Implemen ing Enzyma ic Nume ical P Sys ems
o AI Applica ions by Means o G aphic
P ocessing Uni s
Manuel Ga cı´a–Quismondo, Luis F. Mac´ıas–Ramos, and Ma io J. P´e ez–Jim´enez
Abs ac . A P sys em ep esen s a dis ibu ed and pa allel compu ing model in
which basic da a s uc u es a e, o ins ance, mul ise s and s ings. Enzyma ic Nu-
me ical P Sys ems (ENPS) a e a ype o P sys ems whose basic da a s uc u es
a e se s o nume ical a iables. Sepa a ely, GPGPU (gene al-pu pose compu ing
on g aphics p ocessing uni s) is a no el echnological pa adigm which ocuses on
he de elopmen o ools o g aphic ca ds o sol e gene al pu pose p oblems. This
pape p oposes an ENPS simula o based on GPUs and p esen s gene al concep s
abou i s design and some u u e ideas and pe spec i es.
In oduc ion
Memb ane compu ing is a bio-inspi ed b anch o na u al compu ing, abs ac ing
compu ing models om he s uc u e and unc ioning o li ing cells and om he
o ganiza ion o cells in issues o o he highe o de s uc u es [29]. This b anch
o na u al compu ing s udies he design and p ope ies o memb ane sys ems o P
sys ems. P sys ems a e non–de e minis ic dis ibu ed and pa allel compu ing mod-
els s uc u ed in compa men s known as memb anes. Basic da a s uc u es such as
mul ise s, s ings o nume ical a iables [30] a e associa ed wi h memb anes. Ac-
co ding o he way in which memb anes a e s uc u ed, he e a e se e al ypes o
P sys ems. Fo ins ance, he e exis cell-like P sys ems [29], issue P sys ems [26]
and spiking neu al P sys ems [17], along wi h o he ypes. In P sys ems, memb anes
and hei associa ed da a s uc u es a e p ocessed by means o ew i ing ules o
p og ams associa ed o he cells, in o de o pe o m sequences o con igu a ions
(compu a ions) [29][30]. P sys ems ha e been success ully applied in a wide ange
o domains [4]. Fo ins ance, hey ha e been applied in mic obiological modelling
Manuel Ga cı´a–Quismondo · Luis F. Mac´ıas–Ramos · Ma io J. P´e ez–Jim´enez
Resea ch G oup on Na u al Compu ing, Dp . o Compu e Science and A i icial In elligence,
Uni e si y o Se illa, A da. Reina Me cedes s/n. 41012 Se illa, Spain
e-mail: {mga ciaquismondo,l macias ,ma pe }@us.es
in o de o model phenomena such as quo um sensing in Vib io ische i popula ions
[38] and ecological modelling o p edic he e olu ion o he bea ded ul u e [3]
and he Py enean chamois [9] popula ions in he Ca alan Py enees, as well as im-
age h esholding [7]. Such a e sa ili y makes P sys ems a use ul ool o gaining
knowledge abou a as a ie y o di e en domains, hus p o iding a p omising
ool whi hin he ange o disciplines which composes he ield o s udy o a i icial
in elligence.
A special ype o cell-like P sys ems a e Enzima ic Nume ical P Sys ems
(ENPSs) [36]. ENPSs desc ibe a de e minis ic, maximally–pa allel model in which
he basic da a s uc u es associa ed o memb anes a e nume ical alues which e ol e
by means o p og ams associa ed o hem [30]. In o de o a p og am o be applied,
a ce ain alue o a speci ic a iables (enzyme) may be needed. O he whise, he
p og am canno be applied [36]. This model o compu a ion has al eady been suc-
cess ully used in model obo con olle s, in which a obo needs o a oid obs acles
si ua ed in a closed ci cui [37].
Sepa a ely, GPGPU (gene al-pu pose compu ing on g aphics p ocessing uni s)
is a no el echnological discipline which consis s o he applica ion o g aphic ca ds
(GPUs) in o de o pe o m pa allel, dis ibu ed algo i hms [40]. The basic idea is o
ake ad an age o he pa allel a chi ec u e o GPUs, adi ionally used o g aphics
p ocessing, o execu e algo i hms which can be pe o med in pa allel, hus accel-
e a ing hese algo i hms by di iding hem in concu en asks and execu ing hese
asks in a pa allel mode.
In his pape , we p opose a GPU simula o o ENPSs. The pa allel a chi ec u e o
ENPS makes he simula ions o hei compu a ions a sui able ask o be pa allelized,
hus expec ing an accele a ion in he simula ion imes i compa ed o hei sequen ial
coun e pa s.
This pape is s uc u ed as ollows. Sec ion 10.2.2 p o ides a quick in oduc ion
o Nume ical P Sys ems (NPSs) as a model o compu a ion. Sec ion 10.3 desc ibes
ENPSs as an ex ension o NPSs. Sec ion 10.4 p o ides a gene al o e iew o he cu -
en s a e-o - he-a abou he esul s ob ained by p e ious GPU simula o s wi hin
he ield o memb ane compu ing. Finally, sec ion 10.7 p esen s he conclusions ob-
ained and p oposes some di ec ions o u u e wo k.
2 P elimina ies
2.1 P Sys ems
Memb ane Compu ing is a young and eme gen b anch o Na u al Compu ing in-
oduced by G. P˘aun [30]. I has ecei ed impo an a en ion om he scien i ic
communi y since hen, wi h con ibu ions by compu e scien is s, biologis s, o -
mal linguis s and complexi y heo e icians, en iching each o he s wi h esul s, open
p oblems and p omising new esea ch lines. In ac , memb ane compu ing was se-
lec ed by he Ins i u e o Scien i ic In o ma ion, USA, as a as Eme ging Resea ch
F on in compu e science, and [35] was men ioned in [42] as a highly ci ed pape
in Oc obe 2003. This new model o compu a ion s a s om he obse a ion ha
he cell is he smalles li ing hing, and a he same ime i is a ma ellous iny
machine y, wi h a complex s uc u e, and om he assump ion ha he p ocesses
aking place in he compa men al s uc u e o a li ing cell can be in e p e ed as
compu a ions. The challenge is o ake he cell i sel as a suppo o compu a ions,
o ind in he s uc u e and he unc ioning o he cell seen as a whole hose ele-
men s use ul o compu a ions. Compu a ions in gene al, a he ma hema ical le el,
bu wi h he hope o b ing some hing use ul o p ac ical compu ing, ei he in he
same s yle as gene ic algo i hms and neu al compu ing, o imp o ing he use o he
exis ing compu e s, o p oposing new ypes o elec onic compu e s, o , possibly o
lead o ways o use he cells hemsel es as compu ing suppo s. The de ices o his
model a e called P sys ems. Roughly speaking, a P sys em consis s o a cell-like
memb ane s uc u e, in he compa men s o which one places mul ise s o objec s
which e ol e acco ding o gi en ules.
The main syn ac ic ing edien s o a cell-like memb ane sys em a e he memb ane
s uc u e, he mul ise s o objec s, and he e olu ion ules. A memb ane s uc u e
consis s o se e al memb anes a anged in a hie a chical s uc u e inside a main
memb ane ( he skin), and delimi ing egions ( he space in–be weena memb ane and
he immedia ely inne memb anes, i any). Each memb ane iden i ies a egion inside
he sys em Regions de ined by a memb ane s uc u e con ain objec s co esponding
o chemical subs ances p esen in he compa men s o a cell. The objec s can be
desc ibed by symbols o by s ings o symbols, in such a way ha mul ise o objec s
a e placed in egions o he memb ane s uc u e. The objec s can e ol e acco ding
o gi en e olu ion ules, associa ed wi h he egions (hence, wi h he memb anes).
The seman ics o he cell-like memb ane sys ems is de ined h ough a non de-
e minis ic and synch onous model (in he sense ha a global clock is assumed) as
ollows: A con igu a ion o a cell–like memb ane sys em consis s o a memb ane
s uc u e and a amily o mul ise s o objec s associa ed wi h each egion o he
s uc u e. A he beginning, he e is a con igu a ion called he ini ial con igu a ion
o he sys em. In each ime uni we can ans o m a gi en con igu a ion in ano he
con igu a ion by applying he e olu ion ules o he objec s placed inside he e-
gions o he con igu a ions, in a non-de e minis ic, and maximally pa allel manne
( he ules a e chosen in a non-de e minis ic way, and in each egion all objec s ha
can e ol e mus do i ). In his way, we ge ansi ions om one con igu a ion o he
sys em o he nex one.
In he las yea s, many di e en models o P sys ems ha e been p oposed. In
pa icula , compu a ional de ices inspi ed om he cell in e –communica ion in is-
sues, and adding he ing edien o cell di ision ules o he same o m as in cell–like
memb ane sys ems wi h ac i e memb anes, bu wi hou using pola iza ions. In hese
sys ems, he ules a e used in he non-de e minis ic maximally pa allel way, bu we
suppose ha when a cell is di ided, i s in e ac ion wi h o he cells o wi h he en-
i onmen is blocked; ha is, i a di ision ule is used o di iding a cell, hen his
cell does no pa icipa e in any o he ule, o di ision o communica ion. The se
o communica ion ules implici ely p o ides he g aph associa ed wi h he sys em
h ough he labels o he memb anes. The cells ob ained by di ision ha e he same
labels as he mo he cell, hence he ules o be used o e ol ing hem o hei objec s
a e inhe i ed.
The idea o spiking neu ons, cu en ly an ac i e esea ch opic in neu al compu -
ing (see, e.g., [16], [22], [23]), was ecen ly inco po a ed in memb ane compu ing
(see [18]) – he esul ing o mal sys ems a e called spiking neu al P sys ems, abb e-
ia ed as SN P sys ems. The s uc u e o an SN P sys em has a o m o a di ec ed
g aph wi h nodes ep esen ing neu ons, and edges ep esen ing synapses. The neu-
ons con ain spikes, objec s o a unique ype. A neu on (node) sends signals (spikes)
along i s ou going synapses (edges). Each neu on has i s own ules o ei he send-
ing spikes ( i ing ules) o o in e nally consuming spikes ( o ge ing ules). he
ules o he i s ype consume some spikes and p oduce a new spike, which is sen
o all neu ons linked by a synapse o he neu on whe e he ule was used, while he
o ge ing ules jus emo e spikes om neu ons. In he ini ial con igu a ion a neu-
on s o es he ini ial numbe o spikes, and a any ime momen he cu en ly s o ed
numbe o spikes (cu en con en s) is de e mined by he ini ial con en s and he his-
o y o unc ioning o
σ
( he spikes i ecei ed om o he neu ons, he spikes i sen
ou , and he spikes i in e nally consumed/ o go ). One o he neu ons is he ou pu
one, and i s spikes can also exi in o he en i onmen , hus p o iding a ace o he
sys em e olu ion. Like in neu obiology, we call his ace – sequence o momen s
when a spike exi s he sys em – spike ain.
2.2 Nume ical P Sys ems
As yea s wen by, di e en ypes o P sys ems ha e been in oduced. In he ounda-
ional ansi ion P sys em model, he cell s uc u e consis s o a oo ed ee, in which
each node ep esen s a memb ane o he s uc u e. Edges ep esen he hie a chical
ela ionships be ween memb anes exis en in he s uc u e. Howe e , some models
p opose new ypes o cell s uc u es. Fo ins ance, SN P Sys ems desc ibe an a chi-
ec u e based on a di ec ed g aph, in which cells o neu ons ac as nodes, whe eas
i ing ules ac as a cs. These ules send in o ma ion om one neu on o ano he a -
e a speci ic amoun o ime o delay [19]. Simila ly, in Tissue P sys ems, ins ead o
a hie a chical s uc u e, memb anes a e placed a he nodes o a non-di ec ed g aph.
The edges o he g aph ep esen sympo /an ipo ules which communica e he
memb anes in he g aph, hus mo ing objec s ac oss memb anes [26]. Also, e en
a ian s o hese ones ha e e ol ed. Fo ins ance, in he case o SN P Sys ems, new
ea u es such as SN P Sys ems wi h se e al kinds o spikes [19] and SN P sys ems
wi h neu on di ision and budding [27]. As ega ds o Tissue P Sys ems, he e exis
Tissue P Sys ems wi h cell di ision [33], Tissue P Sys ems wi hou en i onmen [8]
as an example.
Besides, no only ha e memb ane s uc u es e ol ed ac oss he Memb ane Com-
pu ing li e a u e. The da a s uc u es which e ol e by means o applica ions o ules
h ough compu a ion s eps ha e also been a ec ed. As a p oo o ha , in S ing P
Sys ems se s o s ings a e conside ed ins ead o mul ise s o objec s. These s ings
a e ew i en by means o ew i ing–like ules on each compu a ion s ep [6].
Following his end, a new kind o P sys em was in oduced by Gheo ghe and
And ei P˘aun in 2006. In hese P sys ems, known as Nume ical P Sys ems (NPSs
[29]), he adi ional mul ise s o objec s associa ed o memb anes a e eplaced by
se s o nume ical a iables. These a iables e ol e by means o p og ams associa ed
o he memb anes. As in he ounda ional model, he memb ane s uc u e is a ee-
nes ed hie a chy, so no new memb ane a chi ec u e is in oduced in his model.
A nume ical P sys em o deg ee m≥1 is a uple:
Π
=(H,
μ
,(Va 1,P 1,Va 1(0))...(Va m,P m,Va m(0)))
whe e:
•His an alphabe wi h msymbols used as labels o he mmemb anes o he
sys em. The labels con ained in Ha e he labels o he memb anes in
Π
.
•
μ
is a memb ane s uc u e, a oo ed ee wi h m memb anes.
•Va i={x1,i...xki,i}is he se ini e o a iables associa ed wi h compa men i,
(1 ≤i≤m)
•Va i(0)={
λ
1,i...
λ
ki,i}a e nume ical alues ( eal numbe s) o he a iables
in Va i. These alues a e conside ed as ini ial alues; a ime = o he sys em
e olu ion we ha e xj,i=
λ
j,i(1≤i≤m,1≤j≤ki).
•P i=P 1,i...P qi,iis he se o p og ams om compa men io
μ
(1≤
i≤m).Thel- h p og am P l,i om compa men iis o he o m P l,i=
(Fl,i(x1,i,...,xki,i),cl,1| 1+...+cl,ni| ni)whe e Fl,i(x1,i,...,xki,i)is he l- h p o-
duc ion unc ion om compa men iand cl,1| 1+...+cl,ni| ni)desc ibes he
epa i ion p o ocol.
The p oduc ion unc ion Fl,i(x1,i,...,xki,i) om compa men iis a a eal unc ion
ha ing as a iables hose om his compa men . The exp esion cl,1| 1+... +
cl,ni| nidesc ibes he epa i ion p o ocol which has he ollowing meaning: le
1... nibe he se o a iables om compa men i, om he pa en memb ane
o iand o all compa men s co esponding o child en o compa men i. The coe -
icien s cl,1...+cl,nia e na u al numbe s ha speci y he p opo ion o ehe cu en
p oduc ion dis ibu ed o each a iable 1... ni.
Mo e p ecisely, a any ins an ≥0, a p og am P l,ion each se P i(1 ≤i≤m)is
non–de e minis ically chosen. Then, we compu e Fl,i(x1,i( ),...,xki,i( )) and Cl,i=
∑ni
j=1cl,j. The alues o all a iables on which Fl,idepends a e consumed and ese o
0. The alue q=Fl,i(x1,i( ),...,xki,i( ))
Cl,i ep esen s he “uni a y po ion” o be dis ibu ed
o a iables 1,..., ni, acco ding o coe icien s cl,i,...,cl,niin o de o ob ain he
alues o hese a iables a ime +1. Speci ically, a iable l,jwill ecei e q×
cl,j(1≤j≤ni) om compa men i. I a a iable ecei es such “con ibu ions”
om se e al neighbou ing compa men s, hen hey a e added in o de o p oduce
he alue o he a iable a ime +1.
This model o compu a ion was ini ially aimed o cap u e he na u e and be-
ha iou o economic p ocesses [29]. The e had been some p e ious wo ks on he
modelling o economic p ocesses by means o Memb ane Compu ing [34], and his
wo k p oposed some esea ch lines on he applica ion o NPSs o he modelling o
economic phenomena.

3 Enzyma ic Nume ical P Sys ems
3.1 Desc ip ion o Enzyma ic Nume ical P Sys ems
As i is usual on memb ane compu ing models, a new kind o P sys ems has isen
as an ex ension o NPSs. This model is known as Enzyma ic Nume ical P Sys ems
(ENPSs). Al hough his pa allel model o compu a ion has many poin s in common
wi h Nume ical P Sys ems, he e a e some aspec s which di e encia es bo h models.
This way, in con as o Nume ical P Sys ems, Enzyma ic Nume ical P Sys ems de-
sc ibe a de e minis ic model o compu a ion. Thus, ins ead o non–de e minis ically
chosen, he p og ams o be applied a e con olled by speci ic a iables known as
enzyme–like a iables.
An Enzyma ic Nume ical P Sys em o deg ee m≥1 is a uple:
Π
=(H,
μ
,(Va 1,P 1,Va 1(0))...(Va m,P m,Va m(0)))
whe e:
•H,
μ
and (Va 1,Va 1(0))...(Va m,Va m(0)) ha e he same meaning han in Nu-
me ical P Sys ems desc ibed in sec ion 10.2.2.
•P iis he se o p og ams associa ed o memb ane i. Each l- h p og am in se P i
may ha e one o he ollowing o ms:
–P l,i=(Fl,i(x1,i,...,xki,i),cl,1| 1+...+cl,ni| ni)
–P l,i=(Fl,i(x1,i,...,xki,i),(el,i→),cl,1| 1+...+cl,ni| ni)
In bo h o ms, all alues which also appea in sec ion 10.2.2 ha e he same mean-
ing, wi h el,ibeing a a iable in Va i. This a iable is known as he enzyme–like
a iable associa ed o P l,iand i s alue canno be consumed by his p og am.
Enzyme–like a iables a e exclusi e ing edien s o ENPSs. Tha is, hey do no
appea in NPSs.
The main no el y in oduced by ENPSs has o do wi h he use o enzyme–like a i-
ables o con ol he execu ion low o p og ams. This way, each p og am may ha e
an associa ed enzyme–like a iable which con ols i s applica ion. I a p og am is
o be applied a ime , hen his p og am is ac i e a his ime. On each compu a-
ion s ep, all ac i e p og ams in each memb ane a e applied in pa allel. P og ams in
ENPSs a e applied he same way han in NPSs. Howe e , a p og am is ac i e only
in he ollowing cases:
•The p og am does no ha e an associa ed enzyme.
•The p og am has an associa ed enzyme and he alue o his enzyme is g ea e
han he minimum o he alues o he a iables consumed by he p og am.
ENPSs ha e been success ully applied wi hin he ield o obo ics. Fo ins ance,
hey ha e been used o model de e minis ic mobile obo con olle s o obs acle
a oidance. In his model, he speed o he wo obo mo o s is se acco ding o he
alues assigned o wo a iables o he sys em. Thus, he dynamical e olu ion o
hese a iables desc ibes he beha io o he obo h ough a closed ci cui [37].
Mo e in o ma ion abou ENPSs can be ound in [36][37].
Fig. 1 Enzyma ic Nume ical P Sys em
3.2 ENPSs and A i icial In elligence
Mobile obo con ol p oblems, such as obs acle a oidance and odome ic local-
izacion, can be conside ed as a i icial in elligence p oblems. Fo ins ance, obs a-
cle a oidance can be conside ed as a high-le el planning p oblem [21]. In obs acle
a oidance, he objec i e is o ind a sequence o mo emen s in a s a ic o dynamical
en i onmen . The objec i e o his sequence is o obo s which ollow i o a oid
c ashing wi h any obs acles hey migh ind in he en i onmen . The inpu da a is
gi en as a se ies o senso lec u es ob ained om he en i onmen . This ype o pa h
planning p oblems a ising om he ield o obo ics has al eady been a acked by
using a i icial in elligence echniques such as an colony algo i hms [12][11].
Odome ic localiza ion is a widely used me hod o es ima ion o he momen a y
pose o a mobile obo wi h espec o i s s a ing pose [20]. This es ima ion is
a ec ed by se e al e o sou ces, such as imp ecission in he mobile obo kinema ic
pa ame e s and e o s in he senso lec u es [1]. Thus, odome ic localiza ion en ails
an op imiza ion p oblem, i.e., minimizing he global e o in he pose es ima ion.
As an op imiza ion p oblem, odome ic localiza ion has been p e iously ackled by
using well-known a i icial in elligence pa adigms, such as gene ic algo i hms [15]
and a i icial neu al ne wo ks [10]. All in all, ENPs p opose a new amewo k which
can be applied in o de o sol e a i icial in elligence p oblems a ising om obo ics
[37].
3.3 Simula ion o ENPSs
ENPSs desc ibe a pa allel model. The e o e, he huge compu a ional powe equi ed
by ex ensi e models ( o ins ance, hose necessa y o massi e obo swa ms and
obo s wi h complex senso ne wo ks) accoun s o he need o high pe o mance
compu ing pla o ms o simula e hem. Besides, hei pa allel s uc u e makes hem
app op ia e o be simula ed by means o pa allel a chi ec u es such as GPUs, FPGAs
and compu e clus e s.
4 The Compu e Uni ied De ice A chi ec u e (CUDA)
S anda d o GPU Compu ing
4.1 Ou line o he CUDA P og amming Model
Mode n GPUs consis o a la ge numbe o p ocessing uni s. Fo ins ance, Fe mi
ca ds con ain up o 448 p ocesso co es and 1.536 p ocessing uni s pe co e, hus
esul ing in a o al numbe o 448×1.536 = 688.128 h eads [41]. These h eads a e
execu ed in pa allel wi h a ce ain deg ee o dependency om each o he [40].
In o de o make he mos o his massi ely pa allel a chi ec u e, i is necessa y
o make use o s anda ds speci ically designed o hese de ices. Two o hese main
s anda ds in GPGPU a e OpenCL [39] and CUDA [41].
The CUDA p og amming model is an abs ac GPU model p o ided by NVIDIA.
This model is an abs ac ion o he speci ic pa allel de ice whe e he p og am is o be
execu ed. The model de ines a g id. This g id is an abs ac ion o he cu en GPU
ca d whe e he code is o be execu ed. The g id is composed o mul ip ocessing
compu ing de ices known as blocks. Simila ly, each block is composed o se e al
s eam monop ocessing uni s known as h eads(see igu e 10.2). Th eads execu e
pa allel pieces o code o ke nels. On any ins an in he execu ion o a GPU p og am,
he same ke nel is un on e e y h ead a he same ime.
I is con enien o ba ch h eads which pe o m ope a ions in common in he same
block. The eason is ha h eads in he same block can communica e wi h each o he
h ough as on-chip memo y, whe eas h eads in di e en blocks use slow o -chip
memo y o communica e. Thus, i is impo an o minimize he communica ion be-
ween h eads om di e en blocks, u ning i in o communica ion be ween h eads
in he same block when possible. Besides, hey a e allowed o synch onize wi h
each o he ia ba ie s. On he o he hand, he only way o synch onizing h eads o
di e en blocks is by ending he ke nel execu ion. The CUDA p og amming model
equi es h ead blocks in he same ke nel o be independen . I means ha he i-
nal esul o he compu a ion canno depend on he o de in which he blocks a e
execu ed, gi ing he same esul wi hou depending on hei o de o execu ion.
4.2 The CUDA–C P og amming Language
CUDA–C is an ex ension o he C language o wo k agains he CUDA p og amming
model. This language is designed o make he mos o he GPGPU app oach by
enabling p og amme s o encode pa allel applica ions o be un on GPUs [5]. Tha
is, p og amme s a e able o de elop code o be execu ed on each GPU h ead a he
same ime. This way hey can ake ad an age o he GPU pa allel a chi ec u e in
o de o ob ain eno mous speed-up i compa ed o sequen ial e sions o he same
code.
The s uc u e o CUDA-C p og ams consis s o wo main pa s: The hos pa and
he de ice pa . The main di e ence be ween hem consis s o he speci ic de ice in
which hey a e execu ed. Thus, he hos pa is execu ed on he CPU, whils he de-
ice pa is execu ed on he GPU [5]. The hos pa includes calls o ke nels. The
Fig. 2 The CUDA p og amming model
de ice pa is composed o ke nels which de ine he ope a ions o be pe o med in
pa allel. The de elope o ganizes he h eads o execu e he ke nels in wo hie a chi-
cal le els o pa allelism. These le els a e a e lec ion o hose in which he CUDA
p og amming model is o ganized. In o de he o ganize he h eads o execu e he
ke nels, he p og amme de ines he s uc u e o he h ead blocks. This is done by
p og amma ically se ing he numbe o h eads pe block, as well as he o al num-
be o blocks in he g id. This way, bo h pa s o he p og am can coope a e in o de
o ob ain a global esul . Mo e in o ma ion abou he CUDA p og amming model
and he CUDA-C language can be ound on [41][24].
A sample code o a ypical high-pe o mance ope a ion on GPU can be ound
on [4]. In his sample, he summing o he elemen s in wo ec o s is compu ed.
Each pai o elemen s a e assigned o a di e en h ead. The e o e, each pai o
elemen s a e added in a pa allel way. Al hough his example may seem oo simple,
i illus a es qui e well he way in which he CUDA pa allel mode can be applied
o pa allelize ope a ions, hus ob aining a emendous speed-up due o he pa allel
compu ing app oach.
GPGPU and CUDA–C ha e been al eady success ully applied in o de o sim-
ula e di e en kinds o P sys ems. To he bes o ou knowledge, hey ha e been
applied o simula e cell-like objec -based P sys ems [5] and SN P sys ems [2]. Thei
esul s include da a which show no iceable speed-ups in compa ison o hei sequen-
ial coun e pa s. These esul s demons a e he sui abili y o he GPGPU app oach
o simula ing P sys ems in a pa allel mode.
•Check i s posi ion iin P og am applica ions. I he alue o his posi ion is In-
ac i e, do no execu e he ollowing s eps and exi he ke nel. I he alue o his
posi ion is Ac i e, execu e he ollowing s eps.
•Check i i s posi ion iin P oduc ion unc ion node ypes is equal o Va iable.In
o he case, abo he h ead.
•Access i s posi ion iin P oduc ion unc ion a iables.Le jbe he alue o his
coe icien .
•Se posi ion jin Va iables o 0.
5.6 Repa i ion P o ocol Applica ion
The las s ep in he algo i hm consis s on dis ibu ing he esul o he p oduc ion
unc ions. Fo each h ead, his implies eading he alue s o ed in P oduc ion unc-
ion esul s and dis ibu ing i o e i s p og am’s con ibu ed a iables. As he no -
maliza ion o coe icien s is pe o med a he beginning o he algo i hm, his s ep
only en ails mul iplying his ead alue by he associa ed coe icien o each a i-
able in he epa i ion p o ocol and adding he esul o he mul iplica ion o his
a iable. Be o e explaining in de ail he implemen a ion o his p ocess, i is impo -
an o in oduce he da a s uc u es used o ep esen he epa i ion p o ocols o
he simula ed model. Each epa i ion p o ocol is s o ed as a egion in wo a ays.
Thus, each pai coe icien – a iable has an associa ed index, which co esponds o
an associa ed posi ion in each o hese a ays.
Repa i ion p o ocol coe icien s: This a ay con ains he coe icien s associa ed
o each a iable exis ing in epa i ion p o ocols. On he epa i ion p o ocol s ep,
he con en o his a ay is al eady no malized,as i is pe o med a he beginning
o he algo i hm (see subsec ion 10.5.2).
Repa i ion p o ocol a iables: This a ay con ains he indexes o he a iables
o which he epa i ion p o ocols a e con ibu ed. These indexes a e used o ac-
cess he a ay Va iables, in o de o ob ain hei cu en alue.
In e ms o implemen a ion, he dis ibu ion o he esul o he p oduc ion unc ion
o each p og am is pe o med he ollowing way. Each h ead has an associa ed pai
coe icien - a iable assigned. In e ms o implemen a ion, a posi ion iin he epa i-
ion p o ocol a ays is asigned o each h ead. Taking in o accoun his conside a ion,
each h ead pe o ms he ollowing ope a ions:
1. Check i s posi ion iin P og am applica ions. I he alue o his posi ion is In-
ac i e, do no execu e he ollowing s eps and exi he ke nel. I he alue o his
posi ion is Ac i e, execu e he ollowing s eps.
2. Access i s posi ion iin Repa i ion p o ocol coe icien s.Le cbe he alue o his
coe icien .
3. Access he posi ion oo he p og am o i s epa i ion p o oco in P oduc ion
unc ion esul s.Le be he alue o his esul .
4. Pe o m he mul iplca ion o hese alues. Le m=c× .

5. Access i s posi ion iin Repa i ion p o ocol a iables.Le be he alue o his
posi ion.
6. Add m o posi ion in Va iables.
I is impo an o no ice ha , in his s ep, he heo e ical speed-up ac o can be
g ea e han 1, in he case ha he e exis p og ams in he simula ed model in which
he numbe o pai s coe icien – a iable is g ea e han 1. In con as o he case o
p oduc ion unc ions, his is usual in he s udied models [37], so a g ea e heo e ical
speed-up ac o can be ob ained in his s age.
Fig. 7 Da a s uc u e o p oduc ion unc ions
5.7 Execu ion o a Simula ion S ep
As desc ibed in he o me subsec ions, he execu ion o a simula ion s ep consis s
o he checking and applica ion o p og ams o a p ede ined numbe o s eps. This
numbe o s eps, as well as he model o simula e, a e speci ied as inpu s o he
simula o . In he case ha he model simula ed de ines a numbe o s eps, hen his
numbe p e ails o e he one gi en as inpu . The simula ion o a model is pe o med
by execu ing he ollowing s eps:
1. No malize he epa i ion coe icien s, as desc ibed in subsec ion 10.5.2.
2. Fo each simula ion s ep, pe o m he ollowing ope a ions:
a. Assign a p og am o each h ead. This is done by using he indexes o he
h eads in he CUDA p og amming model.
b. Each h ead checks i i s p og am is o be applied, as desc ibed in subsec ion
10.5.3.
c. I i s p og am is o be execu ed, each h ead calcula es i s p oduc ion unc ion,
as desc ibed in subsec ion 10.5.4.2.
d. I i s p og am is o be execu ed,each h ead clea s he alues o hose a iables
which depend on he p oduc ion unc ion o he p og am ( ha is, consumes
i s alues), as desc ibed in subsec ion 10.5.5.
e. Assign a pai coe icien – a iable om each epa i ion p o ocol o each
h ead. This is also done by using he indexes o he h eads in he CUDA
p og amming model.
. I i s epa i ion p o ocol’s p og am is o be execu ed, each h ead dis ibu es
he esul o he co espondingp oduc ion unc ion acco ding o he associa ed
pai , as desc ibed in subsec ion 10.5.6.
5.8 Rema ks on he Simula o
This simula o will be published unde open sou ce license. I can be used o simu-
la ing complex dis ibu ed p ocesses modelled wi h ENPSs. The e o e, se e al obo
beha io s can be simula ed in pa allel ( o example, a obo could a oid obs acles,
ollow ano he obo o look o a a ge a he same ime). The synch oniza ion a
he same ime be ween se e al beha io s o one obo is done by he help o he
enzyme a iables which can be used as s op condi ions [37]. Apa om simula -
ing se e al beha io s o only one obo in pa allel, he simula o could be used o
simula e in e ac ion and coope a ion be ween se e al obo s in complex dis ibu ed
obo ic sys ems.
6 Simula o Pe o mance
6.1 Simula o Wo k low
In o de o ease he simula ion o ENPS models, he simula o akes an inpu ile
desc ibing an ENPS in XML o ma . The XML o ma used is he one accep ed by
SNUPS [36], a p e iously exis en sequen ial simula o o ENPSs. This way, he
eusabili y o he models is imp o ed, as he same ile can be used wi h indepen-
dence o he selec ed simula o , be i SNUPS and on he GPU-based one in oduced,
wi hou any change in he XML ile o ma . Hence, he e is no need o change he
ile o ma , in he case ha he same ENPS is o be simula ed on bo h simula o s.
Thus, in o de o simula e an ENPS, one needs o encode i on he same XML
o ma as i is equi ed on SNUPS. Once his P sys em is encoded, he esul ing
ile can be pa sed by he GPU simula o . A e he pa sing p ocess, he simula ion is
pe o med. E en ually, he in o ma ion is displayed on he command p omp . Figu e
10.6.1 shows a g aphical ep esen a ion o his p ocess.
6.2 Pe o mance Compa ison
All pa allel pa s o he algo i hm a e execu ed wi h a deg ee o pa allelism a leas
equal o he numbe o p og ams o he simula ed model. The deg ee o pa allelism
can be e en g ea e when he epa i ion p o ocol s age is applied. Hence, a heo e -
ical accele a ion o a leas he numbe o p og ams o he model could be eached,
i compa ed o he un ime o sequen ial simula o s. In eal e ms, he simula o
was es ed by using an ENPS model o obs acle a oidance [37] as an example.
These models we e simula ed by using SNUPS [36]. Then, he esul ing un imes
we e compa ed wi h he GPU simula o un imes, in o de o ge an app oxima e
speed-up. In he speci ic case o he obs acle a oidance model, he o al numbe o
p og ams is 41 [37]. Hence, an accele a ion o a leas 41 is heo e ically expec ed
in his case, i compa ed o sequen ial ENPSs simula o s [36].
The no el y o ENPSs as a compu ing model [36] accoun s o he need o gen-
e a e ad-hoc case s udies o he simula o . Tha is, i is no possible o ind an
Fig. 8 Wo k low o he simula o
ex ensi e collec ion o case s udies in he li e a u e. Thus, he au ho s needed o
gene a e hem in o de o measu e he expe imen al pe o mance o he GPU sim-
ula o p oposed. In p ac ice, he simula o pe o mance has been es ed by using
an obs acle a oidance model [37]. Taking his model as a s a ing poin , some case
s udies ha e been gene a ed. All o hem sha e he same p og ams, a iables and
memb ane s uc u e o he obs acle a oidance model p oposed in [37]. Thus, hese
models consis o 9 memb anes, 41 p og ams and 29 a iables each [37]. The only
di e ences be ween hese case s udies consis o he ini ial alues o he a iables
associa ed o he memb anes.
Model numbe SNUPS GPU Accele a ion
136.3702 6.7286 5.4053
214.9084 6.6304 2.2484
314.9040 7.7268 1.9288
426.3204 6.8255 3.8561
515.2276 6.4188 2.3723
618.9548 6.5659 2.8868
730.7377 6.7206 4.5736
827.0497 7.6020 3.5582
915.7529 6.8335 2.3052
10 30.1695 6.6364 4.5460
Fig. 9 Compa ison o execu ion imes o a sequen ial ENPSs simula o (SNUPS) and he
GPU ENPSs simula o p oposed
Fo his pu pose, 10 andomly gene a ed models we e execu ed. Each model was
execu ed o 100 s eps. Figu e 10.9 displays he execu ion imes o hese uns. This
able compa es he execu ion imes o he same models un on SNUPS [36] and he
GPU simula o . The execu ion imes a e gi en in milliseconds. Fo each model, he
accele a ion is gi en as he esul o he di ision SNUPS un ime
GPU un ime .
7 Conclusions
In his pape , a GPU-based simula o o ENPSs. ENPSs desc ibe a pa allel com-
pu ing model wi h applica ions in a i icial in elligence. This simula o migh be
sui able o la ge scale models which can be applied wi hin he ield o obo ics.
The massi ely pa allel en i onmen p o ided by he GPUs is sui able o ENPSs
simula ions. Following his line o wo k, i would be in e es ing o simula e hese
models by means o GPU clus e s o o he pa allel a chi ec u es (such as FPGAs
o compu e clus e s). These sys ems migh be applied o model he beha io o
massi e obo swa ms and complex senso ne wo ks.
ENPSs can be used o model di e en beha iou s, such as ollow he leade ,
obs acle a oidance and wall ollowing (c . Chap. 9 o his book). The esul ing
simula o s could be compa ed in e ms o execu ion ime and pe o mance. This
compa ison could help expe s selec he mos sui able simula o o he ask in
hand, be i wall ollowing, obs acle a oidance,e c.
Ano he in e es ing challenge conce ning he pa allel simula ion o ENPS models
has o do abou explo ing he possibili y o simula ing se e al obo beha io s in
pa allel on GPUs. Tha is, simula ing si ua ions in which obo s need o achie e
mo e han one objec i e a he same ime. These simula ions could help o ec ea e
scena ios in which obo s need o pe o m mul i-objec i e asks.
Ano he impo an open p oblem conce s he in eg a ion o he simula o in o
use -o ien ed so wa e pla o ms. This in eg a ion will ease he use o he simula o
by Memb ane Compu ing expe s, hus imp o ing he human-compu e in e ac ion
expe ience. Some examples o end-use so wa e amewo ks o simula ing P sys-
ems a e SNUPS[36] and P-Lingua[13].
Ano he impo an poin wi h which o deal has o do wi h a mo e exhaus i e e al-
ua ion o he pe o mance o he simula o . Whils he shallow pe o mance e alu-
a ion included in his pape shows an a e age speed-up ac o o 3x i compa ed o
he Ja a simula o SNUPS, in o de o assess he eal speed–up ac o o be eached
by he simula o i is necessa y o de elop la ge models and compa e hei un imes
no only wi h Ja a o o he i ual machine-based p og amming languages, bu also
wi h languages on a lowe le el o abs ac ion, such as C o Fo an.
Ne e heless, he cu en models ha e such a small numbe o p og ams ha hese
low-le el simula o s could yield be e un imes han he GPU simula o , as hey a e
ee om he o e head ega ding he dis ibu ion o asks among he GPU h eads. In
o he wo ds, he GPU simula o is expec ed o yield a be e pe o mance only when
he numbe o p og ams is conside ably high, ha is, abou housands o p og ams
pe model. In o de o asses he pe o mance in hese cases, i is necessa y o ex end
he models cu en ly ound in he li e a u e up o new models wi h housands o
p og ams. On hese models, he GPU simula o is expec ed o yield lowe execu ion
imes no only compa ed o SNUPS execu ion imes, bu also o he imes ob ained
by C and Fo an simula o s.
Acknowledgemen s. Manuel Ga c´ıa-Quismondo, Ma io J. P´e ez-Jim´enez and Luis F.
Mac´ıas Ramos a e suppo ed by p ojec TIN 2009-13192 om “Minis e io de Ciencia e In-
no aci´on” o Spain, co- inanced by FEDER unds. Manuel Ga c´ıa-Quismondo and Ma io J.
P´e ez-Jim´enez a e also suppo ed by “P oyec o de Excelencia con In es igado de Recono-
cida Val´ıa P08-TIC-04200” om Jun a de Andaluc´ıa. Manuel Ga c´ıa-Quismondo is also sup-
po ed by he Na ional FPU G an P og amme om he Spanish Minis y o Educa ion. Ma io
J. P´e ez-Jim´enez is also suppo ed by p ojec TIN2008-04487-E om he Spanish Minis y
o Science and Educa ion h ough he Complemen a y Ac ion TIN2008-0448-E/TIN.
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