Re e sibiliza ion o F ee Will:
Re e sible Local Bounda y Condi ions, KL/B egman Choice
Ope a o s,
and Re e sible Ledge s in Re e sible Cellula Au oma a
Anonymous Au ho
Ve sion: 1.10
Abs ac
We p opose a me hodology o igo ously implemen ing “ ee will” as e e sible lo-
cal bounda y condi ions (RLBC) on ini e domains o e e sible cellula au oma a
(RCA), and placing bo h he in o ma ion ideli y cha ac e iza ion o “why choose his
no ha ” and he choice ope a o in he same e e sible ledge . The co e app oach is:
design each “decision” as a bijec i e upda e on he bounda y s ip, ca ying e e sible e -
idence/ andomness egis e s when necessa y o p ese e la ening and sampling pa hs,
so ha i composes as an o e all bijec ion wi h one-s ep e olu ion o in e nal RCA; he
decision i sel gi en by minimal-KL/I-p ojec ion so selec ion, degene a ing o ha d
selec ion (a gmax) in he Γ-limi , wi h all in o ma ion o e head p ecisely accoun ed o in
Benne e e sible embedding and e e sibly eco e able. Re e sibili y and decidabili y
a e gua an eed by CHL ep esen a ion heo em, Ga den-o -Eden heo em, block/pa i ion
e e sible cellula au oma on s uc u e, and linea -bounda y ma ix c i e ia; he undecid-
abili y o gene al neighbo hoods in wo and highe dimensions is ci cum en ed by selec ing
block pe mu a ion o linea e e sible block e i iable subclasses. This pape also
connec s he bounda y-decision pa adigm wi h he uni ied windowed eadou –phase–
ela i e densi y o s a es–g oup delay calib a ion, p o iding end- o-end e o and s a-
bili y s a emen s.
1 No a ion & Axioms / Con en ions
1. RCA and con igu a ion space. Fini e alphabe A,d≥1, ull space X=AZd. One-
ime-s ep global upda e F:X→Xcon inuous unde Can o opology and commu ing
wi h shi s i and only i i is gi en by ini e- ange local ule (Cu is–Hedlund–Lyndon,
CHL). I Fis bijec i e, called RCA; on ull la ice Zd, Ga den-o -Eden heo em gi es
su jec i e ⇔p e-injec i e (no wins) ⇔no o phans (no p edecesso pa e ns),
and injec i e ⇒su jec i e; hus bijec i e i and only i bo h injec i e and su jec i e
[?].
2. Block/pa i ion e e sibili y and linea e e sibili y. Adop Ma golus pa i ion: i
in a-block ans o ma ion is pe mu a ion, hen global is e e sible; e e se e o-
lu ion implemen ed by in e se pe mu a ion and e e se-o de pa i ion. Re e sibili y
o linea cellula au oma a unde a ious bounda y condi ions (including “in e media e
bounda y”) educes o in e ibili y o ule ma ix and K onecke decomposi ion c i e ion
[?].
3. Gene al undecidabili y and e i iable subclasses. Re e sibili y p oblem o gene al-
neighbo hood CA in wo and highe dimensions is undecidable (Ka i); he e o e his
1
pape ocuses on block pe mu a ion and linea -bounda y ma ix wo e i iable
subclasses [?].
4. Calib a ion iden i y (WSIG–EBOC con en ion). On windowed sca e ing calib a-
ion, adop he iden i y
ρ el(E) = 1
2πi
d
dElog de S(E) = 1
2π Q(E) = 1
2πφ′(E),
whe e Q(E) = −i S†(E)∂ES(E), de S(E) = eiφ(E). Bi man–K e˘ın o mula connec s sca -
e ing phase and spec al shi unc ion, se ing as common mo he scale o ene gy/ ime
eadou and “submission” [?].
5. In o ma ion geome y and choice. “Minimal-KL unde linea consis ency cons ain s
gi es exponen ial amily/so max and sa is ies Py hago ean iden i y and Fenchel duali y”;
“TV–KL” Pinske bound used o empe a u e-pe u ba ion-ji e s abili y es ima ion [?].
6. Re e sible ledge . Only in o ma ion e asu e dissipa es (Landaue lowe bound); logi-
cal e e sibili y can achie e a bi a ily low dissipa ion in he limi ; andom sampling
and la ening e idence accoun ed o h ough e e sible egis e s (Benne ) [?].
2 Model: Re e sible Local Bounda y Condi ions (RLBC) on
Fini e Domain
Le Λ ⋐Zdbe a connec ed ini e domain, ake bounda y s ip ∂Λ o su icien hickness. One-
s ep in e nal e olu ion gi en by some RCA FΛ. De ine bounda y laye ope a o
B:A∂Λ× E −→ A∂Λ× E,
whe e Eis e e sible auxilia y egis e (e idence/ andomness/ la ening labels, e c.); his
pape de aul s Ealphabe ini e, and localizes con igu a ions by bounda y- ouching blocks o
ensu e o e all ini e-alphabe CA amewo k (applicable o CHL/GOE). One- ound e olu ion
de ined as
UFΛ◦B(bounda y i s , hen in e nal).
Call B e e sible local bounda y condi ion (RLBC) i sa is ying:
(R1) Locali y: Bcan ead ini e ou e shell beyond ∂Λ (comp ess obse a ions in o E
i necessa y), bu only w i es ∂Λ and E, no ouching Λ◦;
(R2) Bijec i i y: (b, ϵ)7→ (b′, ϵ′) is a pe mu a ion;
(R3) E idence accoun ing: All andomness, la ening indices, selec ed ac ion index
a⋆(o minimal su icien e idence o econs uc a⋆)and obse a ion e idence used
o decision a e w i en in o E o in e sion eco e y (Benne embedding) [?].
In pa i ion implemen a ion, le all “bounda y- ouching blocks” each unde go in a-block
pe mu a ion; by pa allel di ec p oduc o block pe mu a ions = pe mu a ion,Bau-
oma ically sa is ies (R2). Re e se-o de Ma golus pa i ion and in e se pe mu a ions gi e B−1
[?].
2
3 Cascade Re e sibili y and Decidabili y
P oposi ion 3.1 (Fini e Domain Cascade as Bijec ion).Le global Fbe RCA, and ake i s
local implemen a ion FΛon Λ. I Bis RLBC, and Band FΛadop pa i ion wo-phase
(e.g., Ma golus) implemen a ion such ha bounda y- ouching blocks in each phase
a e pai wise disjoin , and execu e synch onously acco ding o “bounda y phase →
in e nal phase” wo-phase schedule, hen
U=FΛ◦B
is a ini e-domain bijec ion on s a e (x|Λ, b, ϵ), wi h U−1=B−1◦F−1
Λ. When Λand Ba e
applied synch onously o he ull la ice acco ding o he abo e pa i ion iling and wo-phase
schedule, he esul ing global e olu ion is RCA.
P oo . Uis bijec i e wi h U−1=B−1◦F−1
Λ. When Λ and Ba e applied synch onously o ull
la ice ia pa i ion iling, by CHL he co esponding global map is con inuous and commu es
wi h shi s, and i s in e se likewise [?].
P oposi ion 3.2 (Decidable Subclasses o Bounda y-In e nal Uni y).In he ollowing wo
implemen a ion ypes, e e sibili y o Uis decidable and in e sion cons uc ible:
Block/pa i ion RCA: I and only i each block ule is a pe mu a ion [?].
Linea -in e media e bounda y: Re e sibili y educes o in e ibili y o ule ma ix
and i s K onecke decomposi ion; e icien algo i hms a ailable o mul i-dimensional and
in e media e bounda ies [?].
Rema k 3.3 (Gene al Undecidabili y).Re e sibili y o gene al-neighbo hood CA in wo and
highe dimensions is undecidable (Ka i), hence his pape ’s RLBC selec s om e i iable sub-
classes [?].
4 Choice Ope a o : KL/B egman Fideli y (So →Ha d)
Le bounda y easible ac ion se A(b). Gi en baseline q(· | b) and momen cons ain s, in oduce
ea u e map ϕ:A(b)→Rm.
Assump ion (Feasibili y): b⋆∈con {ϕ(a) : a∈ A(b)}.
De ini ion 4.1 (So Selec ion / I-P ojec ion).
p⋆(· | b)∈a g min
p∈∆(A(b)) nDKL(p∥q) : X
a
p(a|b)ϕ(a) = b⋆o,
whose KKT condi ion gi es
p⋆(a|b)∝q(a|b) exp ⟨λ, ϕ(a)⟩,
i.e., exponen ial amily/so max; sa is ies in o ma ion geome y’s Py hago ean iden i y and Fenchel-
Legend e duali y [?].
P oposi ion 4.2 (Robus ness and TV-KL Bound).When empe a u e/ egula iza ion pa ame-
e change in oduces KL e o δ, o al a ia ion de ia ion con olled by Pinske bound
∥p1−p2∥TV ≤q1
2DKL(p1∥p2),
se ing as “ empe a u e–ji e ” uppe bound; B e agnolle–Hube bound a ailable o e inemen
when necessa y [?].
3
Theo em 4.3 (Γ-Limi : So →Ha d, ia En opy/KL Regula iza ion).Le q∈∆(A), cos
c:A → R. Fo τ > 0, le
pτ∈a g min
p∈∆(A)⟨c, p⟩+τDKL(p∥q)
hen pτ(a)∝q(a) exp(−ca/τ), and as τ↓0,pτ⇒δa⋆whe e a⋆∈a g minaca; con e gence
unique i minimize unique. (Wi h linea momen cons ain ) I adding Pap(a)ϕ(a) = b⋆
wi h easibili y (b⋆∈con ϕ(A)), hen he e exis s dual a iable λ(τ)such ha
pτ(a)∝q(a) exp ⟨λ(τ),ϕ(a)⟩−ca
τ.
I minimize in easible se is unique and a poin mass (exis s a⋆∈ A wi h ϕ(a⋆) = b⋆), hen
as τ↓0,pτ⇒δa⋆[?].
P oo ske ch. As τ↓0, en opy/KL e m weigh dec eases, linea objec i e domina es; in expo-
nen ial amily exp ession −ca/τ exponen di e ence ampli ies, concen a ing p obabili y mass on
minimize ; Γ-con e gence and la ge de ia ion p inciple gi e igo ous limi ; momen cons ain
case ia Lag ange mul iplie scale analysis yields same conclusion. Uniqueness and selec ion
s abili y gi en by P oposi ion ??.
5 Re e sible Implemen a ion: Benne Embedding and “Re-
e sible Sampling”
Theo em 5.1 (Re e sible Decide ).Any so /ha d selec ion on ini e ac ion se admi s a e-
e sible ex ension w i ing andomness, la ening e idence, and sampling pa h in o
E:
(b, ϵ)7→ (b′= Sel(b;ϵ), ϵ′),
making his ex ension a pe mu a ion on (b, ϵ); in e sion eco e s sampling ee and la ening
o de om ϵ′and e ases e idence, hence no i e e sible dissipa ion.
P oo . By Benne logical e e sibili y: as long as in e media e in o ma ion is no e ased, can
e e se e ase. Implemen sampling as con olled pe mu a ion o p e ix- ee/alias me hod:
Knu h–Yao DDG- ee gi es en opy-op imal bina y sampling amewo k; Walke /Vose alias
me hod gi es equi alen disc e e sampling s uc u e in cons an amo ized ime. W i ing
ee/ able indices and coin- lip sequences in o Eyields he esul [?].
Rema k 5.2 (Implemen a ion No e (Ve i iable Ope a o Family)).On Ma golus pa i ion’s
bounda y- ouching blocks, apply in pa allel ia “selec ion esul →in a-block pe mu a ion”
o ob ain B; i s su iciency and necessi y as pe mu a ion and in e sion cons uc ion di ec ly
ollow om block e e sibili y [?].
6 Fo mal De ini ions and Main Theo ems
De ini ion 6.1 (RLBC).In one- ound e olu ion, bounda y laye upda e
B(b, ϵ) = b′, ϵ′
sa is ies (R1)–(R3), and in block implemen a ion is di ec p oduc o disjoin in one phase
bounda y- ouching block pe mu a ions.
4
Theo em 6.2 (RLBC ⊗Local RCA ⇒Fini e Domain Bijec ion).I FΛis RCA and Bis
RLBC, hen U=FΛ◦Bis ini e-domain bijec ion; U−1=B−1◦F−1
Λ. I applying isomo phic B
and co esponding FΛ ia pa i ion wo-phase schedule o e e y ansla e copy o he ull
la ice, wi h bounda y- ouching blocks pai wise disjoin in any phase, hen ob ain global RCA,
wi h in e se gi en by e e se phase and in e se pe mu a ion playback [?].
Theo em 6.3 (Choice = I-P ojec ion; So →Ha d).Le Pap(a)ϕ(a) = b⋆be momen con-
s ain wi h easibili y (b⋆∈con ϕ(A)), hen
(i) p⋆= a g min DKL(p∥q)unique and exponen ial amily;
(ii) Fo egula ized amily
pτ∈a g min
p∈∆(A)n⟨c, p⟩+τDKL(p∥q) : X
a
p(a)ϕ(a) = b⋆o,
he e exis s dual a iable λ(τ); and i b⋆is in ela i e in e io o con ϕ(A)(Sla e
condi ion), hen {λ(τ)}is bounded (has clus e poin ). In gene al easible bu non-in e io
case, λ(τ)may di e ge while he ollowing s ill holds:
pτ(a)∝q(a) exp ⟨λ(τ),ϕ(a)⟩−ca
τ.
I minimize unique and a poin mass (exis s a⋆wi h ϕ(a⋆) = b⋆), hen as τ↓0,
pτ⇒δa⋆;
(iii) Ji e s abili y con olled by Pinske /B e agnolle–Hube bounds [?].
Theo em 6.4 (Re e sible Ledge iza ion).Le Sel be he so /ha d selec ion o Theo em ??.
The e exis s a amily o bounda y block-pe mu a ions {Π(a)
block}a∈A and e e sible egis e upda es
such ha
Bθ=Y
bounda y- ouching blocks
Πblocka⋆
θ
cons i u es RLBC, wi h all sampling/ la ening in o ma ion w i en in o Eand e ased back in
B−1
θ, no Landaue cos [?].
Theo em 6.5 (Linea -Bounda y Re e sibili y C i e ion).In linea CA and “in e media e
bounda y” se ings, e e sibili y o Uis equi alen o in e ibili y o co esponding ule ma ix;
ma ix can be educed-dimension es ed ia K onecke decomposi ion [?].
7 Connec ion wi h Windowed Readou –Phase–Densi y o S a es–
G oup Delay Calib a ion
Take he “e idence agg ega ion/consis ency cons ain ” a bounda ies om a class o windowed
spec al eadou s: in he absolu ely con inuous spec um egion, wi h he uni ied calib a ion
ρ el(E) = 1
2πi
d
dElog de S(E) = 1
2π Q(E) = 1
2πφ′(E),
whe e Q=−i S†∂ES, de S(E) = eiφ(E), exp ess “ eadou ” as in eg al unc ional o phase
de i a i e/ ela i e densi y o s a es/g oup delay ace; Bi man–K e˘ın o mula gi es equi alence
o spec al shi and sca e ing phase, enabling explici binding o “submission/decision” con-
sis ency cons ain Pap(a)ϕ(a) = b⋆ o ene gy-phase ledge . Sensi i i y o bounda y-selec ion
so →ha d ansi ion o “ eadou ji e ” con olled by Pinske - ype inequali ies and linea ized
esponse es ima es [?].
5
8 Pa adigm Cons uc ion: Ma golus-Bounda y Re e sible De-
cide
On wo-dimensional Ma golus pa i ion, le ou e ing all be “bounda y- ouching blocks”. Fo
each bounda y- ouching block, gi en ini e ac ion se A ⊂ S(A2×2) (in a-block pe mu a ion
amily) and ea u e map ϕ:A × ∂Λ→Rm. Acco ding o
§
?? I-p ojec ion, o empe a u e
τ > 0 de ine so selec ion dis ibu ion
p⋆
τ,θ(a|b)∝qθ(a|b) exp ⟨λθ, ϕ(a, b)⟩/τ,
whose ha d limi is
a⋆
θ(b)∈a g max
a∈A ⟨λθ, ϕ(a, b)⟩.
De ine
Bθ=Y
bounda y- ouching blocks
Πblocka⋆
θ(b),
wi h e e sible egis e upda e
(b, ϵ)7→ b′= Πblocka⋆
θ(b)(b), ϵ′=ϵ⊕code a⋆
θ(b),
whe e code(·) is e e sible encoding o ac ion index. So case uses e e sible sampling o d aw
a om p⋆
τ,θ(· | b), w i ing sampling pa h and code(a) in o E o gua an ee eplay; ha d case i s
compu es a⋆
θ(b) based on b, w i es code a⋆
θ(b) hen applies pe mu a ion. In e se p ocess eads
encoding, applies Πblock(a⋆)−1and e ases encoding, hus Bθis pe mu a ion on (b, ϵ). A e
cascading wi h in e nal block RCA, Umain ains e e sibili y; ϵaccoun ing gua an ees in e sion
can eco e all andom/e idence pa hs [?].
9 End- o-End Ve i iable Checklis (Theo y-Only, Expe imen -
F ee)
1. Re e sibili y e i ica ion: Block-pe mu a ion o linea -bounda y ma ix me hod; un-
decidable egion o wo-dimensional gene al neighbo hoods no en e ed in o implemen a-
ion ape u e [?].
2. Choice- ideli y: Sol e I-p ojec ion (o i s con ex dual); so /ha d mu ually accessible
unde empe a u e pa ame e uning; ji e -e o ia Pinske /B e agnolle–Hube bounds
[?].
3. Re e sible ledge : Re e sibiliza ion o Knu h–Yao/DDG- ee o alias me hod; andom-
ness and indices w i en in o E; in e sion e ase-back [?].
4. Calib a ion binding: Via ρ el =1
2πφ′=1
2π Qand Bi man–K e˘ın o mula, embed
“ eadou →cons ain ” in uni ied ene gy/phase ledge [?].
Appendix A: Ga den-o -Eden and RLBC Consis ency
On Euclidean la ice, Ga den-o -Eden heo em gi es local p e-injec i i y⇔global su jec-
i i y. RLBC’s block pe mu a ion implemen a ion makes bounda y laye locally injec i e in
i s ac ion domain; in e nal RCA is also globally bijec i e; hei cascade main ains injec i i y and
su jec i i y, hence o e all emains RCA. This a gumen elies on CHL’s con inui y-equi a iance
closedness and GOE’s su jec i i y-p e-injec i i y equi alence (and injec i i y⇒su jec i i y)
[?].
6
Appendix B: Γ-Limi o So →Ha d Selec ion
Le easible se be simplex ∆(A), cos c:A → R, baseline dis ibu ion q∈∆(A). De ine
unc ional
Φτ(p) = ⟨c, p⟩+τDKL(p∥q).
Then pτ∈a gmin Φτgi es pτ(a)∝q(a) exp(−ca/τ). As τ↓0, KL e m weigh ends o ze o, Φτ
Γ-con e ges o linea unc ional Φ0(p) = ⟨c, p⟩, minimized a simplex e ices (poin masses).
I u he assuming equi- igh ness and unique minimize a⋆∈a g minaca, hen minimize
sequence pτcon e ges in weak opology o poin mass: pτ⇒δa⋆. La ge de ia ion p inciple
gua an ees p obabili y mass concen a ion a e a exponen ial scale 1/τ.Momen cons ain
case (assuming easibili y): Adding Pap(a)ϕ(a) = b⋆(assuming b⋆∈con ϕ(A)), he e
exis s dual a iable λ(τ); i b⋆is ela i e in e io (Sla e condi ion), {λ(τ)}is bounded (has
clus e poin ), o he wise i s no m may di e ge while ha d limi s ill equi alen o cons ained
linea p og amming. Exponen ial amily solu ion
pτ(a)∝q(a) exp ⟨λ(τ),ϕ(a)⟩−ca
τ.
As τ↓0, limi p oblem equi alen o min{⟨c, p⟩:Pap(a)ϕ(a) = b⋆}; i minimize a⋆unique
and a poin mass, hen pτ⇒δa⋆[?].
Appendix C: Re e sible Sample Cons uc ion Ou line
DDG- ee (Knu h–Yao): Op imal disc e e sampling wi h andom bi s as sou ce; w i -
ing isi pa h (le - igh b anches) and lea numbe in o Eyields e e sible implemen a ion
[?].
Alias (Walke /Vose): Cons an - ime sampling a e p ep ocessing wo ables; w i ing
able index and h eshold compa ison esul in o E, eplaying in in e sion [?].
Bo h compa ible wi h Benne ’s “sa e-e ase-back” s a egy, hence no i e e sible he mal
lowe bound [?].
Re e ences
[1] Hedlund. Endomo phisms and Au omo phisms o he Shi Dynamical Sys em (CHL cha -
ac e iza ion). h ps://link.sp inge .com/a icle/10.1007/BF01691062
[2] Re e sible cellula au oma on. Wikipedia. h ps://en.wikipedia.o g/wiki/
Re e sible_cellula _au oma on
[3] Ka i. Re e sibili y o 2D cellula au oma a is undecidable.h ps://ui.adsabs.ha a d.
edu/abs/1990PhyD...45..379K/abs ac
[4] Smi h, F.T. Li e ime Ma ix in Collision Theo y. Phys. Re . 1960. h ps://link.aps.
o g/doi/10.1103/PhysRe .118.349
[5] Csisz´a , I. I-Di e gence Geome y o P obabili y Dis ibu ions and Minimiza ion P oblems.
1975. h ps://pages.s e n.nyu.edu/~dbackus/BCZ/en opy/Csisza _geome y_AP_
75.pd
[6] Benne , C.H. Logical e e sibili y o compu a ion. IBM J. 1973. h ps://dl.acm.o g/
doi/10.1147/ d.176.0525
7
[7] Chang e al. Re e sibili y o linea cellula au oma a wi h in e media e bounda y condi ion.
h ps://www.aimsp ess.com/a icle/doi/10.3934/ma h.2024371
[8] Pinske ’s inequali y. Wikipedia. h ps://en.wikipedia.o g/wiki/Pinske ’s_
inequali y
[9] An In oduc ion o Γ-Con e gence. Sp inge . h ps://link.sp inge .com/book/10.
1007/978-1-4612-0327-8
[10] Knu h, Yao. The complexi y o nonuni o m andom num-
be gene a ion.h ps://www.seman icschola .o g/pape /
The-complexi y-o -nonuni o m- andom-numbe -Knu h-Yao/
58 10e b7c76b41a6ddc26 9 94 7 aa1e2e35
[11] Alias me hod. Wikipedia. h ps://en.wikipedia.o g/wiki/Alias_me hod
[12] The Ga den o Eden heo em: old and new. a Xi :1707.08898. h ps://a xi .o g/abs/
1707.08898
8
