder n f eatures of Modelica. In particular , it did no t support much of the
tr ansformations mentioned abo v e, as it w as limited to so-called index-1
sys tems. Also, Mosilab suffered from of a combinatorial explosion of modes,
caused b y t he need to prepare e v er y possible mode: Com posing tw o models
wit h tw o modes each yielded a model wit h four modes, com posing t hree
such models already yielded eight modes, four yielded sixteen, and so on.
Despite t hese cons traints, Mosilab had successfull y demons trated the
pr acticality of structur all y v ariable models and their simulation. The goal of
t he in v estig ation w as to determine, whet her it w ould be possible dev elop a
successor f or Mosilab t hat adds support for higher -index systems, modern
languag e f eatures, and a more g eneral f or m of structur al v ariability , with
man y or ev en infinitel y man y different modes. Ag ain, t he holistic vie w of
t he model tr ansf ormations w as a sev ere problem: While it w as str aight-
f or w ard t o appl y the transf ormations once per mode, t his w as not possible
f or man y or ev en for an infinite amount of modes.
Apparentl y , bo t h ques tions share a common problem. Could it be t hat
bo t h ques tions t hat came up tog et her b y pure chance share t he same answ er?
W ould t he design of a compiler quasi aut omaticall y lead to the support
of unbounded structur al v ariability? This t hesis answ ers t his question
positiv ely . Its main h ypo t hesis can be s tated as follo w s:
The specification of Modelica in t he f orm of an operational se-
mantics and its separate tr anslation enable t he integ ration of
s tructural v ariability into the language in a w a y t hat w or ks w ell
wit h model com position and man y potential modes.
1 . 1 differential algebraic equ a tions
F or a modeling language, the most fundamental aspect of the semantics is
t hat of the modeling domain. Before w e can ev en consider the language
itself, w e ha v e to in v es tigate the requirements t hat originate from t he typical
treatment of its models. Models written in Modelica describe (v ector -v alued)
differential alg ebr aic equations (short: D AE) in an inter v al 𝕀 ⊆ ℝ .
𝐹 (𝑡, 𝑥, 
𝑥, 𝑦) = 0
𝐹
relates t he independent v ariable
𝑡∈𝕀
,
𝑛
v ariables
𝑥∈ℝ 𝑛
(called t he
s t at es ), t heir firs t-or der deriv ativ es 
𝑥 ∈ ℝ 𝑛 and 𝑚 alg ebr aic v ariables 𝑦 ∈ ℝ 𝑚 .
In t he liter ature [K unkel et al., 2001, Cellier, 1996] the definition some-
times also contains exter nal in puts, e.g., in t he case of a control problem
or during op timization. Such extensions are omitted here. Similar ly , t he

1.1 Differential Alg ebraic Equations 15
epon ymous presence of alg ebr aic v ariables pla ys a vital role during effi-
cient simulation (as t he y are nev er subject to integration) but is of little
im portance t hroughout this w or k. Instead, w e view these unknowns as
deriv ativ es wit h non-obser v ed state and combine them with t he original
s tates to a ne w v ector 𝑧 :
𝐹 (𝑡, 𝑥, 
𝑥, 𝑦) = 𝐺(𝑡, 𝑧, 
𝑧), such t hat 𝑧 = (𝑢, 𝑥), 
𝑧 = (𝑦, 
𝑥), 
𝑢 = 𝑦
The introduction of t he unobser v ed states
𝑢
enlarg es t he system, but
it clear l y does not chang e its solutions when no element of
𝑢
appears in
𝐺
. The solution of such an equation is the implicitl y defined continuous
function 𝜉∶𝕀→ℝ 𝑛+𝑚 , such t hat ∀𝑡 ∈ 𝕀 ∶ 𝐺(𝑡, 𝜉(𝑡), 
𝜉(𝑡)) = 0 .
The goal of the simulation is to find t hat solution, and one can appl y
sev er al w ell-es tablished met hods can in its search. F or t he task of t his t hesis,
it is essential to g ain an understanding of the interf ace of t hese met hods, as
t he y im pose requirements on t he im plementation of t he modeling language.
T o that end, t his section giv es a cursor y o v er view of the numerical and
symbolical treatment of differential alg ebr aic equations.
1.1.1 N umerical Int egr ation
Since mos t D AEs cannot be sol v ed anal ytically ,
𝜉
is usuall y appro ximated
b y a numerical met hod. A t discrete points in time
𝑡 𝑜 … 𝑡 𝑘
, t he solution is
appro ximated as
𝑧 𝑖 ≈ 𝜉(𝑡 𝑖 )
. Therefore, the v ariables
𝑧
in
𝐺
represent t he
v alues of a continuous function.
As an exam ple, consider the simple model of a pendulum (as explained
e.g. b y T iller [2001]) 
𝜙 = −𝑔 sin 𝜙
W it h 𝑧 = (𝜙, 
𝜙) t his model can be expressed as a D AE as follo ws:
𝑧 = (𝑧 1 , 𝑧 2 ) 𝐺(𝑡, 𝑧, 
𝑧) = ( 
𝑧 1 − 𝑧 2 , 
𝑧 2 + 𝑔 sin 𝑧 1 𝑔)
The sim ples t numerical sol v er is t he explicit linear Euler met hod, i.e. t he
linear appro ximation of
𝜉(𝑡 𝑖+1 )
from its deriv ativ e and a suitable small s tep
size ℎ ∈ ℝ : 𝜉(𝑡 𝑖+1 ) ≈ 𝜉(𝑡 𝑖 )+ℎ 
𝜉(𝑡 𝑖 ) = 𝜉(𝑡 𝑖 ) + ℎ𝑓 (𝑡 𝑖 , 𝜉(𝑡 𝑖 ))
The solution depends on t he explicit com putation of

𝜉(𝑡 𝑖 )
in terms of
t he already exis ting solution
𝑓 (𝑡 𝑖 , 𝜉(𝑡 𝑖 )
. The function
𝑓
is called t he s tat e-
space-f orm of
𝐺
, a function t hat explicitl y computes the deriv ativ es of t he
model from t he current v alue of t he s tates. It has to be com puted prior to
t he simulation. In t he case of t he pendulum, 𝑓 (𝑡, 𝑧 1 , 𝑧 2 ) = (𝑧 2 , −𝑔 sin 𝑧 1 ) .

16 1 Introduction
While t his method is both easy to understand and t o im plement (cor -
rectl y), it faces a ma jor deficiency when it comes to pr ecision : The computa-
tional effort to sol v e a sys tem in an inter v al
[𝑡 0 , 𝑡 𝑚 ]
depends on t he number
of s teps
𝑚
which is in v ersely proportional t o the step size, i.e.
ℎ = 𝑡 𝑚 −𝑡 0
𝑚
. A
larg er step size increases perf or mance and yields a reduced accur acy . But
ev en t he smalles t step size can yield simulation errors that build up giv en
a long enough simulation: This might lead no t onl y to quantitativ e but
qualitativ e errors (i.e., a completel y wrong beha vior of t he model).
In t he case of the simple pendulum, for ins tance, it is reasonable t o
expect t hat the system does not build up (or lose) momentum o v er time,
i.e.,
𝜙
should periodicall y retur n to
𝜙 0
. Simulating t he pendulum wit h t he
explicit Euler violates t his in v ariance (F igure 1.1a): N o matter ho w small one
chooses t he s tep size, ev entuall y the pendulum will s tart rotating around
its piv ot. Since t his is a qualitativ e chang e, t he solv er is said to be uns table .
−5 0 5 10 15 20 25 30 35 40 45 50 55
0
5
10
15
−5 0 5 10 15 20 25 30 35 40 45 50 55
−0.4
−0.2
0
0.2
0.4
−5 0 5 10 15 20 25 30 35 40 45 50 55
−0.4
−0.2
0
0.2
0.4
𝜙 [ rad ]
−5 0 5 10 15 20 25 30 35 40 45 50 55
−0.4
−0.2
0
0.2
0.4
time [ s ]
a) Explicit Euler
b) Im plicit Euler
c) R ung e-K utta 4
d) Qu alid aes (Cartesian model)
F igure 1.1: Simulating a pendulum wit h different numerical met hods.

1.1 Differential Alg ebraic Equations 17
In or der to retain s tability f or t he giv en model, tw o gener al v ariations of
t he sol v er are possible. F irs t, w e can switch to the implicit Euler method.
𝜉(𝑡 𝑖+1 ) ≈ 𝜉(𝑡 𝑖 )+ℎ 
𝜉(𝑡 𝑖+1 ) = 𝜉(𝑡 𝑖 ) + ℎ𝑓 (𝜉(𝑡 𝑖+1 ))
Here, the appro ximation of
𝜉(𝑡 𝑖+1 )
depends on

𝜉(𝑡 𝑖+1 )
, which is ag ain
defined in terms of t he state-space f orm
𝑓
applied to
𝜉(𝑡 𝑖+1 )
. Since the
solution at point
𝑡 𝑖+1
is no t y et kno wn, a (nonlinear) alg ebraic sys tem of
equations mus t be sol v ed (e.g. using Ne wton ’ s met hod) in ev er y time s tep.
The g ain for this additional effort is an increased stability : U ns table beha vior
is obser v ed for f ew er cases.
Despite t his im prov ement, t he simulated result of our exam ple is still
no t satisfact or y : Ins tead of div erg ence, w e no w obser v e a damping beha vior
(F igure 1.1b). The fact that t he error is limited (i.e., t he sol v er is stable) does
no t suffice for an accur ate simulation: It is also necessar y to im pro v e the
quantitativ e accuracy of the beha vior .
T o achie v e t his impro v ement, it is necessar y to increase t he q uality of t he
appro ximation of t he deriv ativ e. Instead of ev aluating only

𝜉(𝑡 𝑖 )
or

𝜉(𝑡 𝑖+1 )
,
a sol v er might use sev er al intermediate steps in or der to obtain a more
accur ate result. These inter mediate points define a pol ynomial

𝑝
, which
appro ximates t he deriv ativ e

𝜉
. The quality of t he appro ximation rises wit h
t he degree of

𝑝
. The Euler met hods can be inter preted as bor der line cases
wit h pol ynomials of deg ree
0
, i.e., t hey appro ximate t he deriv ativ e as a
cons tant.
One no table exam ple of t his principle is the famil y of Rung e-Kutt a met h-
ods. These met hods directl y calculate t he integ ral of t he pol ynomial

𝑝
as
a linear w eighted sum of sev er al intermediate points
𝑘 𝑖
. These
𝑘 𝑖
can be
com puted eit her f or w ard from
𝑡 𝑖
(in explicit v ariants) or backw ards from
𝑡 𝑖+1
(in im plicit members of t he famil y). F or example, t he famous f ourt h
or der explicit R ung e-K utta met hod is defined as:
𝜉(𝑡 𝑖+1 ) ∧
= 𝜉(𝑡 𝑖 ) + ℎ
6 (𝑘 1 + 2(𝑘 2 + 𝑘 3 )+𝑘 4 )
where 𝑘 𝑖
∧
= 𝑓 (𝑡 1 , 𝜉(𝑡 𝑖 ))
𝑘 2
∧
= 𝑓 (𝑡 𝑖 + ℎ
2 , 𝜉(𝑡 𝑖 ) + ℎ
2 𝑘 1 )
𝑘 3
∧
= 𝑓 (𝑡 𝑖 + ℎ
2 , 𝜉(𝑡 𝑖 ) + ℎ
2 𝑘 2 )
𝑘 4
∧
= 𝑓 (𝑡 𝑖 + ℎ, 𝜉(𝑡 𝑖 ) + ℎ𝑘 3 )
Here,
𝑘 1
is t he increment used in the explicit Euler -met hod,
𝑘 2
is t he
increment obtained from t he midpoint of the slope achiev ed b y
𝑘 1
,
𝑘 3
is a

18 1 Introduction
consecutiv e impro v ement (again from t he midpoint), and
𝑘 4
is t he increment
obtained from t he slope at the end of t his appro ximation. This met hod is
no t onl y s table but yields accurate results ev en in long-ter m simulations of
our exam ple (F igure 1.1c).
This o v er view onl y scratches the surface of t he topic of sol ving differen-
tial equations. There are man y more techniques wit h interesting properties:
e.g., multi-step methods, adaptiv e sol v ers, and ev en t he discretization of
s tate instead of time[Ber g ero et al., 2012], to name onl y a fe w . N otabl y , no
met hod can reliabl y solv e all D AEs. Therefore, t he choice of a solv er is
alw a ys a trade-off betw een com putational effort and precision, restricted
b y t he requirements of t he model at hand.
The im pact on t he languag e semantics is rather sev ere. When t he wrong
numerical met hod is applied, the outcome of a simulation could as w ell be
com pletel y random, but the adequacy of a particular met hod depends on
t he model. A precise description of the results of a simulation is t herefore
pr acticall y unachiev able. An y descrip tion of t he semantics mus t eit her
ignore t he simulation entirel y or someho w incor por ate t his uncertainty .
As w e ha v e seen, t he integ rat or attem p ts to com pute t he deriv ativ es of
a solution. This com putation eit her uses an explicit function
𝑓 (𝑡 𝑖 , 𝜉(𝑡 𝑖 ))
or
relies on t he (numerical) solution of an implicit sys tem of equations. Because
t he concrete im plementation depends on the choice of t he integrator , the
modeling languag e mus t be able to support bo t h t he necessar y reordering
into s tate-space-form and an iterativ e ev aluation of the equations. The
latter point im plies t hat equations can be ev aluated in what appears to be a
r andom order . It is t hus advisable t hat t heir ev aluation is free of side-effects.
1.1.2 Regularization
Aside from t he issues that come wit h t he application of numerical met hods,
modeling a D AE has its intrinsic difficulties: Sys tems of equations described
b y such a model can easil y de viate from t heir ideal f ormulation. They are
often larg er than necessar y , ma y contain redundant information, or ha v e
an opaque s tructure. This deviation is a natural consequence of the higher
lev el of abstr actions, and it motiv ates the need for symbolic met hods that
tr ansform a model into a f or mat t hat is suitable for the numerical simulation.
Like an integrat or , t he symbolical treatment im poses requirements on t he
im plementation of the modeling language.
As an exam ple, consider t he pendulum in its Cartesian formulation
as a firs t-order D AE. Such a formulation might occur more natur all y in
a modeling languag e t han the concise v ariant from abo v e. (In fact, t he

1.1 Differential Alg ebraic Equations 19
sys tems of equations t hat arise from t he usag e of a library for mechanical
modeling are ev en more extensiv e.)

𝑥 = 𝑣 𝑥

𝑦 = 𝑣 𝑦

𝑣 𝑥 = 𝐹 𝑥

𝑣 𝑦 = 𝐹 𝑦 − 𝑔
𝑥 2 + 𝑦 2 = 1
The solution of such nonlinear equations is usuall y achiev ed b y an
iter ativ e process like N ewt on ’ s method. Here, t he zeros of t he implicit
definition
𝐺
are found. Usuall y , the iteration makes use of the in v erse of
t he Jacobian
𝐽 𝐺
−1
. The components of this matrix are t he partial deriv ativ es
of
𝐺
wit h respect to the differential v ariables

𝑧
. For the exam ple abo v e, t he
sol v er needs to determine 
𝑥 , 
𝑦 , 
𝑣 𝑥 , 
𝑣 𝑦 and 𝐹 .
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

𝑣 𝑥 
𝑣 𝑦 
𝑥 
𝑦 𝐹

𝑥−𝑣 𝑥 0 0 1 0 0

𝑦−𝑣 𝑦 0 0 0 1 0

𝑣 𝑥 − 𝐹 𝑥 1 0 0 0 −𝑥

𝑣 𝑦 − 𝐹 𝑦 + 𝑔 0 1 0 0 −𝑦
𝑥 2 + 𝑦 2 − 1 0 0 0 0 0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
The las t ro w contains onl y zero elements. Theref ore t he Jacobian cannot
be in v erted, and hence t he sys tem canno t (easil y) be sol v ed. The cause of
t his problem lies in the fact t hat t he las t ro w describes an alg ebr aic constr aint ,
i.e., an equation t hat onl y relates states. These cons traints do no t directly
contribute to the solution of t he deriv ativ es. U nfortunatel y , they occur
natur all y from t he composition of models.
The solution to this problem stems from the obser v ation t hat the v ari-
ables in t his sys tem of equations are not trul y independent. R ecall t hat each
v ariable represents t he v alue of a continuous function. A cons traint can
t hen be interpreted as a constant univ ariate function
ℎ(𝑡) = 0
. Hence if
ℎ
is differentiable, t hen its to tal deriv ativ e

ℎ(𝑡)
also equals zero. A dding t he
deriv ativ e of an equation does t heref ore no t chang e t he solution of a model.
In t he exam ple, w e can add t he first and second deriv ativ es of t he last ro w

20 1 Introduction
to the system:

𝑥𝑥 + 
𝑦𝑦 = 0

𝑣 𝑥 𝑥+𝑣 2
𝑥 + 
𝑣 𝑦 𝑦+𝑣 2
𝑦 = 0
A dding t hese equations yields an o v erdetermined sys tem. In order t o
sol v e it, one can eliminate eit her
𝑥
or
𝑦
and t heir respectiv e deriv ativ es b y
subs titution. When 𝑦 is eliminated t he resulting system is:

𝑥 = 𝑣 𝑥

𝑣 𝑥 = 𝐹 𝑥
𝑥 
𝑣 𝑥 + 𝑣 2
𝑥 + (𝑥𝑣 𝑥 ) 2
1−𝑥 2 + 𝑔 √ 1−𝑥 2 = 𝐹 (1 − 𝑥 2 )
If t he choice falls on 𝑥 , the resulting system of equations is:

𝑦 = 𝑣 𝑦

𝑣 𝑦 = 𝐹 𝑦 − 𝑔
𝑦 
𝑣 𝑦 + 𝑣 2
𝑦 + (𝑦𝑣 𝑦 ) 2
1−𝑦 2 = 𝐹 (1 − 𝑦 2 )
Bo t h sys tems are sol v able in principle and describe t he desired beha vior .
Bo t h, ho w ev er , are singular in a specific point (either
𝑥 = 0
,
𝑦 = 1
or
𝑥 = 1
,
𝑦 = 0
) due to a division b y zero. This choice is called s tat e selection
1
(see
e.g. [Mattsson et al., 2000]).
The tr ansformation of a differential alg ebr aic equation into such a solu-
ble f orm is called regularization or index reduction (where t he index of an
equation is some measure of t he difficulty of sol ving it). There are sev eral
different definitions of t he index of an equation [Mehrmann, 2015] as w ell as
met hods of regularization. The index
Σ
-met hod dev eloped b y Pr y ce[Pr y ce,
2001], differs from t he more common met hod of P antelides [P antelides,
1988] mos tl y in its concise descrip tion of the algorit hm and its immedi-
ate support for higher -or der differentials. While the latter describes t he
solution finding in detail (and com plexity), the former resorts to a Linear
Prog r amming (LP) descrip tion of the problem t hat is somewhat sim pler
to im plement. Hence w e focus on the
Σ
-met hod. Both met hods impose
similar cons traints on the modeling languag e, t hough.
In t he exam ple abo v e t he differentiation of t he las t equation w as nec-
essar y because the Jacobian w as rank -deficient. In a g raph-theoretical in-
ter pretation this means t hat it w as im possible to match each s tate to an
1 although “state- de selection” w ould be more appropriate

1.1 Differential Alg ebraic Equations 21
equation t hat depends on t he deriv ativ e of t hat s tate. The solution w as to
match t he las t equation to a s tat e instead and differentiate it as needed. In a
certain sense t he requirement f or t he Jacobian to be of full r ank has been
relax ed. This relaxation can be expressed g raphicall y b y t he search for a
highes t-v alue perfect matching where the cost of matching an equation to
a s tate is t he highes t deriv ation deg ree of t he s tate in t he equation (as it is
alw a ys pref erable t o match to a deriv ativ e). Using t his idea, t he
Σ
-met hod
can be summarized as f ollo ws:
1.
Cons truct a Matrix
Σ = (𝜎 𝑖𝑗 ) ∈ ℕ 𝑛,𝑛
of t he model, where ev er y
𝜎 𝑖𝑗
indicates t he highes t deriv ativ e of unkno wn 𝑗 in equation 𝑖
2.
Inter pret
Σ
as t he w eighted biadjacence matrix of a bipartite g raph
wit h
2𝑛
nodes and find a maximum-v alue perfect matching
𝑇
in t hat
g r aph.
3.
Giv en t hat matching, find t he smalles t op timal dual solution
𝑐, 𝑑 ∈ ℕ 𝑛
.
Then 𝑐 𝑖 is t he number of times, equation 𝑖 needs to be differentiated.
The search f or a maximal matching can be defined as a classical Integer
Linear Prog r amming Problem where t he discrete v ariables
𝛼 𝑖𝑗
indicate a
perf ect matching on t he com plete bipartite graph induced b y Σ = (𝜎 𝑖𝑗 ) .
Maximize: 𝑧 = 􏾜 𝜎 𝑖𝑗 𝛼 𝑖𝑗
Subject to: 􏾜
𝑖∈1…𝑛 𝛼 𝑖𝑗 = 1 f or each 𝑗
􏾜
𝑗∈1…𝑛 𝛼 𝑖𝑗 = 1 f or each 𝑖
𝛼 𝑖𝑗 ≥ 0
There are man y algorit hms a v ailable to sol v e t he problem: P articu-
lar l y interes ting are t he classical primal-dual methods, like t he Hung arian
Met hod [K uhn, 1955] or adv anced algorit hms using sophisticated initializa-
tion heuris tics like the Jonker - V olg enant algorit hm [Jonker and V olg enant,
1987]. The dual solution of t he LP has a sur prisingl y elegant interpretation.
When
𝑐 = (𝑐 𝑖 )
is t he v ector that describes ho w often t he
𝑖
-t h equation has to
be differentiated, and
𝑑 = (𝑑 𝑗 )
is t he v ector containing t he maximum degree
of t he deriv ativ es of unkno wn
𝑗
in t he sys tem, t hen each equation im poses
a lo w er limit of t he deg ree of its unkno wns:
𝑑 𝑗 ≥ 𝜎 𝑖𝑗 + 𝑐 𝑖 ∀𝑖, 𝑗 ∈ 1 … 𝑛

22 1 Introduction
The abo v e relation reads as follo ws: The degree of each v ariable is
high enough to fulfill the requirements of all necessar y deriv ations of each
equation it occurs in. A second requirement is t hat index reduction st ops
ear l y : If an unkno wn
𝑗
is com puted b y equation
𝑖
(i.e. the tuple
(𝑖, 𝑗)
is
contained in t he matching
𝑇
), t hen equation
𝑖
is differentiated exactl y as
often as necessar y:
𝑑 𝑗 = 𝜎 𝑖𝑗 + 𝑐 𝑖 ∀(𝑖, 𝑗) ∈ 𝑇
These conditions describe t he dual f easible solution (in t he linear prog ram-
ming sense) for the highest-v alue perfect matching. An y dual solution (as,
e.g., calculated b y t he Hungarian algorithm) fulfills t he abo v e requirements
and t hus also sol v es t he problem of regularization. There are infinitel y man y
f easible solutions. T o find t he smallest , t he second part of the
Σ
-met hod
em plo ys a sim ple fix ed-point iter ation. This iterativ e process is guaranteed
to terminate as long as 𝑇 is a highes t v alue perfect matching of Σ .
There are tw o important consequences of the regularization for the mod-
eling languag e semantics: Firs t, it is necessar y to pro vide a certain structur al
interface t o the equations. In case of the
Σ
-met hod, t his interface has t o
yield t he highes t deriv ativ e of each unkno wn in each equation. Second, un-
kno wns and equations are subject to transf or mation after the equations are
collected and bef or e simulation starts. Hence it is impossible t o decide ho w
t he y will influence t he simulation s tate wit hout kno w ledge of the complete
model.
1.1.3 Sol v er -side S tat e Selection and Quasilinear Equations
Ev en af ter regularization, t he solution of a system might reach a point of
singularity . If t he pendulum is simulated with
𝑥
eliminated, it approaches
an in v alid point when closing in on
(𝑥, 𝑦) = (0, −1)
. The onl y possible
solution to this kind of problems is to switch t o a different state selection,
i.e., to eliminate
𝑦
. This change requires a reliable, dynamic detection of the
singularity , i.e., the com putation of t he eigen v alues of t he system’ s Jacobian
up to a reasonable precision. The most sensible place f or such a computation
is t he numerical sol v er itself, as it (usuall y) already maintains a (numericall y
appro ximated) Jacobian, error appro ximations and controls t he integ ration
s tep size.
Qu alid aes [S teinbrecher, 2006] is an adap tion of t he Radau II-a method
t hat im plements t his detection. Furthermore, Qu alid aes no t onl y detects t he
problematic situation but is also capable of selecting t he new s tates wit hout
furt her inter v ention from t he modeling language. T o do so, how ev er , it

1.1 Differential Alg ebraic Equations 23
requires t he sys tem of equations to be a v ailable in a specific form:
𝐸(𝑡, 𝑧) 
𝑧 = 𝑘(𝑡, 𝑧)
0 = ℎ(𝑡, 𝑧)
In t his f orm,
𝐸
tog et her wit h
𝑘
is called t he q uasilinear representation
of
𝐺
(i.e.
𝐸(𝑡, 𝑥) 
𝑧 = 𝑘(𝑡, 𝑧) ⟺ 𝐺(𝑡, 
𝑧, 𝑧) = 0
), while
ℎ
depicts its hidden
cons tr aints . These constr aints are t he results of regularization: They can be
obtained b y eliminating all deriv ativ es from t he differentiated equations
using t he original equations of the system. In our exam ple, there are tw o
hidden cons traints:
𝑥𝑣 𝑥 + 𝑦𝑣 𝑦 = 0
𝐹 𝑥 2 + 𝑣 2
𝑥 + 𝑦(𝐹 𝑦 − 𝑔) + 𝑣 2
𝑦 = 0
Since t he original equations
𝐺
of t he cartesian pendulum are already
in quasilinear form, Qu alid aes can sol v e t he whole sys tem directl y . A sim-
ulation o v er 50s wit h an initial angle of
𝜙 0 = 0.4
repeatedl y encounters a
possible singularity when
𝜙=0
and
(𝑥, 𝑦) = (1, 0)
. N ev ert heless, Qu ali-
d aes reliabl y pro vides a solution o v er t he whole inter v al wit hout furt her
configur ation(F igure 1.1d).
This particular solv er demonstr ates another challeng e to the specification
of a modeling languag e: Some integ rat ors might require a specific format
of t he equations. T r ansf ormation into t his f orm requires a careful analysis
of its effects on t he regularization (and vice v ersa) and might constr ain t he
semantics of t he languag e.
1.1.4 Hybrid and S tructur all y V ariable Sys t ems
U p until no w , w e ha v e onl y considered systems with one solution
𝜉
o v er an
inter v al
𝕀 ⊆ ℝ
. But models deriv ed from a natural descrip tion of real-w or ld
phenomena are often best described b y multiple solutions
𝜉 1 … 𝜉 𝑠
called
modes .
As a sim ple exam ple, consider the case of a breaking pendulum. After
some time, t he connection betw een t he center of mass and t he joint might
tear apart. The resulting model canno t be described as a pendulum an ymore,
but r at her as a free fall. The length-constr aint and t he v ariable 𝐹 v anish.
𝑣 𝑥 = 
𝑥
𝑣 𝑦 = 
𝑦

𝑣 𝑥 = 0

𝑣 𝑦 = −𝑔

24 1 Introduction
A widel y accepted fr amew or k for such models is the t heor y of Hybrid
A utomata [Henzing er, 2000], dev eloped b y Henzing er . He describes a
h ybrid model as a multig r aph (i.e., a g r aph where more t han one edg e
might connect tw o nodes) t hat consists of control modes
𝑉
(t he v ertices)
and switches
𝐸
(t he edg es). Each control mode consists of a (different)
set of equations. The state of a h ybrid automaton is an element of the set
𝑄 ⊆ 𝑉 × ℝ 𝑛
, consis ting of t he current control mode and t he current v alues
of t he continuous v ariables. A transition relation
𝑞 1 → 𝑞 2
betw een t he states
defines t he semantics of the automat on. A transition might either denote
a successful integ r ation step (called a flow ) or a discrete chang e (called a
jum p ). In t he former case, t he control mode remains unchanged. In the
latter case, a new mode is reached b y a switch.
While t he semantics of h ybrid automata are secondary to this t hesis
(Modelica is not directl y built upon t hem), the question of when to mak e a
jum p is of interes t. A predicate ov er t he model’ s v ariables guar ds a switch.
Such a predicate takes the form of logical (i.e. boolean) combinations of one
or more equality relations, e.g.
𝑡 > 0 ∧ 𝜙 = 0.35
. These propositions onl y
chang e their v alues at t he zero-crossings of t he contained equalities. There-
f ore, each flo w -transition ma y onl y jump t o the next such zero crossing. F or
t he modeling languag e, this somewhat com plicates t he dis tinction betw een
t he descrip tion of a model and other parts of t he languag e. Depending on
its context, a sim ple com parison like the one abo v e might introduce a new
guar d, effectiv el y becoming part of t he model’ s descrip tion.
1 . 2 modelica
The Modelica Association, t he non-profit org anization t hat o wns and main-
tains t he Modelica specification defines its epon ymous modeling language
as f ollo ws:
Modelica is a languag e for modeling of physical sy s tems , designed
to support effectiv e library dev elopment and model ex chang e.
It is a moder n languag e built on acausal modeling with mat hemat-
ical equations and object-orient ed cons tructs to f acilitate reuse of
modeling kno w ledge. (The Modelica Association [Modelica,
2012], italics added)
This s tatement is notable as it explicitl y chooses a rather broad domain
of modeling. This extended scope im plies t hat there is no pref erence for
sol v ers (e.g., ev en t he simple explicit Euler method has a v alid use in some
real-time contexts [Otter et al., 1997]) and contains v er y different kinds of

1.2 Modelica 25
models (e.g. Bondg raphs [Cellier and N ebo t, 2005] or planar mechanical
sys tems[Zimmer, 2012]). This gener ality only em phasizes t he challeng es
t hat come with numerical simulation: An y particular technique for the
solution of differential alg ebr aic equations is sooner or later prone to be
necessar y f or t he simulation of some Modelica models.
1.2.1 Ev aluation vs. Elabor ation
The semantics of Modelica is divided into three parts: Instantiation of a
model from its hierarchic context, cons truction of a system of equations from
such an ins tance, and finall y t he simulation of these equations. The firs t
part, called model evaluation , deals wit h certain features that allo w f or higher -
or der Modeling: These are inheritance, modifications, replaceable classes,
and redeclar ations. In t his context, classes are abstr actions on (collections
of) pyhsical models , capable of creating new v ersions b y redefining constants,
adding new equations or e v en replacing parts of a model-library .
The second aspect is sometimes called flatt ening (in t he Modelica specifi-
cation) or elabor ation (in t he literature, see e.g. [Broman, 2010]) and specified
in a process-oriented manner . Throughout this t hesis, t he latter ter minus
will be preferred. Here, ev aluated models are abstr actions of systems of
equations, i.e., t hey can be used as recipes t o g enerate such systems through
some dynamic com putational process. During elaboration, models can
inter act b y inst antiation , i.e., t hey can add the result of elaborating ano t her
ev aluated model into t he sys tem of equations, and r efer ences , i.e., t hey can
use parts of ano t her model and its ins tances.
The final part deals wit h the equations t hemselv es and t he languag e
f eatures concer ned wit h their symbolical pre-processing (e.g., t he regular -
ization) and numerical simulation. In t hat sense, equations are abs tractions
of numerical pr oblems , i.e., they can be tr anslated into (efficient) programs
t hat com pute a solution for the mat hematical problem t hey represent. This
t hir d aspect is often described as compilation or code g ener ation . T o not con-
fuse it wit h the ov erall com pilation, w e will refer to it as tr ansformation
t hroughout this t hesis.
Since all t hree parts belong to the dynamic semantics of t he modeling
languag e, it is tem pting t o subsume t hem under the ter m ev aluation. W e
will see later , ho w ev er , t hat it is useful to dis tinguish betw een t he first and
t he second part. It will also tur n out t hat the distinction betw een t he second
and t he third is somewhat artificial. Hence will mostl y distinguish betw een
ev aluation and elaboration. Consequentl y , elaboration becomes a part of
ev aluation and transf ormation a part of elaboration.

26 1 Introduction
Definition
(Ev aluation, Elaboration, T ransf or mation)
.
Ev aluation is t he r e-
duction of t erms t o v alues. Elabor ation is t hat part of the ev aluation, that deals wit h
t he construction, inspection, and modification of models. T ransf ormation is t hat
part of elabor ation that applies symbolical manipulations or numerical met hods t o
models.
1.2.2 A causal Equations
The f oundation of Modelica (as w ell as its primar y means of modeling)
is its support for acausal equations. The terminus acausal refers to the
fact that equations do not represent an explicit com putation of a v alue for
an unkno wn — instead, they im ply a relation betw een sev eral simulation
v ariables t hat might be reordered to such a com putation just as needed. As
an exam ple, consider Ohm’ s la w as a Modelica equation:
parameter Real R = 10.0;
equation
u=R*i ;
The equation, can be read as an assignment to com pute t he v oltag e,
u
,
directl y . But it could as w ell be reordered t o compute the current, i :
u/R=i ;
Bo t h f orms are equiv alent, and t here is no preference f or one abo v e t he
o t her . Choosing one particular form is called causalization and perf ormed
automaticall y , alongside o t her optimizations and ref ormulations, b y mos t
im plementations.
A second particularity affects t he ev aluation of t he ter ms that form a
lit er al equation. In t he example abo v e, only the v ariable
2 R
has a v alue in
t he corresponding
3
par ameter declaration. V ariables
u
and
i
are defined
wit hout a v alue and represent unkno wns of t he model. In consequence,
ev aluation of t he (reordered) term requires t he ev aluation (or rather lookup)
of t he corresponding solution 𝜉 𝑖 (𝑡) .
This requirement has a significant im plication f or t he com position of
equations. The meaning of an unknown depends on the mapping of some
v ariables to indices in
𝜉
. An equation can onl y be added to t he sys tem after
such an index exists f or all its unkno wns. The process of assigning a unique
index to each unkno wn is called t he allocation of the unknown. It mus t be
2
During this t hesis, placeholders for v alues inside a ter m are called v ariables, elements
of the solution 𝜉 𝑖 are called unknowns.
3 Ho w exactl y t his correspondence w or ks is a topic of a later chap ter

1.2 Modelica 27
possible to express the ordering betw een t he allocation of unkno wns and
t he elabor ation of an equation.
The elabor ation of equations is not limited t o allocations of unkno wns
and t he collection of liter al equations. Modelica offers tw o sorts of non-
lit er al equations. The first consis ts of what might bes t be described as
factual s tatements. Such statements are written in the syntax of a function
application ins tead of an equation.
equation
u = R*i;
assert (u < 10e6);
Here, t he
assert
-s tatement prev ents t he solution from assuming a too
larg e (and potentiall y un ph ysical) v alue. While t he statement resembles an
assertion kno wn from other languages, its meaning lies in the domain of
numerical simulation (since onl y t hen a v alue can be found f or
u
). Inf ormally ,
one can consider t he model to consis t of one “unobser v ed” equation t hat
terminates t he simulation in case of a violated in v ariance. In a similar
fashion, a
connect
s tatement can contribute to the construction of im plicitl y
defined equations (see Section 3.4.9).
Control s tatements form the second kind of non-literal equations. Loops
allo w gener ating an arbitrary amount of equations (deter mined, for ins tance,
b y some user input). An
if
-equation yields different (or no) equations de-
pending on t he e v aluation of a conditional expression. A g ain, t he outcome
might depend on some user in put. The elabor ation of non-liter al equations
is a com putational process that depends on t he ev aluation semantics.
constant Boolean dbg = true ;
equation
u = R*i;
if dbg then
assert (u < 10e6);
end if ;
In t he abo v e exam ple, an assertion depends on t he v alue of a debug flag.
Onl y if t he flag ev aluates to
true
, t he assertion becomes an element of the
model. N o te, t hat ev en if
dbg
is declared as
constant
, it ma y be altered b y a
modification. Consequentl y , that decision must be part of the elaboration
and canno t be preponed.
1.2.3 Object-Orient ed Cons tructs
Modelica differs from classical object-oriented languag es like Smalltalk or
Ja v a in t he representation of s tat e : The latter describe objects as entities t hat

28 1 Introduction
encapsulate s tate as w ell as functionality (eit her in the form of met hods or
b y reacting to incoming messag es). The state of a simulation is a discrete
solution
𝜉(𝑡)
. Each instantiated model corresponds t o a slice of t hat solution,
but it har dl y encapsulat es it: In fact, t he onl y entity t hat chang es t his state is
t he integrator (and the ev ent handler in case of h ybrid systems). While it
is certainl y possible to someho w reconcile these tw o view s, in t he case of
Modelica, t he terminus “Object-oriented” is more a metaphor to explain
some concep ts of the languag e t han an axiom of its semantics. In order not
to tak e this metaphor too far , it is im portant to ackno w ledg e its limits.
What Modelica does share wit h more common languag es is t he w a y
in which it builds classes. As in Ja v a, Modelica Classes are constructed
wit h com position, inheritance, and o v erriding and deconstructed b y an
in v ocation mechanism t hat implements open r ecursion . As a sim ple example,
reconsider t he pendulum from abo v e. It can be rewritten into a tin y library
as f ollo ws:
package PendulumLibrary
model Constants
constant Real g = 9.81;
end Constants;
model SecondOrderAngular
Real phi;
Real dphi;
end SecondOrderAngular;
model LunarPendulum = AngularPendulum(c.g = 1.62);
model AngularPendulum
Constants c;
extends SecondOrderAngular;
equation
der (phi) = dphi;
der (dphi) + c.g*sin(phi) = 0;
end AngularPendulum;
end PendulumLibrary;
The firs t notable difference is the nesting of the occurring classes. The
liter al models
Constants
,
SecondOrderAngular
, and
AngularPendulum
are
all children of t he packag e
PendulumLibrary
. The short class definition of
as w ell
LunarPendulum
is also a named element of t he packag e. Hier ar chical

1.2 Modelica 29
com position is the principal w a y of org anizing model- libraries in Mod-
elica. Hierarchies can be arbitr aril y deep, and each lev el can contain an
arbitr ar y amount of members. All t he members of one lev el can freel y
ref erence each other as w ell as all members “upw ards” the hierarch y . There
is no particular requirement on t heir or dering (e.g.,
LunarPendulum
ref ers
to AngularPendulum although t he former appears bef ore t he latter).
Models can also be com posed b y inst antiation : The model Const ants is
ins tantiated to a com ponent called
c
. In t he equations, t he compositional
name (
c.g
) t hen accesses the g ra vitational constant. Instantiation forms a
tree where nodes correspond to model ins tances. The root is the instance
of t he model that describes t he simulation experiment, and t he lea v es are
primitiv e components (i.e., the unkno wns).
The t hir d f orm of composition is inherit ance :
AngularPendulum
does no t
contain a declar ation of its com ponents
phi
and
dphi
. Ins tead, t hey are
f ound in t he base class
SecondOrderAngular
. An intuitiv e (but in gener al too
sim plis tic) definition of inheritance is to vie w an
extends
-declar ation as a
shortcut f or a cop y of all declar ations and equations of t he base class. The
deriv ed class t hus inherits appearance as w ell as beha vior .
F inall y , the
LunarPendulum
demons trates Modelica ’ s v ersion of ov errid-
ing: It is declared as a modified v ariant of
AngularPendulum
. This relation
resembles t hat of inheritance (and can be expressed similar l y), but it al-
lo ws t o o v er write an inherited field selectiv el y . Here, t he component
c
is
modified to contain a different gra vitational constant (that of t he moon).
All t hese f eatures are reasonabl y simple t o g r asp and f orm a pow erful
toolset f or t he design and implementation of reusable models. Their combi-
nation leads to r at her complex semantics, though. Consider for ins tance t he
inheritance of a declar ation t hat has a v alue. Some scope contains t he term
t hat represents this v alue. But which one? Since t he deriving class might
modify inherited com ponents, it canno t be t he scope of the base class. But
at t he same time, it might access elements t hat are onl y present in t hat scope
(e.g., a constant element of t he hier archical parent). W e pos tpone an answ er
to this question until Section 2.3, but conclude t hat inter action of Modelica ’ s
classes is a com plicated matter that demands a precise treatment.
1.2.4 Im plementation S tr at egies
An y im plementation of Modelica has to co v er model ev aluation, elaboration
and tr ansformation equall y w ell. There are at least tw o different str ategies
to deal with it:
F irs t, t here is the pragmatic s trategy , which dominates among industrial-
s trengt h im plementations (e.g. OpenModelica[Fritzson et al., 2005], JMod-

30 1 Introduction
elica [Åkesson et al., 2010], Dymola [Br ück et al., 2002]). These imple-
mentations f ocus on t he t hir d aspect of Modelica ’ s semantics and are built
around t he process of tr anslating a set of equations into v er y efficient code.
The y treat t he dynamics of t he firs t aspects (especiall y t he parts concer ned
wit h model ev aluation) like a meta-prog ramming languag e b y r unning an
embedded inter preter bef ore t he tr anslation. This pragmatic s trategy is
so common t hat it has ev en influenced t he Modelica specification: Some
f eatures (like checking the bounds of arra ys depending on user input) are
har d if not pr acticall y im possible to realize in alternativ el y designed im-
plementations, while use cases like v ariable-structure systems are har d to
define in t his w a y .
A different approach to the unification of all aspects is to view Mod-
elica as a particular form of a s tag ed language: The first s tage deals with
ev aluation and elaboration and com put es a prog r am t hat (as t he second
s tag e) im plements t he simulation parts. This approach w or ks w ell wit h
a f ormally defined meta-languag e t hat is capable of staging (as Broman
sho w ed wit h Model yze [Broman and Siek, 2012]). It is also possible to
embed t he second s tag e semantics as a domain-specific-languag e (DSL) into
some exis ting languag e t hat f eatures higher -order functions (as Gior gidze
did wit h Haskell [Gior gidze and Nilsson, 2008]). A stag ed implementation
allo ws t o catch modeling errors during dev elopment (if t he host languag e
is s taticall y typed) and enf orces a clear separation betw een simulation-time
and modeling- time in puts . Gior gidze and Broman both demonstr ated
t he viability of such a rigor ous str ategy f or t he elaboration, translation, and
simulation of an equation based languag e. Ho w ev er , both focus on embed-
ded languag es that do not possess hier archical classes. So far , no rigorous
s trategy has been designed f or t he ev aluation of Modelica-s ty le models.
Com pilation vs. Inter pret ation
Im plementations can also be char acterized b y t he ev aluation time of a model.
Mos t, if not all, exis ting im plementations do no t hesitate to e v aluate expres-
sions bef ore g enerating simulation code. In that regar d, t he y are actuall y
int erpr et ers of the modeling languag e. This point might confuse a reader that
is familiar with t he usual connotation of exis ting implementations as com-
pilers (e.g. t he “Open Modelica Com piler”). While it is tr ue t hat these tools
in some sense tr anslate t heir in put models to the efficient simulation code
and one might call an y such prog r am a com piler , w e reject t his definition
as too broad. F utamur a [1999] makes an essential claim f or the definition of
a (correct) com piler : N amel y t hat t he object prog ram mus t be capable of
yielding t he same output as an inter preter f or all in put dat a . Mos t Modelica

1.2 Modelica 31
im plementations violate this property:
model Fixed
parameter Integer n = fun(1);
Real x;
equation
if n>0 then
x = 0.0;
else
x = time;
end if ;
end Fixed;
When elabor ating t he model abo v e, t hey will yield either a constant or
a linear signal f or
x
, depending on t he result of ev aluating t he function
fun
.
After t hat step, a user cannot chang e t he v alue of
n
, alt hough
parameter
declar ations are considered user input. It is im plicitl y fixed and called a
s tructur al par amet er . There is also no guarantee that t he ev aluation of
fun
will ev er terminate because Modelica ’s algorithms are T uring-complete. All
assertions about the result of such a tr anslation remain under t he reser v ation
of po tential div erg ence – a less t han satisfying situation. For these reasons,
w e consider a compiler a com put able function , i.e., a terminating algorit hm,
which retains t he semantics of the source prog ram. The second condition
will be defined more precisel y t hroughout t his t hesis.
1.2.5 S tructur al V ariability
A rigorous im plementation treats the process of elaboration as a f orm of
ev aluation. The dynamic semantics of t he second stag e (e.g t he integ rat or)
are no t fundamentall y different from the first. This obser v ation leads to
an intriguing ques tion: When t here is no fundamental difference betw een
t he elabor ation of a model and its simulation, t hen wh y should t he f ormer
alw a ys precede t he latter? Without t his res triction, it becomes trivial to
describe im plicit hybrid aut omata , i.e., models that com pute t heir next mode
during simulation. Such automata do no t depend on t he exis tence of a
(predefined) finite set of control modes. Therefore they represent a po tent
f orm of variable s tructure sy st ems .
This paradigm has recentl y attained a lot of interes t[Nytsch-Geusen
and Er ns t, 2005, Zimmer, 2008, Elmqvist et al., 2014], but the pragmatic
im plementations do no t support it. The main reason is of course t hat
Modelica has not y et specified an y means to chang e a model’ s system of
equation during simulation. Currentl y , it is onl y possible to chang e the

32 1 Introduction
v alue of discrete v ariables. But ev en t his limited change is often problematic
f or t he s tructural anal ysis t hat relies on t he symbolic appro ximation of t he
Jacobian.
Man y Modes
But ev en non-standar d implementations f ace a significant problem: While
it is possible to im plement structural v ariable models wit hin a pragmatic
im plementation[Mehlhase, 2012], t he necessity to g enerate code f or each
mode ahead of t he simulation limits this approach to models wit h f ew
modes. If a model has
𝑛
possible modes, it is immediatel y clear t hat the
com position of
𝑘
such models has
𝑛 𝑘
possible modes. This combinatorial
explosion makes e v en moderate le v els of com position unfeasible with t hat
approach.
As an exam ple, consider a continuous-time model of a N and gate. Like
an y logical function, it has tw o output modes: It is eit her activ e (yielding a
high v oltage) or inactiv e (yielding a lo w v oltage). The circuit can be modeled
easil y as a discrete system. In t hat case, all signals w ould pass t hrough it in
no time. But a sim ple RL C circuit can dela y t he output wit h its continuous-
time beha vior . Although t his is a simplified model of the actual electrical
beha vior of a logical com ponent, more realistic modeling of logical circuits
uses t he same principle [Bhushan and K etchen, 2015].
Raising o
−
+
Falling o
High o
−
+
Low o
fall raise
stable
stable
fall raise
F igure 1.2: Modes of t he dela y ed N and g ate

1.3 Detailed R esearch Questions 33
The dela y ed N and gate has f our modes of operation (F igure 1.2): When
t he output is about to chang e from a lo w to a high v oltage, t he model
consis ts of a charging RL C circuit. Similar l y , it uses a dischar ging RL C in
t he opposite case. Whene v er an RLC circuit has s tabilized, it is not necessary
to simulate it (it remains in a s teady state). A constant v oltag e can replace
such a s table com ponent. The gate enters an uns table state when t here is a
chang e in its logical output.
Alt hough it is triviall y possible to im plement each mode as a Modelica
model, t he combinatorial explosion pre v ents its usag e as a s tructurall y
v ariable model: Ev en simple logical functions consis t of dozens, or ev en
hundreds, of such g ates and t he number of their combined possible modes
is f or all practical purposes infinite. Hence w e search for an extension t o
Modelica t hat allo ws the model to com pute its next mode on demand.
1 . 3 det ailed resear ch questions
The anal ysis of Modelica and its modeling domain allow s to break do wn
t he h ypo t hesis into f our separate ques tions. This section giv es a detailed
o v er view of each ques tion and its scope wit hin this t hesis.
1.3.1 Oper ational Semantics of Modelica
The premise of t his thesis is t he feasibility of the definition of an operational
semantics of Modelica. As w e ha v e seen, t his requires both t he co v erag e of
(hier archical) ev aluation and elabor ation. That r aises t he f ollo wing question:
Question 1
What are t he oper ational semantics of Modelica ’ s hierarchical
classes and its elabor ation process?
This ques tion naturall y includes the search for a suitable abs tr act syntax .
T o f ocus t he discussion on t he most interes ting elements, t his t hesis does not
consider a com plete specification of Modelica. Instead, it aims at a minimal
descrip tion of hier archies and elabor ation.
1.3.2 Pr actical Implement ation of t he Modeling Domain
As w e ha v e seen, t he preparation and simulation of differential alg ebraic
equations is a v ast field. Ev en t he simulation of relativ ely sim ple models
requires a larg e machiner y . Its size conflicts wit h t he scope of this w or k.
Hence w e face an im portant question regar ding t he integ ration of this
machiner y :

34 1 Introduction
Question 2
Where lies t he dis tinction betw een Modelica ’ s operational se-
mantics and its symbolical and numerical treatment of equa-
tions and ho w can t he latter be isolated from t he former?
A gain, this t hesis does not giv e a complete answ er , but one that is practi-
cal in its context. It onl y explains t hese parts of t he numerical and symbolical
treatment of D AEs that it requires. In particular , this dissertation does not
examine t he com putation of initial v alues or an y symbolic op timizations of
t he sys tem of equations.
1.3.3 T r anslation t o a Sim pler Calculus
Correct com pilation needs to retain the source languages semantics. Bo th
t he tr anslation and its correctness are nontrivial.
Question 3
Ho w can t he aforementioned abs tr act syntax of hierarchical
classes and elabor ation be translated t o a simpler calculus with-
out changing its meaning?
This ques tion is technicall y the core of t his t hesis and shall be answ ered
t horoughl y . While it is not necessary to repeat t he details of the translation
of s tandard languag e elements, t hat of elabor ation and hierarchies and its
correctness shall be co v ered in depth. The described com piler might not be
com plete with respect to Modelica, but it rests on a solid f oundation.
1.3.4 Int egr ation of S tructur al V ariability
Once it is clear , how a Modelica com piler might be im plemented, it remains
an open ques tion ho w it enables the modeling of unbounded structurall y
v ariable models. This matter has tw o aspects: The abstr act operational
semantics mus t someho w support such models, and t he concrete syntax of
Modelica mus t someho w reflect t hat capability .
Question 4
Which extensions to Modelica and its oper ational semantics are
necessar y to express s tr uctur all y v ariable models, i.e., models
t hat can declare man y different modes, transition betw een
t hem, and be used in t he same w a y as models t hat ha v e onl y a
single mode?
The scope of t his final ques tion is limited to the fundamental means of
modeling. W e lea v e t he dev elopment of more complex applications t o t he
respectiv e domain experts. This w or k refrains from an y quantitativ e assess-
ments of t he perf ormance of simulations or t he quality of t he simulation
results.

1.4 Contributions and Outline 35
1 . 4 c ontributions and outline
The res t of t his thesis is structured as follo ws: This introductory chap ter con-
cludes wit h a discussion of its fundamental methodology and a summar y
of t he no tational con v entions used t hroughout t his w or k. The follo wing
chap ters address the research questions from abo v e in detail.
•
The abs tract syntax and dynamic semantics of the core language,
t he model com position languag e (Mcl), are defined in Chap ter 2. This
pro vides an abstr action of Modelica ’ s hier archical classes and t he
dynamic semantics of model elabor ation.
•
The simulation domain is im plemented in a kernel libr ar y (Chap ter 3).
The libr ar y contains tools f or t he definition, transf or mation, and simu-
lation of h ybrid differential alg ebr aic equations. This im plementation
of a Modelica-sty le simulation en vironment makes a dis tinction be-
tw een t he language and its domain and mak es it much sim pler to
chang e the latter and formalize t he former .
•
Chap ter 4 describes the compilation of Mcl. The front consists of an
encoding of a subset of Modelica into the core languag e. Starting with
t he full-fledg ed Mcl, t he core of t he transf or mation remo v es all t he
f eatures regar ding elaboration and hier archic ev aluation. This second
phase is pro v en correct wit h respect to the residual language. The
resulting languag e resembles a sim ple functional languag e, f or which
furt her com pilation is a standar d matter . The chap ter concludes wit h
a short discussion of t he technicalities of that final translation s tep.
•
The pro to typical front end discussed in Chapter 4 contains an exten-
sion to Modelica that enables t he declaration of s tr uctur all y dynamic
models. Instead of directl y modeling each mode, t he extension giv es
t he modeler some control o v er t he flo w of elaboration and simulation.
This met hod is po w erful y et sim ple: Onl y a handful of new operat ors
and classes allo w t he declaration of v ariable-structure models with
arbitr aril y man y modes.
•
Chap ter 5 contains some illustr ativ e exam ples to demons tr ate t he
application of Mcl to v ariable-s tructure modeling. W e sho w t hat t he
selected approach supports models wit h man y modes and can ev en
be used to im plement a parameter op timization problem directly in
t he extended subset of Modelica.
The t hesis closes with a discussion of its achiev ements and related as
w ell as possible future w or k.

36 1 Introduction
Earlier Publications
This t hesis is the apex of y ear -long research. N aturall y , some of the ideas
laid out in t his w or k ha v e already been published ear lier or are extensions
of such ear lier publications. Our v enture into t he com pilation of Modelica
s tarted wit h an in v estig ation of some relev ant aspects of t he language[Hög er
et al., 2010]. Later w e regar ded t he f easibility of symbolical manipulations
in a com piled setting [Hög er, 2013, 2015, Hög er and S teinbrecher, 2015].
Monadic modeling routines[Hög er, 2016] and t he f ormalization of t he hier -
archical semantics [Hög er, 2017] also ha v e t heir roots in ear lier w or k s.
1 . 5 resear ch method
In t heir report of the T ask F orce on t he Core of Com puter Science, Denning
et al. [1989] identify t hree ma jor par adigms of t he discipline: abstr action,
t heory , and design. They also immediatel y conclude t hat
[…] t he three processes are so intricatel y intertwined t hat it is
irr ational to sa y t hat an y one is fundamental. Instances of theor y
appear at ev er y s tag e of abstr action and design, instances of
modeling at ev er y s tag e of t heory and design, and instances of
design at ev er y s tag e of t heory and abstr action.
This w or k is an exam ple of the fundamental tr ut h of that statement.
The definition of an oper ational semantics can be seen as an application
of abs traction. When f ollo wing t hat paradigm, the scientist f ormulates a
h ypo t hesis, tur ns it into a prediction and tes ts it wit h an experiment. In our
case, t he fundamental h ypo t hesis is that t he hierarchical elements of the
Modelica, as w ell as its elaboration, can be com piled into a sim ple calculus.
Mcl (i.e. the whole Chapter 2) pla ys t he role of t he model and the prediction
is made b y t he formal definition of the translation in Chap ter 4.
A t t hat point, our paradigm shifts seamlessl y to t heory , where according
to Denning et al. the objects of study are in a possible relationship that is
expressed as a t heorem and sho wn to be true b y proof. W e use t his approach
to demons trate that our model indeed predicts correct compilation. T o t hat
end, w e define no t onl y the abstr act syntax and oper ational semantics of
t he source and targ et languag e but also t he inductiv e relation t hat allo ws a
precise f ormulation of a “correct” translation.
The experiment is t he im plementation of t he com piler in Chap ter 4. It
faces a significant pr actical challenge due t o t he symbolical tr ansf ormations
and numerical met hods that t he simulation of ev en simple Modelica models

1.6 N otational Con v entions 37
requires. W e design t he kernel library as a solution to that problem. The
specification s tems not onl y from t he modeling languag e but also from
t he architecture laid out in this t hesis: The central ques tion is ho w a small,
v erified core language can support the v ast f eature set of Modelica and
ideall y decouple t he language semantics from the im plementation of said
tr ansformations and algorithms. This question is answ ered in Chap ter 3.
Design is also applied to v erify our original claim that v ariable s tructure
modeling in such a setting is natur al . The extension proposed in Chapter 4
is sim ple and can be im plemented directl y in t he kernel library . Its incor po-
r ation into the front end is a mere relaxation of some semantic restrictions
and requires no additional processing.
The final v erification of t he results is necessarily reduced t o some selected
exam ple applications. Alt hough a quantitativ e study of this approach is in
principle possible, it is insurmountable (at least in a reasonable amount of
time) in pr actice. The reasons for this are laid out in Chapter 5. Theref ore,
w e resort to some prot o typical exam ples t hat require onl y a small, but
relev ant, subset of t he language and each demons trates an application of
s tructural dynamics. In t hat w a y , the experiment might not be broad, but it
co v ers t he h ypothesis in full dep t h.
1 . 6 no t a tion al c onventions
Languag es are defined in a sim ple BNF-f orm. N onterminals (e.g.
𝑡
,
𝑣
are expressed wit h the same small italic letters as meta-v ariables of t he
corresponding syntactic sort (e.g. w e will use
𝑡
to deno te both t he set of
terms and a v ariable from t hat set). Productions (e.g.
𝑡 ⩴ λ 𝑥 . 𝑡|𝑥
) map
a nonterminal to clauses consis ting of nonter minal and terminal symbols.
Clauses are separated b y a
|
. Each clause is one possible deriv ation of
t he left hand side. T er minal symbols (e.g.
λ , true , if
are written in a non-
proportional font. Sequences are abbreviated in t he s ty le of a v ector or
written as comma-separ ated list of elements e.g.
𝑡
deno tes a sequence of
terms
𝑡 1 , … , 𝑡 𝑛
. W e write
𝑡 ∷ 𝑠
f or t he sequence f ormed b y prepending
𝑡
to
𝑠
,
i.e.
𝑡, 𝑠 1 , … , 𝑠 𝑛
, and
𝑠∷𝑡
f or t he in v erse, i.e.
𝑠 1 , … , 𝑠 𝑛 , 𝑡
. The emp ty sequence
is written ♦ .
The signatures of functions are written using
×
f or Cartesian products
and
→
to dis tinguish domain and co-domain. The domain of a partial func-
tion
𝑝
is written
dom (𝑝)
, t he imag e
img (𝑝)
. P artial functions are indicated
b y a
↪
. F or ins tance, t he successor function on natural numbers and its
(partiall y defined) in v erse are declared as follo w s:
suc c ∶ ℕ → ℕ pr e d ∶ ℕ ↪ ℕ

38 1 Introduction
W e consider partial functions as univ alent relations
{𝑥 ↦ 𝑦}
. These rela-
tions can be augmented using t he relational o v erride oper ator
⊕
, borro w ed
from t he specification languag e Z [Brien et al., 1992]:
𝑝⊕𝑞 ∧
= {𝑥 ↦ 𝑦 | 𝑥 ↦ 𝑦 ∈ 𝑝 and 𝑥 ∉ dom (𝑞)} ∪ 𝑞
Semantic functions are defined using a freel y adapted deno tational sty le:
⟦𝑡⟧ Δ
deno tes t he function
Δ
applied to
𝑡
. In order to enhance readability ,
t hese (recursiv e) functions are defined using patter n matching on t heir
arguments:
⟦𝑡 1 𝑡 2 ⟧ Δ
means t he application of
Δ
to one term formed b y t he
juxtaposition of tw o (possible distinct) terms (i.e. t he ter m representing
t he application of
𝑡 1
to
𝑡 2
). If multiple arguments are passed to a semantic
function, t he y are separ ated b y commas. Meta-v ariables are bound in t he
patter ns or corresponding where-clauses. When necessar y , w e consider
each function as over loaded on different syntactic sorts, i.e. t he function
Δ
can be applied to terms
𝑡
as w ell as top-lev el definitions
𝑇
, where neit her is
a subset of t he o t her .
The oper ator
∗
maps a semantic function on a sequence, e.g.
Δ ∗ 𝑡
yields
a sequence where ev er y element is the result of appl ying
Δ
to the corre-
sponding element in 𝑡 .
∧
=
deno tes meta-lev el equality (e.g. syntactic equality of ter ms), to a v oid
confusion wit h object-lev el equality (e.g. in an equation). W e consider terms
equal modulo
𝛼
-equiv alence, i.e., tw o terms t hat onl y differ in t heir bound
v ariables are considered equal. The same notation defines functions or
oper ators, but both contexts are easy to distinguish. W e write
𝑎 ∧
≠ 𝑏
to
indicate meta-lev el inequality .

2 the model c omposition lan gu a ge
This chap ter introduces the Model Com position Languag e , Mcl. Mcl is an
abs traction or intermediate representation of Modelica and retains onl y
t hose f eatures of Modelica t hat are necessar y to express the com positional
aspects of model ev aluation and elaboration. Despite this reduction, its
syntax and semantics are s till rather larg e. Therefore, t his chapter will
present t he different mechanisms incrementall y one af ter t he o t her , in sty le
com par able to Pierce [2002]. N ew languag e elements will be
highlighted
.
The com plete languag e definition can be found in Appendix B.
2 . 1 b ase lan gu a ge definition
Niklaus W irt h is credited wit h t he s tatement: “The mos t important decision
in languag e design concerns what is to be left out. ” A ccording to this
principle, it seems wise to start with a clean table and onl y add what is
necessar y . In the case of prog ramming languag e design, that clean table
is represented b y Church ’ s
𝜆
-calculus t hat is capable of describing an y
com putable function. The core of Mcl is a (somewhat v erbose) v ariant of
t he
𝜆
-calculus extended wit h fundamental primitiv es and data structures.
These extensions are no t chosen arbitraril y , but reflect their counter parts
in Modelica and represent one possible encoding of Modelica ’ s v alues in a
functional setting.
2.1.1 Synt ax
The syntax is sho wn in Figure 2.1. T erms
𝑡
or alter nativ el y
𝑠
consis t of t he
s tandard elements of the
𝜆
-calculus extended wit h non-s trict conditional
if
and an explicit fix ed-point operator
fix
(which ra ng es o v er multiple,
mutuall y recursiv e functions in
ℱ
). The expression
fix 𝑥 in ℱ
deno tes
t he function
𝑥
from t he group of mutually recursiv e functions
ℱ
. V alues
𝑣 ⊆ 𝑡 (alternativ l y 𝑤 ) are t he ev aluated nor mal forms.
Built-in primitiv es
𝑏
are booleans, rational numbers, integers, and s trings.
The corresponding binar y oper ators are summarized in
∘
(all oper ators
used t hroughout this t hesis ha v e t heir usual arit hmetic meaning). Mcl
dis tinguishes betw een operat ors on integers and r ational numbers. The
latter ones are suffix ed wit h a dot.

40 2 The Model Com position Languag e
Cor e T erms:
𝑡 , 𝑠⩴ 𝑥|𝑣|𝑡𝑡|𝑡 ∘ 𝑡
| if 𝑡 then 𝑡 else 𝑡
| let 𝑥 = 𝑡 in 𝑡
| [ 𝑡 ] | 𝑡 .[ 𝑡 ] | | 𝑡 |
| [ 𝑡 with 𝑡 = 𝑡 ]
| { 𝑓 } | 𝑡 . 𝑙
| { 𝑡 with 𝑙 = 𝑡 }
𝑓 ⩴ 𝑙 = 𝑡
V alues:
𝑣, 𝑤 ⩴ 𝑏 | λ 𝑥 . 𝑡 | fix 𝑥 in ℱ
| [ 𝑣 ] | { fv }
fv ⩴ 𝑙 = 𝑣
ℱ ⩴ 𝑥 = λ 𝑦 . 𝑡
Primitiv es:
∘ ⩴ + | * | < | = | <+
| +. | *. | …
𝑏⩴ 𝑖|𝑞| true | false
F igure 2.1: The core syntax of Mcl
There are also (non-recursiv e) bindings (introduced via
let
), functional
arr a ys
[ 𝑡 ∗ ]
and recor ds
{ 𝑓 ∗ }
of fields
𝑙 = 𝑡
o v er a (finite) set of labels
𝑙
. A
subset of t he labels is reser v ed for tr ansformations and thus considered
special , i.e., t he y are onl y allow ed as the result of semantic functions but not
part of t he concrete syntax. These labels are underlined .
R ecor ds and arr a ys can bo t h be projected (
𝑡 .[ 𝑡 ]
and
𝑡 . 𝑙
) as w ell as func-
tionall y updated (
[ 𝑡 with 𝑡 = 𝑡 ]
and
{ 𝑡 with 𝑙 = 𝑡 }
). The operator
| … |
com putes the lengt h of an arra y . The built-in operat or
<+
appends one arr a y
to ano t her .
W e write
⟦𝑡⟧ fv
to deno te t he set of free v ariables in a ter m. V ariables are
bound b y abstr action,
let
-expressions and t he mutuall y recursiv e functions
(plus t heir arguments) of a fix ed-point (Appendix C.1). Capture-a v oiding
subs titution
[𝑥 ↦ 𝑠]𝑡
of a v ariable
𝑥
wit h a term
𝑠
in a term
𝑡
is defined as
usual (Appendix C.2). As a shortcut, multiple v ariables can be replaced at
once b y a partial function
𝑝∶𝑥↪𝑡
, written as
[𝑝]𝑡
. The usual conditions for
freshness of bound v ariables ha v e to appl y to the codomain of the partial
function.
A context is a term wit h a “hole”, a special v ariable written as

. Plugging
a term
𝑡
into a context
𝑠
is defined b y substitution:
[  ↦ 𝑡]𝑠
. If necessar y ,
multiple holes
 1 , … ,  𝑛
can be dis tinguished b y t heir subscript. By con v en-
tion, no term binds a special v ariable (e.g., as t he v ariable of an abstr action).
W e use
∈
f or syntactical inclusion, i.e.,
𝑡 ∈ 𝑠
if and onl y if
𝑡 ⋆ ∧
= [  ↦ 𝑡]𝑠 ⋆
f or
some 𝑠 ⋆ .

2.1 Base Language Definition 41
Definition (Syntactical Context) . A cont ext is a t erm wit h a free hole:
𝑠 is a context ⟺  ∈ ⟦𝑠⟧ fv
Concrete Syntax
T o enhance readability , w e use a slightl y extended concrete syntax f or exam-
ples of Mcl. A dis tinctiv e f orm of binding, t he common
let rec
, introduces
recursiv e g roups. Multiple arguments of functions can be written sequen-
tiall y (i.e.,
λ 𝑥 𝑦 . 𝑡
ins tead of
λ 𝑥 . λ 𝑦 . 𝑡
). Functions can also be introduced
directl y via
let
(omitting t he
𝜆
). As a special form of recor ds, tuples are
introduced via parent hesis and commas, e.g.,
(1,42.0)
. Fields of tuples are
accessed wit h labels f ormed of natural numbers, e.g., x.2 .
2.1.2 Dynamic Semantics
Ev aluation is defined in big-step (also called natur al [Kahn, 1987]) s ty le: A
term
𝑡
ev aluates (or reduces) to a v alue
𝑣
, iff bo t h are related b y a reduction
relation
𝑡⇓𝑣
. Erroneous ter ms are identified b y not being related to some
v alue. The y are also called “stuck”, although t his terminus is more com-
mon in small-step semantics. All elements of
⇓
are defined inductiv ely b y
inf erence rules.
This s ty le is consider abl y shorter and somewhat easier to read than a
definition in small-s tep sty le while maintaining all the useful aspects of
inductiv e definitions. It also makes it relativ ely sim ple to im plement t he
languag e semantics in an interpreter (b y defining a partial function t hat
selects t he appropriate rule depending on t he form of t he ev aluated ter m
and recursiv ely applies itself t o sub-terms). It is, ho w ev er , impossible t o
express div erg ent beha vior . (In big-step s ty le a div erging term canno t be
dis tinguished from a stuck term without furt her pro visions since both are
no t related to some v alue.)
F igure 2.2 depicts t he basic rules of t he dynamic semantics of Mcl. V al-
ues are already in normal form (r ule V al). Op im plements binar y oper ators
on built-in primitiv es. It is a (larg e) famil y of r ules wit h one element f or
each built-in operat or . W e abbre viate t his larg e famil y b y assuming t hat
all oper ators either ha v e t heir usual arit hmetic meaning or are introduced
bef ore t he y are used.
R ules If-F alse and If- Tr ue implement non-s trict conditionals, while rule
Let allo ws f or con v enient binding of names to v alues. R ule App is t he
s tandard application via subs titution.

42 2 The Model Com position Languag e
Each sequence of fields, terms, or recursiv e functions can be seen as a
partial function t hat maps its left-hand side (or its position) to t he corre-
sponding right-hand side (or t he element). The function
Π
im plements t his
useful con v ersion:
⟦ ♦ ⟧ Π
∧
= ∅
⟦𝑓 ∷ 𝑙 = 𝑡⟧ Π
∧
= ⟦𝑓⟧ Π ⊕ { 𝑙 ↦ 𝑡 }
⟦ ℱ ∷ 𝑥 = 𝑡⟧ Π
∧
= ⟦ ℱ ⟧ Π ⊕ { 𝑥 ↦ 𝑡 }
⟦ [ 𝑣 0 , … , 𝑣 𝑛−1 ] ⟧ Π
∧
= { 0 ↦ 𝑣 0 }⊕…⊕{ 𝑛−1 ↦ 𝑣 𝑛−1 }
The application of one of sev eral mutuall y recursiv e functions is im ple-
mented wit h the substitution of all recursiv e functions. This substitution is
build wit h 𝜇 :
⟦ ℱ ⟧ 𝜇
∧
= { 𝑥 𝑖 ↦ fix 𝑥 𝑖 in ℱ | 𝑥 𝑖 ∈ dom (⟦ ℱ ⟧ Π ) }
F irs t t he argument has t o be ev aluated (similar to rule App). This ar -
gument is t hen subs tituted into the body of t he function, f ollo w ed b y a
subs titution of t he group itself (r ule FixApp). The latter steps enables the
actual recursion.
Mcl contains tw o kinds of data s tructures: F unctional arr a ys are con-
s tructed wit h brack ets
[ … ]
(rule Arra y). Arra y elements can be accessed
b y an integer index s tarting wit h
0
f or t he first element (rule Arra yId x).
An arr a y update yields a new arr a y wit h one element set to a ne w v alue
(rule Arra yUpd). The lengt h of an arra y is the number of its elements (r ule
Arra yLen). As already mentioned, an element can be appended to an arr a y
wit h the builtin operator <+ .
R ecor ds are constructed with cur l y braces
{ … }
(rule Rec ord). V alues
bound to a label
𝑙
can be accessed via projection (rule Pr oject). As wit h
arr a ys, recor ds ma y be constructed as updated v ersions of existing recor ds
(rule Upd a te). In such a case, all existing fields but the updated one are
copied. There is no w a y to add a new label t o a recor d.
2.1.3 S t atic Semantics
Usuall y , t he specification of a languag e contains an in v estig ation of its static
semantics, i.e. a specification of t he properties t hat can be deduced without
resorting to e v aluation. The most common exam ple is a decidable, sound
type sys tem, i.e. t he g ener ation of a mechanical proof about t he absence of
certain errors in a closed expression. This is wit hout doubt a useful tool

2.1 Base Language Definition 43
V al
𝑣 ⇓ 𝑣
Op
𝑣 1 ∘ 𝑣 2 results in 𝑣 3
𝑡 1 ⇓ 𝑣 1 𝑡 2 ⇓ 𝑣 2
𝑡 1 ∘ 𝑡 2 ⇓ 𝑣 3
If- Tr ue
𝑡 1 ⇓ true 𝑡 2 ⇓ 𝑣
if 𝑡 1 then 𝑡 2 else 𝑡 3 ⇓ 𝑣
If-F alse
𝑡 1 ⇓ false 𝑡 3 ⇓ 𝑣
if 𝑡 1 then 𝑡 2 else 𝑡 3 ⇓ 𝑣
Let
𝑡 1 ⇓ 𝑣 1 [𝑥 ↦ 𝑣 1 ]𝑡 2 ⇓ 𝑣 2
let 𝑥 = 𝑡 1 in 𝑡 2 ⇓ 𝑣 2
App
𝑡 2 ⇓ 𝑣 𝑡 1 ⇓ λ 𝑥 . 𝑠
[𝑥 ↦ 𝑣]𝑠 ⇓ 𝑤
𝑡 1 𝑡 2 ⇓ 𝑤
FixApp 𝑡 1 ⇓ fix 𝑥 in ℱ 𝑡 2 ⇓ 𝑣
𝑥 ↦ λ 𝑦 . 𝑠 ∈ ⟦ ℱ ⟧ Π [⟦ ℱ ⟧ 𝜇 ⊕ { 𝑦 ↦ 𝑣 }]𝑠 ⇓ 𝑤
𝑡 1 𝑡 2 ⇓ 𝑤
Arra y
∀𝑖 ∈ 1 … 𝑛 ∶ 𝑡 𝑖 ⇓ 𝑣 𝑖
[ 𝑡 1 , … , 𝑡 𝑛 ] ⇓ [ 𝑣 1 , … , 𝑣 𝑛 ]
Arra yId x
𝑡 1 ⇓ [ 𝑣 ] 𝑛 ↦ 𝑤 ∈ ⟦𝑣⟧ Π 𝑡 2 ⇓ 𝑛
𝑡 1 .[ 𝑡 2 ] ⇓ 𝑤
Arra yLen
𝑡 ⇓ [ 𝑣 1 , … , 𝑣 𝑛 ]
| 𝑡 | ⇓ 𝑛
Arra yUpd
𝑡 1 ⇓ [ 𝑣 ] 𝑡 2 ⇓ 𝑛 𝑡 3 ⇓ 𝑣 𝑛
𝑖 ∈ dom (⟦𝑣⟧ Π ) ⟦ [ 𝑤 ] ⟧ Π
∧
= ⟦ [ 𝑣 ] ⟧ Π ⊕ {𝑖 ↦ 𝑣 𝑛 }
[ 𝑡 1 with 𝑡 2 = 𝑡 3 ] ⇓ [ 𝑤 ]
Rec ord ∀𝑖 ∈ 1 … 𝑛 ∶ 𝑡 𝑖 ⇓ 𝑣 𝑖
{ 𝑙 1 = 𝑡 1 , … , 𝑙 𝑛 = 𝑡 𝑛 } ⇓ { 𝑙 1 = 𝑣 1 , … , 𝑙 𝑛 = 𝑣 𝑛 }
Pr oject
𝑡 ⇓ { fv } 𝑙↦𝑣∈⟦ fv ⟧ Π
𝑡 . 𝑙 ⇓ 𝑣
Upd a te ⟦ gv ⟧ Π
∧
= ⟦ fv ⟧ Π ⊕ { 𝑙 ↦ 𝑤 }
𝑡 2 ⇓ 𝑤 𝑡 1 ⇓ { fv } 𝑙 ∈ dom (⟦ fv ⟧ Π )
{ 𝑡 1 with 𝑙 = 𝑡 2 } ⇓ { gv }
F igure 2.2: Ev aluation semantics of core Mcl

44 2 The Model Com position Languag e
in t he hands of modelers and programmers alike, as it ex cludes a whole
categor y of errors already at com pile time. Of course it comes wit h a price:
The expressiv eness of the languag e in ques tion is limited b y t he po w er of
t he proof g enerator . Some languag e features (e.g. reflection or runtime
metaprog r amming) might eit her no t be allo w ed or (more of ten) ex em pt
from t he guar antees of type-saf ety .
Alt hough Mcl does no t contain an y features that a priori prev ent t he
exis tence of such a type system, w e lea v e it undefined for the follo wing
reasons: The presence of arr a ys and records demand a somewhat elabor ate
type sys tem ev en for the basic elements of t he language. Discussing t he
peculiarities of such a system w ould not onl y s tra y from t he topic of this
t hesis. It w ould also take a lo t of space wit hout adding an y no v el aspects t hat
canno t already be found in the literature. Assigning types to t he language
elements t hat go be y ond t he basics of Mcl almost certainl y does pro vide an
interes ting topic of in v estig ation. This, ho w ev er , w ould be a t hesis on its
o wn right.
Theref ore, during t he course of this w or k , Mcl is considered a type-
agnos tic languag e: A sound, decidable type system ma y exist or no t, but t his
w or k neit her depends on its exis tence or deliberatel y denies it. Section 6.3
contains a sketch on ho w types ma y be assigned to at leas t some elements.
2.1.4 Exam ple
As a sim ple demons tr ation of t he functional aspects of Mcl, consider a pair
of recursiv e functions,
even
and
odd
t hat determine whet her a giv en natur al
number is ev en or not. Such an operation can be triviall y implemented
b y a modulus operat or . How ev er , a recursiv e im plementation is a better
sho w case for the core language.
let rec even = λ n . if n=0 then true else odd (n - 1)
and odd = λ n . if n=0 then false else even (n - 1)
in even 42
The exam ple uses t he concrete syntax of Mcl, ex changing t he somewhat
cumbersome fix ed-point notation with t he more common recursiv e
let
.
The definition is equiv alent to the application
( fix even in ℱ ) 42
when
ℱ is defined as f ollo ws:
ℱ ∧
= even = λ n.if n = 0 then true else odd ( n-1 ) ∷
odd = if n = 0 then false else even ( n-1 ) ∷ ♦
F or t his definition, it follo ws b y r ules FixApp, Op, and If- Tr ue that
( fix even in ℱ ) 0 ⇓ true

2.2 Modeling Routines 45
and ( fix odd in ℱ ) 1 ⇓ true
A s traightf or w ar d inductiv e argument expands t hese statements t o all ev en
natur al numbers
2𝑛
and all odd numbers
2𝑛 + 1
, respectiv ely . Hence, w e
can conclude t hat the example e v aluates to true as expected.
2 . 2 modeling r outines
Elabor ation is t he firs t aspect t hat Mcl inherits from Modelica. This process
can be underst ood as t he gener alization of tw o fundamental modes of
model-com position: A model can ins tantiat e another model and bind t he
result as a component of itself. It might also inherit the beha vior of another
model, in which case it reflects the shape of t he class it ext ends . In both
cases, t here is an inclusion relation betw een t he beha vior of both models
(one contains t he o t her).
2.2.1 A bs tr act Elabor ation
T o g ain an understanding of the abstr act interface of t hese inter actions
betw een models, let us conceptuall y view elabor ation as a function
𝜖
on a
set
𝕄
containing all models that can be elaborated. The definition of t his
function in Mcl will t hen mo tiv ate t he necessar y extensions to syntax and
semantics. Consider a simple Modelica model:
model M
Real x;
end M;
Elabor ation of t his model is equiv alent to the allocation of a single
unkno wn,
𝜉 𝑖
and binding of t hat unkno wn to t he v ariable
x
. The former
is t he elabor ation ’ s beha vior , while t he latter describes its observable r esult .
Im plementation of the beha vior requires some context:
•
Jus t b y looking at
M
, it is im possible to tell the full y qualified name of
t he unkno wn (which is necessar y for the anal ysis of the simulation
results).
•
It is as w ell unclear which part of t he solution (i.e., t he
𝜉 𝑖
)
x
represents
since an arbitr ar y amount of other unkno wns might be part of t he
model.

46 2 The Model Com position Languag e
•
F inall y , it is im possible to decide ho w to deal with
x
during t he sym-
bolic tr ansformation.
It is clear , ho w ev er , that
M
is perf ectl y legal and t hus might be ins tantiated
in sev eral different contexts, which ma y pro vide different answ ers to these
ques tions. Thus, the elaboration of a model
𝑀∈𝕄
needs —as a first
appro ximation— a second argument Γ that contains t he required context.
𝜖∶𝕄→Γ→…
This context has to pro vide answ ers to se v eral v er y different questions,
so it seems natur al to encode it as a (partial) function itself. That w a y , each
such ques tion can be labeled wit h a label
𝑙
, and t he answ er can be encoded
as a v alue 𝑣 . Γ∶𝑙↪𝑣
The question remains ho w to define the image of the function
𝜖
. Elab-
or ation of a com ponent ma y yield more obser v able results t han just an
index
𝑖
f or some unkno wn. Instead, it giv es access to the component’ s inner
s tructure (called an inst ance ). It is possible to access unkno wns, parameters
and cons tants after a model has been instantiated and use them, e.g., in
t he elabor ation of t he next model. Assuming that an y such instance can be
encoded as a v alue 𝑣 , t he “signature” of 𝜖 might look like this:
𝜖∶𝕄→Γ→𝑣
There is, ho w ev er , also a side effect during elabor ation. W e kno w t hat
x
is a new v ariable and t hus it requires a position in t he v ector of unkno wns.
This position must no t y et be occupied and should not be occupied b y an y
subseq uent elabor ations of o t her models. This requires t hat elaboration yields
an updat ed context
Γ 2
∧
= Γ 1 ⊕ {𝑙 ↦ 𝑛 + 1}
, where
𝑙↦𝑛∈Γ 1
is t he old number
of allocated positions. The new context appears on the right-hand side of
t he signature: 𝜖 ∶ 𝕄 → Γ → Γ × 𝑣
In functional prog r amming, such functions, i.e., functions t hat take a
(hidden) s tate and yield a v alue and a new s tate, can be comfortabl y imple-
mented wit h the S tat e Monad (called S tate T r ansformer in [W adler, 1990])
patter n. Its main benefit is t he sim ple com position of monadic operations
wit h the bind operator that a v oids t he exposure of the inter nal state. The
modeling primitiv es in Mcl are closel y related to a comfortable no tion of
such s tate monads.

2.2 Modeling Routines 47
2.2.2 Com position of Submodels
It remains to see ho w t his metamodel of elabor ation fits modeling in t he
s ty le of Modelica. Model instantiation is the simples t form of composition.
As already mentioned, a model
M
can be instantiated b y other models in
different contexts:
model N
M m1, m2;
equation
m1.x = m2.x;
end N;
In t he exam ple abo v e, model
M
is ins tantiated as com ponents
m1
and
m2
.
Elabor ation of t he com posite model N proceeds as f ollo ws:
𝜖⟦ N , Γ⟧ ∧
= ⟨Γ 3 , {m1 = 𝑣 1 ; m2 = 𝑣 2 } ⟩
where ⟨Γ 3 , _ ⟩ ∧
= 𝜖⟦𝑣 1 .x = 𝑣 2 .x , Γ 2 ⟧
⟨Γ 2 , 𝑣 2 ⟩ ∧
= 𝜖⟦ M , Γ 1 ⟧
⟨Γ 1 , 𝑣 1 ⟩ ∧
= 𝜖⟦ M , Γ⟧
Elabor ation of
M
is ex ecuted first with t he original context
Γ
and t hen
wit h the updated context
Γ 1
. The retur ned v alues
𝑣 1
and
𝑣 2
are firs t used to
cons truct and elaborate an equation (lea ving out the details of t hat particular
process f or no w). After w ar ds, t hey cons titute t he instance
{m1 = 𝑣 1 ; m2 = 𝑣 2 }
.
This instance and the new context
Γ 3
are t he final results of elabor ation.
The v alue obtained from elaborating the equation is not used here since
equations are no t first class citizens in Modelica. It w ould, ho w ev er , be
trivial to add them as an obser v able v alue in t his process.
The semantics of elaborating an inherited class can be expressed in a
similar w a y :
model N
extends M;
M m2;
equation
x = m2.x;
end N;
The model abo v e replaces one com ponent wit h an
extends
-clause. This
clause declares a
M
as a base class of
N
. The model has t he same beha vior as
bef ore but exposes a different obser v able result.

48 2 The Model Com position Languag e
𝜖⟦ N , Γ⟧ ∧
= ⟨Γ 3 , {x = 𝑣 1 .x; m2 = 𝑣 2 } ⟩
where ⟨Γ 3 , _ ⟩ ∧
= 𝜖⟦𝑥 = 𝑣 2 .x , Γ 2 ⟧
⟨Γ 2 , 𝑣 2 ⟩ ∧
= 𝜖⟦ M , Γ 1 ⟧
⟨Γ 1 , 𝑣 1 ⟩ ∧
= 𝜖⟦ M , Γ⟧
As in t he case of ins tantiation,
M
is elabor ated twice wit h different con-
texts. The obser v able result
𝑥
of elabor ating t he base class is no w directl y
inserted into the result v alue of t he base class.
V alues:
𝑣, 𝑤 ⩴ 𝑏 | λ 𝑥 . 𝑡 | fix 𝑥 in ℱ
| [ 𝑣 ] | { fv } | 𝑚
fv ⩴ 𝑙 = 𝑣
ℱ ⩴ 𝑥 = λ 𝑦 . 𝑡
𝑘 ⩴  | 𝑥 . 𝑡 ∶∶ 𝑘
Modeling Routines:
𝑚 ⩴ 𝑙 •get | 𝑙 •put 𝑡
| 𝑥 ← 𝑡 ; 𝑡 | return 𝑡
F igure 2.3: Mcl Elaboration Syntax
2.2.3 Elabor ation Syntax
T o im plement elaboration, Mcl contains a small set of operators f or t he
com putation of modeling routines
𝑚
(F igure 2.3). The primitiv e
return
lifts a v alue into an observ able result,
𝑙 •get
obser v es t he elaboration aspect
associated wit h label
𝑙
in t he current s tate and
𝑙 •put 𝑡
sets t hat aspect to
𝑡
. The operat or
𝑥 ← 𝑡 1 ; 𝑡 2
represents t he monadic bind that allo ws f or t he
com position of routines. The v ariable
𝑥
cap tures the obser v able result of
𝑡 1
in t he context of
𝑡 2
. All t hese new v ariants are values . The inductiv e
definition of t he languag e of ter ms
𝑡
remains unchang ed b y t his extension.
R outines are first-class entities (i.e., they can be passed t o and retur ned b y
functions).
The dynamic semantics of elabor ation is defined in a w a y inspired b y t he
continuation passing s ty le [Appel, 1992]. F or this pur pose, a continuation
𝑘
is eit her em p ty (

) or a combination of a v ariable
𝑥
and a term
𝑡
wit h
a f ollo w -up continuation
𝑥 . 𝑡 ∶∶ 𝑘
. Continuations can be seen as a list of
abs tractions. Inf ormall y , this list s tores all future s teps of elaboration.

2.2 Modeling Routines 49
⟦𝑙 •get ⟧ fv
∧
= ∅
⟦𝑙 •put 𝑡⟧ fv
∧
= ⟦𝑡⟧ fv
⟦ return 𝑡⟧ fv
∧
= ⟦𝑡⟧ fv
⟦𝑥 ← 𝑡 1 ; 𝑡 2 ⟧ fv
∧
= ⟦𝑡 1 ⟧ fv ∪ (⟦𝑡 2 ⟧ fv ⧵ { 𝑥 } )
F igure 2.4: Calculating t he free v ariables on routines
The calculation of free v ariables (and t hus t he definition of lexical scopes)
is sho wn in Figure 2.4. The routine
get
can not contain an y v ariables.
return
and
put
do no t introduce a new scopes to their respectiv e enclosed term.
The bind oper ator introduces a ne w lexical scope for its second argument.
2.2.4 Model Elabor ation
As already mentioned, elaboration is specified in a continuation-passing
s ty le to enable fine-grained control o v er the elaboration. This deviation
from t he classical encoding of a s tate monad is justified b y t he fact that
t here is a certain equiv alence betw een CPS and monadic notation as W adler
[1992] points out. The model-elaboration semantics (F igure 2.5) consists
of tw o mutually recursiv e relations: A continuation
𝑘
can pr ogr ess under
an en vironment
Γ
wit h an in put v alue
𝑣
, yielding an output v alue
𝑤
. This
is written
⟨Γ, 𝑣⟩ ⊢ 𝑘 ⇓ k 𝑤
. Execution of a routine
𝑚
to an observ able v alue
𝑣
under an en vironment
Γ
wit h the next continuation
𝑘
is expressed as
⟨Γ, 𝑘⟩ ⊢ 𝑚 ⇓ m 𝑣 .
T w o r ules define progression: R ule St op describes t he end of elaboration.
When t he current continuation is em pty , t he result is t he in put v alue. R ule
C ont describes ho w elaboration proceeds. When the next continuation ev al-
uates to a routine, this routine is ex ecuted with t he remaining continuation
𝑘 and t he current en vironment.
R ules MPut and MGet allo w interaction with parts of t he global state.
Quer ying an aspect of t he s tate is achiev ed b y passing t he resulting v alue
to the current continuation. Setting an aspect requires t he ev aluation of an
argument term. The resulting v alue is used to update the en vironment and
t he current continuation proceeds with t he emp ty record (indicating no
obser v able v alue). R ule MRet describes how t he results of “normal” ev alu-
ation can be used during elaboration b y lif ting t hem via a
return
-s tatement.

50 2 The Model Com position Languag e
St op
⟨Γ, 𝑣⟩ ⊢  ⇓ k 𝑣
C ont
[𝑥 ↦ 𝑣]𝑡 ⇓ 𝑚 ⟨Γ, 𝑘⟩ ⊢ 𝑚 ⇓ m 𝑤
⟨Γ, 𝑣⟩ ⊢ 𝑥 . 𝑡 ∶∶ 𝑘 ⇓ k 𝑤
MPut
𝑡 ⇓ 𝑣 𝑙 ∈ dom (Γ)
⟨Γ ⊕ {𝑙 ↦ 𝑣}, {} ⟩ ⊢ 𝑘 ⇓ k 𝑤
⟨Γ, 𝑘⟩ ⊢ 𝑙 •put 𝑡 ⇓ m 𝑤
MGet
𝑙 ↦ 𝑣 ∈ Γ ⟨Γ, 𝑣⟩ ⊢ 𝑘 ⇓ k 𝑤
⟨Γ, 𝑘⟩ ⊢ 𝑙 •get ⇓ m 𝑤
MRet 𝑡 ⇓ 𝑣
⟨Γ, 𝑣⟩ ⊢ 𝑘 ⇓ k 𝑤
⟨Γ, 𝑘⟩ ⊢ return 𝑡 ⇓ m 𝑤
MBind
𝑡 1 ⇓ 𝑚 ⟨Γ, 𝑥 . 𝑡 2 ∶∶ 𝑘⟩ ⊢ 𝑚 ⇓ m 𝑣
⟨Γ, 𝑘⟩ ⊢ 𝑥 ← 𝑡 1 ; 𝑡 2 ⇓ m 𝑣
F igure 2.5: Elaboration semantics: R outines
The result of ev aluating t he first ar gument is passed to the current continu-
ation f or furt her prog ression. R ule MBind deals wit h t he composition of
monads — it is an adap tion of the monadic bind -oper ator (sometimes also
called flatmap ) to the meta-lev el: T o elaborate tw o composed model-routines,
t he firs t routine has to be elabor ated with t he second routine becoming t he
next continuation. Thus, changes in the state during the elaboration of the
firs t routine as w ell as the resulting v alue are passed to the second routine
via rule C ont.
2.2.5 Executing Modeling Routines
As already mentioned, Modelica does not offer an y language elements to
perf orm a simulation directl y . Instead, the setup of parameters, sol v er set-
tings, simulation output, etc. is left to t he im plementation (with t he notable
ex ception of the standar dized
experiment
-anno tation, which pro vides an
op tional entry point).
Since Mcl shall not rel y on an external setup, it is necessar y to add t hese
f eatures as w ell. N aturall y , t his takes the form of additional elaboration
primitiv es. F ollo wing t he design of t he elaboration primitiv es introduced
abo v e, t he languag e elements f or t his task are as g eneral as possible. Since
simulation, tr ansformation, and elabor ation are all expressible as modeling
routines, the start of a simulation can be achie v ed b y t he ex ecution of a
modeling routine wit h an initial s tate. Such an initial state can be defined

2.2 Modeling Routines 51
V alues:
𝑣, 𝑤 ⩴ 𝑏 | λ 𝑥 . 𝑡 | fix 𝑥 in ℱ
| [ 𝑣 ] | { fv } |𝑚| < fv >
fv ⩴ 𝑙 = 𝑣
ℱ ⩴ 𝑥 = λ 𝑦 . 𝑡
𝑘 ⩴  | 𝑥 . 𝑡 ∶∶ 𝑘
Cor e T erms:
𝑡 , 𝑠⩴ 𝑥|𝑣|𝑡𝑡|𝑡 ∘ 𝑡
| …
| < 𝑓 >
| exec 𝑡 with 𝑡
F igure 2.6: Mcl wit h ex ecution
in a w a y similar to a recor d (Figure 2.6), using angular br ackets t o mar k t he
dis tinction. The primitiv e operator
exec 𝑡 1 with 𝑡 2
ex ecutes t he modeling
routine obtained from the ev aluation of
𝑡 1
.
𝑡 2
yields t he necessary initial
elabor ation context. Since t his pro vides an entr y point for the simulation of
a model, t he oper ator is itself a t erm . Hence, t he ev aluation and elaboration
are mutuall y recursiv e processes.
St a te
∀𝑙 𝑖 = 𝑡 𝑖 ∈ fv ∶ 𝑡 𝑖 ⇓ 𝑣 𝑖
< 𝑓 > ⇓ < 𝑙 𝑖 = 𝑣 𝑖 >
R un
𝑡 1 ⇓ 𝑚 𝑡 2 ⇓ < fv > ⟨⟦ fv ⟧ Π ,  ⟩ ⊢ 𝑚 ⇓ m 𝑣
exec 𝑡 1 with 𝑡 2 ⇓ 𝑣
F igure 2.7: Elaboration Semantics, Ex ecution
Ev aluation of a state expression (rule St a te) proceeds similar l y to a
recor d. If t he right-hand side of ev er y field
𝑙 𝑖 = 𝑡 𝑖
ev aluates to a v alue
𝑣 𝑖
, t he
s tate expression ev aluates to a s tate v alue. R ule R un combines ev aluation
and elabor ation. The premise consists of t hree parts: The first argument
𝑡 1
needs to e v aluate to a model routine
𝑚
. The second argument
𝑡 2
shall
ev aluate to a state
< fv >
. This state is then tur ned into an elabor ation en viron-
ment obtained b y constructing a partial function as the union of all fields.
If t he ex ecution of
𝑚
succeeds under t his new en vironment and t he emp ty
continuation, t hen the ter m
exec 𝑡 1 with 𝑡 2
ev aluates to t he obser v able
v alue of t hat elabor ation. This is t he onl y w a y to “escape” from the monadic
routines.

52 2 The Model Com position Languag e
2.2.6 Exam ple
Alt hough elabor ation is intended to cons tr uct a system of equations, its
im plementation in Mcl is quite flexible. There are no constr aints on t he
s tructure of the elaboration context, and it can be (ab-)used for arbitr ar y
com putations. As an example, consider ag ain t he distinction betw een ev en
and odd numbers, t his time in an im perativ e v ariant:
let rec even = do n ← number • get ;
if n=0 then return true
else do
number • put (n - 1) ;
odd
and odd = do n ← number • get ;
if n=0 then return false
else do
number • put (n - 1) ;
even
in exec even with number = 42
This exam ple ag ain uses a more concrete syntax. The ke yw ord
do
in-
troduces com posite routines, and w e omit t he v ariable in bindings when
it is unused. The result of t he elaboration is a s tate with only one aspect,
number
. Its final v alue is
0
. The obser v able v alue is
true
. This can be sho wn
similar l y to Section 2.1.4.
2 . 3 classes and object -oriented modeling
One of t he mos t im portant benefits of Modelica is arguabl y its flexibility:
Libr aries can co v er v er y different modeling domains (e.g., electrical or me-
chanical engineering), support different applications of simulation results
(e.g., op timization or real-time control), and build upon different lev els of
abs traction (e.g., lum ped or distributed elements). Models from different
libr aries can be combined to meet man y potential requirements.
This flexibility results from t he adop tion of f eatures usuall y associated
wit h object-oriented languag es. Syntacticall y , Modelica ’ s classes are com-
posed in hier archies. Each class (ex cep t f or t he unnamed top-le v el class) is a
member of, and has access to, its parent. A parent can also access elements
defined inside its children, using t he notation of com pound names sepa-
r ated b y a dot. Classes and models can also ext end each o t her via inheritance.
In addition classes can modify or r edeclar e elements of other classes to create

2.3 Classes and Object-Oriented Modeling 53
Lis ting 2.1: Hierarchical Lookup
class C
constant Integer y = B.z;
class B
class A
constant Integer x = y + z;
end A;
constant Integer z = 21;
end B;
end C;
C.B.A.x
z
B.z y
a new v ersion of an existing model. This section defines t hese im portant
aspects as t he object-oriented parts of Mcl.
Modelica is no t a classical object-oriented languag e, ho w ev er . Its classes
share a lo t (and ev en sur pass some) of t he f eatures found in common lan-
guag es like Ja v a or C++, but t he similarities end when it comes to objects. In
Section 2.2.1 it w as assumed t hat Modelica ’ s objects can be represented b y
Mcl’ s v alues, in particular its recor ds. The reason f or t his is t hat Modelica ’ s
objects do not encapsulate beha vior , as “normal” objects do. Instead, a class
contributes to a global model via elabor ation while t he resulting object
merel y structures t he resulting v alues. Modelica ’ s objects are immutable .
There is no w a y to chang e an instance or equation after elaboration. F or
t his reason, Mcl does not require an y notion of objects in the classical sense.
Ins tead, it solel y deals wit h t he composition of classes.
2.3.1 Scoping and Hier ar c hies
In its sim ples t f orm, a Modelica class ser v es as a container for a sequence
of declar ations . These ma y introduce constants, par ameters, unkno wns or
declare com ponents that are instances of o t her classes which then ser v e as
sub-containers. Some declarations —lik e equations— are unnamed. The
elabor ation semantics of such a sequence is already co v ered quite w ell b y
Mcl’ s monadic routines. What is not co v ered is t he relation betw een t hese
declar ations and t he syntactical v ariables used on t heir right-hand side.
Mcl uses a relativ ely s traight-f or w ard lexical scoping. Modelica, on t he
o t her hand, features a some what more com plicated binding of names as
t he exam ple in Listing 2.1 sho ws.
Here, t he declar ation of t he cons tant
x
in class
A
ref ers to tw o free v ari-
ables,
y
and
z
. Both are not declared inside the class
A
itself but outside of it.

54 2 The Model Com position Languag e
Class
A
is a child of class
B
, which is in tur n a child of
C
in t he class hier ar chy .
Hence it “sees” all declarations
1
of its parent classes. In t he classical sense,
B
is part of
A
’ s lexical scope. Therefore,
z
is f ound directl y in the surrounding
scope. N ote that t he definition of constant
z
(t he liter al
21
) is syntacticall y
placed after
A
. The scope of a binding is independent of declaration or der .
It co v ers its whole surrounding class.
V ariable
y
is no t defined inside
B
. The next candidate is
C
. There, it
is f ound, defined as
B.z
. Such composite names, separ ated b y dots, giv e
access to elements down war ds t he hierarch y . In a firs t s tep,
B
is found as
bef ore in t he scope of
C
. The result of t his search is t hen used to search f or
z
, which is defined as
21
. Hence t he result of ev aluating
C.B.A.x
yields
42
.
2.3.2 Inherit ance and Modifications
This scoping might seem pretty standar d (af ter all similar rules exist in mos t
languag es with classes). Ho w ev er , t here is a subtle difficulty embedded in
t his seemingl y simple principle: Modelica does not differentiate betw een
looking up a class (e.g.,
B
) and its fields (e.g.,
z
). This design decision might
seem like an eleg ant unification, but it turns out to be a source of significant
com plication in combination with inheritance because base classes need to
be looked up t o determine t he set of visible names in a class. This ambiguity
can make it some what challenging to find the source of a definition.
class D
extends C.B(z=2);
constant Integer x = A.x;
end D;
class C … end C;
In t he class
D
abo v e, wit h
C
left unchanged from Lis ting 2.1, what is t he
v alue of
D.x
? Since
A
is inherited in
D
from
C.B
, it is temp ting to assume
t he answ er is, again,
42
. This is no t t he case
2
: The retur ned v alue is
23
.
U nf ortunatel y , at t he time of writing this t hesis, t he Modelica Specification
makes the reason for this somewhat har d to unders tand:
The f ound base class is flattened wit h a new en vironment and
t he partiall y flattened enclosing class of t he extends-clause.[…]
The elements of t he flattened base class become elements of
t he flattened enclosing class, and are added at the place of t he
extends-clause. (Modelica Specification Chapter 7, Section 1)
1 A t least the ones with t he proper v ariability
2 as discussed in https://trac.modelica.or g/Modelica/ticket/2013

2.3 Classes and Object-Oriented Modeling 55
Lis ting 2.2: Hierarchical lookup with inheritance
class D
extends C.B(z=2);
constant Integer x = A.x;
end D;
class C
constant Integer y = B.z;
class B
class A
constant Integer x = y + z;
end A;
constant Integer z = 21;
end B;
end C;
D.x
A.x
z
B.z y
It is easy to see that t he “f ound base class” is
C.B
, and t he enclosing class
of t he extends-clause is natur all y
D
, but t he ke yw or ds here are “ partiall y flat-
t ened enclosing class” and “ flatt ened base class”. The specification does not
giv e a precise definition of what a (partiall y) flattened class looks like, but it
is no t just a liter al cop y of t he class’ definition since the original definitions
retain t heir lexical scope. It is also not jus t an agg regation since modifica-
tions and redeclar ations in t he extending class might effect definitions in
t he base class.
Bef ore w e in v estig ate a more precise definition, let us ha v e a look at ho w
lookup actuall y proceeds (Listing 2.2, accor ding to the consent of different
im plementations and discussions in the Modelica Design Group). In the
firs t step,
A
is f ound, inherited from t he base class
C.B
. Hence,
A.x
is resol v ed
b y looking for
x
in
C.B.A
. This class contains t he same definition of
x
as
bef ore. A ccor dingl y , the same v ariables need to be looked up. V ariable
z
is
ag ain looked up in its immediatel y enclosing class. This time, ho w ev er , t his
class is not
A
, but b y the sub-class
D
(as determined b y t he specification).
Theref ore t he resulting v alue is
2
. In an interesting twis t,
y
is no t subject to
t his modification. Its lookup passes t hrough
C
and onl y t hen retur ns to the
definition of
z
. The inheritance is discarded. The v alue
21
is f ound in t he
lexical scope of A . The o v er all ev aluation yields 23 .
This exam ple demons tr ates an im portant fact about classes in Modelica:
The site of t he definition of a free v ariable is not a syntactic property of the
class. Ins tead, it depends on t he cont ext in which this class is used. F or a
separ ate com pilation, it is t hus necessar y to express this dependency in t he

56 2 The Model Com position Languag e
abs tract syntax in such a w a y t hat t he com pilation preser v es its meaning in
all possible contexts.
2.3.3 The Principle of Open Recursion
This context is t he result of ev aluating all relev ant base classes. Therefore,
t he definition of lookup has to be part of the ev aluation of classes and vice
v ersa. In Modelica, t here is no explicit ordering betw een declarations. Due
to the existence of inheritance and because classes are look ed up in the
same w a y as o t her declar ations, such an ordering canno t be found without
kno w ledge of the context of t he class. Both t he construction of t he context
and t he ev aluation of class references are recursiv el y linked.
In classical object-oriented languag es, this principle is called open re-
cursion [Aldrich and Donnell y, 2004]: Each met hod has access to a special
v ariable (of ten called
this
or
self
). Methods are alw a ys in v oked from a
concrete object (sometimes called t he receiv er of a message). This object
t hen becomes the definition of t he special v ariable during ev aluation of
t he method’s body (the special v ariable is late bound ). Free v ariables in
t he method are usually interpreted as t he in v ocation of a met hod on t he
special v ariable. This principle yields an im plementation of r ecursion since
t he method itself is an element of t he receiving object. It is open because
t he method might be part of different concrete objects (and in v oke different
siblings on each). It is possible to chang e t he beha vior of all met hods of an
object b y ex changing onl y one met hod. The same concep t can be used to
explain t he lookup inside Modelica ’ s classes when it is applied not onl y to
one but possibl y man y special objects (Listing 2.3).
In a certain sense, t he v ariables
y
and
z
are dynamicall y bound, and
t heir v alue depends on t heir hierarchical context. In the abov e listing,
t he ref erences to that context ha v e been codified b y tw o kinds of special
v ariables.
this
deno tes a reference t o t he immediatel y enclosing class, while
up(
i
)
expresses access to the
𝑖
-t h enclosing class (hence
up(
0
)
is t he same
as
this
, but less readable). Such a form eliminates all free v ariables, and
once it is found b y a static anal ysis (which will be sketched in Section 4.1),
it can be used in all contexts.
W e also mo v ed t he modification into t he scope of t he extending class.
Modifications are resol v ed in t he scope of t he modifying class. Hence, t his
merel y clarifies t he lexical position of t he definition of
z
. It also illus trates t he
relationship betw een t he modification of an extends-clause in Modelica and
t he o v erriding of met hods in more con v entional object-oriented languages
like Ja v a.

2.3 Classes and Object-Oriented Modeling 57
Lis ting 2.3: Hierarchical lookup with special v ariables
class D
constant Integer z = 2;
//conceptual modification
extends up(1).C.B;
constant Integer x = this.A.x;
end D;
class C
constant Integer y = this.B.z;
class B
class A
constant Integer x = up(2).y + up(1).z;
end A;
constant Integer z = 21;
end B;
end C;
2.3.4 The Hier ar c hical Envir onment
The exam ple in Lis ting 2.2 demonstr ates t hat t he property of being an
enclosing class is no t transitiv e: The enclosing class of
D.A
’ s immediate
parent
D
— t he unnamed roo t class— is different from
D.A
’ s next ancestor ,
C
.
T o unders tand t his complication, w e resort ag ain to an inf or mal deno tational
in v estig ation of the matter . F or reasons t hat will become clear in a moment,
w e call an ev aluated class a node of t he syntactic sort
𝜈
and deno te t he sort
of en vironments wit h
ℰ
. Then t he operations can be giv en t he (informal)
signatures:
ℰ ⊆ ℕ ↪ 𝜈
no de ∶ 𝐶 × ℰ → 𝜈
env ∶ 𝜈 → ℰ
class ∶ 𝜈 → 𝐶
N odes
𝜈
are created b y t he combination of a literal class
𝐶
wit h an
en vironment (function
no de
). En vironments map natural numbers t o nodes.
A node is created b y t he function
no de
. Its class is accessible wit h
class
. It
is possible to obtain the en vironment inside a node (i.e., t he en vironment

58 2 The Model Com position Languag e
t hat is v alid for its fields).
⟦⟦𝐶, ℰ ⟧ no de ⟧ class
∧
= 𝐶
⟦⟦𝐶, ℰ ⟧ no de ⟧ env
∧
= { 0 ↦ ⟦𝐶, ℰ ⟧ no de } ∪ { 𝑛 + 1 ↦ 𝜈 | 𝑛 ↦ 𝜈 ∈ ℰ }
The en vironment of a node is t he cap tured en vironment with one additional
entr y , mapping
0
to the node itself. All other entries link back to the original
en vironment (jus t one lev el higher). In a certain sense, this definition forms
an in v erted view of t he syntax tree, as each node giv es access to an ordered
set of children (which ma y be its parents in t he syntax tree). En vironments
are f orests of such trees, and literal classes are node-labels. Hence t he name
node f or t he elements of t his s tructure.
In t hat deno tational view , t he ev aluation of hier archical terms can be de-
fined b y t he resolution of special v ariables and t he t hree mutuall y recursiv e
oper ations, select , sear ch and ev al . During ev aluation, a special v ariable
𝑖
is
r esol ved in a giv en en vironment ℰ to the 𝑖 -t h entr y of t he en vironment.
⟦ ℰ , up( 𝑛 ) ⟧ eval
∧
= ℰ (𝑛)
The ev aluation of a literal class with a giv en en vironment yields a context.
It is created b y appending t hat liter al class to the current en vironment. W e
ignore t he ev aluation of proper ter ms f or no w .
⟦ ℰ , 𝐶⟧ eval
∧
= ⟦𝐶, ℰ ⟧ no de
⟦ ℰ , 𝑡⟧ eval
∧
= …
A reference consis ts of a hierarchical term
ℎ
and a label
𝐿
— w e will
define t hese syntactical sorts precisel y later . Such a reference (e.g.
up(2).z
)
are ev aluated from lef t to right b y selecting the label:
⟦ ℰ , ℎ . 𝐿⟧ eval
∧
= ⟦⟦ ℰ , ℎ⟧ eval , 𝐿⟧ sele ct
The definition of a field has to be f ound in t he class or a base class,
to select it. The f ound term is t hen ev aluated under a new en vironment,
obtained via
env
from
𝜈 ′
. Setting t he
0
-t h en vironment entr y to t he receiv er
of t he selection giv es t he special v ariable this a new meaning.
⟦𝜈, 𝐿⟧ sele ct
∧
= ⟦⟦𝜈 ′ ⟧ env ⊕ {0 ↦ 𝜈}, ℎ⟧ eval
when ⟦𝜈, 𝐿⟧ se ar ch
∧
= 𝜈 ′ , ℎ

2.3 Classes and Object-Oriented Modeling 59
N ote, that t he result of searching t he label is ev aluated af ter it is found
in t he en vironment. In a certain sense, hier archical terms are ev aluated
wit h a non-s trict call-b y -name ev aluation str ategy , contr ar y to core terms
t hat f ollo w a strict call-b y -v alue approach. This distinction will become
im portant in Chap ter 4, when t he former are translated int o the latter .
A definition is sear ched recursiv el y: If t he field is a literal child, its right-
hand side is searched. If it is for w arded t o a base class, t he base class is
ev aluated and t he search continues t here.
⟦𝜈, 𝐿⟧ se ar ch
∧
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
𝜈, ℎ if 𝐿 = ℎ ∈ ⟦𝜈⟧ class
𝜈 ′ , ℎ ′ if in ⟦𝜈⟧ class , 𝐿 forw ar ds to base class ℎ 𝑆
⟦ℎ 𝑆 , ⟦𝜈⟧ env ⟧ eval
∧
= 𝜈 𝑆
⟦𝜈 𝑆 , 𝐿⟧ se ar ch
∧
= 𝜈 ′ , ℎ ′
Graphical Inter pret ation
As an exam ple, consider t he classes
C
and
D
from abo v e and t he ev aluation
of
D.x
. Since all classes in t hat exam ple ha v e a unique name, this name is
used in abbreviations as a subscrip t, i.e.,
𝜈 𝐷
is t he obtained from ev aluating
class D , ℰ 𝐷 is its en vironment and 𝐶 𝐷 deno tes its literal class.
Each node can be inter preted as a directed graph with edge labels t o
indicate t he or dering of its enclosing classes. A node label indicates t he
liter al class. Graphs with tw o kinds of v ertices displa y en vironments. The
number of each entr y is sho wn in a circular node t hat is labeled wit h a
natur al number and has exactl y one outgoing edg e to t he corresponding
element of t he en vironment. The top-lev el en vironment has onl y one entr y ,
pointing to the unnamed root class:
(1) ℰ 𝑟𝑜𝑜𝑡
∧
= . 0
Creating new nodes from a liter al classes inside an en vironment is achiev ed
b y adding a new node, replacing the special nodes wit h edges from that
node to their respectiv e targ et and increasing t heir label b y one:
(2) ⦃𝐶 𝐷 in ℰ 𝑟𝑜𝑜𝑡 ⦄ ∧
= 𝜈 𝐷
∧
= D .
1
A node pro vides t he en vironment for its children (function
env
). This
en vironment is obtained b y mapping t he node itself to
0
and adding an

60 2 The Model Com position Languag e
entr y f or each outgoing edg e:
(3) ⟦𝜈 𝐷 ⟧ env
∧
= ℰ 𝐷
∧
= D .
0 1
1
Classes C and B are ev aluated in a similar sty le to D in (2):
(4) 𝜈 𝐶
∧
= ⦃𝐶 𝐶 in ℰ 𝑟𝑜𝑜𝑡 ⦄ ∧
= C .
1
(5)
𝜈 𝐵
∧
= ⦃𝐶 𝐵 in ℰ 𝐶 ⦄ ∧
= B C .
1
2
1
Using t hese classes, a sim ple ev aluation can be processed as follo ws:
⟦𝜈 𝐶 , y ⟧ sele ct
∧
= ⟦ ℰ 𝐶 , up( 0 ).B.z ⟧ eval
∧
= ⟦⟦𝜈 𝐶 , B ⟧ sele ct , z ⟧ sele ct
∧
= ⟦𝜈 𝐵 , z ⟧ sele ct
∧
= ⟦ ℰ 𝐵 , 21 ⟧ eval
∧
= 21
(6)
Similar l y , t he base class of D can be ev aluated:
⟦ ℰ 𝐷 , up( 1 ).C.B ⟧ eval
∧
= ⟦⟦𝜈 𝑟𝑜𝑜𝑡 , C ⟧ sele ct , B ⟧ sele ct
∧
= ⟦𝜈 𝐶 , B ⟧ sele ct
∧
= 𝜈 𝐵
(7)
W it h t hese obser v ations it is no w possible to unders tand t he ev aluation
of D.x . In a first s tep, the field is found in the node itself:
⟦𝜈 𝐷 , x ⟧ se ar ch
∧
= 𝜈 𝐷 , up(0).A.x (8)
There is no field
A
in
𝐶 𝐷
. The search continues in t he base class,
𝜈 𝐵
. Here,
t he field is f ound:
(9) ⟦𝜈 𝐷 , A ⟧ se ar ch
∧
= ⟦𝜈 𝐵 , A ⟧ se ar ch b y (7)
∧
= 𝜈 𝐵 , 𝐶 𝐴
𝐶 𝐴
is f ound in t he base class
𝜈 𝐵
. The ev aluation en vironment is con-
s tructed from both t he base class as w ell as t he extending class:
(10)
ℰ 𝑆
∧
= ⟦𝜈 𝐵 ⟧ env ⊕ {0 ↦ 𝜈 𝐷 } ∧
= D
0
C
1
.
2
1
1

2.3 Classes and Object-Oriented Modeling 61
Ev aluation of t he literal class t hen appends
𝐶 𝐴
to the nodes of t he en viron-
ment. This node, in tur n, has its en vironment as usual:
(11)
ℰ 𝐴𝑆
∧
= ⟦𝜈 𝐴𝑆 ⟧ env
∧
= A
0
D
1
C
2
.
3
1
2
3
1
1
R esol ving
up(2)
and
up(1)
in
ℰ 𝐴𝑆
yields
𝜈 𝐶
and
𝜈 𝐷
, respectiv ely . Hence,
y and z can be selected:
⟦ ℰ 𝐴𝑆 , up( 2 ).y ⟧ eval
∧
= ⟦𝜈 𝐶 , y ⟧ sele ct b y (11)
∧
= 21 b y (6)
(12)
⟦ ℰ 𝐴𝑆 , up( 1 ).z ⟧ eval
∧
= ⟦𝜈 𝐷 , z ⟧ sele ct b y (11)
∧
= ⟦ ℰ 𝐷 , 2 ⟧ eval
∧
= 2
(13)
The modification in
𝐶 𝐷
has affected t he ev aluation as intended. The in-
heritance is “f orgo tten” when t he lookup passes t he corresponding lexical
scope. The ev aluation of
y
takes place in
𝜈 𝐶
wit hout further changes. In
conclusion, t he result is 23 , as expected:
⟦𝜈 𝐷 , x ⟧ sele ct
∧
= ⟦ ℰ 𝐷 , this.A.x ⟧ eval b y (8)
∧
= ⟦⟦𝜈 𝐷 , A ⟧ sele ct , x ⟧ sele ct b y (3)
∧
= ⟦⟦ ℰ 𝑆 , 𝐶 𝐴 ⟧ eval , x ⟧ sele ct b y (9)
∧
= ⟦ ℰ 𝐴𝑆 , up( 2 ).y + up( 1 ).z ⟧ eval b y 𝐶 𝐴
∧
= ⟦ ℰ 𝐴𝑆 , 21 + 2 ⟧ eval
∧
= 23 b y (12,13)
2.3.5 Synt ax
It remains to define a syntactic representation of the en vironment and its
contents as w ell as an operational view of the functions used abo v e. T o
t his end, w e introduce a new syntactic sort, called hierarchical term,
ℎ
(F igure 2.8).
There are fiv e v ariants of hierarchical terms. The special v ariables
up( 𝑖 )
ref er to t he
𝑖
-t h enclosing class. Liter al classes
𝐶
are a list of fields
𝐹
br acketed
in special cur l y br aces:
⦃𝐹⦄
. A class field can eit her contain a hierarchical
class (e.g. a child class) (
𝐿 = ℎ
) or a v alue (
𝑙 = 𝑡
). W e presume t hat class labels

62 2 The Model Com position Languag e
Cor e T erms:
𝑡, 𝑠 ⩴ 𝑥 | 𝑣 | 𝑡 𝑡 | 𝑡 ∘ 𝑡
| …
| 𝑟
𝑓 ⩴ 𝑙 = 𝑡
𝑟 ⩴ ℎ .super( ℎ ). 𝑙
rv ⩴ 𝜈 .super( 𝜈 ). 𝑙
Hier ar c hical T erms:
ℎ, 𝑔 ⩴ up( 𝑖 ) |𝐶|𝜈|𝑅
| ℎ with 𝐹
𝑅 ⩴ ℎ .super( ℎ ). 𝐿
𝜈, 𝜋 ⩴ ⦃𝐹 in 𝜈⦄
𝐶 ⩴ ⦃𝐹⦄
𝐹 ⩴ 𝐿 = ℎ|𝑙 = 𝑡
F igure 2.8: Mcl classes and hierarchical terms
can be dis tinguished from v alue labels (i.e.
𝐿 ∩ 𝑙 ∧
= ∅
). A hierarchical node
𝑣
(or
𝜋
) deno tes a class containing t he fields
𝐹
as a hier archical child of t he
enclosing class denoted b y
𝜋
, written
⦃𝐹 in 𝜋⦄
(t he en vironment is t hus
encoded as a list of nodes and each node contains its o wn en vironment).
Explicit Modifications
ℎ with 𝐹
o v erride t he fields defined in t he class
described b y
ℎ
wit h the fields in
𝐹
. This mechanism essentially g eneralizes
t he o v erriding of fields b y decoupling it from open recursion.
A ccess to the field
𝐿
of a class requires an explicit notion of the corre-
sponding base class (often also called t he “super” class, a ter m which is
used f or hist orical reasons in Mcl). In a reference like
ℎ 1 .super( ℎ 2 ). 𝐿
,
ℎ 1
ref ers to some class that extends
ℎ 2
which in tur n contains the declaration
labeled wit h
𝐿
. The access reads as: “Get the field labeled wit h
𝐿
in t he class
ℎ 2
extended b y
ℎ 1
”. Instead of one
extend
-s tatement t hat pulls all fields
of t he base class int o t he current class, Mcl requires one explicit f or w ard-
ref erence for each inherited element. This sty le makes the interface of a
class immediatel y visible (since all inherited fields ha v e to be defined lo-
call y as for w ard-ref erences), a necessar y precondition for a syntax-directed
semantics.
As an exam ple, consider the class
D
from Listing 2.2. Jus t b y looking at
its definition, onl y t he fields
x
and
z
are visible. This changes in its Mcl
encoding, where also t he inherited class A is explicitl y mentioned:
⦃ A = up( 0 ).super(up( 1 ).C.B).A , z = 2 , x = up( 0 ).A.x ⦄
Often, no base class shall be referenced directl y , and bo t h hier archical
terms are equal. Since t his is a common case, w e introduce t he abbreviation
ℎ . 𝐿 ∧
= ℎ .super( ℎ ). 𝐿 t o enhance t he readability of the examples.

2.3 Classes and Object-Oriented Modeling 63
Because classes are not firs t-class citizens, t hey do no t contain free v ari-
ables. Neither substitution nor free v ariables need to be extended to reflect
t he exis tence of hier archical terms. Special v ariables are bound im plicitl y
b y t he hierarchical en vironment. Hence all special v ariables in a term are
“free”. They are collected b y the function FSV (Appendix C.1).
Besides t he introduction of
ℎ
, an extension to the proper ter ms
𝑡
is neces-
sar y to actuall y access v alues bound in classes (highlighted in Figure 2.8). A
v alue field can be accessed from a ter m
𝑡
wit h a no tation similar to the class
selection. A reference
𝑟 ∧
= ℎ 1 .super( ℎ 2 ). 𝑙
ref ers to the field labeled wit h
𝑙
in class
ℎ 2
extended b y
ℎ 1
. A gain w e allo w t he con v enient abbreviation
ℎ . 𝑙 ∧
= ℎ .super( ℎ ). 𝑙
f or references that do not explicitl y ref er to a base class.
Alt hough hier archical ref erences are not firs t-class citizens, it will become
useful later to dis tinguish references wit h full y ev aluated hierarchical parts.
These ref erences “v alues” are abbreviated as
rv
. A reference has no free
v ariables and is nev er subject to subs titution.
2.3.6 Semantics
Ev aluation of a hierarchical term
ℎ
to a node
𝜈 ∧
= ⦃𝐹 in 𝜋⦄
wit h parents
𝜋 1 … 𝜋 𝑛
is written
ℎ ⇓ h 𝜈
. This relation is again giv en in a big-step s ty le
(F igure 2.9). R ule Select augments t he ev aluation-relation ⇓ .
Hier archical nodes are already in normal form (r ule N ode). R oo t classes
are wr apped into an em pty en vironment (R oot). Selection of a child class
via HSelect or a field via Select relies on t he search of the corresponding
definition in t he full y ev aluated context. W e abbreviate t his operation
wit h the relation
𝜙
− →
and
Φ
− →
. N o tabl y , t hese relations do no t depend on t he
(hier archical) ev aluation. Hence t hey correspond t o a single step of the
search oper ation. Searching t hrough multiple lev els of inheritance relies on
t he encoding of inherited fields as ref erences from
this
to the corresponding
base class and is reflected b y multiple applications of eit her HSelect or
Select.
F olding t he En vironment
In or der to adhere t o t he principle of open recursion betw een fields of a
class, t he special v ariables in a selected (hier archical) term are resolv ed
using t he ternar y fold-oper ators
Φ ∶ ℎ × 𝜈 × ℎ → ℎ
and
𝜙 ∶ ℎ × 𝜈 × 𝑡 → 𝑡
after a successful search and before e v aluation continues. These mutuall y
recursiv e functions (Figure 2.10) tak e t hree arguments: An ev aluated class
𝜈
represents t he tip of the en vironment (i.e. the self -instance). The second

64 2 The Model Com position Languag e
Lifting classes:
⟦ ♦ ⟧ Π
∧
= ∅
⟦𝐹 ∷ 𝑙 = 𝑡⟧ Π
∧
= ⟦𝐹⟧ Π ⊕ {𝑙 ↦ 𝑡}
⟦𝐿 = ℎ ∷ 𝐹⟧ Π
∧
= ⟦𝐹⟧ Π ⊕ {𝐿 ↦ ℎ}
Look up of fields:
𝜈 .super( ⦃𝐹 in 𝜋⦄ ). 𝑙 𝜙
− → ⟦𝜈, 𝜋, 𝑡⟧ 𝜙 ⟺ 𝑙 ↦ 𝑡 ∈ ⟦𝐹⟧ Π
𝜈 .super( ⦃𝐹 in 𝜋⦄ ). 𝐿 Φ
− → ⟦𝜈, 𝜋, ℎ⟧ Φ ⟺ 𝐿 ↦ ℎ ∈ ⟦𝐹⟧ Π
R ules of Evaluation:
HSelect
ℎ 1 ⇓ h 𝜈 1 ℎ 2 ⇓ h 𝜈 2 𝑔 ⇓ h 𝜋
𝜈 1 .super( 𝜈 2 ). 𝐿 Φ
− → 𝑔
ℎ 1 .super( ℎ 2 ). 𝐿 ⇓ h 𝜋
Select
ℎ 1 ⇓ h 𝜈 1 ℎ 2 ⇓ h 𝜈 2 𝑡 ⇓ 𝑣
𝜈 1 .super( 𝜈 2 ). 𝑙 𝜙
− → 𝑡
ℎ 1 .super( ℎ 2 ). 𝑙 ⇓ 𝑣
N ode
𝜈 ⇓ h 𝜈
Mod
ℎ ⇓ h ⦃𝐹 1 in 𝜋⦄ ⟦𝐹 2 ⟧ FS V
∧
= ∅
dom (𝐹 2 ) ⊆ dom (𝐹 1 )
⟦𝐹 3 ⟧ Π
∧
= ⟦𝐹 1 ⟧ Π ⊕ ⟦𝐹 2 ⟧ Π
ℎ with 𝐹 2 ⇓ h ⦃𝐹 3 in 𝜋⦄
R oot
⦃𝐹⦄ ⇓ h ⦃𝐹 in ♦ ⦄
F igure 2.9: Hierarchical Ev aluation Semantics
argument
𝜋
is t he lis t of hier archical parents of t he base class containing
t he definition of the current ter m. The t hird ar gument is the input that is
being f olded:
Φ
expects and retur ns an hier archical term
ℎ
while
𝜙
w or ks
on plain terms
𝑡
. In t he case of plain terms, the result of f olding is sim ple.
Ex cept hier archical v alue-references,
𝜙
applies itself on the sub-ter ms or
retur ns its in put unchang ed if t he argument is primitiv e. V alue-ref erences
are processed b y folding the hierarchical sub-terms wit h Φ .
F olding hierarchical terms is somewhat more in v olv ed. Special v ari-
ables
up( 𝑖 )
are resol v ed via substitution: The special v ariable
up( 0 )
(t he
this -v ariable) is replaced wit h t he self -ins tance (t he firs t entr y of the en vi-
ronment). Ot her special v ariables are folded recursiv el y: If the en vironment
is em p ty , t he result is left unresolv ed. Ot her wise, t he first entr y is retur ned

2.3 Classes and Object-Oriented Modeling 65
⟦𝜈, 𝜋, up( 0 ) ⟧ Φ
∧
= 𝜈
⟦ _ , (𝜈 ∷ 𝜋), up( 1 ) ⟧ Φ
∧
= 𝜈
⟦𝜈, ( _ ∷ 𝜋), up( 𝑛+2 ) ⟧ Φ
∧
= ⟦𝜈, 𝜋, up( 𝑛+1 ) ⟧ Φ
⟦ _ , ♦ , up( 𝑛+1 ) ⟧ Φ
∧
= up( 𝑛+1 )
⟦𝜈, 𝜋, ℎ 1 .super( ℎ 2 ). 𝐿⟧ Φ
∧
= ⟦𝜈, 𝜋, ℎ 1 ⟧ Φ .super( ⟦𝜈, 𝜋, ℎ 2 ⟧ Φ ). 𝐿
⟦𝜈, 𝜋, ℎ with 𝐹⟧ Φ
∧
= ⟦𝜈, 𝜋, ℎ⟧ Φ with 𝑓 ∗ 𝐹
where 𝑓 ⟦𝐿 = ℎ ⋆ ⟧ ∧
= 𝐿 = ⟦𝜈, 𝜋, ℎ ⋆ ⟧ Φ
𝑓 ⟦𝑙 = 𝑡 ⋆ ⟧ ∧
= 𝑙 = ⟦𝜈, 𝜋, 𝑡⟧ 𝜙
⟦ _ , _ , 𝜈⟧ Φ
∧
= 𝜈
⟦𝜈, 𝜋, ⦃𝐹⦄⟧ Φ
∧
= ⦃𝐹 in 𝜈 ∷ 𝜋⦄
⟦𝜈, 𝜋, 𝑥⟧ 𝜙
∧
= 𝑥
⟦𝜈, 𝜋, λ 𝑥 . 𝑡⟧ 𝜙
∧
= λ 𝑥 . ⟦𝜈, 𝜋, 𝑡⟧ 𝜙
⟦𝜈, 𝜋, 𝑡 1 𝑡 2 ⟧ 𝜙
∧
= ⟦𝜈, 𝜋, 𝑡 1 ⟧ 𝜙 ⟦𝜈, 𝜋, 𝑡 2 ⟧ 𝜙
⟦𝜈, 𝜋, ℎ 1 .super( ℎ 2 ). 𝑙⟧ 𝜙
∧
= ⟦𝜈, 𝜋, ℎ 1 ⟧ Φ .super( ⟦𝜈, 𝜋, ℎ 2 ⟧ Φ ). 𝑙
… (See Appendix C.2.3)
F igure 2.10: Resolution of open recursion with hierarchical f olding
(in case of an upw ard ref erence of lev el 1) or recursiv el y resol v ed wit h t he
remaining en vironment (in case of higher ref erences). The result of folding
hier archical references and modified classes is the result of folding their
corresponding sub-terms: In t he case of modifications, t his ensures that a
modification is ev aluated in t he context of t he modification site (and not in
t he context of the modified class). Literal classes are turned into nodes b y
s toring the en vironment alongside t heir fields. Contrary to modifications,
t he fields are no t subject to f olding in t his case. This excep tion ensures t hat
child-classes retain t heir context: All special v ariables remain unresolv ed
until a field is selected from the child-class. N odes are lef t unchang ed b y
t he f old-function.

66 2 The Model Com position Languag e
Modifications
Modifications in Modelica differ from classical o v erriding in t heir scope. In
classical object-oriented languag es (and in Mcl) t he field of an extending
class might chang e a field of t he base class from the inside b y o v erriding it
(a mechanism t hat exis ts in Modelica onl y via explicit redeclarations). A
modification, in contras t, liv es outside of t he modified class. F unction
𝜙
im plements this difference. A second specialty that distinguishes modi-
fications from classical inheritance is t he fact that it is not possible t o add
a field via a modification. Each modified field must already exis t in t he
modified class. F or t hese reasons, modifications require special treatment
in t he syntax and semantics of Mcl.
The modification of a class
ℎ
wit h a sequence of fields
𝐹
results in a class
containing t he modified fields merg ed wit h the result of t he ev aluation
(rule Mod in Figure 2.9). Merging is im plemented b y lif ting t he fields into
partial functions, augmenting t he original function wit h t he new fields, and
using t he result as a new sequence of fields. The modifying declar ations
mus t not contain an y special v ariables. This pro vision ensures t hat they
retain t heir original scope, similar to modifications in Modelica.
2.3.7 Exam ple
Open recursion can replace classical recursion. Our pair of exam ple func-
tions,
even
and
odd
can be written in a hierarchical sty le as follo ws. The
concrete syntax deviates ag ain from t he abstr act definition: By a syntactical
con v ention, all classes are hierarchical, i.e., each liter al class in t he source
code is alw a ys im plicitl y surrounded b y t he root operat or .
{| val even = λ n . if n=0 then true else this.odd (n - 1) ;
val odd = λ n . if n=0 then false else this.even (n - 1)
|}.even 42
The ev aluation proceeds similar y to Section 2.1.4, but requires r ules
Select and App ins tead of FixApp.
2 . 4 v ariable-str ucture modelin g
Modelica is a h ybrid languag e and capable of changing t he simulation
beha vior depending on t he occurrence of ev ents during t he simulation
itself. Their effects are limited, ho w ev er . In Modelica, it is onl y possible to
affect t he current v alue of a state v ariable (using t he
reinit
-oper ator) and

2.4 V ariable-Structure Modeling 67
discrete v ariables (wit h a discrete-time equation). N either equations nor
com ponents can be remo v ed or added during simulation.
The current state of research is considerabl y more adv anced than t hat.
Sev er al researchers ha v e put much effort into the modeling and simulation
of so-called variable-s tructur e sys t ems [Mattsson et al., 2015, P epper et al.,
2011, Mehlhase, 2012, Nytsch-Geusen and Er ns t, 2005]. Alt hough all t hese
w or ks differ in t he details of t heir semantics, t here is a consensus on t he
intuition of viewing a v ariable-structure model as a h ybrid automaton. Each
of its modes corresponds to a classical s tatic-structure model and ev ents
control t he tr ansitions. Usuall y , the number of modes is considered to
be finite and small in practice. Such finite-state aut omata can readil y be
im plemented with Mcl b y giving each mode a number (e.g., b y s toring
t hem in an arr a y) and switching betw een t hem when an ev ent requires it
(t he aspects of ev ent handling are explained in Chap ter 3).
There are experimental languag es[Giorgidze and Nilsson, 2009, Zim-
mer, 2010] t hat allo w for automata with man y modes, but it remains an
open ques tion ho w Modelica could support such models. In a first s tep,
a small extension to Mcl enables the dynamic computation and ex ecution
of a new mode during simulation. W it h t his addition and a little support
from t he kernel library highl y dynamic systems can be modeled and simu-
lated easil y in t he core language. This brings t hem one s tep closer to their
im plementation in Modelica itself.
2.4.1 Explicit Contr ol f or S tructur al Chang es
Com puting a new mode is, in principle, a sim ple matter . A com ponent can
be designed to im plement different beha vior , depending on its input ; e.g., a
switch might be open when a boolean parameter is tr ue and closed if it is
false. The subtle problem arises due to t he questions of when t o attem pt
t his chang e, how t o send the new in put v alue and ho w to im plement t he
actual switch betw een old and new mode.
The las t question is the sim plest t o answ er . Assume, a new model
receiv es a new in put v alue; t hen elaboration can com pute t he new mode
as usual when t he elabor ation state is r eset to the v alue it had just bef ore
t he las t instantiation of the com ponent. In a br ute-f orce approach, it ev en
suffices to res tart elaboration from scr atch wit h an initial s tate. A routine
mus t hence be capable of setting t he elaboration s tate to some cap tur ed v alue.
Since t he v alue of a state is already part of a languag e, t his requires onl y
tw o new primitiv es: One for obtaining the current state as a whole and one
f or setting it.

68 2 The Model Com position Languag e
The firs t and second questions ha v e a common, slightl y more com plex
answ er . Usuall y , t he simulation engine will answ er the question f or t he
when of a s tructural chang e, e.g., b y t he iterativ e solution of a guarding equa-
tion. Since t his happens after the model instance is com pletel y elaborated,
sending t he new in put in a sense requires sending a messag e back in time.
T o make this possible, Mcl borro ws a f eature from Lisp and its dialects:
These languag es offer the operator
call − with − current − continuation
or short
call/cc
. With t his oper ator , a prog r am gains access t o continu-
ations as firs t-class v alues. In turn, sophisticated control s tr uctures (e.g.,
co-routines [Ha ynes et al., 1984]) can be gener alized. N o t quite incidentall y ,
t he elabor ation semantics is already defined in a continuation-passing s ty le,
and continuations are already syntacticall y specified. Theref ore, it remains
to im plement primitiv es to obtain the current continuation and pass control
to it ag ain.
2.4.2 Synt ax
The manag ement of the control flo w requires t hree, management of s tate
needs tw o additional elements (highlighted in Figure 2.11): Continuations
𝑘
become firs t-class citizens (i.e., they are proper v alues). The primitiv e
routine
continue 𝑡 1 with 𝑡 2
retur ns the control flow to the continuation
deno ted b y
𝑡 1
wit h
𝑡 2
as in put. In v ocation of
call/cc 𝑡
passes t he current
continuation to
𝑡
and continues elabor ation wit h t he result. The routine
get
retur ns the current elaboration s tate, while
put 𝑡
sets it to the state obtained
from ev aluating 𝑡 .
V alues:
𝑣, 𝑤 ⩴ 𝑏 | λ 𝑥 . 𝑡 | fix 𝑥 in ℱ
| [ 𝑣 ] | { fv } |𝑚| < fv > | 𝑘
fv ⩴ 𝑙 = 𝑣
ℱ ⩴ 𝑥 = λ 𝑦 . 𝑡
𝑘 ⩴  | 𝑥 . 𝑡 ∶∶ 𝑘
Modeling R outines:
𝑚⩴𝑙 •get | 𝑙 •put 𝑡
| 𝑥 ← 𝑡 ; 𝑡 | return 𝑡
| get | put 𝑡
| continue 𝑡 with 𝑡
| call/cc 𝑡
F igure 2.11: Syntax for control flo w and state manag ement
Scoping of t he new elements is sho wn in F igure 2.12. Most of the new
cons tructs do not bind v ariables. Only continuations bind their in put v ari-
able to their body , but no t their follo w -up continuation.

2.4 V ariable-Structure Modeling 69
⟦ put 𝑡⟧ fv
∧
= ⟦𝑡⟧ fv
⟦ get ⟧ fv
∧
= ∅
⟦ call/cc 𝑡⟧ fv
∧
= ⟦𝑡⟧ fv
⟦ continue 𝑡 1 with 𝑡 2 ⟧ fv
∧
= ⟦𝑡 1 ⟧ fv ∪ ⟦𝑡 2 ⟧ fv
⟦  ⟧ fv
∧
= ∅
⟦𝑥 . 𝑡 ∶∶ 𝑘⟧ fv
∧
= ( ⟦𝑡⟧ fv ⧵ { 𝑥 } ) ∪ ⟦𝑘⟧ fv
F igure 2.12: Free v ariables of routines t hat handle control flo w
2.4.3 Semantics
The syntactic extensions are joined b y four ne w rules for the elaboration
relation (F igure 2.13). R ules MPutGlob al and MGetGl ob al are str aightfor -
w ard. In a w a y similar to MPut and MGet, t hey alter the global state or pass
it to the current continuation. The new s tate is created from or inter preted
as a s tate v alue wit h t he function Π .
MPutGlob al
𝑡 ⇓ < fv > Γ ∧
= ⟦ fv ⟧ Π ⟨Γ, {} ⟩ ⊢ 𝑘 ⇓ k 𝑣
⟨Γ, 𝑘⟩ ⊢ put 𝑡 ⇓ m 𝑣
MGetGlob al
Γ ∧
= ⟦ fv ⟧ Π ⟨Γ, < fv > ⟩ ⊢ 𝑘 ⇓ k 𝑣
⟨Γ, 𝑘⟩ ⊢ get ⇓ m 𝑣
MCall/CC
𝑡 𝑘 ⇓ 𝑚 ⟨Γ, 𝑘⟩ ⊢ 𝑚 ⇓ m 𝑣
⟨Γ, 𝑘⟩ ⊢ call/cc 𝑡 ⇓ m 𝑣
MC ontinue
𝑡 1 ⇓ 𝑘 1 𝑡 2 ⇓ 𝑤 ⟨Γ, 𝑤⟩ ⊢ 𝑘 1 ⇓ k 𝑣
⟨Γ, 𝑘⟩ ⊢ continue 𝑡 1 with 𝑡 2 ⇓ m 𝑣
F igure 2.13: Extended Elaboration semantics
Continuation handling is somewhat more intricate than t hese simple
rules. The current continuation is accessed via
call/cc 𝑡
(rule MCall/CC).
Since a continuation is a firs t-class citizen, it can be passed as an argument
to the function denoted b y
𝑡
. When t his application yields a routine, elab-
or ation continues wit h t his routine and the current continuation. In that
w a y , it becomes possible t o sa v e t he current point of elaboration. In or der

70 2 The Model Com position Languag e
to return to such a sa v ed point, t he routine
continue 𝑡 1 with 𝑡 2
passes t he
result of ev aluating
𝑡 2
to the continuation obtained via
𝑡 1
and discards the
old continuation 𝑘 .
2.4.4 Exam ple
Our no w familiar exam ple of determining whether a natural number is
ev en or odd can be im plemented as a trivial loop. The features that enable
s tructural dynamics are a g ener alization of such a loop. As w e ha v e already
seen in Section 2.2.6, elaboration is a special f or m of g eneral com putation.
Hence, it is possible to com pute t his loop as a model routine:
let even = do
loop ← call/cc ( λ k. return k) ;
n←n•g e t;
if n <= 0 then return true else
if n=1 then return false else
do
p u tn=n-2 ;
continue loop with loop
in
exec even with {n = 42}
The deriv ation tree t hat deter mines the semantics of t he abo v e exam ple
is akin to a unrolling of the loop: In ev er y iter ation, t he current continuation
(i.e., t he v ariable
loop
and t he term starting with
x ← get
) is cap tured b y a
combination of rules MCall/CC and MRet. MGetGl ob al and MPutGlob al
ensure t hat the loop v ariable is updated at ev er y s tep. The loop repeats
via MC ontinue. R ecursion is replaced wit h the explicit capture of the
continuation in its o wn body .
While t he exam ple is small enough to demons trate the concept of firs t-
class continuations, it takes some f antasy to relate it t o a v ariable-structure
model. Later exam ples will demonstr ate t his relationship but depend on
some higher -lev el abstr actions. N ev ert heless, a v ariable-structure simu-
lation is im plemented in a similar pattern: A multi-mode model creates
a checkpoint (
loop
) during elaboration and enters the simulation phase
(
x ← get
). Depending on t he results of t he simulation, it can retur n to the
elabor ation after modifying t he global state (
put
,
continue
). The implemen-
tation of t he different modes is the sole responsibility of t he model.

2.5 T op-Lev el Entities: P ackag es and Modules 71
2 . 5 top -level entities: p a c ka ges and modules
Separ ate com pilation ob viousl y depends on t he definition of “separate”
entities. Usually , t hese entities are bo t h represented in t he source languag e,
e.g., as modules or top-le v el declarations, and easy to dis tinguish from each
o t her b y t he user and exter nal tools, e.g., as separ ate files or directories.
Modelica packag es can be mapped to a hier archical file system accor ding
to the specification [see Modelica, 2012, chap ter 13].
The application of t his mapping is, ho w ev er , somewhat arbitrary . Mod-
elica does no t distinguish betw een classes residing in a file and classes that
are nes ted inside other classes. A user might as w ell merg e a complete
hier arch y into one file, map each class into a single file, or do an yt hing
in betw een t hese extremes. Because of t his and t he lookup semantics, it
is im possible to or der source files b y t heir dependencies without actually
processing t hem. Therefore, in Mcl w e consider onl y t op-lev el classes as
units of com pilation. F or t hem, t he user or an exter nal tool giv e t he order . In
t he Modelica ecosys tem, t his corresponds to different libr aries , which ha v e
explicit dependency anno tations. Dependency anal ysis for these libraries is
t he task of a packag e manager (e.g. im pact[T iller and W inkler, 2015]).
Alongside top-le v el classes, Mcl supports simple modules. These en-
tities do not ha v e a correspondence in Modelica. Their addition to the
languag e is jus tified b y t he fact that t he compilation tar get shall be a subset
of Mcl. They also serv e as an entr y -point that trigg ers the ev aluation of a
class. Contrary to classes, modules are ev aluated eag er l y . Hence, each defi-
nition can trigg er t he hier archical ev aluation, elaboration, and simulation
of a class.
2.5.1 Synt ax
A top-le v el ter m
𝑇
(F igure 2.14) consis ts of a sequence of top-le v el definitions
𝐷
. Each such definition binds eit her a module term
𝑀
or a class term
𝐶
to a t op-lev el identifier
𝑋
. W e presume
𝑥∩𝑋 ∧
= ∅
. Classes are defined
as bef ore, wit h t he additional v ariant of using onl y the name of a top-
lev el class. Modules are sim ple ML-like s tr uctures[MacQueen, 1984]. They
consis t of a series of module-lev el definitions
d
. Each of t hese definitions
binds an identifier
𝑥
to a term
𝑡
. Analogous to sim ple terms, top-le v el ter ms
ha v e nor mal forms. Module ter ms ev aluate to s tr uctures
MV
, consis ting
of ev aluated definitions
dv
, each binding an identifier to a v alue. Classes
are already in normal form. T op-lev el definitions ev aluate to a binding of
top-le v el identifiers to eit her a structure or a class. T op-lev el v alues are
sequences of such ev aluated definitions.

72 2 The Model Com position Languag e
Cor e T erms:
𝑡, 𝑠 ⩴ 𝑋 : 𝑥 |𝑥|𝑣|𝑡𝑡|𝑡 ∘ 𝑡
| …
Hier ar c hical T erms:
ℎ, 𝑔 ⩴ up( 𝑖 ) |𝐶|𝜈|𝑅
| ℎ with 𝐹
𝑅 ⩴ ℎ .super( ℎ ). 𝐿
𝜈, 𝜋 ⩴ ⦃𝐹 in 𝜈⦄
𝐶 ⩴ 𝑋 | ⦃𝐹⦄
𝐹 ⩴ 𝐿 = ℎ|𝑙 = 𝑡
T op-lev el T erms:
𝑇 ⩴ 𝐷
𝐷 ⩴ 𝑋 = 𝑀
| 𝑋 = 𝐶
𝑀 ⩴ 𝑋 | struct d end
d ⩴ 𝑥 = 𝑡
T op-lev el V alues:
MV ⩴ struct dv end
D V ⩴ 𝑋 = ( MV | 𝐶)
𝑉 , 𝑊 ⩴ D V
dv ⩴ 𝑥 = 𝑣
F igure 2.14: Mcl top-lev el entities
As already mentioned, hier archical terms can use top-lev el classes b y
name ins tead of literal classes. T erms are augmented b y a new kind of
v ariable
𝑋 : 𝑥
, called long . Such a long v ariable refers to the definition of
𝑥
inside t he s tructure referred to b y
𝑋
. U nlike normal v ariables, long v ariables
can be substituted e v en in hierarchical terms. The procedure is trivial as
new long v ariables can only be introduced at t he top-le v el. The subs titu-
tion of a partial function
𝑃
containing top-le v el identifiers, written
[𝑃]
is a
s traightf or w ar d matter (Appendix C.2). F ree long v ariables are determined
b y FL V (Appendix C.1).
Concrete Syntax
The necessar y lexical dis tinction betw een top-le v el and ter m-lev el v ariables
is made b y t heir first letter . Capital v ariables belong to t he f ormer categor y ,
lo w er case v ariables to t he latter . The same str ategy applies to class labels
and v alue labels. The library consis ts of sev eral modules. These modules
consis t of sequences of definitions. Each definition takes t he f orm of a
let
or let rec binding.

2.5 T op-Lev el Entities: P ackag es and Modules 73
2.5.2 Semantics
⟦ ♦ ⟧ Π
∧
= ∅
⟦ D V ∷ 𝑋 = 𝐶⟧ Π
∧
= ⟦ D V ⟧ Π ⊕ {𝑋 ↦ 𝐶}
⟦ dv ∷ 𝑥 = 𝑣⟧ Π
∧
= ⟦ dv ⟧ Π ⊕ {𝑥 ↦ 𝑣}
⟦ D V ∷ 𝑋 = MV ⟧ Π
∧
= ⟦ D V ⟧ Π ⊕ {𝑋 ↦ MV }
TopEmpty
♦ ⇓ T ♦
TopModule
𝑇 ⇓ T 𝑉 ⟦𝑀⟧ FL V ⊆ dom (⟦𝑉 ⟧ Π )
[⟦𝑉 ⟧ Π ]𝑀 ⇓ M MV
𝑇 ∷ 𝑋 = 𝑀 ⇓ T 𝑉 ∷ 𝑋 = MV
TopClass
𝑇 ⇓ T 𝑉 ⟦𝐶⟧ FL V ⊆ dom (⟦𝑉 ⟧ Π )
𝑇 ∷ 𝑋 = 𝐶 ⇓ T 𝑉 ∷ 𝑋 = [⟦𝑉 ⟧ Π ]𝐶
Str uctEmpty
struct ♦ end ⇓ M struct ♦ end
Str uct
struct d end ⇓ M struct dv end [⟦ dv ⟧ Π ]𝑡 ⇓ 𝑣
struct d ∷ 𝑥 = 𝑡 end ⇓ M struct dv ∷ 𝑥 = 𝑣 end
F igure 2.15: Mcl top-lev el ev aluation
Ev aluation of top-lev el terms follo ws the design of a sim ple module
languag e. The big-step relation
𝑇 ⇓ T 𝑉
in F igure 2.15 describes t his process.
An em p ty sequence of top-lev el declarations e v aluates to an em p ty sequence
(rule TopEmpty). N on-emp ty sequences are processed from right to left: If
t he las t element of a non-em p ty sequence is a module declar ation
𝑋 = 𝑀
and t he prefix
𝑇
of t he sequence ev aluates to a top-le v el v alue
𝑉
, t hen the
module
𝑀
is subject to subs titution b y the so obtained structures and classes.
If t he result of this substitution ev aluates to a s tructure, t he corresponding
binding is appended to the top-le v el v alue. N o te t hat the inter pretation of
𝑉
as a partial function via
Π
ensures shadowing . If a top-le v el v ariable is
bound twice, t he later binding dominates the ear lier one in all declar ations
t hat come after w ar d.
Ev aluation of classes is sim pler t han that of modules. Since a class does

74 2 The Model Com position Languag e
no t ev aluate furt her , a class-declaration e v aluates b y appending its right-
hand side to the result of ev aluating t he prefix (rule TopClass). Ev aluation
of modules is defined via
𝑀 ⇓ M MV
. Onl y literal modules ev aluate; free
top-le v el v ariables remain s tuck. An em p ty s tructure is already in normal
f orm (r ule Str uctEmpty). Definitions proceed from t he right to the lef t,
similar to t op-lev el declarations. If the prefix ev aluates to a s tructure, it is
used to subs titute free v ariables in t he bound term. If t he result ev aluates,
its v alue is appended to the structure. Definitions ha v e a str aightforw ard
lexical scope.
2.5.3 Exam ple
Classes and modules can be defined separatel y at t he top le v el. Open
recursion can replace classical recursion. Our pair of example functions,
even
and
odd
f orms a simple class called
Example
. Since they are bo t h
members of t he same class, t he y can refer t o each o ther wit h t he special
v ariable this .
(* Example.mcl.class *)
even = λ n . if n=0 then true else this.odd (n - 1) ;
odd = λ n . if n=0 then false else this.even (n - 1)
On its o wn, a class is meaningless because its members do not e v aluate.
A module t hat ser v es as t he necessar y entr y point can ref er to
Example
as
f ollo ws. The class t hat replaces t he top-le v el automaticall y becomes a roo t
node.
(* Test.mcl.module *)
let test = Example.even 42
When a top-le v el ter m consists of the declarations of Example and Test
(in t hat or der), it can be ev aluated. The deriv ation requires an application
of T opModule and TopClass and o t her wise follo ws the patter n laid out in
Section 2.3.7. It yields a top-lev el v alue consisting of the class and a module
v alue. The latter binds t he field test to true , as expected.

3 modelin g & simula tion as a librar y
U p to this point, Mcl does not pro vide an y means for numerical model-
ing and simulation (aside from basic arit hmetic operations). As already
mentioned, t he reason f or t his is the complexity and div ersity of Modelica ’ s
tr ansformation and simulation semantics. This problem is w ell known and
has been recognized b y different aut hors:
The definition of semantics is in contr ast far more problem-
atic. From Dymola t o Modelica, semantics ha v e alw a ys been
described in an inf ormal manner .[…] This is because a formal
definition of semantics requires a commonl y abstr acted and
w ell defined computational fr amew or k. […] F or equation-based
languag es, such a common com putational fr amew or k does not
exis t in g ener al. (Zimmer [2010])
[…]what is actuall y t he transf or mation process? What is t he
expected result of t he ex ecution? W e w ould argue t hat the
answ er to t hese questions w ould differ depending on who y ou
ask because t he current specification is open f or inter pretation.
(Broman [2010])
As already noted in the outline of t his t hesis, w e attemp t to sol v e t his
issue b y implementing t he aspects of Modelica t hat reg ar d transf or mation
and simulation in a kernel -libr ar y . In this w a y , our inter pretation of t he
specification is codified as an individual com putational frame w or k. This
decision is no sil v er bullet, t he semantics of models and t heir simulation
s till require a considerable effort. It is ho w ev er consider abl y simpler t o
maintain and extend a library in a prog r amming languag e t han to do the
same wit h the informall y specified semantics of a modeling languag e. In a
certain sense, t he kernel library is a middle g round betw een a full-fledg ed
f ormal specification and a prosaic description.
3 . 1 outline
The final pur pose of modeling is a numerical simulation. In t he end, ev er y
part of t he kernel library exists solel y f or t his reason. Therefore, its im ple-
mentation is the most significant aspect of the library . Hence, to mo tiv ate

76 3 Modeling & Simulation as a Libr ar y
t he o t her parts, w e prepone its discussion to t his section; later sections will
discuss its requirements.
3.1.1 The Simulation Loop
As already mentioned, simulation consis ts of a repetitiv e in v ocation of t he
integ r ator in what is often called t he simulation loop . A t each step of this
loop, t he simulation s t at e (T able 3.1) is updated accor dingl y . The simulation
s tate consists of the current simulation time, t he current solution (
reals
,
discrete
) and some meta data (
layout
). It can link to an older state, if nec-
essar y (
pre
). The solv er can indicate found roo ts, and (discrete) equations
might cause discrete-time effects, e.g., mode chang es.
Aspect Descrip tion Initial
time Current simulation time []
layout
Map of unique identifiers to indices of unkno wns
[]
reals Current v alues of t he unkno wns []
pre F or mer Simulation S tate O:none
roots Zero-crossing indicator []
discrete Current discrete simulation s tate see T able 3.7
effects Current discrete effects see T able 3.8
T able 3.1: Simulation State
In a functional languag e like Mcl, a loop takes the form of a tail-recursiv e
function. It expects t he prepared model
prep
(i.e., t he sys tem of equations
after symbolical processing), a descrip tion of t he experiment’ s par ame-
ters
exp
(containing t he s tart and st op time), a list of t he model’ s ev ents,
t he integrator callbacks
rout
and
fout
(see Section 3.3.2), and t he current
simulation s tate
s
. The loop terminates, when the stop time of the giv en
experiment is reached, or termination is explicitly required b y an equation.
If a tr ansition is required, it is directl y ex ecuted (effectiv el y lea ving t he loop
b y a continuation). In an y other case, t he sol v er updates t he simulation
s tate ( Prim:qualidaes ). The result of t he loop is t he final simulation state.

3.1 Outline 77
let rec loop prep exp events root_detect fout s =
if s.time >= exp.stop then
(* Normal end of simulation *)
let _ = Printf:printf "Simulation finished at: %f\n" s.time in
return {s with pre = O:none}
else if s.effects.terminate <> O:none then
(* Simulation stopped *)
let _ = Printf:printf "Simulation terminated at: %f\n'%s'\n"
s.time (O:grab s.effects.terminate) in
return {s with pre = O:none}
else if s.effects.transition >= 0 then
(* Transition requested *)
let _ = fout s in
let _ = Printf:printf "OUT:\n" in
Struct:switch s
else
(* Continue integration *)
let s = {s with pre = O:none} in
loop prep exp events root_detect fout
(Prim:qualidaes s exp.stop [] events
prep.dynamics prep.constraints
root_detect fout exp.interval )
The simulation s tate depends on some support from t he kernel library:
Ob viousl y , it depends on t he definition of t he integ r ator ,
qualidaes
. Its
in put
prep
consis ts of t he quasilinear descrip tion of t he model in the form
of t he dynamic equations and hidden cons traints. These ter ms need to
be encoded in a format suitable f or t he integrator and tr ansformed from
t heir Modelica representation. Similar requirements hold for the discrete
parts of t he model, encoded in t he sol v er output function
root_detect
.
S tructural v ariability is expected in t he f orm of explicit transitions. All these
prerequisites are dealt wit h b y t he kernel library and will be discussed in
t he f ollo wing sections.
3.1.2 S tructur e of t he Libr ar y
The f oundation of t he kernel library is a small set of ubiquitous data s truc-
tures. The actual modeling tools are im plemented on top of it in t hree
dis tinct parts: N umerical and symbolical primitiv es are exposed as built-in
functions in t he firs t part. The second part contains data structures and
algorit hms f or t he cons truction and transf ormation of equations, signals,
and ev ents. Basic modeling routines are defined in t he t hird part which
concludes wit h the in v ocation of t he simulation loop.

78 3 Modeling & Simulation as a Libr ar y
3 . 2 fund ament al d a t a s tr uctures
T w o ubiquitous data s tructures pro vide t he foundation of t he kernel library:
op tional v alues and a g eneric map im plemented as a binar y tree. While t he y
ha v e nothing to do wit h modeling per se, t he f ollo wing implementations
w ould be much more complicated without t heir support. Less common,
but particular l y im portant f or t he com position of Modelica models are
equiv alence relations. A t hird data s tructure supports t hem.
3.2.1 Op tionals
One of t he mos t im portant aspects of functional programming is the ability
to express the absence of t he result of a computation. An em pty arr a y can
encode an absent v alue. Present v alues can be wrapped as one-element
arr a ys:
let none = []
let some x = [x]
let grab o = o.[0]
The cons tructor
none
yields t he absent v alue,
some
wr aps its argument,
and
grab
un wr aps a present v alue. Application of
grab
to
none
results in
a run-time error . The module is a v ailable in t he kernel library under t he
top-le v el name
O
. It also contains t he standar d functions
map
and
flatmap
and is sho wn in Appendix F .1.
3.2.2 Sets and Maps
A second im portant class of v alues deals wit h the collection of data. Both
sets and maps, i.e., collections of v alues allo wing for reasonabl y efficient
lookup and insertion, are used throughout t he ker nel libr ar y and are essen-
tial f or man y symbolical algorit hms and met hods. Sets and maps share t he
same im plementation —balanced binary trees— since sets can be seen as
maps to the unit v alue. The absent node represents t he em p ty tree:
(** Create an empty tree *)
let empty = O:none
N odes contain tw o subtrees, a ke y , a v alue, and t he height of t he larg est
subtree. A tree containing a single node can be created wit h
singleton
.
Insertion of a new v alue f or a giv en ke y wit h
insert
creates a new tree.
This oper ation maintains t he desired ordering of elements and balance of
subtrees.
cardinal
counts t he nodes of a tree,
find
searches t he v alue for

3.3 Quasi-Primitiv e Operations 79
a giv en key , and
delete
remo v es it.
first
and
last
yield t he smalles t and
larg est elements, respectiv el y .
pop
remo v es t he first element. T ransf or ma-
tions are supported wit h
map
and
fold
. All operations that use com parisons
require t he or dering of elements in t he f orm of t he additional argument
ops . The implementation can be f ound in Appendix F .1.
3.2.3 Eq uivalence Relations
Im plicit equations (Section 3.4.9) require t he cons truction of equiv alence
relations betw een connector s . The kernel library implements the classical
disjoint set [T ar jan, 1975] data structure on top of an immutable map. Each
ke y in t hat map points to a recor d with a set of references t o children and an
op tional ref erence to a parent. An update to that map emulates mutation.
The representativ e of a key is either t he representativ e of t he parent
or t he ke y itself if no parent is set for that key . T w o ke ys are equiv alent
if t heir representativ es are equal (function
equiv
). The function
add
joins
t he equiv alence classes of its arguments. All equiv alence classes can be
enumer ated b y first finding all nodes that are t heir o wn representativ e and
t hen collecting their (disjoint) sets of children (function
classes
). The full
im plementation of the module can be found in Appendix F .1.
3 . 3 qu asi-primitive opera tions
The tr ansformation and the simulation of differential equations depend
on symbolical and numerical algorit hms. It is an essential design decision,
ho w to embed t hese oper ations within a modeling language. It might seem
natur al to define them as primitiv e operations, pro vide special syntax, and
define ev aluation or elaboration rules in t he core languag e. W e do no t
f ollo w t his route, t hough, for the follo wing reasons.
N umerical simulation is, b y definition, an appro ximation. Contrary to
t he precise definitions of discrete mathematics used so far , an y implemen-
tation will div ert more or less from an ideal semantics. The anal ysis of t his
deviation and its specification and their im plications for the correctness
properties are sim pl y bey ond t he scope of t his t hesis, which f ocuses on t he
com positional and oper ational aspects of modeling.
The s tructural anal ysis of differential algebr aic equations on t he other
hand is a problem wit hin the domain of discrete mathematics, and undoubt-
edl y could be specified precisely . Ho w ev er , t hese oper ations are not w ell
posed f or t he kernel library , either . N ot onl y is t heir im plementation r at her

80 3 Modeling & Simulation as a Libr ar y
com plex and writing or reading it in pure Mcl w ould be tedious. It is also
no t t he topic of this t hesis.
F or t hese practical consider ations, t he integ rat or and the structural anal y -
sis are linked into the kernel library wit h an ad-hoc foreign function interface
(see Section 4.8 for a short explanation). Thus, f or practical purposes, t he y
can be considered primitives of t he languag e. But, from a rigorous point of
view , t hey are part of the kernel library . In t he follo wing sections, w e shall
discuss t he interface of these quasi-primitiv e operations.
3.3.1 Regularization
The firs t quasi-primitiv e is the structural anal ysis of ordinary differential
equations. The function
sigma_method
expects an arra y of maps from in-
teg er v alues to integ er v alues, t he sparse ro ws of the
Σ
-matrix. It yields
a function t hat maps a ro w index to its dual in the solution v ector . Each
application of t he resulting function maps an equation index t o t he number
of differentiations t hat is necessary for regularization. See [Höger, 2014] f or
a detailed descrip tion of the implementation.
As an exam ple consider t he index-reduction of the cartesian pendulum
from Section 1.1. The unkno wns
𝑥
,
𝑦
and
𝐹
are represented b y t he indices,
0, 1, and 2. Each equation has an entr y in the arra y of ro ws of
Σ
. The result
contains f or each equation t he number of to tal deriv ations t hat need to be ap-
plied to regularize the system: The first equation needs t o be differentiated
twice and t he second and third equations remain unmodified:
let x=0
let y=1
let f=2
(* Create the
Σ
-matrix: *)
let sigma_pend = [
(*
𝑥 2 + 𝑦 2 = 1
*)
T:insert T:int_map x 0 (T:insert T:int_map y 0 T:empty) ;
(*

𝑥 = 𝐹 𝑥
*)
T:insert T:int_map x 2 (T:insert T:int_map f 0 T:empty) ;
(*

𝑦 = 𝐹 𝑦 − 𝑔
*)
T:insert T:int_map y 2 (T:insert T:int_map f 0 T:empty) ]
(* Perform regularization *)
let result = Prim:sigma_method sigma_pend
let sol = [ result 0; result 1; result 2 ]
(* == [2;0;0] *)

3.3 Quasi-Primitiv e Operations 81
This exam ple sho ws that t he regularization imposes tw o requirements
on t he encoding of or dinary differential equations: An equation needs to
pro vide t he necessar y information to calculate its ro w in
Σ
and it must
support an arbitr ar y number of differentiations.
3.3.2 Int egr ation
The second quasi-primitiv e giv es access to t he integrator Qu alid aes [Stein-
brecher, 2006]. In v ocations of
qualidaes
require sev eral ar guments (T a-
ble 3.2). As shown in Section 1.1.3, t he model is passed to the sol v er in
t hree parts: The first is the quasilinear representation of t he system of equa-
tions.
qls
contains quasilinear terms (see Section 3.4.4),
constrs
t he hidden
cons traints in the form of alg ebraic coefficients (Section 3.4.3).
Argument Descrip tion
start Initial simulation s tate
t1 End time of simulation as a floating point number
initial Initial equations as arr a y of alg ebraic coefficients
roots Zero-crossings of ev ents as arra y of alg ebraic coefficients
qls
Quasilinear dynamics
𝐸(𝑡, 𝑥) 
𝑥 = 𝑓 (𝑡, 𝑥)
as arra y of quasilinear
equations
constrs
(Hidden) constr aints
𝐻 (𝑡, 𝑥) = 0
as arra y of alg ebr aic con-
s traints
rout Ev ent combinator as a transf or mer of the simulation state
fout Output combinator as a tr ansformer of t he simulation state
interval
Maximum size of desired output s teps as floating-point num-
ber
T able 3.2: Qu alid aes arguments
Qu alid aes attem pts t o find a consis tent initial solution before each sim-
ulation. The second part of t he input consists of additional initial equations
initial
f or t his task. The t hird part,
roots
, contains ev ents in t he f orm of
zero-crossing functions. Both arguments are arr a ys of algebr aic coefficients.
The solution inter v al
𝕀
is encoded as t he time of the initial simulation state
and an additional explicit s top time t1 .
N aturall y , t he simulated solution needs to be f ed back to the user some-
ho w: Since t he ideal definition of Mcl’ s ev aluation semantics does not allo w

82 3 Modeling & Simulation as a Libr ar y
f or side effects, t his has to happen in a functional s tyle: Qu alid aes expects
tw o operators
fout
and
rout
. The sol v er acts like a catamorphism (a
fold
function) o v er t he sequence of time steps. It maintains an internal simula-
tion state, initialized t o
start
. Whenev er a roo t is found, the operat or
rout
is applied to the current simulation state (containing the current time and
solution). The result of this application replaces t he inter nal simulation state
(t hus, chang es of t he discrete state can tak e effect). If no root is f ound in an
inter v al specified via
interval
, t he sol v er applies
fout
in a similar manner .
Since t he integrator relies on the encoding of equations, w e postpone an
exam ple until Section 3.4.4.
3 . 4 embedding equ a tions
As sho wn in Section 1.1, equations in Modelica are gener al nonlinear first-
or der ordinary differential equations and need to be tr ansformed into the
quasi-linear representation required b y Qu alid aes. N ot all equations of a
model are kno wn at compile time. Their representation, t heir com putation,
and t heir tr ansf ormation into quasi-linear form need to be part of the kernel
libr ar y . The library also contains some fundamental cons tructors for discrete
equations and signals, which are equall y im portant but f ollo w a much more
s traightf or w ar d design.
3.4.1 Shallow v s. Deep Embedding
This design, i.e., t he definition and cons truction of numerical expressions
in a libr ar y , resembles the embedding of a domain-specific languag e (DSL),
where a hos t languag e (Mcl) is used to define expressions of the object
languag e (equations). There are tw o s trategies for the embedding of a
domain specific languag e, dis tinguished b y t he encoding of t he embedded
terms.
In a deep embedding expressions of t he object languag e are data in the
hos t languag e. It usuall y takes t he f orm of algebr aic data types. Inspection
and tr ansformation of object terms are hence as sim ple and efficient as t he
manipulation of an y o t her data in t he hos t languag e. This solution implies
flexibility : The object language is no t necessarily subject t o t he same static
semantics as t he hos t languag e. Hence it becomes possible to trespass the
limits im posed on t he hos t languag e b y its static semantics. It is, f or exam ple,
possible to im plement an inter preter for full Sys tem-F in a language with a
Hindle y -Milner type system.

3.4 Embedding Equations 83
The price f or t his flexibility is a certain inter pretativ e ov er head and t he
loss of s tatic safety . Since expressions are data , an interpreter has to dispatc h
based on t heir categories (usuall y implemented b y inspecting the tag of
a tagg ed union). A dditionall y , whene v er t he computation mak es use of a
hos t-languag e term, t he argument(s) ha v e to be dynamicall y checked, and
t he result needs to be lifted into a v alue of t he object language (b y appl ying
t he appropriate tag). The inter pretativ e ov er head is mos tl y t he result of
managing t hese tags.
A hos t languag e can not type-check an object languag e t hat is more
expressiv e t han itself. Ev en if t here is a decidable static semantics f or t he
object languag e, it can b y design onl y take effect after t he object language’ s
term has been computed. A t t hat point, static saf ety might come too late
1
.
When using a deep embedding, the inter pretation of an object ter m might
g et s tuck , or errors are be detected late.
The second s trategy , ap tl y called shallow embedding, does not represent
object ter ms as data, but as expressions, usually in the f or m of first-class func-
tions. These functions im plement t he object-languag e semantics t hrough
t he means of the host languag e, e.g., b y using t he same mechanisms f or
v alues, binding of v ariables or function composition. Such an embedding
requires no dispatch for ex ecution and no lifting of v alues. This design
decidedl y reduces t he inter pretativ e o v er head. A dditionall y , t he embedding
inherits all t he saf ety guarantees of the host languag e. If t he host languag e’ s
type sys tem is sound, no typed object ter m will ev er g et s tuck. On t he other
hand, t he static semantics of the host languag e limit t he allo w ed expressions
in t he object languag e. It is also not easil y possible to inspect or tr ansform
object terms because functions cannot usuall y be “opened” during ev alua-
tion. This limitation makes it difficult to im plement new oper ations o v er
terms of t he embedded language.
3.4.2 Shallow Embedding in Mcl
The kernel library contains a shallo w embedding of or dinar y differential
equations f or t he f ollo wing reasons:
•
Equations are arguabl y t he mos t crucial aspect of a modeling lan-
guag e. The performance of ev aluating them is alw a ys of high rele-
v ance. Therefore, an y unnecessar y o v er head should be a v oided.
•
Equations are also t he centr al syntactical element of models. Being
unable to appl y static assertions on them, w ould make a type-saf e
1
This does of course not hold if the static semantics co v ers more t han a classical type-
system, e.g., absence of div erg ence or inference of effects

84 3 Modeling & Simulation as a Libr ar y
im plementation of Modelica more or less pointless. Recall that Mcl is
agnos tic reg ar ding a type system. The decision for a deep embedding
w ould put a hea vy burden on t he later introduction of such a facility .
•
The number of oper ations t hat deal wit h equations is small: Modelica
itself does not contain f eatures for the introspection or transf ormation
of equations. Thus, t here are no transf or mations aside from t he ones
required b y t he specification, and t hese are onl y a handful. Therefore,
t he flexibility g ained b y a deep embedding is not needed.
•
Ev en if t his w ould turn out to be a bad design decision in t he future, the
choice of an embedding s trategy is no t irrev ocable: Since equations are
part of t he kernel library , it is alw a ys possible to switch to a different
encoding, should t he need occur .
Using a shallo w embedding presents one significant challenge, though:
T r ansformation of t he g ener ated equations is an essential part of the elab-
or ation. A dditionall y , t he domain of numeric modeling demands high
perf ormance and efficiency while t he mandator y tr ansf ormations, e.g., sym-
bolic differentiation, are prone to yield inefficient and unnecessaril y com-
plicated results. The difference in ex ecution time betw een optimized and
non-op timized differentials can reach sev er al orders of magnitude f or real-
w or ld applications [Guenter, 2007]. Hence a language that is unable to
tr ansform, optimize and possibl y jus t-in-time-com pile t he equation lan-
guag e w ould fall behind t he s tate-of-t he-art.
The solution to t his challeng e is presented b y Carette et al. [2009]. Instead
of directl y encoding t he dynamic semantics of t he object language as hos t-
languag e functions, the implementation essentiall y abstr acts o v er a f old -
oper ator f or t he language using Hask ell’ s type classes or OCaml’s funct ors.
Since Mcl has neit her type-classes nor functors, a more clumsy s tyle has t o
be used here but t he pattern still applies:
let rec c 1 a 1 … a 𝑘 = {
op 1 = λ … ;
…
op 𝑚 = λ … ;
}
…
and c 𝑛 b 1 … b 𝑙 = {
op 1 = λ … ;
…
op 𝑚 = λ … ;
}

3.4 Embedding Equations 85
N ame Descrip tion Argument
simplify
Arit hmetic sim plification; yields a (optional)
kno wn result and t he simplified term Em p ty record
string T extual representation of t he equation Em p ty record
eval
Ev aluation under t he current simulation state
Simulation s tate
T able 3.3: Operations on Alg ebraic Equations
F or a finite set of operations
op 1 … op 𝑛
all syntactical v ariants of t he DSL
c 1 … c 𝑚
are encoded as (mutuall y recursiv e) constructors yielding recor ds
of t he shape sho wn abo v e. Because the constructors are mutuall y recursiv e
(and due to lexical scoping), all operations can create arbitr ar y terms of
t he languag e. The downside of this patter n is that t he addition of a new
oper ation requires t he chang e of the original im plementation (and spans
o v er sev eral cons tructors). In contr ast, it is sim ply possible t o deriv e a
slightl y extended language b y adding new constructors. This dichot om y is
a long-kno wn fact coined t he expr ession-pr oblem b y W adler [W adler, 1998].
In t he case of this ker nel libr ar y t his does no t pose a problem: As already
mentioned, t he set of operations is finite (and r at her small) but adding
new cons tr uctors, e.g., trigonometric functions, might be necessary for later
v ersions of t he library . Thus, t he selection of a shallo w embedding fits nicel y
into the domain of t he library .
3.4.3 Alg ebr aic Cons tr aints
The fundamental embedded terms represent algebr aic coefficients (T a-
ble 3.4), i.e., equations t hat do not contain deriv ativ es. The set of con-
s tructors is intentionall y minimal. Subtraction, f or instance, is encoded as
addition of a negated term. There are constructors f or real-v alued cons tants
constant
and t he independent v ariable
time
. The
𝑖
-t h element of the state
v ector is represented b y
unknown
. Operat ors include addition of ter ms
add
,
multiplication mul and t he multiplicativ e in v erse inv .
Each record im plements t hree basic operations (T able 3.3):
eval
ev al-
uates t he term under a giv en simulation state (see T able 3.1 for a detailed
descrip tion),
simplify
sim plifies the term b y ev aluating all constants and
appl ying some arit hmetic identities and
string
yields a textual represen-
tation of t he term. The module
Alg
contains t he full im plementation of all
v ariants and t heir operations (Appendix F .3).

86 3 Modeling & Simulation as a Libr ar y
Cons tructor Descrip tion Argument(s)
const A cons tant function 𝑓 (𝑡, 𝑥) = 𝑐 Constant 𝑐
time
A function t hat alw a ys yields t he v alue
of t he independent v ariable 𝑓 (𝑡, 𝑥) = 𝑡 Emp ty record
unknown An unkno wn 𝑓 (𝑡, 𝑥) = 𝑥 𝑖 Index 𝑖
add
A ddition of tw o continuous terms
𝑓 (𝑡, 𝑥) = 𝑎(𝑡, 𝑥) + 𝑏(𝑡, 𝑥) Summands 𝑎 and 𝑏
mult
Multiplication of tw o continuous ter ms
𝑓 (𝑡, 𝑥) = 𝑎(𝑡, 𝑥)𝑏(𝑡, 𝑥) F actors 𝑎 and 𝑏
inv
Multiplicativ e in v ersion of a continu-
ous term 𝑓 (𝑡, 𝑥) = 1
𝑔(𝑡,𝑥)
Coefficient 𝑔
signal
Lifts a discrete-time signal into a
continuous-time equation The signal recor d
T able 3.4: Constructors of Alg ebraic Equations
As an exam ple, consider t he multiplication of tw o ter ms. It takes tw o
arguments,
a
and
b
. T o ev aluate the product of tw o algebr aic ter ms, it
suffices to e v aluate t he fact ors and multipl y t he result. Alt hough it is unused
in t hat sim ple case, t he argument
s
is no t arbitrary but contains t he current
simulation s tate. Alg ebraic coefficients that represent unknowns or the
current simulation time need it to look up their current v alue.
and mult a b = {
eval = λ s. (a.eval s) *. (b.eval s) ;
…
}
Sim plification maps each term to a term
dynamic
and an op tional floating
point number
static
. The former contains t he term af ter sim plifying it as
much as possible. The latter contains a floating-point number if t he term
can be com pletel y ev aluated to that number or none other wise.
simplify = λ _.
let a = a.simplify {} in let b = b.simplify {} in
if (a.static = O:some 0.0) || (b.static = O:some 0.0) then
( *a*0=0*b*0* )
{static = O:some 0.0; dynamic = const 0.0}
else if a.static = O:some 1.0 then b
( *1*b=b* )
else if b.static = O:some 1.0 then a
( *a*1=a* )

3.4 Embedding Equations 87
else if b.static <> O:none && a.static <> O:none then
(* Constant folding *)
let prod = (O:grab a.static) *. (O:grab b.static) in
{static = O:some prod ; dynamic = const prod}
else {static = O:none; dynamic = mult a.dynamic b.dynamic};
The sim plification of a product proceeds b y sim plifying both arguments.
If eit her of the tw o simplifies to zero, t he whole term is zero. Ot her wise,
if one of t he arguments sim plifies to one, the other argument is retur ned.
If neit her sim plifies to zero or one, but both sim plify to some v alue, t he
product of t hese v alues is t he result. In an y other case, t here is no furt her
sim plification.
3.4.4 Quasi-Linear Eq uations
The s tructure of a quasilinear term is str aightf or w ard. Coefficients
𝑓 𝑖 (𝑡, 𝑥)
are (sparsel y) mapped to deriv ativ es using t he binary tree defined ear lier
and combined wit h an alg ebr aic coefficient
𝑐(𝑡, 𝑥)
. Ev aluation depends on
t he deriv ativ es and is t heref ore part of Qu alid aes, not t he kernel library .
Intuitiv ely , it is t he im plementation of a sum of products:
𝑞(𝑡, 𝑥, 
𝑥) = 𝑛
􏾜
𝑖=1 𝑓 𝑖 (𝑡, 𝑥) 
𝑥 𝑖 + 𝑐(𝑡, 𝑥)
Operations
The kernel library restricts itself t o operations that are required for the trans-
f ormation from ordinary to quasi-linear differential equations. Primitiv e
terms are con v erted triviall y to their quasi-linear representation. Oper ations
are transf or med b y transf or ming t heir respectiv e arguments recursiv el y
and combining t he results using the r ules of calculus. T ake, for exam ple,
t he addition of tw o quasilinear ter ms (𝑓 , 𝑐) and (𝑔, 𝑑) :
(𝑓 (𝑡, 𝑥) ⋅ 
𝑥 + 𝑐(𝑡, 𝑥)) + (𝑔(𝑡, 𝑥) ⋅ 
𝑥 + 𝑑(𝑡, 𝑥)) ∧
= ℎ(𝑡, 𝑥) ⋅ 
𝑥 + 𝑏(𝑡, 𝑥)
where ℎ 𝑖 (𝑥, 𝑡) ∧
= 𝑓 𝑖 (𝑥, 𝑡) + 𝑔 𝑖 (𝑥, 𝑡)
𝑏(𝑥, 𝑡) ∧
= 𝑐(𝑥, 𝑡) + 𝑑(𝑥, 𝑡)
This can be directl y implemented as a function of quasilinear terms.
Each coefficient is t he alg ebr aic sum of t he corresponding coefficients of t he
summands. The onl y particularity lies in t he application of t his construction
to the tw o sparsel y populated in put v ectors
ℎ
and
𝑓
: This transf or mation
tr anslates directl y to Mcl. Each coefficient is t he alg ebraic sum of t he

88 3 Modeling & Simulation as a Libr ar y
corresponding coefficients of t he summands. The onl y particularity lies
in t he application of this construction to the tw o sparsel y populated in put
v ectors ℎ and 𝑓 :
(** Sum up two ql-terms by summing up all coefficients *)
let sum a b =
let add_coeff k v f =
let g = T:find T:int_map k f in
let new = O:safe v (O:map (Alg:add v) g)
in T:insert T:int_map k new f
in
{ c = Alg:add a.c b.c ; f = T:fold add_coeff a.f b.f }
The set of quasi-linear terms is not closed under basic arithmetic oper -
ations (e.g. t he multiplication of tw o quasi-linear ter ms is not itself quasi-
linear). In some cases, t he result depends on t he con v ersion of a quasi-linear
term into an alg ebr aic coefficient. This is possible, when additional equa-
tions 
𝑥 𝑗 = 𝑥 𝑖 f or new unkno wns 𝑥 𝑖 are introduced:
(** convert a ql-term into a constraint-term:
create an identity for all non-zero coefficients
and sum up the resulting constraints *)
let to_constr state ql =
let add_alg k v r =
(* Obtain identity for der(x_k) *)
let fresh = obtain_id r.state k in
{state = fresh.state;
result = Alg:add r.result (Alg:mult v (Alg:unknown
fresh.result))}
in
T:fold add_alg {state = state; result = ql.c} ql.f
The function
obtain_id
yields such an identity . The identities are man-
ag ed b y t he state s tored in
r.state
. The update to this state is threaded
t hrough the fold function of t he binar y tree module.
Exam ple
As an exam ple, consider t he differential of t he exponential

𝑥 = 𝑥
. This sim ple
ODE can be integ r ated as sho wn in Listing 3.1. The equation is already
in quasilinear f orm and can be encoded wit h t he kernel module
Ql
(see
Section 3.4.4). The initial conditions are encoded as an alg ebr aic coefficient
and an em p ty simulation s tate. There are no ev ents nor hidden constr aints.

3.4 Embedding Equations 89
Lis ting 3.1: Integ ration Exam ple
let x=0
(*

𝑥 = 𝑥
*)
let qls = [
Ql:sum (Ql:der x (Alg:const 1.0))
(Ql:of_constr (Alg:mult (Alg:unknown x) (Alg:const (-1.0))))
]
(* initial:
𝑥 = 1
*)
let initial = [Alg:add (Alg:unknown x) (Alg:const -1.0)]
(* initial state *)
let start = {Sim:default_sim_state with reals = [1.0]}
let fout state =
(* Output the current state, using OCaml directly *)
let _ = Printf:printf "%f\n" state.reals.[x] in state
let integrate =
Prim:qualidaes start 1.0 initial [] qls [] ( λ s.s) fout 0.25
(* outputs 1.000000 1.284076 1.648692 2.117011 2.718290 *)
A sim ple output routine
fout
is im plemented b y directl y in v oking t he built-
in routine Printf:printf t hat the targ et languag e, OCaml, pro vides.
3.4.5 Or dinar y Differ ential Equations
The embedded languag e of nonlinear first-or der ordinary differential equa-
tions is im plemented similar l y to the alg ebraic coefficients. As an addition,
deriv ativ es can be expressed wit h t he cons tructor
der
(T able 3.5). Differen-
tial equations support oper ations for textual representation and sim plifica-
tion as sho wn in T able 3.6. There is no operation f or t he ev aluation since
t he v alues of t he deriv ativ es onl y exist inside the integ rat or . As required b y
t he s tructural anal ysis, operation sigma fills a ro w of t he Σ matrix.
Equations are not closed under differentiation with respect to the in-
dependent v ariable since t he deriv ativ e of t he first-or der differential ter m

𝑥(𝑡)
w ould yield a second-order term. In t hat case, t he differentiation can
create artificial equations, similar to the identities introduced when trans-
f orming quasi-linear ter ms to alg ebraic cons tr aints. This “side effect” of

90 3 Modeling & Simulation as a Libr ar y
N ame Descrip tion Argument(s)
const Cons tant term 𝑓 (𝑡, 𝑥, 
𝑥) = 𝑐 Constant 𝑐
signal A (discrete-time) signal The signal
time The independent v ariable 𝑓 (𝑡, 𝑥, 
𝑥) = 𝑡 Emp ty record
unknown An unkno wn 𝑓 (𝑡, 𝑥, 
𝑥) = 𝑥 𝑖 Index 𝑖
der A deriv ativ e 𝑓 (𝑡, 𝑥, 
𝑥) = 
𝑥 𝑖 Index 𝑖
add A ddition 𝑓 (𝑡, 𝑥, 
𝑥) = 𝑎(𝑡, 𝑥, 
𝑥) + 𝑏(𝑡, 𝑥, 
𝑥) Summands 𝑎 and 𝑏
mult Multiplication 𝑓 (𝑡, 𝑥) = 𝑎(𝑡, 𝑥, 
𝑥)𝑏(𝑡, 𝑥, 
𝑥) F actors 𝑎 and 𝑏
inv Multiplicativ e in v erse 𝑓 (𝑡, 𝑥, 
𝑥) = 1
𝑔(𝑡,𝑥, 
𝑥) Coefficient 𝑔
T able 3.5: Constructors of Differential Equations
adding new equations can be im plemented as an operation that takes a
differentiation s tate and yields a new s tate and bo t h t he differentiated term
and its deriv ativ e b y combining differentiated sub ter ms according t o t he
rules of calculus.
diff = λ s.
let da = a.diff s in
let db = b.diff da.state in
{state = db.state;
result = add (mult da.result b) (mult db.result a) } ;
The second addition to the operations on alg ebraic terms is t he tr ans-
f ormation of an ordinary differential equation into quasi-linear f orm. This
oper ation im plements t he algorit hm sketched in ear lier w or k [Hög er and
S teinbrecher, 2015] and uses the transf ormation of quasi-linear ter ms into
alg ebr aic cons traints, Ql.to_constr that w as introduced in Section 3.4.4.
ql = λ ts.
let qla = a.ql ts in
let qlb = b.ql qla.state in
let ca = T:cardinal qla.result.f in
let cb = T:cardinal qlb.result.f in
let ql = if ca > cb then qla.result else qlb.result in
let cnstr =
Ql:to_constr qlb.state ( if ca > cb then qlb.result else
qla.result)
in
{state=cnstr.state; result = Ql:mult_c ql cnstr.result};

3.4 Embedding Equations 91
In t he case of the multiplication, t he transf ormation t hreads t he giv en
s tate
ts
t hrough the transf ormation of t he sub-ter ms
a
and
b
. The smaller of
t he tw o resulting quasi-linear ter ms is con v erted into an alg ebraic coefficient
and multiplied wit h the larg er one. The size is measured wit h
cardinal
as
t he number of non-zero coefficients, i.e., t he number of nodes in t he tree
t hat represents the under l ying map. Multiplication of QL-ter ms wit h an
alg ebr aic coefficient is ag ain a str aightfor w ar d operation.
N ame Descrip tion Argument(s)
simplify
Arit hmetic sim plification; yields a
recor d of t he (op tional) kno wn result
and t he sim plified ter m
Em p ty record
string T extual representation of t he equation Emp ty record
diff
Differentiation; yields t he deriv ativ e of
t he equation Differentiation s tate
ql
yields t he quasi-linear representation of
t he equation T ransf ormation state
sigma
U pdates t he giv en row with t he struc-
tur al signature of t he term. A ro w of Σ
T able 3.6: Operations on Differential Equations
3.4.6 Hidden Cons tr aints
A quasi-linear equation wit h an em p ty v ector of coefficients, i.e., all deriv a-
tiv es are multiplied wit h zero, is a visible constr aint. Hidden constr aints
are no t so easy to spo t (hence the name). In gener al, all constr aints are
visible when t he sys tem is in lo w er -echelon form, i.e., it consists of
𝑛
s tates,
𝑛 equations wit h non-zero coefficients and 𝑚 cons tr aints.
Gaussian Elimination
A symbolical Gaussian elimination process can obtain this format: F or each
s tate
𝑥 𝑗
(i.e. column in t he matrix of quasilinear equations), select a ro w
𝑓 (𝑡, 𝑥) ⋅ 
𝑥 + 𝑐(𝑡, 𝑥)
t hat has a non-zero coefficient
𝑓 𝑗 (𝑡, 𝑥) ∧
≠ 0
f or t hat column
and eliminate t he s tate from all remaining ro ws
𝑔(𝑡, 𝑥) ⋅ 
𝑥 + 𝑑(𝑡, 𝑥)
b y appl ying
a s tandard piv o t operation (i.e. subtr act 𝑔 𝑗 (𝑡,𝑥)
𝑓 𝑗 (𝑡,𝑥) (𝑓 (𝑡, 𝑥) ⋅ 
𝑥 + 𝑐(𝑡, 𝑥) )).

92 3 Modeling & Simulation as a Libr ar y
This tr ansformation is problematic when t he coefficient is close to zero
(
𝑓 𝑗 (𝑡, 𝑥) ≈ 0
). As t he dynamic beha vior of t he transf or med system is a-priori
unkno wn, all non-constant coefficients ha v e to be considered as potential
zeros. F or t his reason, t he piv ot oper ation should onl y divide ro ws b y trul y
cons tant coefficients (i.e. coefficients created wit h
Alg:constant
). If no row
wit h such a coefficient exis ts for a giv en column
𝑗
, t he no w familiar patter n
of adding a new equation is applied: A new equation

𝑥 𝑗 = 𝑥 𝑘
wit h a new
column
𝑘
is added and all occurrences of

𝑥 𝑗
are replaced b y
𝑥 𝑘
, effectiv ely
eliminating t he column 𝑗 .
Exam ple
In t he case of t he pendulum, onl y Equation (5) is ob viousl y a constr aint. As
sho wn in Section 1.1, t he hidden constr aints can be obtained b y eliminating

𝑥 and 
𝑦 from Equation (6) and 
𝑣 𝑥 and 
𝑣 𝑦 from Equation (7).

𝑥 = 𝑣 𝑥
(1)

𝑦 = 𝑣 𝑦
(2)

𝑣 𝑥 = 𝐹 𝑥 (3)

𝑣 𝑦 = 𝐹 𝑦 − 𝑔 (4)
𝑥 2 + 𝑦 2 = 1 (5)

𝑥𝑥 + 
𝑦𝑦 = 0 (6)

𝑣 𝑥 𝑥+𝑣 2
𝑥 + 
𝑣 𝑦 𝑦+𝑣 2
𝑦 = 0 (7)
In t his exam ple, onl y non-zero coefficients are used. Equation (6) is
tur ned into a cons traint b y subtracting Equation (1) multiplied with t he
piv ot fact or
𝑥
and Equation (2) multiplied wit h
𝑦
. Similar l y , Equation (7) is
tr ansformed b y subtracting Equation (3) multiplied with
𝑥
and Equation (4)
multiplied wit h 𝑦 .
Im plement ation
The im plementation of t his tr ansf ormation is contained in the module
Gaussian
(Appendix F .6). It demonstr ates an interesting particularity re-
g arding the inspection of equations. As mentioned abo v e, t he transf or -
mation is unsafe in case of a non-cons tant piv ot f actor . This case can be
detected wit hout resorting to a deep embedding: Onl y if t he simplification
yields a s taticall y known result, the alg ebr aic coefficient is a constant.

3.4 Embedding Equations 93
F ield Descrip tion Initial
ints Discrete Integ ers []
reals Discrete R eals []
bools Discrete Booleans []
int_layout Integ er la y out T:empty
real_layout R eal la y out T:empty
bool_layout Boolean la y out T:empty
T able 3.7: Discrete elements of t he simulation state
3.4.7 Signals & Ev ents
Discrete Modeling in Modelica resembles t he no tion of functional reactiv e
prog r amming[W an and Hudak, 2000, Nilsson et al., 2002]. Signals (or
Beha viors) are functions t hat retur n v alues depending on some abstr act
domain of time
𝑇
. Ev ents are signals of possibl y undefined (i.e. op tional)
v alue. Both entities are first class v alues in FRP as w ell as in the kernel
libr ar y .
The kernel library deviates from the classical implementation of func-
tional reactiv e prog ramming in tw o w a ys: Because t he integ r ator controls
t he detection of ev ent guar ds (Section 3.3.2) t he continuous simulation
driv es t he discrete parts. In contras t, W an and Hudak [2000] describe an
integ r ator driv en b y a signal. Due to the technical properties of Qu alid aes,
t he kernel library is tied to the former str ategy . The second difference con-
cer ns the computation of a signal’ s v alue. Where classical FRP -signals are
pure functions wit h a defined in put and output, signals in Modelica are
determined b y discrete-time v ariables. The v alue of such a discrete v ariable
is determined b y a global fix ed-point (specified as ev ent iteration[Modelica,
2012, Chap ter 3]) o v er all discrete-time equations (Section 3.4.8). F or t his
reason, signals in t he kernel library are im plemented no t as stream tr ans-
f ormers, but as functions t hat transf orm t he simulation state (see T able 3.1).
P art of t hat state is the v alue of t he discrete v ariables of t he model (T a-
ble 3.7). It consists of a flat representation of the discrete memor y , giving
access to s tored integ ers, floating point v alues, and boolean v ariables. A d-
ditionall y , t he la y out, i.e., a map from unique identifiers to arr a y indices, is
maintained f or each kind of v ariable. An y v alue can be lif ted into a signal:

94 3 Modeling & Simulation as a Libr ar y
(** lift a value into a signal *)
let pure x = { read = λ s. x ; write = pass }
(** apply the value of a signal to another signal *)
let fmap f a = { read = λ s. (f.read s) (a.read s) ; write = pass }
If t he v alue of a signal is a function, it can be applied to ano ther signal in
a v er y natural w a y using
fmap
. The presence of both functions means t hat
Mcl’ s signals are an applicativ e functor [McBride and P aterson, 2008]. This
obser v ation might seem like a purel y academic remar k, but it im plies t hat
all oper ators and functions defined in Mcl can be lifted into the domain of
signals. Such a lif ting can alw a ys be inserted b y t he front-end directl y but
is sometimes r at her cumbersome to read. The rest of the kernel library t hus
defines primitiv es for the access t o discrete v ariables and some common
oper ators mainl y to enhance the readability of Mcl-models.
Some signals can be modified wit h an additional method,
write
. It
yields an updated simulation s tate, wit h t he new v alue of t he v ariable and
t he ev ent flag. Read-onl y signals return t he state unmodified.
(* A discrete Integer variable *)
let int j = { read = λ s. s.discrete.ints.[j] ;
write = λ x s.
let new_ints = [s.discrete.ints with j = x] in
let new_event = s.effects.event
|| s.discrete.ints.[j] <> x
in
{s with
discrete = {s.discrete with ints = new_ints };
effects = {s.effects with event = new_event }
}
}
3.4.8 Discr ete Eq uations & Event It er ation
Discrete equations chang e the discrete state of a simulation. Such equations
relate a discrete v ariable wit h a term ov er o t her discrete v ariables t hat
determines its output. Despite t his syntactic adherence to the Modelica ’ s
sing le-assignment rule, t heir effects are not causal, i.e., t hey do no t form a
global ex ecution order . The specification of Modelica does not pro vide a
met hod to sol v e all discrete equations, and wit hout furt her pro visions, such
a met hod canno t exis t, because it is possible to model arbitr ar y Diophantine
equations. The solution of t hese equations, (Hilbert’s 10t h problem), is
famousl y undecidable [Matiy asevich, 1993].

3.4 Embedding Equations 95
T o no t inflate t he libr ar y with an attemp t to sol v e a subset of t hese
equations, w e restr ain the domain to piecewise causal discrete equations, i.e.,
recursiv e definitions of an element of t he new discrete state. Therefore,
t he search f or a solution of t he discrete sys tem of equations is an iter ativ e
process, which runs until it reaches a fix ed point. Ensuring t he exis tence of
such a fix ed point lies in t he responsibility of t he modeler .
In each s tep of t hat iter ation t he global set of discrete equations
eqs
calculates a chang ed simulation state. This calculation is defined b y t he
sequential application of t heir
write
met hods. As a f eedback to the iteration
loop, t hese methods ha v e to update t he
event
flag of t he s tate when t he y
chang e t he v alue of one or more discrete v ariables. Iteration then contin-
ues until t he
event
flag is not set. The implementation can be f ound in
Appendix F .7.
Mos t discrete equations are part of a
when
-equation, i.e., t he y are onl y
activ e, when t he Boolean signal used as guard chang es from
false
to
true
.
Ev ent guards, in tur n, can refer t o an y continuous or discrete v ariables.
Each discrete equation ma y t hus trigg er ev ents.
(* Pre-signal *)
let pre signal = {
read = λ s.(signal.read (O:grab s.pre));
write = pass ;
}
(* edge() operator *)
let edge signal = {
read = λ s.(signal.read s) && (not ((pre signal).read s)) ;
write = pass ;
}
Each iteration s tep updates t he field
pre
of t he simulation s tate. It
consis ts of a cop y of t he simulation s tate from t he las t iteration. Signals g ain
access to that “memor y” of t he iter ation wit h t he
pre
oper ator (im plemented
as t he higher -order signal
pre
). The ev ent g enerat or
edge
makes use of this
capability to find the rising flank in a Boolean signal.
This oper ator allo ws the definition of a Modelica-sty le
when
-equation.
Such a discrete equation wraps a signal (the guard) and an arra y of equations.
Its effect is t he activ ation of all wrapped equations when t he guard becomes
true. Modelica supports
else
-clauses f or
when
equations. In our library ,
t he y are additional arguments in an arr a y of discrete equations, elsewhen :

96 3 Modeling & Simulation as a Libr ar y
let when signal eqs elsewhen = {
read = λ s . {};
write = λ _ s.
if (edge signal).read s then
write_all eqs 0 s
else
O:safe s (O:map ( λ w. w.write {} s) elsewhen)
}
During ev ent iteration, effects might become activ e. These are indicated
b y flags in t he
effects
recor d of t he simulation s tate (T able 3.8). The itera-
tion sets t he
event
to
true
and continues. If a special operat or reques ts a
tr ansition or termination, t he simulation loop reacts accordingl y .
Flag Descrip tion Initial
event Indicates t he chang e of a discrete v ariable false
transition Index of t he reques ted mode tr ansition -1
terminate Indicates termination of simulation false
T able 3.8: Elements of t he simulation state related t o discrete effects
3.4.9 Im plicit Equations
Modelica supports equations t hat are im plicitl y created during elaboration.
These
connect
-equations mimic ph ysical connections betw een parts of t he
model. The sim plest such equations are inspired b y Kirchhoff’ s la w s and
used to con v enientl y model electrical circuits. (The same mechanism also
applies to different domains but loses some of its intuition.) F or t he sake
of brevity , w e ignore t he more elaborate o v ercons tr ained and stream con-
nectors. Their im plementation follo ws a similar pattern but is considerabl y
more in v olv ed.
N ame Descrip tion Initial
potential Equiv alence classes of potential quantities Equiv:empty
flow Equiv alence classes of flo w quantities Equiv:empty
asserts assert -s tatements []
T able 3.9: Implicit Equation S tate

3.4 Embedding Equations 97
In t his elementary case, a
connect
-s tatement introduces a relation be-
tw een a pair of unkno wns. Bo th unknowns are labeled eit her flow or po t ential
quantities. A t elaboration time the equiv alence class of each such relation
is used to create a single sum-t o-zero equation f or t he flo ws:
let connect_flow a ao b bo = do
implicits ← implicits • get ;
implicits • put {implicits with flows =
Equiv:add T:int_map implicits.flows
( if ao then -a else a)
( if bo then -b else b)}
A flo w connection combines t he flo w v ariables
a
and
b
and t heir respec-
tiv e outside flags [see Modelica, 2012, Chapter 9]
ao
and
bo
into an integ er .
These flags control t he “direction” of a connection and hence the sign of
t he associated v ariable inside t he g ener ated equations.
let flatten_flows =
let rec create_sum y _ r =
if (y < 0) then
(Ode:diff r (Ode:unknown (-1 * y)))
else
(Ode:add r (Ode:unknown y))
in
let flatten_flow_set x set r =
let rest = T:delete T:int_map x set in
let base =
if (x < 0) then
(Ode:mult (Ode:const -1.0) (Ode:unknown (-1 * x)))
else
(Ode:unknown x)
in do
Basic:equation (T:fold create_sum base rest) ; r
in do
implicits ← implicits • get ;
let classes = Equiv:classes T:int_map implicits.flows in
T:fold flatten_flow_set (return {}) classes
F unction
flatten_flow_set
creates a new routine. Elabor ation of t hat
routine yields t he sum-to-zero equation. The sets are enumerated with t he
module
Equiv
and elabor ated wit h t he
fold
oper ation from t he module
T
.
Flattening po tentials w or k s similar l y but g ener ates
𝑛−1
equalities ins tead
of one sum-to-zero equation.

98 3 Modeling & Simulation as a Libr ar y
3 . 5 elabora tion interf a ce
W it h the embedded equations it is in principle possible to express t he system
of equations of a Modelica-model. Using t he quasi-primitiv e oper ations,
it is also possible to regularize and simulate it. This approach is ho w ev er
no t onl y v erbose but also quite repetitiv e and hardl y compositional. This
section adds t he missing pieces b y pro viding some fundamental elaboration
routines.
3.5.1 Chec kpoints and T r ansitions
The languag e means introduced in Section 2.4 f or t he modeling of v ariable-
s tructure systems are v er y abstr act and do no t immediatel y lead to an
eleg ant f or mulation of such models. As discussed in Section 1.1.4, a common
meta-model f or t his kind of application is t hat of a h ybrid automat on, wit h
t he significant dr a wback t hat modes and transitions are second-class citizens
and t hus the number of possible modes is limited in practice.
Experiment
Ins tantiate
𝐴
𝐵 1 / 𝐵 2 𝐶 1 / 𝐶 2
Simulate
Process Ev ents
F igure 3.1: Control Flo w of a Mode Chang e
Com puting new modes on-the-fly allo ws f or an infinite number of po ten-
tial modes (since onl y one mode is in existence at an y giv en time). But t heir
com position causes a challeng e when it comes to the control flo w of a simu-
lation. Consider a model
𝐴
wit h tw o components
𝐵
and
𝐶
(F igure 3.1).
𝐵
and
𝐶
each consis t of tw o modes
𝐵 1,2
and
𝐶 1,2
. The global mode can be com-
puted b y fact orization.
𝐴
im plicitl y consists of f our modes:
(𝐵 1 , 𝐶 1 )
,
(𝐵 1 , 𝐶 2 )
,
(𝐵 2 , 𝐶 1 )
and
(𝐵 2 , 𝐶 2 )
(one f or each possible combination of t he modes of its
com ponents).
But ho w does a transition, sa y
𝐵 1 ⟶ 𝐵 2
obtain t he new mode of
𝐴
?
T o process the change, it does no t suffice to onl y implement t he difference
betw een
𝐵 1
and
𝐵 2
, since
𝐴
(and its parents) are also po tentiall y in v alid.
The y might, for ins tance, refer t o an unkno wn that is allocated in different
positions in
𝐵 1
and
𝐵 2
. Ev er y step of the elaboration that has been taken after

3.5 Elaboration Interface 99
t he elabor ation of 𝐵 1 must thus be repeated. The det ection of t he tr ansition
happens during simulation time. It is a part of t he ev ent handling. Theref ore,
it is necessar y to r eturn from there to the point at which t he transition w as
declared and appl y t he transition ’s effect. F rom t here on,
𝐶
can com pute its
new mode, and simulation can continue.
This process resembles t he mechanism of ex ceptions kno wn from classi-
cal prog r amming languag es in t he sense that it exposes a non-local change
in t he control-flo w . The languag e means necessar y to express such a change
ha v e been introduced in Section 2.4 wit h t he routines
continue
and
call/cc
.
R esetting the elaboration s tate to a f ormer l y reached v alue is possible wit h
get
and
put
. The kernel library combines t hese tw o features to pro vide a
c hec kpointing facility :
(** Create a checkpoint and store it
in the modeling state *)
let checkpoint = do
state ← get ;
c ← call/cc ( λ k. return {routine = k;
state = state}) ;
cs ← checkpoints • get ;
checkpoints • put (cs <+ c) ;
return |cs|
checkpoint
s tores the current elaboration s tate and the current continua-
tion in an arr a y of checkpoints. The actual chang e of control flo w can t hen be
ex ecuted via
continue
after obtaining t he targ et from t he field
transition
of
t he current effects s tored in the simulation state. A corresponding discrete
equation, t he switch, has to determine these effects.
let switch pre =
let t = pre.effects.transition in
do
cs ← checkpoints • get ;
put cs.[t].state ;
sim ← sim_state • get ;
sim_state • put (forward sim pre) ;
initial_equations • put [] ;
continue cs.[t].routine with cs.[t]
When ex ecuting such a transition, t he numerical results are rest ored
depending on t heir allocation time. The current simulation state is used
to “f as t-forw ard” the v alues of t he discrete and continuous simulation
v ariables t hat w ere allocated befor e the declaration of the transition. T o
achiev e t his despite a probable chang e in t he la y out of t he solution v ectors,

100 3 Modeling & Simulation as a Libr ar y
each v ariable needs a unique identity , e.g., its instance name. U nkno wns
t hat are allocated after t he transition, e.g., in a parent model, are no t present
in t he res tored s tate. They obtain their start-v alues from the old simulation
s tate when t hey are allocated. T o t his end, t he
pre
-element of t he new
simulation s tate keeps the old state until simulation res tarts.
3.5.2 Basic Routines
The mos t basic aspect of elaboration is the allocation of unkno wns and equa-
tions. These operations are im plemented as modeling routines in t he kernel
libr ar y . They incrementall y create a model b y successiv e modification of
t he elabor ation state. The com plete elabor ation state is sho wn in T able 3.10.
An initial state (i.e. all aspects set to t heir respectiv e initial v alues) can be
accessed via
Sim:initial_state
. The allocation of a new equation is im ple-
mented b y sim pl y appending it to t he end of t he arr a y of current equations
(continuous, discrete, and initial equations are all handled similar l y):
let equation eq = do
eqs ← equations • get ;
equations • put (eqs <+ eq)
Allocations of unkno wns and discrete v ariables need to consider the
possible presence of a transition. The tr ansition might w ant to chang e t he
v alue of a v ariable depending on t he last simulation s tate. Therefore, the
allocation of a simulation v ariable is significantl y more in v olv ed:
let real uid start restart =
let read_reals s = s.reals in
let write_reals s reals = {s with reals = reals} in
do
log_real uid ;
allocate_variable read_reals write_reals uid start restart
The routine
allocate_variable
(see Appendix F .9) expects a reader and
a writer , i.e., functions t hat can eit her g et a v alue from a simulation state
or update it (creating a new s tate). Depending on the content of t hese
arguments, the routine adds a discrete v ariable of an y type or an unknown
to the model. The unique identifier (
uid
) of a v ariable is used to determine
its position in t he f ormer simulation state. If t hat
pre
s tate does not exis t,
t he v ariable has not been allocated in it, or
restart
is
true
) t he v ariable is
set to its
start
v alue. This setting enables initial allocation, allocation of
new v ariables af ter a mode chang e, and reinitialization wit h a new v alue

3.5 Elaboration Interface 101
Aspect Descrip tion Initial
equations ODEs of t he model []
initial_equations
Initial (alg ebr aic) equations of t he
model
[]
discrete_equations Discrete equations of t he model []
events Zero crossings []
checkpoints Checkpoints f or mode-chang es []
sim_state Current simulation s tate see T able 3.1
experiment Experiment Setup 𝕀 = [ 0, 1 ]
outputs Obser v ed v ariables []
implicits Current Im plicit Equations T able 3.9
T able 3.10: Model elaboration s tate
when a tr ansition demands it. Ot her wise, a v ariable retains its v alue from a
f ormer state.
It remains to sho w ho w the elaboration initiates t he simulation. The
simulate
routine expects a single argument
m
(t he roo t-model). This argu-
ment is elabor ated, and t he required aspects of t he model are queried from
t he s tate:
let simulate m =
do m;
exp ← experiment • get ;
os ← outputs • get ;
ieqs ← initial_equations • get ;
eqs ← equations • get ;
deqs ← discrete_equations • get ;
events ← events • get ;
sim ← sim_state • get ;
flatten_potentials ;
flatten_flows ;
sim ← sim_state • get ;
eqs ← equations • get ;
let no_root = [ false | (|events|)] in
let root_detect = root_detect no_root deqs in
let prep = System:prepare eqs in
The resulting sys tem is t hen regularized and tr ansf ormed (function

102 3 Modeling & Simulation as a Libr ar y
System:prepare
), t he identities are added to the global list of equations
(
alloc_ids
), t he sol v er arguments f or output, root-detection, and ev ent-
iter ation are set up, and t he initial simulation state is com puted. Finall y ,
control is passed to the simulation loop:
do
alloc_ids prep.state ;
s ← sim_state • get ;
( if (s.pre = O:none) then
start_output os
else return {}) ;
let fout s = output os s in
(* allocate root-finding memory *)
let s = if (s.pre = O:none) then {s with roots = no_root} else s
in
let initial_state = initialize prep s.pre exp
ieqs events
root_detect fout s
in
loop prep exp events root_detect fout initial_state
3 . 6 example
The kernel library offers t he means to im plement and simulate “flat” models,
i.e., models t hat do not use inheritance nor depend on their hierarchical
context. Thus, in a first s tep to w ar ds com pilation, it is possible to manuall y
tr anslate models from t heir Modelica f ormulation into Mcl. Recall our
running example from Section 1.1.2, the cartesian pendulum:
let pendulum = do
(* Components *)
x ← Basic:real "x" 0.76 ; Basic:log_real "x" ;
y ← Basic:real "y" 0.76 ; Basic:log_real "y" ;
vx ← Basic:real "vx" 0.0 ;
vy ← Basic:real "vy" 0.0 ;
_F ← Basic:real "F" 0.0 ;
experiment • put {start=0.0; stop = 50.; interval=0.05} ;
(*

𝑥 = 𝑣𝑥
*)
Basic:equation (Ode:diff (Ode:der x) (Ode:unknown vx)) ;
(*

𝑦 = 𝑣𝑦
*)
Basic:equation (Ode:diff (Ode:der y) (Ode:unknown vy)) ;

3.6 Exam ple 103
(*

𝑣𝑥 + 𝐹 𝑥 = 0
*)
Basic:equation (Ode:add (Ode:der vx) (Ode:mult (Ode:unknown _F)
(Ode:unknown x))) ;
(*

𝑣𝑦 + 𝑔 + 𝐹 𝑦 = 0
*)
Basic:equation (Ode:add (Ode:der vy)
(Ode:add (Ode:const 9.81)
(Ode:mult (Ode:unknown _F)
(Ode:unknown y)))) ;
(*
𝑥 2 + 𝑦 2 = 1
*)
Basic:equation
(Ode:add (Ode:add
(Ode:mult (Ode:unknown x) (Ode:unknown x))
(Ode:mult (Ode:unknown y) (Ode:unknown y)))
(Ode:const -1.0));
in exec (Sim:simulate pendulum) with Sim:initial_state
The model consists of tw o parts: Com ponents, i.e., t he model’ s un-
kno wns, are allocated via
Basic:real
. Equations are added wit h the rou-
tine
equation
from t he module
Basic
and constructed via the operations
defined in
Ode
. Equalities are expressed as differences. The initial simula-
tion s tate contains t he start v alues of t he allocated v ariables; initialization is
no t necessar y . The simulation is started b y t he ex ecution of
Sim:simulate
.
R esults are sho wn in Figure 1.1d. T o com pare t he simulation to the other
integ r ators, the result is con v erted to an angle 𝜙 .
This sim ple exam ple demons trates that t he quasi-primitiv es and rou-
tines f or regularization and simulation are in principle capable of processing
Modelica models. Moreo v er , ev en if t he resulting Mcl code ma y be some-
what noisier it s till resembles t he s tructure of its Modelica counter part.

4 fr om modelica via mcl t o ocaml
W it h t he kernel library as a foundation, it remains to sho w , ho w Modelica
tr anslates to Mcl and ho w Mcl can be translated from its current high-
lev el shape into a languag e suitable for ex ecution. The com piler follo w s
tr aditional architecture and is org anized in phases which might be furt her
subdivided into s tag es (F igure 4.1). The first phase, called t he front end
tr anslates a model from concrete syntax into Mcl. The second phase
1
carries t he bulk of the transf or mations and t he t hir d phase, t he back end ,
encodes t he result of the transf ormations into t he targ et languag e.
Modelica Mcl Mcl OCaml
F ro nt E nd T ra n s fo rma tion s Back E nd
F igure 4.1: Compiler Phases
A t t he core of t he second phase lies t he idea of r eduction t o a known
f oundation . Instead of pro viding a direct mapping from Mcl to a static-single-
assignment f orm, assembl y or ev en machine-code, t his chap ter f ocuses on
t he tr anslation of t he f eatures t hat are special to Mcl. T o maintain a concise
presentation of t he result of this translation, a subset of Mcl itself (namel y
its functional core) is used as t he encoding of the targ et languag e. The
furt her tr anslation of such a standar d language is a w ell-kno wn process
and described in g reat detail in t he s tandard liter ature[Appel, 1997]. Hence,
onl y a fe w peculiarities of t his process shall be discussed in Section 4.8.
4.1 f ro n t e n d
In t he firs t phase, lexical and syntactical anal ysis creates a Modelica class
from its textual representation. The resulting abstr act syntax tree is anal yzed
and encoded as an Mcl class. This process is discussed ex emplary in t he
f ollo wing section.
1 Sometimes called the “middle end” in an oxymoronic manner

106 4 F rom Modelica via Mcl to OCaml
Source AST Scopes & Shapes 𝑇
P arsing A nalysis E ncoding
F igure 4.2: Front End
4.1.1 Illus tr ative Exam ple
W e illustr ate t he front end wit h an exam ple model that implements a sim ple
electrical circuit. This resist or -based digital to analog con v erter (F igure 4.3,
t he im plementation is found in Section 4.1.2) does no t contain a single
differential equation and could be considered trivial from t he perspectiv e
of simulation. Its regularity and t he mixture of analog and digital signals
tur n it into an interes ting sho w case for the modeling languag e, ho w ev er .
𝑥 1 𝑥 2 𝑥 𝑛−1 𝑥 𝑛
output
…
2R
2R R
2R 2R 2R
R
F igure 4.3: Analog-to-digital con v ersion b y a regular resist or ladder
The con v erter can be scaled to an arbitr ar y amount of input signals
𝑥 1 … 𝑥 𝑛
— higher numbers f or
𝑛
yield a larg er system of equations and
increase t he in put resolution exponentially . Due to its regularity , indicated
in t he schema b y t he ellipsis, t he descrip tion of t he circuit is independent of
𝑛
. The ability t o cap ture such eleg ant abbreviations is one of the hallmar k s of
Modelica and one of t he core reasons t o use a high-lev el modeling languag e
in t he firs t place. The consequences for the compiler will be discussed in
t he f ollo wing sections.
On a more technical side, t his circuit freely mix es analog and digital
signals. The inputs
𝑥 𝑖
supposedl y implement an
𝑛
-ar y par allel signal. Each
in put alternates betw een a high and a lo w v alue and an exter nal clock
synchronize t hese chang es. Such a beha vior lies clear l y in t he domain of

4.1 Front End 107
discr ete modeling. The output of t he circuit, on t he other hand, depends on
sim ple continuous equations, in particular , Ohm’ s and Kirchhoff’ s la ws. The
im plementation is a hybrid model.
4.1.2 N ano Modelica
The sheer size of Modelica (not to mention t he com plexity of some of its
parts) makes it a futile attem pt t o giv e a com plete account in a document
like this. In line wit h es tablished tradition[Satabin et al., 2015, Ber g ero et al.,
2012] , t he front end onl y unders tands a manag eable subset of t he languag e.
This subset is called N ano Modelica or short nModelica.
N ano Modelica has t he same structure as its mature cousin: Classes can
be arbitr aril y nested and are org anized in hierarchies. Each class consists of
child classes, short class declar ations (which can be modified), com ponent
declar ations,
extends
-s tatements, and (initial) equations. Com ponents can
ha v e a right-hand side definition and a condition statement. N ano Modelica
supports cons tants as w ell as discrete and continuous v ariables, but onl y a
limited f orm of parameters (no par ameter equations and no modifications
of par ameters in classes are allo w ed). Equations consist of sim ple equalities,
when
-equations,
if
-equations and im plicit equations (
connect
and
assert
s tatements). The expressions are limited to literal cons tants, t he application
of built-in functions, binar y oper ators, and ref erences.
V ar iable-str ucture modeling
nModelica is a proper syntactical subset of Modelica. It also extends t he
modeling languag e with support for v ariable-structure modeling. As w e
ha v e seen in Section 2.4, Mcl supports direct control o v er the control flo w of
t he elabor ation process. This enables g reat flexibility and is a gener alization
of concep ts like loop equations, but it does not fit the im plicit design of Mod-
elica ’ s flattening. In Section 3.5.1 w e introduced t he concep t of a chec kpoint
t hat cap tures t he current state of elabor ation and can be resumed . W e also
already considered t he possible presence of tr ansitions f or t he allocation of
unkno wns in Section 3.5.2. W it h t hese pro visions, t he front end onl y needs
to pro vide t he means to:
• declare checkpoints
• declare t he resump tion to a checkpoint
• distinguish modes from each o ther and t he initial mode
• configure t he beha vior of unkno wns in case of a tr ansition

108 4 F rom Modelica via Mcl to OCaml
Sim ple additions to the language fulfill these requirements. The class
Checkpoint
is introduced to declare checkpoints. It has no visible elements.
Each ins tance of t he class represents a checkpoint t hat is created at the
point of elabor ating t he class. The corresponding operat or
resume
retur ns
t he control flo w to an existing checkpoint. Its sole argument must be an
ins tance of t he class
CheckPoint
. It is im plemented as a discrete operat or
in t he kernel library , similar l y to
reinit
. If a discrete equation in v okes it
during t he ev ent iter ation, t he simulation engine exits the simulation loop
and perf orms t he transition. Af ter such a tr ansition, t he last simulation
s tate is maintained in t he
pre
field. The builtin function
resuming
uses
t his fact t o decide whether t he current elaboration s tems from a specific
tr ansition. It takes a checkpoint as an ar gument and returns
true
if t he
las t simulation w as ter minated b y resuming to t hat checkpoint and
false
o t her wise. This introspection allo w s t he modeler to br anch to a different
model after a transition.
Modeling t he ladder in nModelica
nModelica lacks arra ys and functions. Alt hough t hese are important as-
pects of Modelica, t heir treatment w ould not contribute significantl y to t his
t hesis and require a s tupendous additional v olume. Some aspects of t heir
com pilation will be discussed in Section 6.3.1. Since arra ys cannot be used
to im plement t he regular structure of t he resist or ladder , w e resort to a
recursiv e formulation:
model RecLadder
constant Real R=8e3;
parameter Integer nr = 4;
Electrical.Sources.ClockDividerPulse
sigvolt(period=integer(2^(nr-1)));
Electrical.Sources.Ground ground;
Electrical.Basic.Resistor r1(R=2*R);
Electrical.Basic.Resistor r2(R= if nr == 1 then 2*R else R);
RecLadder next(nr = nr - 1) if nr > 1;
equation
if nr <= 1 then connect (r2.p, ground.p); end if ;
connect (r1.n, r2.n);
connect (ground.p, sigvolt.p);
connect (sigvolt.n, r1.p);
connect (r2.p, next.r2.n);
end RecLadder;
1
2
3

4.1 Front End 109
Each rung of t he ladder has an input signal (anno tation 1 ) which flips
after a giv en amount of clock cy cles (t he
period
par ameter). In the
𝑛
-t h rung,
t he in put period is t he
𝑛−1
-t h po w er of
2
, yielding a binar y counter from
0
to
2 𝑛−1
. This dis tinction demons trates the modification of a parameter .
N ote, that neit her t he instantiation nor the surrounding model mentions
t he cloc k of t he in put signals.
The recursiv e instantiation of t he class (anno tation
2
) also explicitl y
sets a parameter (the number
nr
of t he current rung). A t t hat point, it is
im portant to mak e a dis tinction betw een cons tants and parameters. The
f ormer are bound in classes and can be accessed from each com ponent,
while t he latter are part of t he ins tance and can onl y be accessed during
elabor ation. In our example, this allo w s to giv e t he same class a different
beha vior : If
nr
is less or equal to
1
, the recursion st ops.
next
is called a
conditional component of
RecLadder
. The first rung of t he ladder circuit is
g rounded on its left port. This special case is implemented as a conditional
eq uation (annotation
3
). If
nr
is less t han or equal to
1
(i.e. in t he leas t
significant rung), t he second resist or is g rounded.
The digital in put signals demons trate t he use of s tructural v ariability .
Depending on an exter nal counter that increases at each clock cy cle, t he y
yield an high or lo w output. T w o modes im plement t his beha vior , each
wit h a different equation. The output t hen merel y con v erts t his digital state
to an analog v oltag e.
model ClockDividerPulse
outer discrete Integer clock;
Checkpoint cp;
parameter Integer period=1;
parameter Boolean hi = mod(div(clock, period), 2) == 0;
extends Interfaces.TwoPins
(u(start= if hi then 0.1 else 3.3, restart=resuming(cp)));
equation
if hi then u = 0.1; else u = 3.3; end if ;
when mod(div(clock, period), 2) == 0 then resume(cp); end when ;
when mod(div(clock, period), 2) == 1 then resume(cp); end when ;
end ClockDividerPulse;
4
5
6
7
8
9
10
The class declares its requirement for an external
clock
wit h the ke y -
w ord
outer
(anno tation
4
). This dependency spreads upw ards through t he
ins tance hierarch y . Alt hough its declar ation does not reflect it,
RecLadder

110 4 F rom Modelica via Mcl to OCaml
requires t he same argument and im plicitl y passes it to all its descendants.
A t some point in t he instance hier arch y , a corresponding
inner
declar ation
has to pro vide t he argument. All
outer
com ponents in t he scope of that
declar ation are t hen im plicitl y identified. In our case, t his enables synchro-
nization since ev ents t hat are equal but no t identical to each o t her might not
be detected simultaneousl y . The compiler has t o detect and cope with t hese
dynamicall y scoped arguments. N o tabl y , t his met hod is t he onl y w a y to
pass non-par ameters as arguments betw een models and obtain an identity
betw een t hem.
A combination of checkpointing,
when
-equations, and
resume
im ple-
ments s tructural v ariability . The checkpoint (annotation
5
) indicates t hat
t he elabor ation of t his class ma y be repeated during t he simulation. T r an-
sitions are a consequence of discrete ev ents t hat use t he
resume
oper ator
in t he body of a
when
-equation. Depending on t he v alue of
clock
, t here is
a tr ansition from t he high to the lo w state and vice v ersa (annotations
9
and
10
). The different modes are im plemented wit h a standar d
if
-equation
(anno tation
8
). Depending on t he v alue of
hi
, t he output is either a high or
lo w v oltag e.
In nModelica, expressions of t he v ariability
discrete
can determine
t he v alue of parameters. The y yield their
start
v alue in t hat case. This
principle allo ws it t o set t he par ameter
hi
wit h a discrete time v alue during
elabor ation (annotation 6 ).
The class is a standar d electrical component and inherits the usual
TwoPins
interface (anno tation
7
). T o pro vide consis tent simulation results,
t he
start
v alue of t he v oltag e
u
is explicitl y set to t he analog counter part of
hi
. Since it depends on t he actual mode, t he component’ s
restart
attribute
ensures t hat the start v alue is re-ev aluated af ter each transition.
4.1.3 Anal ysis
The firs t task of t he front end is t he resolution of scope-bound names, i.e., t he
assignment of a special v ariable
up( 𝑖 )
to each v ariable in the Modelica model.
The scoping rules of Modelica ’s inheritance complicate t his task. Each
extends
-s tatement “pulls” a set of names into its scope. Lookup t hrough a
scope wit h such a s tatement can onl y continue when all inherited names
are kno wn. This set is called t he shape of t he inherited class.
Shapes
Besides holding t he name of each element, t he shape distinguishes betw een
t heir kind , i.e., whet her an element is a class or a component. For each

4.1 Front End 111
com ponent, the follo wing properties are inferred:
•
The v ariability determines whet her a component is a cons tant, a dis-
crete or continuous time v ariable or a parameter
•
The connectivity determines whet her a continuous time v ariable is a
flo w or a potential
•
The op tionality determines whether a component is conditional, i.e.
might be absent from t he model due to user in put.
•
The ownership determines whet her a com ponent is defined locall y or
required as an im plicit argument (via outer )
•
The f orm of an element is eit her one of t he builtin primitiv e types
(boolean 𝔹 , reals ℝ or integ ers 𝕀 ) or an ins tance of a class
During t he ev aluation, a class might be used in sev eral different contexts,
while t he front end onl y ev er sees t he context of its original definition.
Theref ore, t he results of t he anal ysis need to be in variant with respect to the
context. In particular , no modification nor redeclar ation is allo w ed t o chang e
t he shape of an inherited class. This is reflected in t he Modelica Specification
b y t he requirement t hat base classes be tr ansitivel y nonr eplaceable .
Shape Inf erence
The result of t he anal ysis is t he assignment of a shape to all
extends
s tate-
ments and a scope to all v ariables. Such a solution is consistent when, after
encoding, each base class ev aluates to a node which has t he assigned shape.
A t t he same time, t he special v ariables need to point t o t he firs t occurrence of
t he name in the en vironment. The search for such a solution can be difficult,
when sev eral
extends
s tatements depend on each other , as t he f ollo wing
exam ple sho ws:
model A
constant Real a = 42.0;
model B
extends C;
constant Real b = a;
model C
extends D;
end C;
end B;
model D constant Real a = 0.0; end D;
end A;

112 4 F rom Modelica via Mcl to OCaml
Here, t he scope of t he v ariable
C
in model
B
can be immediatel y resolv ed
to
0
. The calculation of its shape depends on t he shape of t he base class
D
in
model
C
. Such circular dependencies render a sim ple brute-force approach
unsuitable, as pat hological exam ples can yield non-termination. It is neces-
sar y to anal yze t he dependencies betw een t he not y et resolv ed shapes and
v ariables and proceed wit h t he leafs of t he induced dependency graph.
Cy cles can be solv ed b y the application of an additional special rule of t he
Modelica semantics. The specification demands t hat no
extends
-clause ma y
influence t he lookup of ano t her such clause of the same class: “and a name
used f or a base-class ma y not be inherited from an y base-class” [Modelica,
2012, section 5.6.1]. F ollo wing t hat rule, it is legal to break an y circle b y
remo ving t he dependency from a shape to the v ariable of its
extends
-clause.
If a circle does no t contain such an edg e, i.e., it consis ts solel y of shapes to
be calculated, w e consider it an error in t he model. In t he exam ple abo v e,
t he solution is t hus to ignore the shape of
C
when looking up
D
(resulting in
2). Af ter w ards, all shapes can be com puted in order .
The im plementation of this process is based on t he abstr act ev aluation
of hier archical terms. Each built-in and encoded class declaration contains
its shape as a pseudo-v alue and t he shape of a not y et translated nModelica
class can be obtained b y recursiv el y ev aluating t he shape of its (already
encoded) base classes and com ponents. Besides t he encoding, which will
be explained belo w , the only requirement is an inter preter f or hier archical
terms. Such an ev aluation function is presented in Section 4.6.
Exam ple Shape
In t he more sim ple exam ple, the analysis is r at her str aightfor w ard. After
ev aluating t he shape of
TwoPins
, it comes up wit h the result shown in
T able 4.1. The
Electrical
libr ar y needed f or t his is presented in Section 5.2.
There are t hree sub-com ponents: The checkpoint
cp
and t he pins
p
and
n
. One v ariable,
clock
, is declared as
outer
, tw o (
u
and
i
) are declared
locall y . F inall y , t here are tw o par ameters,
hi
and
period
. All com ponents
ha v e potential connectivity and are mandator y .
4.1.4 Encoding nModelica in Mcl
The encoding is a recursiv e process t hat deals wit h classes, their com ponents,
and equations. In or der not t o inflate this chapter more than necessar y ,
w e spare t he reader t he technical details and resort to a more concep tual
o v er view .

4.1 Front End 113
Element V ariability Connect Op tional Owner Form
clock discrete po tential
mandatory
outer 𝕀
cp
continuous
po tential
mandatory
local com ponent
period par ameter potential
mandatory
local 𝕀
hi par ameter potential
mandatory
local 𝔹
i
continuous
po tential
mandatory
local ℝ
n
continuous
po tential
mandatory
local com ponent
p
continuous
po tential
mandatory
local com ponent
u
continuous
po tential
mandatory
local ℝ
T able 4.1: Shape anal ysis results f or t he ClockPulseDivider class
Classes and Com ponents
The encoding s tarts wit h t he top most class. Its encoding successiv el y
cons tructs an Mcl class, t he elaboration routine, and the model’ s instance.
The process can be summarized as f ollo ws:
• Encode all child classes
• Encode all class-kinded elements
• Encode all base classes
• Encode all component-kinded elements
• Create t he pa y load as field elaborate
Child classes are encoded recursiv el y . Class elements, i.e., short class
definitions, are encoded as hierarchical terms. They consis t of names and
possible modifications, which are encoded directl y into the respectiv e Mcl
terms. Modifications appl y to constants or par ameters and are encoded
as hier archical v alues modifications or as arguments to the elaboration,
respectiv ely . nModelica does no t support modification of continuous nor
discrete elements. Equations must be used ins tead. nModelica does no t
make a distinction betw een redeclarations and nor mal short class definitions
- bo t h o v erride potentiall y inherited elements. Hence, redeclarations can be
directl y encoded.

114 4 F rom Modelica via Mcl to OCaml
Elaboration
Ev er y class can be elaborated. This process is im plemented b y t he special
field
elaborate
. It contains a routine wit h three arguments: A string
uid
contains a human-readable descrip tion —e.g., the component name— of
t he com ponent,
outer
consis ts of all implicit ar guments from the parent,
and
params
contains all modifications to the parameters of the component.
The body of t he routine in v okes the follo wing effects:
• Elaborate all base classes
• Elaborate all non-cons tant com ponents
• Elaborate all equations
• Create and retur n an instance
The elabor ation routine is created during t he translation of (inherited)
com ponents. Inheritance is translated b y for w arding all inherited fields.
Each base class is processed as f ollo ws.
•
Create a fresh base class field:
𝑆 𝑖 = ℎ 𝑖
, where
ℎ 𝑖
is t he tr anslation of t he
base class and § 𝑖 is a fresh class label.
• F or each inherited element:
– Encode t he name as 𝑥 , 𝑙 and 𝐿
–
If it is a cons tant, add
𝑙 = this.super( 𝑆 𝑖 ). 𝑙
to the class. The con-
s tant can t hen be ref erenced b y a hierarchical term.
–
In each case, add
𝐿 = this.super( 𝑆 𝑖 ). 𝐿
. This ensures t hat f or
each com ponent t here is alw a ys a specific declar ation class t hat
acts as a targ et of class-lev el modifications t hat shall onl y affect
a single com ponent.
– If it is declared as outer , add t he binding
let 𝑥 = outer. 𝑙 in …
to the elaboration.
–
Ot her wise, if it is a discrete or continuous v ariable, add t he
binding 𝑥 ← ( this. 𝐿) .elaborate 𝑢 𝑥 𝑜 𝑥 𝑝 𝑥 ; …
to the elaboration and the field
𝑙 = 𝑥
to the instance, where
𝑢 𝑥
is
t he unique identifier of the component,
𝑜 𝑥
is a recor d containing
t he v alues of all
outer
elements and
𝑝 𝑥
is t he par ameter record
of t he com ponent.

4.1 Front End 115
– Ot her wise, if it is a par ameter , add t he binding
let 𝑥 = params. 𝑙 in …
to the elaboration.
Local com ponents are encoded similar l y , but instead of a base class, t he
special v ariable
this
is used. When a sub-component
𝑥
(whet her inherited
or no t) is elaborated, the unique identifier
𝑢 𝑥
, t he par ameter modifications
𝑝 𝑥
and t he im plicit arguments
𝑜 𝑥
are com puted from the declaration and
t he shape of the declaration class.
The par ameters of a com ponent are set via (possibl y nes ted) par ameter
modifications. The parameter object is a recor d consisting of an op tional for
each par ameter and a nested op tional parameter recor d f or each com ponent
of t he modified model. Modifications of par ameters are encoded as a simple
update of t hat par ameter record. The v alues of parameters are ev aluated
during elabor ation.
Exam ples of Inst antiation
Consider t he declar ation of
sigvolt
in anno tation
1
. It consis ts t he modifi-
cation
period=integer(2^(nr-1))
. Its encoding becomes an element of t he
elabor ation routine:
(* excerpt from class RecLadder *)
val elaborate = λ uid. λ outer. λ params. do
(* ... *)
ground ←
(* ... *)
let nr = O:safe 4 (params.nr) in
do
sigvolt ← this.C_sigvolt.elaborate (uid ^ ".sigvolt" ) outer
{(params.sigvolt {}) with period =
O:merge (int_of_float ((float_of_int 2) **
(float_of_int (nr - 1))))
(params.sigvolt {}).period
};
11
12
Elabor ation is forw ar ded to the com ponent’ s declared class (anno ta-
tion
11
), which is a local class to allo w for the modification of constants.
The unique id of t he com ponent is cons tructed b y appending t he s tring
".sigvolt"
to the parent’ s unique id. There are no implicit ar guments.
Hence t he
outer
recor d is passed unmodified. The nested par ameter recor d

116 4 F rom Modelica via Mcl to OCaml
(anno tation
12
) is modified wit h the help of t he utility function
O:merge
.
The modification to
period
is firs t com piled to a floating-point expression
(because Modelica ’ s
^
-oper ator is defined on
Real
-typed arguments) and ex-
plicitl y con v erted to an integ er af ter w ard. The con v ersion functions as w ell
as t he oper ator names (
**
f or exponentiation and
^
f or string concatenation)
are inherited from t he targ et languag e.
The recursiv e declaration of
RecLader
requires a s top condition. This
condition is reached if t he par ameter
nr
is smaller t han or equal to
1
. The
field
next
is t hus conditional . Its possible absence is implemented b y wrap-
ping t he ins tance into an op tional v alue.
(* excerpt from class RecLadder *)
C_next = this
val elaborate = λ uid. λ outer. λ params. do
(* ... *)
let nr = O:safe 4 params.nr in
(* ... *)
next ← O:elab (
if nr > 1 then
(O:some (this.C_next.elaborate (uid ^ ".next" ) outer
{(params.next {}) with nr =
O:merge (nr - 1) (params.next {}).nr}))
else O:none);
The ex cer pt abo v e contains t he compiled s tatement in annotation
2
:
next(nr = nr - 1) if nr > 1
. It proceeds similar to the instantiation of
sigvolt
, but wraps the recursiv e elabor ation into an op tional and for w ards
it to the library routine
O:elab
, which retur ns
O:none
, when its argument is
O:none .
Outer com ponents are no t instantiated in the class t hat contains t heir
declar ation. Instead, t he y are directl y taken from the
outer
recor d and
lifted into t he elabor ation. The follo wing snippet sho ws the translation of
anno tation
4
in model
ClockDividerPulse
. As sho wn bef ore, t he im plicit
arguments are no t modified in
RecLadder
. Hence, t he instantiation lies in
t he responsibility of the model t hat instantiates the whole ladder (expressed
b y a corresponding inner declaration).
(* excerpt from class ClockDividerPulse *)
val elaborate = λ uid. λ outer. λ params.
(* ... *)
clock ← return (outer.clock);

4.1 Front End 117
As an exam ple f or an inherited com ponent, consider t he extends s tate-
ment (anno tation
7
) in
ClockDividerPulse
. Here, t he elabor ation is for -
w arded t o t he single base class. The modified par ameters is taken into
account.
(* excerpt from class ClockDividerPulse *)
Super_0 = up(2).Interfaces.TwoPins
val elaborate = λ uid. λ outer. λ params. do
super_0 ← this.super(this.Super_0).elaborate uid outer
{ params with u =
( λ _. { {(params.u {}) with start =
O:merge
( if hi then 1.000000e-01 else 3.300000e+00)
(params.u {}).start}
with restart =
(* ... *)
let u = super_0.u in
let p = super_0.p in
let n = super_0.n in
let i = super_0.i in
(* ... *)
Since t he class contains no
inner
declar ations, t he outer arguments
are ag ain not modified. The par ameter recor d is treated differentl y from
a sim ple ins tantiation: Because a class and its base class share t he same
par ameters, potential modifications ha v e to be mer g ed into the components
o wn set of parameters. This expression sets t he record field
start
in t he
field
u
of t he par ameter recor d (or creates it, if it is not giv en b y t he outside).
F inall y , the inherited components are made a v ailable in the local scope.
F or m, Effect, and Operator Resolution
The encoding of an expression requires t he kno w ledge of some properties
of its subexpressions. These properties are inferred using the shape of t he
surrounding class. They are hence as w ell in v ariant with respect to t he
context.
•
While mos t operat ors are pure and do not im pose an effect on t he
model, some need to be ev aluated as part of t he elaboration. One com-
mon exam ple is t hat of continuous-time com parisons: The languag e
specification demands t hat each such com parison g ener ate an ev ent
t hat can be tr acked b y t he integ rat or .

118 4 F rom Modelica via Mcl to OCaml
Oper ator Effect V ariability F or m R esolution
+ pure continuous ℝ Ode:add
+ pure discrete ℝ Discrete:fmap2 ( + .)
+ pure parameter or cons tant ℝ (+.)
+ pure discrete 𝕀 Discrete:fmap2 ( +)
+ pure parameter or cons tant 𝕀 (+)
< elab continuous ℝ Basic:lt
< pure discrete an y Discrete:fmap2 ( <)
< pure parameter or cons tant an y (<)
T able 4.2: Resolution of Oper ators (ex cer pt)
•
The v ariability is is usuall y sim ply the highest of the respectiv e subex-
pressions. F or instance, t he addition of a parameter to a discrete
v ariable yields a discrete expression. Some built-in operat ors ha v e
specific rules regar ding t heir v ariability . For ins tance t he
resuming
oper ator expects a continuous-time com ponent and yields a discrete
signal.
•
The f orm is inferred similar l y to t he v ariability and deter mines t he
definition of oper ators, e.g., whether it is an integer - or real-v alued
addition. This dis tinction is necessar y due to the o v er loading of built-
in oper ators in Modelica, which is no t reflected in Mcl.
•
The r esolution of an operator determines its im plementation in the
kernel library . Usuall y , each operator is directl y forw arded to a k ernel
libr ar y function. In some cases, an Mcl definition might also be
directl y inlined.
Expressions
The resolution of builtin operat ors is t he first s tep when encoding expres-
sions. It is defined b y a combination of t he f orm and v ariability of each
oper ator’ s arguments. As an example, consider the operat ors for addition
(T able 4.2). The expression might encode a discrete signal, a differential
equation, or a plain (constant or par ameter) ter m. Onl y in t he latter case,
t he corresponding Mcl oper ator is used, in the other cases, an appropriate
embedding is selected from t he kernel library .

4.1 Front End 119
Depending on t he context of t he tr anslated expression, it might be nec-
essar y to lift the resulting ter m. When for ins tance a parameter is used
in t he context of a discrete signal, it mus t be con v erted wit h the function
Discrete:pure . The desired lev el of v ariability is in put to t he tr anslation.
As a furt her com plication, each subexpression might introduce anon y-
mous ev ents . They occur when a com parison betw een continuous-time
expressions is tr anslated and are used to signal possible discontinuities
to the integ rat or . From the perspectiv e of t he translat or , these ev ents take
t he f orm of routines and need to be appended to the elaboration of the
surrounding model. Hence, ev en during t he tr anslation of an arbitrary
expression, t here might be a chang e in t he model’ s instantiation.
Eq uations and Connections
Equations fall int o tw o categories. Sim ple equations are encodings of im-
plicit, or dinar y differential, or discrete equations and cons tructed b y lifting
a tr anslated expression into the required v ariability and allocating t he result
wit h the appropriate routine from t he ker nel libr ar y’ s Basic module.
F or instance, t he con v ersion betw een t he discrete signal
out
and t he
analog v oltage
u
in t he model
ClockDividerPulse
(anno tation
8
) requires
tw o steps of con v ersion. F irs t, t he high and lo w v oltag e constants ha v e to be
lifted into t he discrete domain. The surrounding
if
expression is translated
into a conditional discrete signal. The resulting discrete signal is lifted to
an ODE via Ode:signal and added as a sim ple equation:
Basic:equation
(Ode:diff (Ode:unknown u)
(Ode:signal (Discrete:conditional (Discrete:bool out)
(Discrete:pure 3.300000e+00)
(Discrete:pure 1.000000e-01)
)
)
)
In a certain sense, t he abo v e examples are lit er al equations, compar a-
ble to numerical or s tructural cons tants in t he model. A second categor y
builds on top of these primitiv es and combines t hem to larg er systems b y a
com putational process that decides which equations are part of t he model.
Alt hough Modelica is somewhat limited in this regar d as it onl y supports
loops and conditionals, it is nev ert heless necessary to handle t hese “meta
equations” during tr anslation.
The modeling routines in Mcl are not quite incidentall y firs t-class citi-
zens of t he languag e. A conditional equation is str aightforw ar d to im ple-

120 4 F rom Modelica via Mcl to OCaml
ment as a conditional that selects betw een sev eral different routines. The
onl y difficulty in t hat process is actuall y to pick the correct translation from
t he sev er al different syntactical v ariations t hat Modelica offers. In our run-
ning exam ple, t he onl y conditional equation is used in t he model
RecLadder
to ground t he least significant rung (anno tation
3
). It is translated int o the
f ollo wing form:
if (nr <= 1) then
do
Basic:connect this.C_r2.C_p.connect false
this.C_ground.C_p.connect false
r2.p ground.p ;
return {}
else return {}
The lis ting abo v e also show s ho w im plicit equations (in t he f orm of
connect
) s tatements are translated: Each class contains an iterat or
connect
t hat recursiv el y w alk s t hrough all com ponents of tw o instances of the class
and creates connections betw een t he primitiv e components. The kernel
libr ar y routine
Basic:connect
ensures symmetr y b y connecting both direc-
tions. Its arguments are t he iterat ors, t he instances and an indicator f or t he
relativ e location of t he connection (called inside or outside connection in t he
Modelica specification and used f or determining t he sign of a flo w).
4 . 2 the c ompila tion pipeline
𝐶 𝛽 …
… 𝑇 𝑇 𝛼 𝑀 𝛽 𝑀 𝛾 …
(F ro nte nd) De suga ring Re fac to ring L ookup (Back end)
F igure 4.4: Compilation Pipeline
The essential com pilation takes place in the second phase and consists of
t hree s tag es, called
𝛼
,
𝛽
and
𝛾
. The first s tag e tr ansforms modeling routines
into more primitiv e functions and applications. The second stag e,
𝛽
yields
a normal form of t he hier archical terms and splits t he implementation of
met hods from their position in t he hierarch y (called t he skelet on of a class).

4.2 The Compilation Pipeline 121
F inall y , these implementations are com piled to core-terms in
𝛾
b y partiall y
ev aluating t he hierarchical ref erences. This stag e is t he first time that con-
textual inf ormation about classes from other com pilation units comes into
pla y (in t he form of t heir skelet ons). Thus, t he pipeline offers genuinel y
separ ate com pilation but depends on some in put from its dependencies.
The t hir d phase, t he Back End , is a mere encoding: It is technicall y similar
to the
𝛼
-s tag e of t he second phase, but inherentl y simpler (as it introduces
no additional logic). The precise output is also highly specific t o t he chosen
targ et languag e. For these reasons, t he Back End shall onl y be sketched in
Section 4.8.
4.2.1 Illus tr ative Exam ple
The size and com plexity of actual models make it har d to unders tand t he
results of t heir tr ansf ormation. Therefore w e resort to a sim pler example
f or t he illus tration of the second phase. By merging the techniques from
Sections 2.3.7 and 2.4.4 w e can create an implementation of our sim ple
ev en/odd function t hat uses man y f eatures of Mcl:
(* top-level class EvenOdd *)
val do_even =
( do
loop ← call/cc ( λ k. return k) ;
n ← n • get ;
if n <= 0 then return true else
if n=1 then return false else
do
n • put (n - 2);
continue loop with loop) ;
val even = λ n. exec this.do_even with n = n ;
val odd = λ n. if n=0 then false else this.even (n - 1)
13
14
15
16
17
The class
EvenOdd
sho wn abo v e uses open recursion (e.g. anno tation
16
),
s tarts elaboration from within a core ter m (anno tation
13
), interacts with t he
modeling state (e.g. anno tation
15
), s tores a continuation (anno tation
14
),
and resumes it (anno tation
17
). T o actuall y use t he class, a module is
required as an entr y point f or t he ev aluation. This module is compar ativ el y
unremar kable. It onl y requires one reference t o t he class (anno tation
18
belo w):

122 4 F rom Modelica via Mcl to OCaml
(* top-level module Test *)
let _ =
let test = EvenOdd.odd 42 in
Printf:printf "42 odd? %b\n" test
18
4.2.2 S t ag es
The output of each s tag e is a v ariant of t he original Mcl, annotated with
t he corresponding s tag e symbol:
𝑡 α
is a term from t he first s tag e,
ℎ β
is a
hier archical-term from t he second stag e and so on. All t hese v ariants are
proper subsets of t heir counterparts in t he ear lier s tag es, i.e.
𝑡 β ⊆ 𝑡 α ⊆ 𝑡
, etc.
The im plementation of the Second Phase is defined via t hree functions:
α ∶ 𝑇 → 𝑇 α ∪ { }
β ∶ 𝑇 α → 𝑇 β ∪ { }
γ ∶ 𝑇 β → 𝑇 γ ∪ { }
Com pilation can fail at an y stag e of t he second phase. The error mar ker
in t he co-domain of the functions indicates t his possibility .
Definition
(Guar ded T r ansformations, Composition)
.
T r ansf ormations t hat
po t entiall y yield an error ar e called guar ded. The (lef t t o right) composition of suc h
functions is expr essed wit h t he ▶ oper at or :
⟦𝑥⟧ 𝑓 ▶𝑔
∧
=
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
when ⟦𝑥⟧ 𝑓
∧
=
⟦⟦𝑥⟧ 𝑓 ⟧ 𝑔 o t her wise
(The second function, 𝑔 , is no t necessaril y guar ded.)
The second phase is t he com position of its t hree stag es, defined as
α ▶ β ▶ γ .
S tage Cor rectness
There are tw o desirable properties f or a transf or mation: F irst, it is funda-
mental t hat it can be im plemented at all, i.e., its function is defined b y an
algorit hm that terminat es . Second, t he g ener ated prog r am should relate to
t he source program reasonabl y . T er mination is a r at her s traightf or w ar d

4.2 The Compilation Pipeline 123
requirement t hat can be guar anteed directl y b y each s tag e, but t he pre-
cise definition of a “reasonable” relation betw een source and output is no t
ob vious.
In an ideal w or ld, all transf or mations w ould map terms to equiv alent
counter parts, f or some equiv alence relation ≡ 𝑇 :
⟦𝑇 ⟧ 𝑓 ≡ 𝑇 𝑇
Or , more directl y ,
𝑓 ⊆ (≡ 𝑇 )
. But t hat is im practical f or man y interesting
tr ansformations. Consider for ins tance t he Church encoding of t he natur al
numbers: ⟦0⟧ church
∧
= λ 𝑓 . λ 𝑥 . 𝑥
Of course, such a tr ansformation is correct in the sense t hat it implements
t he expected beha vior . Considering it as part of an equiv alence relation
has some irritating consequences, ho w ev er . Intuitiv el y , one w ould expect
equiv alent ter ms to be e x c hang eable in an y context (t he relation should be
cong ruent), so t he expression
1 + 0
w ould be equiv alent to
1 + ⟦0⟧ church
, but
t he latter expression is meaningless, unless t he source languag e already
contains pro visions for the Church-encoding of numerals. A non-cong ruent
tr ansformation requires a different no tion of correctness.
In a first appro ximation, one could argue t hat a tr anslated term should
ev aluate to t he same result (modulo translation), before and after tr anslation:
𝑇 ⇓ T 𝑉 ⟹ ⟦𝑇 ⟧ 𝑓 ⇓ T 𝑊 ∧ 𝑊 ∧
= ⟦𝑉 ⟧ 𝑓
This requirement, often called pr eservation of semantics , is intuitiv e but
too s trict. The s tatement abo v e does not take int o account t he possible
failure of the transf or mation. In such a case, the abov e conclusion does
no t hold. Hence, t he success of t he transf or mation needs to be part of the
premise:
⟦𝑇 ⟧ 𝑓
∧
≠ ∧ ⟦𝑉 ⟧ 𝑓
∧
≠ ∧ 𝑇 ⇓ T 𝑉 ⟹ ⟦𝑇 ⟧ 𝑓 ⇓ T 𝑊 ∧ 𝑊 ∧
= ⟦𝑉 ⟧ 𝑓
This somewhat relax ed requirement is still t oo strict. The right-hand side
of t he im plication uses syntactic equality of terms. If a compiler renames
bound v ariables, t he resulting terms are then
𝛼
-equiv alent but ob viousl y
no t syntacticall y equal. Similar l y , the compiler might reor der expressions or
partiall y ev aluate a ter m. All t hese operations should result in semanticall y
equiv alent, but not necessaril y equal terms.
As t he name sugg ests, semantic equiv alence — written
≡ 𝑇
,
≡ ℎ
,
≡ 𝑀
,
≡
—
is a famil y of equiv alence relations betw een elements of t he same syntactic
sort. W e pos tpone t he precise definition of these relations and conclude t he
dev elopment of t he correctness property of a s tag e:

124 4 F rom Modelica via Mcl to OCaml
Definition
(S tag e Consis tency)
.
A s t ag e is consis t ent if it f ollows fr om t he
ev aluation of its input t o a t op-level v alue that its output ev aluates t o anot her
t op-lev el v alue t hat is eq uiv alent t o t he firs t v alue.
⟦𝑇 ⟧ 𝑓
∧
≠ ∧ ⟦𝑉 ⟧ 𝑓
∧
≠ ∧ 𝑇 ⇓ T 𝑉 ⟹ ⟦𝑇 ⟧ 𝑓 ⇓ T 𝑊 ∧ 𝑊 ≡ 𝑇 ⟦𝑉 ⟧ 𝑓
There is one las t issue wit h t he abo v e definition: T ransf or mations should
be com posable. In particular , a tr ansf ormation should maintain t he equiv a-
lence of terms: If a ter m is equiv alent to another and both can be transf or med,
t he results of tr ansf orming both should be equiv alent as w ell.
Definition
(Preser v ation of Equiv alence)
.
A s t ag e
𝑓
pr eserves eq uiv alence, if
t he property holds aft er successful tr ansformation of bo th t erms, i.e.:
⟦𝑇 ⟧ 𝑓
∧
≠ ∧ ⟦𝑆⟧ 𝑓
∧
≠ ∧ 𝑇 ≡ 𝑇 𝑆 ⟹ ⟦𝑇 ⟧ 𝑓 ≡ 𝑇 ⟦𝑆⟧ 𝑓
Bo t h properties combined are required f or t he no tion of correctness:
Definition
(S tag e Correctness)
.
A s t ag e
𝑓
is corr ect, if it is bot h consis t ent and
pr eserves eq uiv alence.
N aturall y , this property shall appl y to the whole second phase of t he
com piler . T o s tructure t his requirement, it comes in handy t hat correctness
is maintained b y com position:
Corollar y 1
(Com position of Correct S tag es)
.
If
𝑓
and
𝑔
ar e correct s tag es,
t hen 𝑓 ▶𝑔 is a corr ect s t ag e.
Pr oof. F irst w e sho w t hat preser v ation is compositional:
• Assume ⟦𝑇 ⟧ 𝑓 ▶𝑔
∧
≠ , ⟦𝑆⟧ 𝑓 ▶𝑔
∧
≠ , and 𝑇 ≡ 𝑇 𝑆 .
• From the and definition of ▶ it follo w s t hat ⟦𝑇 ⟧ 𝑓
∧
≠ and ⟦𝑆⟧ 𝑓
∧
≠ .
Theref ore, b y preser v ation of 𝑓 : ⟦𝑇 ⟧ 𝑓 ≡ 𝑇 ⟦𝑆⟧ 𝑓
• By preser v ation of 𝑔 it f ollo ws directl y t hat: ⟦𝑇 ⟧ 𝑓 ▶𝑔 ≡ 𝑇 ⟦𝑆⟧ 𝑓 ▶𝑔 .
Second, w e show that 𝑓 ▶𝑔 is also consis tent:
• Assume t hat ⟦𝑇 ⟧ 𝑓 ▶𝑔
∧
≠ , ⟦𝑉 ⟧ 𝑓
∧
≠ , ⟦𝑉 ⟧ 𝑓 ▶ , and 𝑇 ⇓ T 𝑉 .
•
F rom the and definition of
▶
it follo ws t hat
⟦𝑇 ⟧ 𝑓
∧
≠
and
⟦𝑉 ⟧ 𝑓
∧
≠
.
Consequentl y , b y consistency of 𝑓 : ⟦𝑇 ⟧ 𝑓 ⇓ T 𝑊 𝑓 and 𝑊 𝑓 ≡ 𝑇 ⟦𝑉 ⟧ 𝑓 .

4.2 The Compilation Pipeline 125
•
Similar l y , b y consistency of
𝑔
, w e conclude t hat
⟦⟦𝑇 ⟧ 𝑓 ⟧ 𝑔 ⇓ T 𝑊 𝑔
and
𝑊 𝑔 ≡ 𝑇 ⟦𝑊 𝑓 ⟧ 𝑔 .
• Finall y , b y preser v ation of 𝑔 it follo ws, that:
⟦𝑊 𝑓 ⟧ 𝑔 ≡ 𝑇 ⟦⟦𝑉 ⟧ 𝑓 ⟧ 𝑔
W it h t his frame w or k , w e can pro v e t he correctness of t he tr anslation
b y pro ving correctness of each part separatel y . But before w e in v estig ate
t he s tag es in detail, it is necessar y to catch up with t he definition of t he
equiv alence relation t hat w e ha v e skipped ear lier .
4.2.3 T erm Eq uiv alence
Equiv alence betw een terms has an intuitiv e meaning: T er ms are equiv alent
if t he y yield t he same result in all possible contexts. Specific f or ms of
equiv alence are w ell-kno wn in t he lambda calculus. Bound v ariables can be
renamed arbitr aril y (
𝛼
-equiv alence) and arguments can be f olded into the
body of functions (
𝛽
-equiv alence) or expanded around partial applications
(
𝜂
-equiv alence). Mcl has more syntactic forms and t hus more f orms of
equiv alent ter ms. F or ins tance, t he addition
1+1
is ob viousl y equiv alent to
2+0
,
0+2
and so on,
𝛽
-equiv alence also applies to recursiv e functions, and
selecting an updated field from a recor d is equiv alent to the new l y updated
v alue. The equiv alence betw een functions is intuitiv el y approached b y
considering all possible contexts: T w o functions are equiv alent if t heir
usag e in an y closed context yields the same result.
This intuitiv e no tion of cont extual eq uivalence is hard to w or k wit h because
it is too g eneral. T o obtain a more accessible definition of equiv alence, w e
f ollo w the design laid out b y Pitts[Pierce, 2005, chap ter 7]. There, the
equiv alence relation is defined to be t he larg est relation that is transitiv e,
reflexiv e, symmetric, compatible with t he structure of terms, substitutiv e
and adequate f or t he obser v ation of ev aluation.
T r ansitivity , reflexivity , and symmetry are naturall y required from an
equiv alence relation. Compatibility means that equiv alence of tw o ter ms
f ollo ws from the equiv alence of corresponding sub-terms, e.g., if
𝑡 ≡ 𝑠
, t hen
also
λ 𝑥 . 𝑡 ≡ λ 𝑥 . 𝑠
. Subs titutivity is t he property t hat replacing v ariables
wit h equiv alent terms in a context yields equiv alent terms, e.g. if
𝑡 1 ≡ 𝑡 2
,
t hen
[𝑥 ↦ 𝑡 1 ]𝑠 ≡ [𝑥 ↦ 𝑡 2 ]𝑠
. F or Pitts, adequacy onl y requires t hat from t he
termination of one ter m it f ollo w s t hat all equiv alent terms ter minate. In

126 4 F rom Modelica via Mcl to OCaml
our case, w e are also interested in the resulting v alues (see Lemma 1 for a
precise definition).
T o obtain a more concrete definition of equiv alence, w e do no t require t he
larg est adequate relation, but use the axiomatic definition of a pr ecise subset
t hat is sufficient f or our case. This choice simplifies matters significantl y:
Some rules can be merg ed, e.g., betw een compatibility and tr ansitivity ,
subs titutivity becomes a corollar y of t he definition, and all equiv alent terms
can easil y be enumerated since our subset of equiv alence is a com pletel y
defined inductiv ely .
Eq uiv alence of par tial functions
The no tion of equiv alence naturall y expands to partial functions. Equiv a-
lence f or t he domain determines t he equiv alence for the function. In each
case, t he same lef t-hand side element must be mapped t o an equiv alent
right-hand side.
Definition
(Equiv alence of partial functions)
.
T wo partial functions ar e equiv a-
lent, writt en
𝑃 ≡ 𝑄
iff t he y ar e mutuall y compatible, i.e.
𝑄≡𝑃⟺𝑃⊒𝑄 ∧ 𝑄⊒𝑃
A partial function
𝑄
is com patible t o
𝑃
, writt en
𝑄⊒𝑃
, iff all of t he following
s t at ements hold:
(i) 𝑙↦𝑡∈𝑃⟹𝑙↦𝑠∈𝑄 ∧ 𝑡≡𝑠
(ii) 𝐿 ↦ ℎ ∈ 𝑃 ⟹ 𝑙 ↦ 𝑠 ∈ 𝑄 ∧ ℎ ≡ ℎ 𝑔
(iii) 𝑥↦𝑡∈𝑃⟹𝑥↦𝑠∈𝑄 ∧ 𝑠≡𝑡
(iv) 𝑋 ↦ 𝑀 ∈ 𝑃 ⟹ 𝑋 ↦ 𝑁 ∈ 𝑄 ∧ 𝑀 ≡ 𝑀 𝑁
(v) 𝑋 ↦ 𝐶 ∈ 𝑃 ⟹ 𝑋 ↦ 𝐷 ∈ 𝑄 ∧ 𝐶 ≡ ℎ 𝐷
Eq uiv alence of hierarchical ter ms
F igure 4.5 sho ws the equiv alence relation
≡ ℎ
of Mcl’ s hierarchical terms:
R eflexivity , tr ansitivity and symmetr y are defined as usual (via induction
rules). Compatibility ensures that t he equiv alence is a cong ruence relation
wit h respect to the construction of terms: N odes are equiv alent if all fields
and all parent nodes are equiv alent (HEq-Node). Classes are equiv alent, if
all fields are equiv alent (r ule HEq-Class). Class references are equiv alent
if bo t h t heir subclasses and base classes are equiv alent and modifications
are equiv alent if both t he modified classes and t he modifying fields are so
(rules HEq-Ref and HEq-Mod).

4.2 The Compilation Pipeline 127
Reflexivity , Symmetr y and T r ansitivity :
HEq-Refl
ℎ ∧
= 𝑔
ℎ ≡ ℎ 𝑔
HEq-Symm
ℎ ≡ ℎ 𝑔
𝑔 ≡ ℎ ℎ
HEq- Trans
ℎ 1 ≡ ℎ ℎ 2 ℎ 2 ≡ ℎ ℎ 3
ℎ 1 ≡ ℎ ℎ 3
Com patibility :
HEq-N ode
⟦𝐹⟧ Π ≡ ⟦𝐺⟧ Π ∀ 𝑖 ∶ 𝜈 𝑖 ≡ ℎ 𝜋 𝑖
⦃𝐹 in 𝜋⦄ ≡ ℎ ⦃𝐺 in 𝜈⦄
HEq-Class
⟦𝐹⟧ Π ≡ ⟦𝐺⟧ Π
⦃𝐹⦄ ≡ ℎ ⦃𝐺⦄
HEq-Ref ℎ 1 ≡ 𝑔 1 ℎ 2 ≡ 𝑔 2
ℎ 1 .super( ℎ 2 ). 𝐿 ≡ ℎ 𝑔 1 .super( 𝑔 2 ). 𝐿
HEq-Mod
ℎ ≡ 𝑔 ⟦𝐹⟧ Π ≡ ⟦𝐺⟧ Π
ℎ with 𝐹 ≡ ℎ 𝑔 with 𝐺
F igure 4.5: R ules for the equiv alence of hierarchical terms
Eq uiv alence of proper and top-lev el ter ms
The equiv alence relation
≡
betw een proper ter ms follo w s t he same design.
Due to its size, the r ules are found in the appendix (F igure C.3) A g ain, t he
basic definitions f or transitivity , symmetr y , and reflexivity are giv en. The
rules for com patibility are again purel y structur al. Equiv alence for top-le v el
terms and modules is defined similar l y (Figure C.2). Bo t h
≡ 𝑇
and
≡ 𝑀
are
reflexiv e, transitiv e, symmetric and compatible relations on their respectiv e
terms.
Reduction-related eq uivalence
A t t he core of t he equiv alence relation lie t he reduction-related rules (Fig-
ure 4.6). They f or mulate equiv alence betw een certain terms t hat are guaran-
teed to e v aluate to the same v alue. The r ule for the reduction-equiv alence
of hier archical terms is sim ple: Whenev er a hier archical term can ev aluate
to a node, node and term are equiv alent to each other (rule HEq-red). The
equiv alence from lambda calculus motiv ates rule Eq-Bet a. R enaming of
bound v ariables and application of constant ar guments yield equiv alent
terms. MEq-Bet a co v ers a special case of
𝛽
-equiv alence for modules. A
module t hat has t he same name as a f ollo wing class can directl y be applied
to the class and remo v ed from t he top le v el. R ule TEq-Sw ap allo ws classes
and modules to be reor dered, as long as t he class is substituted int o an y

128 4 F rom Modelica via Mcl to OCaml
HEq-red
ℎ ⇓ h 𝜈
ℎ ≡ ℎ 𝜈
Eq-Bet a
[𝑥 ↦ 𝑣]𝑠 ≡ ( λ 𝑥 . 𝑠) 𝑣
MEq-Bet a
𝑃 ∧
= { 𝑋 ↦ MV } 𝑋 ∉ ⟦ MV ⟧ FL V
𝑇 ∷ 𝑋 = [𝑃]𝐶 ≡ 𝑇 𝑇 ∷ 𝑋 = MV ∷ 𝑋 = 𝐶
TEq-Sw ap
𝑃 ∧
= { 𝑋 ↦ 𝐶 } 𝑌 ∉ ⟦𝐶⟧ FL V
𝑇 ∷ 𝑋 = 𝐶 ∷ 𝑌 = 𝑀 ≡ 𝑇 𝑇 ∷ 𝑌 = [𝑃]𝑀 ∷ 𝑋 = 𝐶
F igure 4.6: R ules for the A dequacy of Equiv alence
previousl y defined module.
Flattening of ref erences
Mcl supports tw o different kinds of recursiv e ter ms: Classical recursiv e
functions, using a fix ed-point operat or and mutuall y recursiv e definitions
inside hier archical classes. Since both kinds can be used to express a recur -
siv e computation, it is natur al to introduce an equiv alence betw een the tw o.
W e sa y t hat t he function flatt ens t he ref erence.
F or an intuitiv e understanding of this equiv alence, com pare t he func-
tions in Section 2.1.4 and Section 2.3.7. Each reference has a definition t hat
can be obtained b y ev aluating t he ref erence and searching its label. The
equiv alent function is structur all y similar to t hat definition, ex cept f or refer -
ences t hat are replaced b y an application of recursiv ely bound definitions.
Definition
(Flattening)
.
A map
𝑀
and a gr oup of r ecursiv e functions
ℱ
flatt en
a (number ed) group of hier ar chical r ef er ences
𝜎
, writt en
𝑀 ⊢ ℱ flattens 𝜎
if t he
f ollowing conditions hold:
• 𝜎
is a or der ed set of hier ar c hical r efer ences, taken fr om
𝑛
holes in some
arbitr ar y conte xt i.e.:
𝜎 ∧
= {  𝑖 ↦ 𝑟 𝑖 | 𝑖 ∈ 1 … 𝑛 }
• 𝑀
is a function t hat maps indices of these r ef er ences to t heir r espectiv e
definitions:
dom (𝑀) ∧
= dom (𝜎)
(𝑖 ↦ 𝑘) ∈ 𝑀 ⟺ 𝑟 𝑖 ⇓ h rv 𝑘 ∧ rv 𝑘
𝜙
− → [𝜎]𝑡 𝑘

4.2 The Compilation Pipeline 129
• 𝜌 r eplaces t he holes wit h variables accor ding t o 𝑀 :
𝜌 ∧
= {  𝑖 ↦ 𝑥 𝑘 {} | 𝑖 ∈ 1 … 𝑛, 𝑖 ↦ 𝑘 ∈ 𝑀 }
• ℱ is t he sequence of t he r ecursiv e, closed t hunks of t he definitions:
⟦ ℱ ⟧ Π
∧
= { 𝑥 𝑘 ↦ λ 𝑦 𝑘 . [𝜌]𝑡 𝑘 | 𝑘 ∈ img (𝑀) }
⟦𝑡 𝑗 ⟧ fv ⊆ {  𝑖 | 𝑖 ∈ 1 … 𝑛 }
The definition abo v e g ener alizes t he relationship, as it does no t assume
a one-to-one relation betw een a reference and its definition. Instead, it is
possible t hat man y different ref erences share t he same definition. This
slight com plication will make it easier t o use t he definition in t he actual
com pilation process.
The mos t im portant aspect of t his equiv alence lies in its relation to the
ev aluation of t he equiv alent terms. If t he reference can be e v aluated t han
t he same holds f or t he recursiv e function and vice v ersa. What is more, t he
s tructural similarity betw een t he definition and t he function also applies to
t he ev aluation results.
Corollar y 2
(Ev aluation of flattened references)
.
When a mapping
𝑀
and a
gr oup of r ecursive functions
ℱ
flatt en a set of hier ar c hical r efer ences
𝜎
, t hen t he
r ecursiv e gr oup can r eplace t he hier ar chical r efer ences in an y evaluation cont ext.
I.e.: 𝑀 ⊢ ℱ flattens 𝜎 ∧ [𝜎]𝑡 ⇓ [𝜎]𝑣 ⟹ [𝜔 ]𝑡 ⇓ [𝜔 ]𝑣
Wher e
𝜔
is a shortcut for t he com position of t he subs titutions of t he fixed-point
of ℱ and 𝜌 :
𝜔 ∧
= {  𝑖 ↦ ( fix 𝑥 𝑘 in ℱ ) {} |  𝑖 ∈ dom (𝜎), 𝑖 ↦ 𝑘 ∈ 𝑀 }
The in ver se dir ection also holds:
𝑀 ⊢ ℱ flattens 𝜎 ∧ [𝜔 ]𝑡 ⇓ [𝜔 ]𝑣 ⟹ [𝜎]𝑡 ⇓ [𝜎]𝑣
Pr oof.
By induction on t he deriv ation
[𝜎]𝑡 ⇓ [𝜎]𝑣
or
[𝜔 ]𝑡 ⇓ [𝜔 ]𝑣
respec-
tiv ely . The most interes ting case for the first direction of the corollar y is
t he ev aluation wit h r ule R -Ref, when
𝑡 ∧
=  𝑖
f or some
𝑖 ∈ 1 … 𝑛
. Similar l y , t he
case of FixApp is t he mos t com plicated case for the second direction. See
Appendix D.1 f or a sketch of the proof.
This equiv alence betw een open and proper recursion is cap tured b y rule
Eq-RefFla t (Figure 4.7). When a g roup of recursiv e functions flattens a set
of ref erences, t hen each ref erence in t hat set is equiv alent to the in v ocation
of its corresponding recursiv e function.

130 4 F rom Modelica via Mcl to OCaml
Eq-RefFla t
𝑀 ⊢ ℱ flattens 𝜎  𝑖 ↦ 𝑟 𝑖 ∈ 𝜎 𝑖 ↦ 𝑘 ∈ 𝑀 𝑥 𝑘 ∈ dom (⟦ ℱ ⟧ Π )
fix ℱ in 𝑥 𝑖 {} ≡ 𝑟 𝑖
F igure 4.7: Equiv alence of flattened references
Substitutivity
Subs titutivity , i.e., t he fact that equiv alent terms can be substituted f or
equiv alent ter ms in the same context, is a corollar y of equiv alence. This
subsumes (top-le v el) substitution as w ell as f olding o v er hierarchical and
flat terms.
Corollar y 3
(Subs titutivity of t he equiv alence relations)
.
The f ollowing impli-
cations hold:
(i) 𝑡 ≡ 𝑠 ∧ 𝜎 ≡ 𝜌 ⟹ [𝜎]𝑡 ≡ [𝜌]𝑠
(ii) 𝑡 ≡ 𝑠 ∧ 𝑃 ≡ 𝑄 ⟹ [𝑃]𝑡 ≡ [𝑄]𝑠
(iii) ℎ ≡ ℎ 𝑔 ∧ 𝑃 ≡ 𝑄 ⟹ [𝑃]ℎ ≡ [𝑄]𝑔
(iv) ℎ ≡ ℎ 𝑔 ∧ ⟦𝐹⟧ Π ≡ ⟦𝐺⟧ Π ∧ 𝜈 ≡ ℎ 𝜈 ′ ∧ ∀𝜋 𝑖 ∈ 𝜋 ∶ 𝜋 𝑖 ≡ ℎ 𝜋 ′
𝑖 ⟹ ⟦𝜈, 𝜋, ℎ⟧ Φ ≡ ℎ
⟦𝜈 ′ , 𝜋 ′ , 𝑔⟧ Φ
(v) 𝑡 ≡ ℎ 𝑠 ∧ ⟦𝐹⟧ Π ≡ ⟦𝐺⟧ Π ∧ 𝜈 ≡ ℎ 𝜈 ′ ∧ ∀𝜋 𝑖 ∈ 𝜋 ∶ 𝜋 𝑖 ≡ ℎ 𝜋 ′
𝑖 ⟹ ⟦𝜈, 𝜋, 𝑡⟧ 𝜙 ≡ ℎ
⟦𝜈 ′ , 𝜋 ′ , 𝑠⟧ 𝜙
Pr oof.
By induction o v er
𝑡 ≡ 𝑠
or
ℎ ≡ ℎ 𝑔
. The case of deriv ation b y transitiv -
ity , symmetry , and com patibility f ollo ws immediatel y from t he induction
h ypo t hesis. The r ules f or adequacy are also s traightf or w ar d. The case for
Eq-RefFla t depends on t he fact that t he flattened definitions are closed.
The basic case is t he deriv ation b y reflexivity . Here, t he simpler s tatements
(f or
𝑠 ∧
= 𝑡
or
ℎ ∧
= 𝑔
) are pro v en b y structural induction o v er
𝑡
or
ℎ
, respectiv ely .
The nested base cases (primitiv es, (special) v ariables and ter ms untouched
b y folding) then follo w from t he tr ansitivity property .
In v er sion of eq uiv alence der iv ations
The equiv alence relation is not directl y in v ertible, i.e., for an y giv en ter m
t here are (po tentiall y infinitel y) man y different equiv alent ter ms and deriv a-
tions of t hat equiv alence. This property makes it com plicated to pro v e

4.2 The Compilation Pipeline 131
certain properties b y equiv alence. T o fix t his issue, the follo wing obser v a-
tions are helpful.
Corollar y 4
(In v ersion Properties of Equiv alent T erms)
.
The following s tat e-
ments hold:
(i) 𝑉 ≡ 𝑇 𝑊 ⟹ ⟦𝑉 ⟧ Π ≡ ⟦𝑊 ⟧ Π
(ii) 𝑣 ≡ λ 𝑥 . 𝑡 ⟹ 𝑣 ∧
= λ 𝑥 . 𝑠 such t hat 𝑡 ≡ 𝑠
(iii) 𝑣 ≡ [ 𝑣 ] ⟹ 𝑣 ∧
= [ 𝑤 ] such t hat ⟦𝑣⟧ Π ≡ ⟦𝑤⟧ Π
(iv) 𝑣 ≡ { fv } ⟹ 𝑣 ∧
= { gv } such t hat ⟦ fv ⟧ Π ≡ ⟦ gv ⟧ Π
(v) 𝑣 ≡ < fv > ⟹ 𝑣 ∧
= < gv > such t hat ⟦ fv ⟧ Π ≡ ⟦ gv ⟧ Π
(vi) 𝑣 ≡ fix 𝑥 in ℱ ⟹ 𝑣 ∧
= fix 𝑥 in ℱ ⋆ such t hat ⟦ ℱ ⟧ Π ≡ ⟦ ℱ ⋆ ⟧ Π
Similar l y , t he symmetric counterparts (i.e. t he st atements obt ained by exc hang-
ing t he ar guments of ≡ above) also hold.
Pr oof.
These properties follo w via str aightf or w ard induction o v er t he equiv -
alence s tatement.
Ev aluation b y equiv alence
This extensiv e definition ser v es but one pur pose: If eit her one of tw o equiv -
alent terms ev aluates, t his should be adequate to conclude that t he other
does so as w ell. The ev aluation of t he second term should yield a v alue
equiv alent to the result of t he first. This property is called adequacy b y Pitts
and w e follo w his design. T o sim plify t he application of t his property , w e
f ormulate its premise as a ter nar y relation and define it inductiv el y for
hier archical, module-lev el, top-le v el, and proper terms:
Definition
(Obser v ational T riples)
.
If a t erm
𝑡
is eq uivalent t o
𝑠
and
𝑠
ev aluat es
t o a v alue
𝑣
, t hen
𝑠
is called a witness for t he evaluation of
𝑡
, writt en
𝑡 ≡ 𝑠 ⇓ 𝑣
.

132 4 F rom Modelica via Mcl to OCaml
Similar ly f or hier ar c hical t erms, modules and t op-lev el t erms.
L- Wit
𝑡 ≡ 𝑠 𝑠 ⇓ 𝑣
𝑡≡𝑠⇓𝑣
R - Wit
𝑠 ≡ 𝑡 𝑠 ⇓ 𝑣
𝑡≡𝑠⇓𝑣
L-HWit
ℎ ≡ 𝑔 𝑔 ⇓ h 𝜈
ℎ ≡ ℎ 𝑔 ⇓ h 𝜈
R -HWit
𝑔 ≡ ℎ 𝑔 ⇓ h 𝜈
ℎ ≡ ℎ 𝑔 ⇓ h 𝜈
L-ModWit
𝑀 ≡ 𝑀 𝑁 𝑁 ⇓ M MV
𝑀 ≡ 𝑀 𝑁 ⇓ M MV
R -ModWit
𝑁 ≡ 𝑀 𝑀 𝑁 ⇓ M MV
𝑀 ≡ 𝑀 𝑁 ⇓ M MV
L- TopWit
𝑇 ≡ 𝑇 𝑆 𝑆 ⇓ T 𝑉
𝑇 ≡ 𝑇 𝑆 ⇓ T 𝑉
R - TopWit
𝑆 ≡ 𝑇 𝑇 𝑆 ⇓ T 𝑉
𝑇 ≡ 𝑇 𝑆 ⇓ T 𝑉
The inductiv e definition of t he premise cov ers bo t h sides of t he equiv a-
lence relation. The inductiv e definition of t he relation is not recursiv e and
alw a ys depends on a (hierarchical, t op-lev el, module-lev el) equiv alence and
a (hier archical, top-le v el, module-lev el) ev aluation relation.
The conclusion of t he adequacy can be expressed as t he logical conjunc-
tion of ev aluation and equiv alence of t he results.
Lemma 1
(A dequacy for the Obser v ation of Ev aluation)
.
Eq uivalent (t op-
lev el, module-level, hier ar c hical) t erms ar e adequat e witnesses for t he observation
of (t op-lev el, module-level, hier ar c hical) evaluation, i.e. t he following im plication
holds:
(i) ℎ ≡ ℎ 𝑔 ⇓ h 𝜋 ⟹ ℎ ⇓ h 𝜈 ∧ 𝜈 ≡ ℎ 𝜋
(ii) 𝑡≡𝑠⇓𝑤⟹𝑡⇓𝑣 ∧ 𝑣≡ ℎ 𝑤
(iii) 𝑀 ≡ 𝑀 𝑁 ⇓ M NV ⟹ 𝑀 ⇓ M MV ∧ MV ≡ 𝑀 NV
(iv) 𝑇 ≡ 𝑇 𝑆 ⇓ T 𝑊 ⟹ 𝑇 ⇓ T 𝑉 ∧ 𝑉 ≡ 𝑇 𝑊
Pr oof.
By induction o v er t he deriv ation of equiv alence. See Appendix D.2
f or a sketch.
4 . 3 desugarin g model r outines
Desug aring[Landin, 1964] is t he process of remo ving parts of a languag e
b y replacing certain languag e elements (usuall y of higher abstr action) with

4.3 Desugaring Model R outines 133
more basic but semanticall y equiv alent elements. One exam ple for such
a desug aring is t he replacement of t he binding
let 𝑥 = 𝑡 1 in 𝑡 2
wit h the
semanticall y equiv alent application
( λ 𝑥 . 𝑡 2 ) 𝑡 1
. A desugared languag e main-
tains t he semantics of its extended v ersion while being smaller and t hus
easier to deal with.
In t he case of Mcl, t he goal is to explain the modeling semantics via “s tan-
dar d” ev aluation semantics. Hence, all monadic routines
𝑚
are remo v ed in
t he firs t stag e, while constructs like primitiv es, arra ys, and
let
-expressions
remain in t heir readable f orm.
4.3.1 Synt ax
F igure 4.8 depicts the result of t he first s tag e:
𝑇 𝛼
is t he result of remo v -
ing modeling routines
𝑚
, continuations, and t he explicit modeling state
< 𝑓 >
from
𝑇
. As already mentioned, the difference is mar ked with t he
α -anno tation on t he meta-v ariables.
4.3.2 T r ansf ormation
The classical definition of a desug aring is t hat of a cont ext-fr ee rewriting, al-
t hough current research expands this scope considerabl y , e.g., on com plete
type deriv ations[Lorenzen and Erdw eg, 2016]. The definition of
α
f ollo w s
t he classical s ty le. An in put term is desugared in a recursiv e process b y
desug aring its subterms and combining t he results in a w a y t hat encodes
t he semantics of the remov ed languag e elements. This process is ob viousl y
terminating as t he depth of t he in put term is finite. Hence it is onl y inter -
es ting, ho w each resulting term is encoded. F or a first intuition, recall the
abs tract vie w on elaboration from Section 2.2:
Γ ∶ 𝑙 ↪ 𝑣
𝜖 ∶ 𝕄 → Γ → Γ × 𝑣
If a denotational interpretation
𝜖
exis ts t hat interprets each modeling
routine, the desugaring could use an encoding of
𝜖
and
𝕄
in t he targ et
languag e and encode each primitiv e routine as an application of t he model to
its elabor ation function. The com position of routines can t hen be delegated
to function com position. The sequential composition, f or instance, can be
expressed as a function
seq
t hat expects tw o (desugared) routines and an
elabor ation en vironment. The en vironment could t hen passed to the first
routine t hat yields a new s tate. This new elabor ation en vironment could
t hen be handed o v er to the second routine:

134 4 F rom Modelica via Mcl to OCaml
Cor e T erms:
𝑡 𝛼 ⩴ 𝑋 : 𝑥|𝑥|𝑣 𝛼
| 𝑡 𝛼 𝑡 𝛼 | 𝑡 𝛼 ∘ 𝑡 𝛼
| if 𝑡 𝛼 then 𝑡 𝛼 else 𝑡 𝛼
| let 𝑥 = 𝑡 𝛼 in 𝑡 𝛼
| [ 𝑡 𝛼 ] | 𝑡 𝛼 .[ 𝑡 𝛼 ] | | 𝑡 𝛼 |
| [ 𝑡 𝛼 with 𝑡 𝛼 = 𝑡 𝛼 ]
| { 𝑓 𝛼 } | 𝑡 𝛼 . 𝑙
| { 𝑡 𝛼 with 𝑙 = 𝑡 𝛼 }
| 𝑟 𝛼
𝑓 𝛼 ⩴ 𝑙 = 𝑡 𝛼
𝑟 𝛼 ⩴ ℎ 𝛼 .super( ℎ 𝛼 ). 𝑙
V alues:
𝑣 𝛼 ⩴ 𝑏 | λ 𝑥 . 𝑡 𝛼
| fix 𝑥 in ℱ 𝛼
| [ 𝑣 𝛼 ] | { fv 𝛼 }
fv 𝛼 ⩴ 𝑙 = 𝑣 𝛼
ℱ 𝛼 ⩴ 𝑥 = λ 𝑦 . 𝑡 𝛼
rv 𝛼 ⩴ 𝜈 𝛼 .super( 𝜈 𝛼 ). 𝑙
Hier ar c hical T erms:
ℎ 𝛼 ⩴ up( 𝑖 ) | 𝐶 𝛼 | 𝜈 𝛼 | 𝑅
| ℎ 𝛼 with 𝐹 𝛼
𝑅 𝛼 ⩴ ℎ 𝛼 .super( ℎ 𝛼 ). 𝐿
𝜈 𝛼 , 𝜋 𝛼 ⩴ ⦃𝐹 𝛼 in 𝜈 𝛼 ⦄
𝐶 𝛼 ⩴ 𝑋 | ⦃𝐹 𝛼 ⦄
𝐹 𝛼 ⩴ 𝐿 = ℎ 𝛼 | 𝑙 = 𝑡 𝛼
T op-lev el T erms:
𝑇 𝛼 ⩴ 𝐷 𝛼
𝐷 𝛼 ⩴ 𝑋 = 𝑀 𝛼
| 𝑋 = 𝐶 𝛼
𝑀 𝛼 ⩴ 𝑋 | struct d 𝛼 end
T op-lev el V alues:
MV 𝛼 ⩴ struct dv 𝛼 end
D V 𝛼 ⩴ 𝑋 = ( MV | 𝐶 𝛼 )
𝑉 𝛼 , 𝑊 𝛼 ⩴ D V 𝛼
dv 𝛼 ⩴ 𝑥 = 𝑣 𝛼
F igure 4.8: Output syntax of t he first s tage

4.3 Desugaring Model R outines 135
let seq r1 r2 env = r2 (r1 env).1
This encoding is of course not correct (as it ignores continuations), but
giv es a useful intuition on how t he desugaring will w or k. For each com pos-
ite routine, function application can be used to im plement t he elaboration
or der . The desugared routine is a function that expects a v alue which repre-
sents t he current elabor ation state. Similar l y , t he current continuation is an
argument of that function. There is some deg ree of freedom in t he concrete
encoding, as long as it is consistent. Therefore the first task is t o decide
on an encoding of t he elabor ation state, continuations, and the primitiv e
routines as v alues of t he residual languag e. The s trategy f or composite
routines t hen f ollo ws natur all y .
Elaboration S tate
The elabor ation state is s tr ucturall y similar to a recor d. Both map labels to
terms and both are ev aluated when all t heir fields are ev aluated. Hence, t he
firs t decision is ob vious. A literal elabor ation state is expressed as a recor d
t hat maps each elabor ation aspect to its defining term. The terms st ored in
t he s tate are processed recursiv ely .
⟦𝑙 = 𝑡⟧ α
∧
= 𝑙 = ⟦𝑡⟧ α
⟦ < 𝑓 > ⟧ α
∧
= { α ∗ 𝑓 𝛼 }
Continuations
A continuation is essentiall y a list of elements. The shape of each element
resembles t hat of a
𝜆
-abs traction; bo t h consist of a v ariable and a body . An
em p ty continuation immediately yields its in put v alue (rule St op), when it is
elabor ated. A non-emp ty continuation evaluat es t he body of its first element
wit h the input v alue and elabor ates the resulting routine wit h t he in put state
and t he f ollo wing continuation (r ule C ont). Therefore, the encoding of a
continuation requires tw o parameters: the in put v alue
𝑥
and t he elabor ation
s tate 𝑦 𝑒𝑛𝑣 .
⟦  ⟧ α
∧
= {continue = λ 𝑥 . λ 𝑦 Γ . 𝑥 }
⟦𝑥 . 𝑡 ∶∶ 𝑘⟧ α
∧
= {continue = λ 𝑥 . λ 𝑦 Γ . (⟦𝑡⟧ α ⟦𝑘⟧ α ) 𝑦 Γ }
where 𝑦 Γ ∉ ⟦⟦𝑡⟧ α ⟧ fv ∪ ⟦⟦𝑘⟧ α ⟧ fv and 𝑥 ∉ ⟦𝑘⟧ fv

136 4 F rom Modelica via Mcl to OCaml
The g ener ated function is wrapped in a recor d wit h t he special field
continue
(see Section 4.8 for t he reason). The em p ty continuation ignores
t he s tate and yields t he in put v alue; it pro vides a w a y (t he onl y w a y , in fact)
to “escape” from the elaboration. A non-em p ty continuation passes its tail
and t he elabor ation s tate to the body of its first element. The input v alue is
bound to the v ariable. If t he in put v ariable occurs free in t he tail, it needs
to be replaced with a
𝛼
-con v ersion. The v ariable name f or the state is fresh.
Managing t he State
As seen abo v e, an encoded non-em p ty continuation passes its encoded tail
and t he elabor ation s tate to the encoding of its body . A consistent encoding
of routines has to adhere t o t his interface. Therefore, routines are encoded as
abs tractions o v er both t he next continuation
𝑥 𝑘
and t he current elabor ation
s tate 𝑦 Γ .
⟦𝑙 •put 𝑡⟧ α
∧
= λ 𝑥 𝑘 . λ 𝑦 Γ . (𝑥 𝑘 .continue {} { 𝑦 Γ with 𝑙 = ⟦𝑡⟧ α } )
where 𝑥 𝑘 , 𝑦 Γ ∉ ⟦⟦𝑡⟧ α ⟧ fv
⟦𝑙 •get ⟧ α
∧
= λ 𝑥 𝑘 . λ 𝑦 Γ . (𝑥 𝑘 .continue (𝑦 Γ . 𝑙) 𝑦 Γ )
⟦ put 𝑡⟧ α
∧
= λ 𝑥 𝑘 . λ 𝑦 Γ . (𝑥 𝑘 .continue {} ⟦𝑡⟧ α )
where 𝑥 𝑘 , 𝑦 Γ ∉ ⟦⟦𝑡⟧ α ⟧ fv
⟦ get ⟧ α
∧
= λ 𝑥 𝑘 . λ 𝑦 Γ . (𝑥 𝑘 .continue 𝑦 Γ 𝑦 Γ )
The encoding of
put
and
get
resemble t he classical im plementation
of a s tate monad, written in continuation passing sty le. F rom an existing
elabor ation state
𝑦 Γ
, a new s tate and a obser v ed v alue are calculated and
passed to the continuation
𝑥 𝑘
, in t he order required b y the encoding of
t he continuations. Modification of just a single aspect
𝑙
is im plemented as
a recor d update, inspection of t he aspect becomes record projection, and
t he global s tate is directl y tur ned into t he obser v able v alue or set from t he
argument.
Control Flo w
Com positional routines im pose order on the elaboration. The bind operator
or ders tw o routines sequentiall y . A sequence is ter minated b y retur ning a
v alue. N on-sequential elaboration is achie v ed via continuing wit h a contin-
uation t hat has been sa v ed wit h
call/cc
. The encoding im plements t his
control flo w b y creating the respectiv e next continuation.

4.3 Desugaring Model R outines 137
⟦𝑧 ← 𝑡 ; 𝑠⟧ α
∧
= λ 𝑥 𝑘 . λ 𝑦 Γ . (⟦𝑡⟧ α {continue = λ 𝑧 . ⟦𝑠⟧ α 𝑥 𝑘 } 𝑦 Γ )
where 𝑥 𝑘 , 𝑦 Γ ∉ ⟦⟦𝑡⟧ α ⟧ fv ∪ ⟦⟦𝑠⟧ α ⟧ fv
⟦ return 𝑡⟧ α
∧
= λ 𝑥 𝑘 . λ 𝑦 Γ . (𝑥 𝑘 .continue ⟦𝑡⟧ α 𝑦 Γ )
where 𝑥 𝑘 , 𝑦 Γ ∉ ⟦⟦𝑡⟧ α ⟧ fv
⟦ continue 𝑡 with 𝑠⟧ α
∧
= λ 𝑥 𝑘 . λ 𝑦 Γ . (⟦𝑡⟧ α .continue ⟦𝑠⟧ α 𝑦 Γ )
where 𝑥 𝑘 , 𝑦 Γ ∉ ⟦⟦𝑡⟧ α ⟧ fv ∪ ⟦⟦𝑠⟧ α ⟧ fv
⟦ call/cc 𝑡⟧ α
∧
= λ 𝑥 𝑘 . λ 𝑦 Γ . ((⟦𝑡⟧ α 𝑥 𝑘 ) 𝑥 𝑘 𝑦 Γ )
where 𝑥 𝑘 , 𝑦 Γ ∉ ⟦⟦𝑡⟧ α ⟧ fv
Binding of tw o routines is implemented b y the creation of a continuation.
The body of t his continuation expects t he bound v ariable
𝑧
and passes t he
outer continuation
𝑥 𝑘
to the encoding of t he second routine
𝑠
. This contin-
uation is passed as an argument t o the first routine
𝑡
. This patter n again
resembles t he classical monad im plementation, expressed in continuation-
passing s ty le. The
return
-routine is desug ared into a function that directl y
passes t he argument term to t he next continuation
𝑥 𝑘
. A continuation re-
tains control via
continue
b y for g etting the current continuation
𝑥 𝑘
and
passing t he argument
𝑡
to the continuation obtained from
𝑠
. The current
continuation is st ored via
call/cc
b y passing it twice as an argument to
t he called routine
𝑡
: The first time it represents the actual v alue and t he
second time guar antees t hat control is retur ned to the current point af ter
𝑡
is finished. In each case, t he new l y introduced v ariables are fresh, and t he
s tate is passed as t he second argument.
T ransf or mation of T er ms
These definitions suffice to explain the transf or mation
α
. Dealing wit h t he
res t of t he in put languag e is sim pl y a recursiv e tr a v ersal. This scaffolding is
sho wn in Appendix C.3. The onl y interesting term transf ormation regar ds
t he entry point of elaboration. The ex ecution of a routine is implemented b y
passing a special continuation t hat ignores its s tate argument
𝑦 Γ
and yields
its injected v alue 𝑥 follo w ed b y t he tr ansf ormed state 𝑠 .
⟦ exec 𝑡 with 𝑠⟧ α
∧
= ⟦𝑡⟧ α {continue = λ 𝑥 . λ 𝑦 Γ . 𝑥 } ⟦𝑠⟧ α

138 4 F rom Modelica via Mcl to OCaml
4.3.3 Exam ple
. When
α
is applied liter all y , t he desugared routines become r ather long and
are har d to read. Hence t he exam ple below is no t t he literal output of the
tr ansformation but has been op timized f or readability . Redundant v ariables
ha v e been remov ed via eta-reduction, some applications ha v e been inlined
and resulting redundant expressions ha v e been hoisted.
The firs t stag e does not chang e t he test module. The class
EvenOdd
contains t he same three fields as before, but t heir definitions no w onl y
consis t of t he firs t stag e syntax:
(* class EvenOdd after first stage transformation *)
val do_even = λ k. λ s.
let cont =
{_continue_ =
λ loop. λ s_0.
let n = s_0.n in
if (n <= 0) then k._continue_ true s0
else if (n = 1) then k._continue_ false s0
else loop._continue_ loop {s0 with n=n-2 }
}
in cont._continue_ cont s;
val even = λ n. this.do_even {_continue_ = λ x. λ s.x} {n = n};
val odd = λ n. if (n = 0) then false else this.even (n - 1)
19
20
21
22
Each top-le v el routine (e.g. t he definition of
do_even
) is tr anslated into a
function t hat expects a continuation
k
and a s tate
s
. How ev er , when literal
routines are transf or med, it is much simpler t o inline the corresponding
applications directl y . F or ins tance annotation
20
sho ws the translation of
t he
get
routine from anno tation
15
. Instead of cons tructing a continuation
recor d and an abstr action, t he returned v alue
n
is directl y bound wit h a
let
-expression. The result of transf or ming t he bind routine results in an ap-
plication of a continuation. That continuation is immediatel y decons tructed
b y t he transf or med
get
routine. The additional rewriting from an applica-
tion to a let -expression is trivial and merel y increases t he readability .
The tr anslation of t he
call/cc
routine from annotation
14
f ollo w s in a
similar l y op timized manner (anno tation
21
). The next elaboration s teps are
a v ailable in t he form of liter al routines and are consequentl y inlined in t he

4.3 Desugaring Model R outines 139
tr anslation. The resulting continuation is bound to the artificial v ariable
cont to a v oid code duplication.
The
continue
routine from annotation
17
is tr anslated in a str aightfor -
w ard manner . The loop continues onl y if
n
is g reater than
1
. In t his case,
t he v ariable
loop
is in v oked wit h itself as first ar gument and a new state
(inlined from t he corresponding
put
routine) (anno tation
22
). Similar l y , the
ex ecution of t he monadic loop from annotation
13
is relativ ely sim ple: It
onl y requires t he trivial translation of the state t o a record and the insertion
of t he “s top” continuation that extracts the obser v able v alue and discards
t he elabor ation s tate annotation 19 .
4.3.4 T ermination & Corr ectness
Since
α
is a sim ple syntactic tr ansformation, there should be no reasonable
doubt t hat it is bo t h a function and terminates. Reg ar ding correctness,
t he definition of
𝛼
immediatel y leads to tw o useful properties. F irs t, t he
application of α dis tributes o v er substitution:
Corollar y 5
(Subs titutivity of t he F irst S tag e)
.
The firs t st ag e dis tribut es ov er
subs titutions:
(i) ⟦[𝜎]𝑡⟧ α ≡ [ α ∘ 𝜎]⟦𝑡⟧ α
(ii) ⟦[𝑃]𝑡⟧ α ≡ [ α ∘ 𝑃]⟦𝑡⟧ α
(iii) ⟦⟦𝜈, 𝜋, 𝑡⟧ 𝜙 ⟧ α ≡ ⟦⟦𝜈⟧ α , α ∗ 𝜋, ⟦𝑡⟧ α ⟧ 𝜙
(iv) ⟦⟦𝜈, 𝜋, ℎ⟧ Φ ⟧ α ≡ ℎ ⟦⟦𝜈⟧ α , α ∗ 𝜋, ⟦ℎ⟧ α ⟧ Φ
Pr oof.
By a s traightf or w ar d s tructural induction on
𝑡
, or
ℎ
, respectiv el y . The
base cases are t he (special) v ariables.
The second property reg ards the equiv alence relation. Alt hough
α
is
no t an equiv alent transf ormation, it retains t he relationship when applied
to tw o (top-le v el, hierarchical, module-lev el or proper) terms:
Corollar y 6
(Preser v ation of t he F irst S tag e)
.
The firs t st ag e pr eserves eq uiv a-
lence, i.e.
(i) 𝑡 ≡ 𝑠 ⟹ ⟦𝑡⟧ α ≡ ⟦𝑠⟧ α
(ii) ℎ ≡ ℎ 𝑔 ⟹ ⟦ℎ⟧ α ≡ ℎ ⟦𝑔⟧ α
(iii) 𝑀 ≡ 𝑀 𝑁 ⟹ ⟦𝑀⟧ α ≡ 𝑀 ⟦𝑁 ⟧ α

140 4 F rom Modelica via Mcl to OCaml
(iv) 𝑇 ≡ 𝑇 𝑆 ⟹ ⟦𝑇 ⟧ α ≡ ℎ ⟦𝑆⟧ α
Pr oof. By a s tr aight forw ar d induction on t he equiv alence deriv ation.
Since Mcl consists of se v eral, mutuall y recursiv e, syntactic categories,
consis tency of top-le v el ter ms depends on sev eral, mutuall y recursiv e, state-
ments of consis tency , one f or each syntactic categor y .
Consider , for ins tance, Mcl’ s continuations. What makes
α
consis tent
concer ning this particular sort of v alues? The onl y use of a continuation
inside t he ev aluation is t he adv ance relation
⇓ m
depicted in Section 2.2. This
relation links four elements: A state
Γ
, a v alue
𝑣
, t he continuation
𝑘
and t he
result v alue
𝑤
. When t he transf or mation is consistent, these four elements
are also related in t he resulting prog r am. In particular , t he continuation
corresponds to an application of their respectiv e desug ared counter parts.
The tr ansformation is consis tent if t his ev aluation yields a v alue t hat is
equiv alent to 𝑤 (modulo tr ansf ormation).
Lemma 2
(Consis tency of t he Firs t Stag e)
.
The firs t s tag e is a consis t ent encoding
of t he elabor ation semantics.
⟦𝑇 ⟧ α
∧
≠ ∧ ⟦𝑉 ⟧ α
∧
≠ ∧ 𝑇 ⇓ T 𝑉 ⟹ ⟦𝑇 ⟧ α ⇓ T 𝑊 ∧ 𝑊 ≡ 𝑇 ⟦𝑉 ⟧ α
Pr oof.
The proof follo w s from t he follo wing s tatements b y mutually recur -
siv e induction on t he deriv ations. See Appendix D.3 for a sketch of the
proof. Consistency is a direct consequence of S tatement 6.
ℎ ⇓ h 𝜈 ⟹ ⟦ℎ⟧ α ⇓ h 𝜈 𝛼 ≡ ⟦𝜈⟧ α
(1)
𝑡 ⇓ 𝑣 ⟹ ⟦𝑡⟧ α ⇓ 𝑤 𝛼 ≡ ⟦𝑣⟧ α
(2)
⟨Γ, 𝑘⟩ ⊢ 𝑚 ⇓ m 𝑣 ⟹ ⟦𝑚⟧ α ⟦𝑘⟧ α ⟦Γ⟧ α ⇓ 𝑣 𝛼 ≡ ⟦𝑣⟧ α
(3)
⟨Γ, 𝑣⟩ ⊢ 𝑘 ⇓ k 𝑤 ⟹ ⟦𝑘⟧ α .continue ⟦𝑣⟧ α ⟦Γ⟧ α ⇓ 𝑤 𝛼 ≡ ⟦𝑤⟧ α
(4)
𝑀 ⇓ M MV ⟹ ⟦𝑀⟧ α ⇓ M MV 𝛼 ≡ 𝑇 ⟦ MV ⟧ α
(5)
𝑇 ⇓ T 𝑉 ⟹ ⟦𝑇 ⟧ α ⇓ T 𝑉 𝛼 ≡ 𝑇 ⟦𝑉 ⟧ α
(6)
4 . 4 sep ara tin g terms and hierar chy
In Mcl, t he s tructure of a class is a requirement f or t he com pilation of its
dependents. If
A
extends
B
, f or exam ple, t hen t he com pilation of
A
requires
kno w ledge about the shape of all components declared or inherited in
B
.

4.4 Separating T erms and Hierarch y 141
Com pilation thus needs to r etain this structure. A t t he same time, w e w ant
to r emov e it, t hough, because it has no counter part in the targ et languag e.
This apparent par ado x is resolv ed b y a separation of concerns betw een
t he s tructure of top-lev el classes and t heir im plementation. Recall that
t he front end assigns the special v ariables inside a class at com pilation
time. The shape of a compiled class ne v er changes. Onl y the values of its
com ponents depend on the context due to possibl y different results for the
lookup of hierarchic references. Thus, t he v alue of each component is an
abs tr action o v er t he possible v alues of these references. Such an abstr action
can be expressed in t he targ et languag e as a function and the second stag e
is responsible f or finding precisel y such a function for each field of a class.
T echnicall y , t his process is a special form of
𝛽
-expansion, t he in v erse
oper ation to
𝛽
-reduction from t he lambda calculus. F or ins tance, t he sim ple
term
this.foo + 2
can be expressed as t he more com plex, but equiv alent
term
( λ 𝑥 ℎ . 𝑥 ℎ + 2 ) this.foo
. Here, t he reference
this.foo
has been replaced
b y a v ariable
𝑥 ℎ
and is placed into its f or mer context via application to a
new l y created function.
Ano t her useful v ariant of t he original term is
𝑋 : 𝑥 𝑓 this.foo
, when
t he module
𝑋
contains t he definition
𝑥 𝑓 = λ 𝑥 ℎ . 𝑥 ℎ + 2
. W e call the former
expression skelet on and t he latter its im plement ation . The skeleton contains
onl y a symbolic reference to the im plementation applied to a (possibl y
em p ty) list of hier archical ref erences.
The im plementation does no t contain an y hierarchical term whatsoev er .
Modules are a common element of functional languag es and exis t in t he
targ et language. Theref ore, t his transf ormation achiev es t he separation of
concer ns mentioned abo v e. The implementation can be tr anslated furt her ,
while t he skelet on describes how o t her units of compilation can use it. The
second s tag e applies t his tr ansf ormation to all top-le v el classes and t hereb y
enables separ ate com pilation.
4.4.1 Synt ax
The output of t he second s tag e res tricts hier archical terms. The right-hand
side of fields ma y only contain a specific form, t he skelet on
𝑎
(F igure 4.9).
A skelet on consis ts eit her of t he application of a long identifier to an em p ty
recor d or t he application of a reference t o ano t her skelet on. The references
are wr apped into functions with an unused argument. Proper ter ms 𝑡 𝛽 do
no t contain an y references. Module definitions are not tr ansf ormed and
ma y still contain arbitr ar y hier archical terms.

142 4 F rom Modelica via Mcl to OCaml
Cor e T erms:
𝑡 𝛽 ⩴ 𝑋 : 𝑥 |𝑥|𝑣 𝛽
| 𝑡 𝛽 𝑡 𝛽 | 𝑡 𝛽 ∘ 𝑡 𝛽
| if 𝑡 𝛽 then 𝑡 𝛽 else 𝑡 𝛽
| let 𝑥 = 𝑡 𝛽 in 𝑡 𝛽
| [ 𝑡 𝛽 ] | 𝑡 𝛽 .[ 𝑡 𝛽 ] | | 𝑡 𝛽 |
| [ 𝑡 𝛽 with 𝑡 𝛽 = 𝑡 𝛽 ]
| { 𝑓 𝛽 } | 𝑡 𝛽 . 𝑙
| { 𝑡 𝛽 with 𝑙 = 𝑡 𝛽 }
𝑓 𝛽 ⩴ 𝑙 = 𝑡 𝛽
V alues:
𝑣 𝛽 ⩴ 𝑏 | λ 𝑥 . 𝑡 𝛽
| fix 𝑥 in ℱ 𝛽
| [ 𝑣 𝛽 ] | { fv 𝛽 }
fv 𝛽 ⩴ 𝑙 = 𝑣 𝛽
ℱ 𝛽 ⩴ 𝑥 = λ 𝑦 . 𝑡 𝛽
Hier ar c hical T erms:
ℎ 𝛽 , 𝑔 𝛽 ⩴ up( 𝑖 )
| ℎ 𝛽 .super( ℎ 𝛽 ). 𝐿
| 𝜈 𝛽 | ℎ 𝛽 with 𝐹 𝛽 | 𝐶 𝛽
𝜈 𝛽 , 𝜋 𝛽 ⩴ ⦃𝐹 𝛽 in 𝜈 𝛽 ⦄
𝐹 𝛽 ⩴ 𝐿 = ℎ 𝛽
| 𝑙 = 𝑎
𝑎 ⩴ 𝑋 : 𝑥 {}
| 𝑎 ( λ 𝑥 . 𝑟 𝛽 )
𝑟 𝛽 ⩴ λ 𝑥 . (ℎ 𝛽 .super( ℎ 𝛽 ). 𝑙)
T op-lev el entities:
𝑇 𝛽 ⩴ 𝐷 𝛽
𝐷 𝛽 ⩴ 𝑋 = 𝑀 𝛼
| 𝑋 = 𝐶 𝛽
𝐶 𝛽 ⩴ 𝑋 | ⦃𝐹 𝛽 ⦄
F igure 4.9: Output syntax of t he second stag e

4.4 Separating T erms and Hierarch y 143
4.4.2 T r ansf ormation
The second s tag e is an in v erse subs titution: F ind all references in a term,
tr ansform them recursiv el y , replace t he transf or med results wit h fresh v ari-
ables and retur n an abs tr action o v er t hese v ariables. While t his is s tr aight-
f or w ard t o im plement, its bott om-up nature makes it some what tedious to
describe as a function. Therefore w e describe it as an (algorit hmic) relation
betw een t he transf or med ter ms and the transf ormation substitution.
Definition
(
𝛽
-refact oring)
.
A partial function
𝑞 ∶ 𝑥 ↪ 𝑟 𝛽
is said t o be a
𝛽
-r efact oring of a hierar chical t erm
ℎ 𝛼
int o an im plementation
𝑋
and a skelet on
ℎ 𝛽
,
iff
𝑞 ⊢ ℎ 𝛼 ≡ 𝑋 ℎ 𝛽
. Similar ly f or terms
𝑡 𝛼
and
𝑡 𝛽
. This quat ernar y r elation is defined
inductiv el y as follow s:
R - V ar
∅ ⊢ 𝑌 ≡ 𝑋 𝑌
R -CFld
𝑞 ⊢ ℎ 𝛽 ≡ 𝑋 𝑔 𝛼
𝑞 ⊢ (𝐿 = ℎ 𝛽 ) ≡ 𝑋 (𝐿 = 𝑔 𝛼 )
R -Class
∀𝑖 ∶ 𝑞 ⊢ 𝐹 𝛽
𝑖 ≡ 𝑋 𝐺 𝛼
𝑖
𝑞 ⊢ ⦃𝐹 𝛽 ⦄ ≡ 𝑋 ⦃𝐺 𝛼 ⦄
R -CRef
𝑞 ⊢ ℎ 𝛽
1 ≡ 𝑋 𝑔 𝛼
1 𝑞 ⊢ ℎ 𝛽
2 ≡ 𝑋 𝑔 𝛼
2
𝑞 ⊢ ℎ 𝛽
1 .super( ℎ 𝛽
2 ). 𝐿 ≡ 𝑋 𝑔 𝛼
1 .super( 𝑔 𝛼
2 ). 𝐿
R -Mod
𝑞 ⊢ ℎ 𝛽 ≡ 𝑋 𝑔 𝛼 ∀𝑖 ∶ 𝑞 ⊢ 𝐹 𝛽
𝑖 ≡ 𝑋 𝐺 𝛼
𝑖
𝑞 ⊢ ℎ 𝛽 with 𝐹 𝛽 ≡ 𝑋 𝑔 𝛼 with 𝐺 𝛼
R -Node
∀𝑖 ∶ 𝑞 ⊢ 𝐹 𝛽
𝑖 ≡ 𝑋 𝐺 𝛼
𝑖 ∀𝑖 ∶ 𝑞 ⊢ 𝜈 𝛽
𝑖 ≡ 𝑋 𝜋 𝛼
𝑖
𝑞 ⊢ ⦃𝐹 𝛽 in 𝜈 𝛽 ⦄ ≡ 𝑋 ⦃𝐺 𝛼 in 𝜋 𝛼 ⦄
R -Fld
𝑞 ⊢ 𝑡 𝛼 ≡ 𝑋 𝑎
𝑞 ⊢ 𝑙 = 𝑎 ≡ 𝑋 𝑙 = 𝑡 𝛼
R -Ref 𝑞 ⊢ ℎ 𝛽
1 ≡ 𝑋 𝑔 𝛼
1 𝑞 ⊢ ℎ 𝛽
2 ≡ 𝑋 𝑔 𝛼
2
𝑞 ⊢ λ 𝑥 . ℎ 𝛽
1 .super( ℎ 𝛽
2 ). 𝑙 ≡ 𝑋 𝑔 𝛼
1 .super( 𝑔 𝛼
2 ). 𝑙
R - Term 𝑞 ⊢ 𝑎 ≡ 𝑋 ( λ 𝑥 . [  ↦ 𝑥 {} ]𝑠 𝛼 )
𝑠 𝛼 ∧
≠  𝑞 ⊢ 𝑟 𝛽 ≡ 𝑋 𝑟 𝛼 𝑥 fresh in 𝑠 𝛼
𝑞 ⊢ 𝑎 𝑟 𝛽 ≡ 𝑋 [  ↦ 𝑟 𝛼 ]𝑠 𝛼
R -Mo ve 𝑦 ∉ ⟦𝑡 𝛽 ⟧ fv
𝑞 ⊕ { 𝑧 ↦ λ 𝑦 . 𝑡 𝛽 } ⊢ 𝑋 : 𝑧 {} ≡ 𝑋 𝑡 𝛽

144 4 F rom Modelica via Mcl to OCaml
A refact oring
𝑞
is a
𝑡 𝛽
-subs titution, i.e., a partial functions from short
v ariables to second-s tag e terms. T op-lev el v ariables are not subject to ref ac-
toring (rule R - V ar). R efact oring distributes o v er hierarchical terms (r ules
R -CFld, R -Class, R -N ode, R -CRef, and R -Mod). V alue fields are refact ored
b y replacing t he right-hand side wit h a skelet on that is obtained b y refact or -
ing t he original term (r ule R -Fld). Ref erences are refact ored b y refact oring
t he hier archical subterms (r ule R -Ref).
A t t he core of a refactoring lies the rewriting of a firs t-stag e term
𝑡 𝛼
to
a second-stag e skelet on
𝑎
(rule R - Term). If a term contains a reference
𝑟 𝛼
,
it is extr acted and replaced wit h an application of a fresh v ariable to the
em p ty argument
{}
. The reference itself is ref actored and appended as an
argument t o the skelet on. A ter m t hat does no t contain an y references is a
second-s tag e term. It is stored in the partial function and replaced b y an
application of a new v ariable to an em p ty recor d (rule R -Mo ve).
Corollar y 7
(Application of R efact oring)
.
Subs titution of a r efact oring is an
eq uivalent tr ansf ormation.
(i) ⟦ dv ⟧ Π ⊢ ℎ ≡ 𝑋 ℎ 𝛽 ⟹ [{ 𝑋 ↦ struct dv end }]ℎ 𝛽 ≡ ℎ ℎ
(ii) ⟦ dv ⟧ Π ⊢ 𝑡 ≡ 𝑋 𝑡 𝛽 ⟹ [{ 𝑋 ↦ struct dv end }]𝑡 𝛽 ≡ ℎ 𝑡
Pr oof.
By structur al induction o v er t he deriv ation of t he refact oring relation.
R ules R - V ar, R -CFld, R -Class, R -N ode, R -CRef, R -Mod, R -Fld, and R -Ref
are s traightf or w ar d. R ules R - Term and R -Mo ve follo w from Eq-Bet a.
The second s tag e builds on t his refact oring and applies it recursiv el y to
all top-le v el classes. The second-stag e output of a first-s tage t op-lev el class
𝐶 𝛼 is t he skelet on 𝐶 𝛽 and t he implementation dv 𝛼 . β nev er yields an error .
Definition
(Second S tag e)
.
The second s t ag e ref actor s classes and ignor es mod-
ules. It puts t he implement ation of each class in fr ont of t he skeleton.
⟦ ♦ ⟧ β
∧
= ♦
⟦𝑇 𝛼 ∷ 𝑋 = 𝑀 𝛼 ⟧ β
∧
= ⟦𝑇 ⟧ β ∷ 𝑋 = 𝑀 𝛼
⟦𝑇 𝛼 ∷ 𝑋 = 𝐶 𝛼 ⟧ β
∧
= ⟦𝑇 𝛼 ⟧ β ∷ 𝑋 = struct dv 𝛽 end ∷ 𝑋 = 𝐶 𝛽
wher e ⟦ dv 𝛽 ⟧ Π ⊢ 𝐶 𝛽 ≡ 𝑋 𝐶 𝛼

4.4 Separating T erms and Hierarch y 145
4.4.3 Exam ple
R efact oring the class
EvenOdd
yields its im plementation and its skelet on.
The tes t module is again no t changed b y t he transf ormation. The skelet on
f or w ards the elaboration t o t he “flat” representation in the implementation.
All ref erences are tur ned into ar guments to the corresponding functions
from t he im plementation:
(* Skeleton of EvenOdd after second-stage transformation *)
val do_even = EvenOdd:flat_do_even {};
val even = EvenOdd:flat_even {} ( λ _. this.do_even);
val odd = EvenOdd:flat_odd {} ( λ _. this.even)
23
The lis ting abo v e sho w s t hat ref erences like the one from annotation
16
are wr apped into thunk s (annotation
23
). The rest of the im plementation
has been mo v ed to the im plementation. The implementation contains the
definitions, also wrapped int o t hunk s. There, all t he references ha v e been
replaced wit h additional arguments. These arguments are un wrapped
when needed (anno tation 24 belo w).
(* Implementation of EvenOdd after second-stage transformation *)
let flat_odd =
λ _. λ _harg0. λ n. if (n = 0) then false else _harg0 {} (n - 1)
let flat_even =
λ _. λ _harg0. λ n. _harg0 {} {_continue_ = λ x. λ s.x} {n = n}
let flat_do_even = λ _. λ k. λ s.
let cont =
{_continue_ =
λ loop. λ s_0.
let n = s_0.n in
if (n <= 0) then k._continue_ true s0
else if (n = 1) then k._continue_ false s0
else loop._continue_ loop {s0 with n=n-2 }
}
in cont._continue_ cont s
24

146 4 F rom Modelica via Mcl to OCaml
4.4.4 T ermination & Corr ectness
T ermination of t he second stag e depends on t he search for a ref actoring. The
refact oring relation is algorit hmic. The distinction betw een first-s tage and
second-s tag e syntax, i.e., whet her to appl y rule R -Mo ve or R - Term, requires
a search f or references. This search requires as a simple tr a v ersal of t he
syntax. All expressions are visited onl y once, and a refact oring can alw a ys
be f ound. Hence, ter mination is guaranteed b y construction.
The correctness f ollo w s directl y from t he obser v ation that a refact oring
is a subset of the equiv alence relation. The re-application of a refact oring
via subs titution yields an equiv alent ter m (Corollar y 7), and t his allo ws t o
conclude t hat refact ored top-lev el terms are equiv alent to their origin:
Lemma 3
(Equiv alence of t he Second Stag e)
.
The second s t ag e tr ansforms
t erms int o equiv alent v ariants, i.e:
𝑇 ≡ 𝑇 ⟦𝑇 ⟧ β
Pr oof.
By s tructural induction o v er
𝑇
. The base case,
𝑇 ∧
= ♦
is trivial (b y
reflexivity of
≡ 𝑇
). So is t he case of a module definition (b y induction, com-
patibility of
≡ 𝑇
and reflexivity of
≡ 𝑀
). The case of top-lev el class follo ws
from rule MEq-Bet a, Corollar y 7, and compatibility and tr ansitivity of
≡ 𝑇 .
4 . 5 ev al u a tin g hierar chic al terms
Bef ore w e tur n to the final stag e, w e ha v e to remo v e a major obs tacle to
its im plementation. R eplacing a reference with its definition requires t he
ev aluation of t he subclass and t he base class (which are of ten t he same).
In o t her w ords, w e need to interpret a part of t he language, namel y t he
hier archic semantics, during t he tr anslation.
The definition of
⇓ h
directl y leads to an inter preter . A naiv e imple-
mentation of t his interpreter will not alw a ys ter minate, t hough, due to the
recursiv e nature of hierarchical terms. Ev aluating t he subterms of a class
ref erence might require t he ev aluation of t he same class reference ag ain:
{| class A = this; x = this.A.x |}.x
In t he abo v e exam ple, a naiv e inter preter might repeatedl y attemp t to
ev aluate t he class-reference
this.A.x
. Therefore, it is necessary to res trict
t he ev aluation process in a w a y t hat guarantees termination and retains a
v alid result for a meaningful subset of the ter minating nodes.

4.5 Ev aluating Hier archical T er ms 147
A s traightf or w ar d str ategy is to no t consider an y lookup, i.e., a combi-
nation of an en vironment and a label, twice. This met hod is sufficient if
t he set of v alid en vironments is finite. Then t he ev aluation can guarantee
termination b y remembering “visited” labels f or each en vironment. If a
label is encountered twice in t he same en vironment, ev aluation can abort
wit h an error .
In t his section, w e restrict the ev aluation of Mcl to a finite set of en vi-
ronments. On the basis of t his restriction, w e dev elop a ter minating and
correct algorit hm f or t he ev aluation of hierarchical terms.
4.5.1 Hier ar c hical Nodes as Multigr aphs
W e assume t hat t he function
𝐿
obtains a unique identifier from a set
𝕃⊆ℕ
f or each literal class. As a shortcut, w e write
𝐹 𝑖
when
⟦𝐹 𝑖 ⟧ 𝐿
∧
= 𝑖
. The function
dep t h
yields t he synt actic dep th of a literal class from its label; i.e., it counts
t he number of syntacticall y visible enclosing classes. It is easy to see t hat
t his number is in v ariant during ev aluation (o t her wise, special v ariables
might be in v alidated). Each node is only v alid if it has t he correct amount
of enclosing classes. A node t hat fulfills t his requirement is cont ext-corr ect :
Definition
(Context-Correctness of N odes)
.
The set of cont ext-corr ect nodes is
inductiv el y defined. A node
𝜈
is cont ext-corr ect, writt en
𝜈 cor rect
, if it is in t hat
set. In particular , a node wit h an emp ty en vir onment is onl y conte xt-correct, if t he
dep t h of its liter al class is zer o.
⟦𝑖⟧ dept h
∧
= | 𝜋 | ∀𝜋 𝑖 ∈ 𝜋 ∶ 𝜋 𝑖 cor rect
⦃𝐹 𝑖 in 𝜋⦄ cor rect
This res triction is reasonable, but not sufficient. The set of context correct
nodes is not finite, because a class might “point back t o itself”. Consider
tw o distinct liter al classes
𝐹 1
and
𝐹 2
wit h
⟦1⟧ dept h
∧
= 0
and
⟦2⟧ dept h
∧
= 1
, t hen
⦃𝐹 2 in ⦃𝐹 1 in ♦ ⦄⦄ is context-correct. But so is
⦃𝐹 2 in ⦃𝐹 2 in ⦃𝐹 1 in ♦ ⦄⦄⦄
⦃𝐹 2 in ⦃𝐹 2 in ⦃𝐹 2 in ⦃𝐹 1 in ♦ ⦄⦄⦄⦄
⦃𝐹 2 in ⦃𝐹 2 in ⦃𝐹 2 in ⦃𝐹 2 in ⦃𝐹 1 in ♦ ⦄⦄⦄⦄⦄
and so on. Intuitiv ely , hierarchies with t he repeated occur ance of t he same
liter al class are problematic. Such structures, ho w ev er , do not ha v e a mean-
ingful application in Modelica (and hence no t in Mcl). It is t hus necessar y
to find a criterion t o rule out such str ange loops.

148 4 F rom Modelica via Mcl to OCaml
The directed g r aphs used in Section 2.3.4 are obtained b y representing
liter al classes as labeled v ertices and adding an edg e labeled wit h
𝑛
to the
𝑛
-t h parent of a node. This notation can be f or malized as a directed multi-
g r aph
(𝑉 , 𝐸)
wit h v ertices
𝑉 ⊆ 𝕃
represented b y t he labels of literal classes
and edg es as triples of one outgoing and one incoming v ertex tog et her wit h
a natural number
𝐸 ⊆ 𝕃 × 𝕃 × ℕ
. The identity of an edge is defined b y its
source, its des tination, and its number . The usual ter ms from (multi) g raph
t heory (reachability , cy cles, etc.) appl y .
Definition
(Gr aph R epresentation of N odes and En vironments)
.
The (di-
r ected multi) gr aph of a node is t he v ert ex labeled wit h t he lit eral class of t he node
linked t o t he gr aphs of all par ents by or der ed edg es. The graph of an en vir onment is
t he union of the gr aphs of eac h node (wher e t he union of gr aphs is t he gr aph formed
by t he union of t heir components).
⟦⦃𝐹 𝑢 in 𝜋⦄⟧ 𝑔𝑟
∧
= ({ 𝑢 } ∪ 𝑉 , 𝑃 ∪ 𝐸)
where 𝑃 ∧
= { (𝑢, 𝑝 𝑖 , 𝑖) | 𝑝 𝑖
∧
= ⟦𝜋 𝑖 ⟧ 𝐿 }
(𝑉 , 𝐸) ∧
= 􏾌
𝑖∈1… | 𝜋 | ⟦𝜋 𝑖 ⟧ 𝑔𝑟
The set of possible results of t his tr ansf ormation is finite if t here is a
finite set of labels
𝕃
and an upper bound for the number of all edg es. Both
conditions are triviall y fulfilled b y graphs created from context-correct
nodes since each node is in itself a finite structure, t he syntactic dep th of
each node is limited b y t he syntactic structure of t he source program and
each source prog r am is labeled b y a finite set of labels.
Context-correctness extends natur all y to t he g r aph representation:
Definition
(Context-Correctness of Gr aphs)
.
A multigr aph is conte xt-corr ect
if all its nodes ha v e one outgoing edg e f or eac h synt acticall y enclosing class of t he
lit er al class labeled wit h t hat node and all its edges link one node t o a corr esponding
enclosing class.
(𝑉 , 𝐸) cor rect iff ∀𝑝, 𝑖 ∶ (𝑝 ∈ 𝑉 ∧ 1 ≤ 𝑖 ≤ ⟦𝑝⟧ dep t h ⟺ ∃𝑞 s.t. (𝑝, 𝑞, 𝑖) ∈ 𝐸)
Context-correctness allo ws recons tructing a node from wit hin a g r aph.
This in v ersion is in particular possible due to t he one to one relationship be-
tw een t he outgoing edges of each v ertex and t he order of the corresponding
enclosing classes.

4.5 Ev aluating Hier archical T er ms 149
Lemma 4
(R econs truction of N odes in a Gr aph)
.
If a cont ext-corr ect, r oo t ed
multigr aph is the gr aph repr esent ation of a cont ext-corr ect node, t hen t his r epr esen-
t ation can be positivel y r econstruct ed:
Let t he injective function 𝑛𝑑 be defined as follow s,
⟦𝐺⟧ 𝑛𝑑
∧
= ⦃𝐹 𝑟 in 𝜋⦄
where 𝑟 ∧
= t he roo t of 𝐺
{ (𝑟, 𝑝 𝑖 , 𝑖) } ∧
= all outgoing edg es from 𝑟 in 𝐺
𝐺 𝑖
∧
= t he subtr ee wit h roo t 𝑝 𝑖
𝜋 𝑖
∧
= ⟦𝐺 𝑖 ⟧ 𝑛𝑑
t hen
𝑛𝑑
is t he in ver se of
𝑔𝑟
, for nodes wit h a cont ext-corr ect, r oo t ed gr aph r epr esen-
t ation: ⟦𝜈⟧ 𝑔𝑟 r oo t ed ∧ ⟦𝜈⟧ 𝑔𝑟 cor rect ⟹ ⟦⟦𝜈⟧ 𝑔𝑟 ⟧ 𝑛𝑑
∧
= 𝜈
Pr oof.
By s tructural induction o v er nodes
𝜈
, using t he contextual correctness
and t he fact that 𝐺 does not contain an y cy cles.
A node can alw a ys be represented as a g r aph. Context-correctness of
t his representation can be checked b y looking at all nodes, and it is sim ple
to decide whether a giv en multig r aph is roo ted.
4.5.2 Ev aluation of Hier ar c hical T erms
W it h t his prerequisites, w e can res trict t he lookup to an admissible in put.
This res triction builds upon t he interpretation of nodes (and en vironments)
as multig r aphs:
Definition
(A dmissible Lookups)
.
A cont ext-corr ect node is admissible, if and
onl y if its graph r epr esentation is a cont ext-corr ect, roo t ed multigraph. An en vi-
r onment
𝜋
is admissible, if and onl y if all cont ained nodes ar e admissible and t he
look up of a label
𝐿
in node
𝜈
wit h an en vir onment
𝜋
is admissible if and onl y if t he
en vironment is admissible:
(i) 𝜈 admissible ⟺ 𝜈 cor rect ∧ ⟦𝜈⟧ 𝑔𝑟 cor rect ∧ ⟦𝜈⟧ 𝑔𝑟 rooted
(ii) 𝜋 admissible ⟺ ∀𝑖 ∈ 1 … | 𝜋 | ∶ 𝜋 𝑖 admissible
(iii) ⟨𝜈 ∷ 𝜋 ⋅ 𝐿⟩ admissible ⟺ 𝜈 admissible ∧ 𝜋 admissible
(iv) ⟨𝜈 ∷ 𝜋 ⋅ 𝑙⟩ admissible ⟺ 𝜈 admissible ∧ 𝜋 admissible

150 4 F rom Modelica via Mcl to OCaml
F or a finite set of labels of literal classes
𝕃
, t he set of admissible nodes
is finite because t here is onl y a finite set of multig raphs f or a finite set of
v ertices. The same property holds for en vironments (and lookups) under
t he pro vision t hat t here are no infinitel y larg e en vironments, which is a
corollar y of the definition of dept h .
Corollar y 8
(F initeness of A dmissible Lookups)
.
F or any giv en finite labeling
of lit er al classes, and a finite maximal dep t h of classes t he set of admissible lookups
is finit e.
Pr oof.
The set of admissible nodes is finite due to the injectiv e mapping
of nodes to a context-correct, rooted multigraph o v er t he (finite) labeled
v ertices. A dmissible (finite) en vironments and lookups are products of
finite sets.
F or t his final set of nodes, w e can define t he ev aluation of hierarchical
terms in a ter minating, to tal function ℋ :
⟦𝐺, 𝜈⟧ ℋ
∧
= 𝜈
⟦𝐺, up( 𝑖 ) ⟧ ℋ
∧
=
⟦𝐺, ⦃𝐹⦄⟧ ℋ
∧
= ⦃𝐹 in ♦ ⦄
⟦𝐺, ℎ with 𝐹 2 ⟧ ℋ
∧
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⦃𝐹 3 in 𝜋⦄ if ⟦𝐺, ℎ⟧ ℋ
∧
= ⦃𝐹 1 in 𝜋⦄
dom (𝐹 2 ) ⊆ dom (𝐹 1 )
⟦𝐹 3 ⟧ Π
∧
= ⟦𝐹 1 ⟧ Π ⊕ ⟦𝐹 2 ⟧ Π
o t her wise
⟦𝐺, ℎ 1 .super( ℎ 2 ). 𝐿⟧ ℋ
∧
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⟦𝐺 ′ , 𝑔⟧ ℋ if ⟦𝐺, ℎ 1 ⟧ ℋ
∧
= 𝜈 1
⟦𝐺, ℎ 2 ⟧ ℋ
∧
= ⦃𝐹 in 𝜋⦄
𝐺 ⊢ 𝜈 1 ∷ 𝜋
𝐿 ↦ ℎ 𝐿 ∈ ⟦𝐹⟧ Π
⟨𝜈 1 ∷ 𝜋 ⋅ 𝐿⟩ admissible
⟨𝜈 1 ∷ 𝜋 ⋅ 𝐿⟩ ∉ 𝐺
𝐺 ′ ∧
= 𝐺 ∪ { ⟨𝜈 1 ∷ 𝜋 ⋅ 𝐿⟩ }
𝑔 ∧
= ⟦𝜈 1 , 𝜋, ℎ 𝐿 ⟧ Φ
o t her wise

4.6 Resol ving Ref erences 151
This function f ollo ws the definition of
⇓ ℎ
. It is correct b y construction, i.e.,
⟦𝐺, ℎ⟧ ℋ
∧
=𝜈⟹ℎ⇓ h 𝜈
and consequentl y
𝜈 ≡ ℎ ℎ
. The “memor y”
𝐺
contains
a set of admissible lookups and ensures termination.
In each recursiv e application, eit her
𝐺
g ro ws, or the function tra v erses
a liter al subter m. W e can hence or der t he arguments of applications of
ℋ
wit h a combination of t he subset and subterm relations. The recursiv e steps
of
ℋ
are s trictl y mono tone on that order . Hence each chain of recursiv e
applications ev entuall y reaches t he larg es t possible his tory , i.e., the set t hat
contains all admissable lookups, or a primitiv e ter m. Consequently , t he
function alw a ys terminates.
4 . 6 resol vin g references
The t hir d s tag e uses t he ev aluation of hierarchical terms to resol v e all refer -
ences wit h their definitions. While t his is a str aightf or w ard proposition, it is
necessar y to maintain the order of e v aluation to no t introduce subtle errors
in t he g enerated code. It is also necessar y to maintain possible recursiv e
dependencies betw een fields of classes. This section deals wit h t hese issues.
4.6.1 Looking up Ref er ences
The definition of
ℋ
immediatel y yields an algorit hm for the lookup of
ref erences,
ℒ
. A lookup is successful when both t he subclass and its base
class can be ev aluated successfully , t he resulting en vironment is admissible,
and t he tip class contains a matching element.
ℒ
retur ns the ev aluated
ref erence mapped to the folded result term.
⟦ℎ 1 .super( ℎ 2 ). 𝑙⟧ ℒ
∧
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
rv ↦ 𝑡 if ⟦∅, ℎ 1 ⟧ ℋ
∧
= 𝜈 1
⟦∅, ℎ 2 ⟧ ℋ
∧
= 𝜈 2
rv ∧
= 𝜈 1 .super( 𝜈 2 ). 𝑙
rv 𝜙
− → 𝑡
o t her wise
This function can be im plemented directl y wit h t he help of
ℋ
: F irs t,
ev aluate t he base and subclass, t hen look up t he definition and finall y fold
t he resulting field. If one of the steps f ails, retur n an error . Since
ℋ
is
correct b y construction, w e can conclude that t he result of
ℒ
is v alid, i.e.,
⟦𝑟⟧ ℒ
∧
= ( rv ↦ 𝑡) ⟹ 𝑟 ≡ rv ∧ rv 𝜙
− → 𝑡 .

152 4 F rom Modelica via Mcl to OCaml
4.6.2 Exhaus tiv e Look up
The t hir d stag e,
γ
, shall replace all references in a ter m. This requirement
includes ref erences t hat occur in t he result of a successful lookup (and in
t he result of looking up t hese ref erences and so on). Such an exhaus tive
sear c h is achiev ed b y repeated applications of t he one-step search
𝒢
to an
intermediate result set 𝑅 :
⟦ ⟧ 𝒢
∧
=
⟦𝑅⟧ 𝒢
∧
=
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
if [  ↦ 𝑟]𝑠 ∈ img (𝑅)
such t hat ⟦𝑟⟧ ℒ
∧
=
𝑅 ∪ { ⟦𝑟⟧ ℒ | [  ↦ 𝑟]𝑠 ∈ img (𝑅) } o t her wise
𝒢
looks up all references in the image of the result set
𝑅
. If one of t hese
oper ations yields an error , t he o v erall result is an error . Ot her wise, t he
results are merg ed wit h t he old result set. The exhaustiv e search ter minates,
when
𝒢
reaches a fix ed point, i.e.,
⟦𝑅⟧ 𝒢
∧
= 𝑅
. It alw a ys reaches such a point
after a finite amount of repeated applications, because t he set of admissible
lookups is finite:
Lemma 5
(F ix ed P oint of
𝒢
)
.
The ascending Kleene c hain of
𝒢
has a leas t fixed
point.
Pr oof.
The partial functions (and error mar ker) obtained b y
𝒢
f orm a com-
plete partial or der under t he subset relation, i.e. for
𝑅 1,2
∧
≠ 𝑅 1 ≤ 𝑅 2 ⟺
𝑅 1 ⊆ 𝑅 2
wit h

as top element, i.e.
𝑅 ≤
, because t he set of admissible
en vironments (t he domain of each
𝑅
) is finite and adding a top element t o
a cpo yields a cpo.
F unction
𝒢
is also Sco tt-continuous: Let
𝐾
be directed subset of t he cpo,
t hen either ∈ 𝐾 and consequentl y eit her :
⊔𝐾 ∧
= ∧
= ⊔( 𝒢 ∗ 𝐾 )
or : 𝐾 ∧
= { 𝑅 1 … 𝑅 𝑛 } and ⊔ 𝐾 ∧
= 􏾌 𝑅 𝑖
In t he latter case, if there is at least one 𝑅 𝑖 with ⟦𝑅 𝑖 ⟧ 𝒢
∧
= , t hen also
⟦ 􏾌 𝑅 𝑖 ⟧ 𝒢
∧
= ∧
= ⊔( 𝒢 ∗ { 𝑅 1 … 𝑅 𝑖 … 𝑅 𝑛 })
If no such 𝑅 𝑖 exis ts,
⟦⊔𝐾 ⟧ 𝒢
∧
= ⟦ 􏾌 𝑅 𝑖 ⟧ 𝒢
∧
= 􏾌 ⟦𝑅 𝑖 ⟧ 𝒢
∧
= ⊔( 𝒢 ∗ 𝐾 )

4.6 Resol ving Ref erences 153
𝒢
is also inflationar y . This guarantees the existence of the conditional
fix ed point when starting from a set
𝑅 0
(see [see P epper and Hofstedt, 2006,
chap ter 10]).
4.6.3 Look up Closur e
The definition of a ref erence might (after sev er al steps of lookup) contain
t he ref erence itself. This implicit recursion has t o be transf or med into an
explicit application of t he fix ed-point operat or of t he core languag e. F or
t his tr ansf ormation, all definitions ha v e to be reg arded as functions, since
Mcl does not allo w f or an y other recursiv e v alues. This is easil y achiev ed
b y lif ting definitions to “thunks”, i.e. abstr actions
λ 𝑥 . 𝑡
where
𝑥 ∉ ⟦𝑡⟧ fv
. A
t hunk can be f or ced to e v aluate t hrough t he application of an arbitrary v alue
(here w e are using t he emp ty record
{}
). The recursiv e g roup can t hen
be f ormed b y lifting all definitions, naming t hem uniquel y and replacing
each ref erence wit h a term t hat forces the corresponding name (i.e., the
name of t he definition of t he v alue reference it e v aluates to). This process
of ev aluation, lif ting and transf or mation is called t he closur e of t he search
result.
Definition
(Lookup Closure)
.
The closur e of a lookup
𝑅
, defined by function
𝒞
,
r eplaces each r ef er ence with t he name of its corr esponding definition. All definitions
ar e lif t ed t o t hunks. W e assume t hat the look up r esult is arbitr aril y or der ed.
⟦{ rv 𝑗 ↦ 𝑡 𝑗 | 𝑗 ∈ 1 … 𝑚 }⟧ 𝒞
∧
= ℱ
where 𝜎 ∧
= {  𝑖 ↦ 𝑟 𝑖 | 𝑖 ∈ 1 … 𝑛 }
[𝜎]𝑠 𝑖
∧
= 𝑡 𝑖
𝑀 ∧
= { 𝑖 ↦ 𝑗 | ⟦𝑟 𝑖 ⟧ ℒ
∧
= ( rv 𝑗 ↦ 𝑡 𝑗 ) }
𝜌 ∧
= {  𝑖 ↦ 𝑥 𝑗 {} | 𝑖 ↦ 𝑗 ∈ 𝑀 }
⟦ ℱ ⟧ Π
∧
= { 𝑥 𝑖 ↦ λ 𝑦 𝑖 . [𝜌]𝑠 𝑖 | 𝑖 ∈ 1 … 𝑛 }
∀𝑖 ∈ 1 … 𝑛 ∶ 𝑥 𝑖 , 𝑦 𝑖 fresh
The closure of t he result of an exhaus tiv e lookup
𝑅
cons tructs a flattening
of its references. The recursiv e g roup contains one uniquel y named term
𝑥 𝑖 ↦ λ 𝑦 𝑖 . [𝜌]𝑠 𝑖
f or each found definition
𝑡 𝑖
in
𝑅
. The ordering is maintained
(i.e., 𝑥 𝑖 maps to the closure of t he 𝑖 -t h definition).
The tr ansformation of each definition depends on tw o substitutions and
an intermediate ter m: F irst, all ref erences in
𝑡 𝑖
are extr acted to a subs titution

154 4 F rom Modelica via Mcl to OCaml
𝜎
. The inter mediate term
𝑠
does no t contain an y furt her ref erences. A place-
holder
 𝑗
takes the place of t he
𝑗
-t h ref erence
𝑟 𝑗
. The second substitution,
𝜌
is
cons tructed as follo ws: Lookup
ℒ
determines t he corresponding definition
𝑡 𝑗
f or each placeholder . The placeholder is t hen mapped to the application
of t he name of the definition
𝑥 𝑗
to the em p ty record. This application forces
t he corresponding thunk to ev aluate.
Corollar y 9
(Equiv alence of
𝒞
)
.
The application of eac h function in t he fixed point
cr eated fr om t he closur e of an exhaus tiv e lookup is eq uivalent t o t he corr esponding
r efer ence in t he look up:
When 𝑅 is an exhaus tive look up wit h arbitr arily or der ed r esults, i.e.:
⟦𝑅⟧ 𝒢
∧
= 𝑅 ∧
= { rv 𝑖 ↦ 𝑡 𝑅
𝑖 | 𝑖 ∈ 1 … 𝑛 }
And t he closur e of 𝑅 forms a fixed-point :
⟦𝑅⟧ 𝒞
∧
= ⟦ ℱ
𝑅 ⟧ Π
∧
= { 𝑥 𝑖 ↦ 𝑡 𝐶
𝑖 | 𝑖 ∈ 1 … 𝑛 }
Then t he application of any of t he r ecursiv e functions in
ℱ
𝑅
t o an em pty r ecor d is
eq uivalent t o t he corresponding r ef er ence value:
∀𝑖 ∈ 1 … 𝑛 ∶ rv 𝑖 ≡ ( fix 𝑥 𝑖 in ℱ
𝑅 ) {}
Pr oof.
By application of rule Eq-RefFla t. By correctness of
ℒ
, it is easy to
see t hat
⟦𝑅⟧ 𝒞
flattens
𝑅
. N ote, that
ℒ
can nev er fail inside
𝒞
when
𝑅
is a
fix ed point.
The tr ansformation directl y implements this correspondence betw een
hier archical and proper terms. Each reference in a tr ansf ormed ter m is
replaced b y t he application of t he corresponding recursiv e function in its
lookup closure. A gain, po tential errors are forw ar ded.
Definition
(Thir d S tag e T ransf ormation)
.
The t hir d s tag e r eplaces all r efer ences
wit h their definition. It iter ates ov er modules and r eor ders t op-level classes. The
r eor dering finds a t op-lev el t erm, wit h all classes mov ed to t he right-hand side. (See
Appendix C.4 f or t he full definition of the tr ansf ormation.)
⟦𝑇 ∷ 𝑋 = 𝑀⟧ γ
∧
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
𝑇 𝛾 ∷ 𝑋 = 𝑀 𝛾 if ∄𝑆 ∶ 𝑆 ≺ 𝑇 𝑇 ∷ 𝑋 = 𝑀
⟦𝑇 ⟧ γ
∧
= 𝑇 𝛾 ∧
≠
⟦𝑀⟧ γ
∧
= 𝑀 𝛾 ∧
≠
⟦𝑆⟧ γ if 𝑆 ≺ 𝑇 𝑇 ∷ 𝑋 = 𝑀
o t her wise

4.6 Resol ving Ref erences 155
⟦[  ↦ 𝑟]𝑠⟧ γ
∧
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⟦[  ↦ 𝑡 𝑟 ]𝑠⟧ γ if ⟦𝑟⟧ ℒ
∧
= rv 1 ↦ 𝑡 1
𝑅 ∧
= ⟦𝑅⟧ 𝒢
𝑅 ∧
= ⟦{ rv 1 ↦ 𝑡 1 }⟧ 𝒢 (𝑛)
⟦𝑅⟧ 𝒞
∧
= ⟦ ℱ
𝑅 ⟧ Π
𝑡 𝑟
∧
= ( fix 𝑥 1 in ℱ
𝑅 ) {}
o t her wise
The r eor dering r elation ≺ 𝑇 is defined inductivel y as follow s:
𝑆 ≺ 𝑇 𝑇
𝑇 ∷ 𝑋 = 𝑀 ≺ 𝑇 𝑆 ∷ 𝑋 = 𝑀 𝑆 ≺ 𝑇 𝑇
𝑇 ∷ 𝑋 = 𝐶 ≺ 𝑇 𝑆 ∷ 𝑋 = 𝐶
𝑌 ∉ ⟦𝐶⟧ FL V
(𝑇 ∷ 𝑌 = [{ 𝑋 ↦ 𝐶 }]𝑀 ∷ 𝑋 = 𝐶) ≺ 𝑇 (𝑇 ∷ 𝑋 = 𝐶 ∷ 𝑌 = 𝑀)
T erms t hat do not contain ref erences (i.e., ter ms t hat are already in t he
f orm of t he second stag e) are retur ned unmodified. A ter m t hat contains
a ref erence is transf or med as f ollo ws. F irst, the reference is look ed up via
ℒ
. The result
rv 1
used as a starting point for the exhaustiv e search. A fix ed
point is reached after
𝑛
applications of
𝒢
. The exhaustiv e result is closed
and tur ned into a group of recursiv e functions
ℱ
𝑅
. The original reference
𝑟
is t hen replaced b y t he application of the t hunk
𝑥 1
t hat corresponds to its
definition in
𝑅
. The replacement is free of references. The tr ansformation
t hen proceeds with t he rest of the ter m. If one of t he abo v e steps does no t
succeed, t he tr ansf ormation yields an error .
Modules are tr ansformed similar l y to
α
, but top-lev el ter ms require
special treatment f or practical reasons. Whenev er
ℋ
encounters a top-le v el
v ariable, it yields an error . Hence t hese names need to be resolv ed before
γ
can be applied. This ordering is achie v ed b y loading classes into the
modules bef ore t he actual transf or mation. The binar y relation
≺ 𝑇
mo v es
all classes to the right-hand side of t he top-le v el ter m. T w o top-le v el terms
are related when t he smaller one is obtained b y mo ving one top-lev el class
pas t one top-le v el module to its right. The minimal (i.e., sorted) ter m is
t he one with all classes mo v ed to the right and modules to the lef t. The
tr ansitiv e closure of
≺ 𝑇
is s trongl y confluent. Therefore
γ
terminates and is
a w ell-defined function.
4.6.4 Exam ple
The t hir d s tag e does not affect the im plementation of a class (as it is already
free of ref erences) nor its skelet on (since t he s tag e ignores classes). Instead,

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Why institutions use Plag.ai for originality review, entry 85

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