Tenth Brainstorming Week on Membrane Computing Sevilla, January 30 – February 3, 2012 Volume I : RGNC REPORT 1/2012

Martínez del Amor, Miguel Ángel; Paun, Gheorghe; Pérez Hurtado de Mendoza, Ignacio; Romero Campero, Francisco José
Tenth Brainstorming Week on Membrane Computing Sevilla, January 30 – February 3, 2012 Volume I : RGNC REPORT 1/2012

Author: Martínez del Amor, Miguel Ángel; Paun, Gheorghe; Pérez Hurtado de Mendoza, Ignacio; Romero Campero, Francisco José
Publisher: Fénix Editora
Year: 2012
Source: https://idus.us.es/bitstreams/42a8da42-c8c4-4de2-9cd3-96124aae12f5/download
Ten h B ains o ming Week
on Memb ane Compu ing
Se illa, Janua y 30 – Feb ua y 3, 2012
Volume I
Miguel A. Ma ´ınez-del-Amo
Gheo ghe P˘aun
Ignacio P´e ez-Hu ado
F ancisco J. Rome o-Campe o
Edi o s
Ten h B ains o ming Week
on Memb ane Compu ing
Se illa, Janua y 30 – Feb ua y 3, 2012
Volume I
Miguel ´
Angel Ma ´ınez del Amo
Gheo ghe P˘aun
Ignacio P´e ez Hu ado de Mendoza
F ancisco Jos´e Rome o Campe o
Edi o s
RGNC REPORT 1/2012
Resea ch G oup on Na u al Compu ing
Se illa Uni e si y
F´enix Edi o a, Se illa, 2012
c
Au o es
ISBN: ??????
Dep´osi o Legal: SE-????–06
Edi a: F´enix Edi o a
A da. de C´adiz, 7 – 1C
41004 Se illa
enixedi o a@ ele onica.ne
Tel . 954 41 29 91
P e ace
These p oceedings, consis ing in wo olumes, con ain he pape s eme ged om he
Ten h B ains o ming Week on Memb ane Compu ing (BWMC), held in Se illa,
om Janua y 30 o Feb ua y 3, 2012, in he o ganiza ion o he Resea ch G oup
on Na u al Compu ing om he Depa men o Compu e Science and A i icial
In elligence o Se illa Uni e si y. The i s edi ion o BWMC was o ganized a he
beginning o Feb ua y 2003 in Ro i a i Vi gili Uni e si y, Ta agona, and all he
nex edi ions ook place in Se illa a he beginning o Feb ua y, each yea .
The 2012 edi ion o BWMC was o ganized in conjunc ion wi h he Fi s In e -
na ional Con e ence on De elopmen s in Memb ane Compu ing (ICDMC2012).
In he s yle o p e ious mee ings in his se ies, he en h BWMC was con-
cei ed as a pe iod o ac i e in e ac ion among he pa icipan s, wi h he emphasis
on exchanging ideas and coope a ion. Se e al “p o oca i e” alks we e deli e ed,
mainly de o ed o open p oblems, esea ch opics, conjec u es wai ing o p oo s,
ollowed by an in ense coope a ion among he 40 pa icipan s – see he lis in he
end o his p e ace. The e iciency o his ype o mee ings was again p o ed o be
e y high and he p esen olumes illus a e his asse ion.
Sligh ly di e en om he p e ious mee ings was he combina ion wi h he
ICDMC2012, in he sense ha se e al alks also had he s yle o a con e ence:
mo e ime dedica ed o p esen ing achie emen s ob ained by he esea ch g oups
om whe e he pa icipan s came om, bu also con e ging owa ds he s yle o
he b ains o ming, i.e., p esen ing on ie esul s, esea ch opics, ongoing appli-
ca ions.
The pape s included in hese olumes, a anged in he alphabe ic o de o he
au ho s, we e collec ed in he o m a ailable a a sho ime a e he b ains o m-
ing; se e al o hem a e s ill unde elabo a ion. The idea is ha he p oceedings a e
a wo king ins umen , pa o he in e ac ion s a ed du ing he s ay o au ho s in
Se illa, mean o make possible a u he coope a ion, his ime ha ing a w i en
suppo .

i P e ace
Selec ions o he pape s om hese olumes will be conside ed o publica ion
in special issues o Theo e ical Compu e Science and o In e na ional Jou nal o
Compu e Ma hema ics.
A e each BWMC, one o wo special issues o a ious in e na ional jou nals
we e published. He e is hei lis :
•BWMC 2003: Na u al Compu ing – olume 2, numbe 3, 2003, and New Gen-
e a ion Compu ing – olume 22, numbe 4, 2004;
•BWMC 2004: Jou nal o Uni e sal Compu e Science – olume 10, numbe 5,
2004, and So Compu ing – olume 9, numbe 5, 2005;
•BWMC 2005: In e na ional Jou nal o Founda ions o Compu e Science –
olume 17, numbe 1, 2006);
•BWMC 2006: Theo e ical Compu e Science – olume 372, numbe s 2-3, 2007;
•BWMC 2007: In e na ional Jou nal o Uncon en ional Compu ing – olume 5,
numbe 5, 2009;
•BWMC 2008: Fundamen a In o ma icae – olume 87, numbe 1, 2008;
•BWMC 2009: In e na ional Jou nal o Compu e s, Con ol and Communica-
ion – olume 4, numbe 3, 2009;
•BWMC 2010: Romanian Jou nal o In o ma ion Science and Technology –
olume 13, numbe 2, 2010;
•BWMC 2011: In e na ional Jou nal o Na u al Compu ing Resea ch – olume
2, numbe s 2-3, 2011.
O he pape s elabo a ed du ing he en h BWMC will be submi ed o o he
jou nals o o sui able con e ences. The eade in e es ed in he inal e sion o
hese pape s is ad ised o check he cu en bibliog aphy o memb ane compu ing
a ailable in he domain websi e h p://ppage.psys ems.eu.
***
The lis o pa icipan s as well as hei email add esses a e gi en below, wi h
he aim o acili a ing he u he communica ion and in e ac ion:
1. A iom Alhazo , Uni e si y o Milano - Bicocca, I aly, aa [email protected]
2. Ioan A delean, Ins i u e o Biology o he Romanian Academy, Bucha es ,
Romania, ioan.a [email protected]
3. Ma i Angels Colome Cuga , Uni e si y o Lleida, Spain,
colome @ma ema ica.udl.ca
4. E zs´ebe Csuhaj-Va j´u, Facul y o In o ma ics, E¨o ¨os Lo ´and Uni e si y,
Budapes , Hunga y, [email p o ec ed]l e.hu
5. Rudol F eund, Technological Uni e si y o Vienna, Aus ia,
udi [email protected]
6. Manuel Ga c´ıa-Quismondo Fe n´andez, Uni e si y o Se ille, Spain,
mga [email p o ec ed]
7. Ma ian Gheo ghe, Uni e si y o She ield, Uni ed Kingdom,
m.gheo ghe@she ield.ac.uk
8. Ca men G aciani D´ıaz, Uni e si y o Se ille, Spain, c[email p o ec ed]
P e ace ii
9. Miguel A. Gu i´e ez Na anjo, Uni e si y o Se ille, Spain, [email p o ec ed]
10. Flo en in Ipa e, Uni e si y o Pi e¸s i, Romania, lo en in.ip[email p o ec ed]o
11. Joze Kelemen, Silesian Uni e si y, Opa a, Czech Republic,
joze .kelemen@ p .slu.cz
12. Abhay K ishna, CABIMER, Se ille, Spain, Abhay.K ishan@cabime .es
13. Raluca Le ica u, Uni e si y o Pi e¸s i, Romania, aluca.le ic[email p o ec ed]om
14. Albe o Lepo a i, Uni e si y o Milano - Bicocca, I aly,
lepo a [email protected]
15. Luis Felipe Mac´ıas Ramos, Uni e si y o Se ille, Spain, l mac[email p o ec ed]
16. Vincenzo Manca, Uni e si y o Ve ona, I aly, incenzo.manca@uni .i
17. Luca Ma che i, Uni e si y o Ve ona, I aly, luca.ma che i@uni .i
18. Miguel A. Ma ´ınez del Amo , Uni e si y o Se ille, Spain, [email p o ec ed]
19. Gianca lo Mau i, Uni e si y o Milano - Bicocca, I aly, mau [email protected]
20. Adam Ob ulowicz, Polish Academy o sciences, Wa saw, Poland,
A.Ob [email protected]
21. Ana B ˆandu¸sa Pa el, Poli ehnica Uni e si y o Bucha es , Romania,
anab [email p o ec ed]om
22. Gheo ghe P˘aun, Romanian Academy, Bucha es , Romania, and Uni e si y
o Se ille, Spain, [email protected]
23. Hong Peng, School o Ma hema ics and Compu e Enginee ing, Xihua
Uni e si y, China, ph.xhu@ho mail.com
24. Ignacio P´e ez Hu ado de Mendoza, Uni e si y o Se ille, Spain, pe [email p o ec ed]
25. Ma io de J. P´e ez Jim´enez, Uni e si y o Se ille, Spain, ma pe @us.es
26. An onio En ico Po eca, Uni e si y o Milano - Bicocca, I aly,
po [email protected]
27. Ra´ul Reina Molina, Uni e si y o Se ille, Spain, [email protected]
28. Agus ´ın Riscos N´u˜nez, Uni e si y o Se ille, Spain, a [email protected]
29. Iu ie Rogojin, Ins i u e o Ma hema ics and Compu e Science o he Academy
o Sciences o Moldo a, Chi¸sin˘au, Moldo a, y [email protected]
30. ´
Al a o Rome o Jim´enez, Uni e si y o Se ille, Spain, ome o.al a [email protected]
31. F ancisco Jos´e Rome o Campe o, Uni e si y o Se ille, Spain, [email protected]
32. Jose Ma ´ıa Sempe e Luna, Poly echnical Uni e si y o Valencia, Spain,
jsempe [email p o ec ed]
33. Pe Sos´ık, Silesian Uni e si y, Opa a, Czech Republic, and Uni e sidad
Poli ´ecnica de Mad id, Spain, p[email p o ec ed]z
34. C is ian S¸ e an, Uni e si y o Pi e¸s i, Romania, li iu.s e [email protected]
35. Luis Valencia Cab e a, Uni e si y o Se ille, Spain, [email p o ec ed]
36. Gy¨o gy Vaszil, Facul y o In o ma ics, Uni e si y o Deb ecen, Hunga y,
aszil.gyo [email p o ec ed].hu
37. Se ghei Ve lan, Uni e si y o Pa is Es , F ance, e lan@uni -pa is12.
38. Jun Wang, Elec ical and In o ma ion Enginee ing, Xihua Uni e si y, China,
wj.xhu@ho mail.com
39. Claudio Zand on, Uni e si y o Milano - Bicocca, I aly,
zand [email protected]
iii P e ace
40. Gexiang Zhang, School o Elec ical Enginee ing, Sou hwes Jiao ong
Uni e si y, China, [email protected]
As men ioned abo e, he mee ing was o ganized by he Resea ch G oup on
Na u al Compu ing om Se illa Uni e si y (h p://www.gcn.us.es)– and all
he membe s o his g oup we e en husias ically in ol ed in his (no always easy)
wo k.
The mee ing was suppo ed om a ious sou ces: (i) P oyec o de Excelencia
con in es igado de econocida al´ıa, de la Jun a de Andaluc´ıa, g an P08 – TIC
04200, (ii) P oyec o del Minis e io de Educaci´on y Ciencia, g an TIN2009 – 13192,
(iii) Ins i u o de Ma em´a icas de la Uni e sidad de Se illa (IMUS), (i ) Conseje ´ıa
de Inno acion, Ciencia y Emp esas de la Jun a de Andaluc´ıa, as well as by he De-
pa men o Compu e Science and A i icial In elligence om Se illa Uni e si y.
Gheo ghe P˘aun
Ma io de Jes´us P´e ez-Jim´enez
(Se illa, May 3, 2012)
Con en s
Sel -S abiliza ion in Memb ane Sys ems
A. Alhazo , M. An onio i, R. F eund, A. Lepo a i, G. Mau i ............ 1
Cha ac e izing he Compu a ional Powe o Ene gy-Based P Sys ems
A. Alhazo , M. An onio i, A. Lepo a i.................................. 11
Asynch onous and Maximally Pa allel De e minis ic Con olled
Non-Coope a i e P Sys ems Cha ac e ize N F IN ∪coN F IN
A. Alhazo , R. F eund .................................................. 25
The Compu a ional Powe o Exponen ial-Space
P Sys ems wi h Ac i e Memb anes
A. Alhazo , A. Lepo a i, G. Mau i, A.E. Po eca, C. Zand on ........... 35
The Powe o Sympo -3 wi h Few Ex a Symbols
A. Alhazo , Y. Rogozhin ................................................ 61
Coun ing Cells wi h Tissue-like P Sys ems
I. A delean, D. D´ıaz-Pe nil, M.A. Gu i´e ez-Na anjo,
F. Pe˜na-Can illana, R. Reina-Molina, I. Sa chizian .................... 69
Gene al Topologies and P Sys ems
E. Csuhaj-Va j´u, M. Gheo ghe, M. S anne ............................ 79
Skele onizing Images by Using Spiking Neu al P Sys ems
D. D´ıaz-Pe nil, F. Pe˜na-Can illana, M.A. Gu i´e ez-Na anjo ........... 91
A Fo mal F amewo k o P Sys ems wi h Dynamic S uc u e
R. F eund, I. P´e ez-Hu ado, A. Riscos-N´u˜nez, S. Ve lan .............. 111
P Sys ems wi h Minimal Le and Righ Inse ion and Dele ion
R. F eund, Y. Rogozhin, S. Ve lan ..................................... 123
Simula ing La ge-Scale ENPS Models by Means o GPU
M. Ga c´ıa–Quismondo, A.B. Pa el, M.J. P´e ez–Jim´enez ............... 137
4 A. Alhazo e al.
De ini ion 3. A P sys em Πis possibly closed wi h espec o a ﬁni e se Siﬀ
o e e y non-hal ing conﬁgu a ion C∈S he e exis s a conﬁgu a ion C′∈Ssuch
ha C⇒C′.
De ini ion 4. A P sys em Πis closed wi h espec o a ﬁni e se Siﬀ o e e y
non-hal ing conﬁgu a ion C∈S C ⇒C′implies C′∈S.
We say ha a sys em is (possibly, almos su ely) se -con e ging i
i is (possibly, almos su ely, espec i ely) con e ging o some ﬁni e se o
conﬁgu a ions.
We say ha a sys em is possibly sel -s abilizing i i is possibly con-
e ging o some ﬁni e se So conﬁgu a ions and i i is possibly closed
wi h espec o S.
We say ha a sys em is (almos su ely) sel -s abilizing i i is (almos
su ely, espec i ely) con e ging o some ﬁni e se So conﬁgu a ions and
i i is closed wi h espec o S.
The examina ion o compu a ional aspec s o hese p ope ies mo i a es us o
add “weakly” o he p ope ies p oposed in [1] – (possibly, almos su ely) con e g-
ing, (possibly) closed, (possibly, almos su ely) se -con e ging, (possibly, almos
su ely) sel -s abilizing – i he co esponding condi ions o e conﬁgu a ions Conly
spans he eachable non-hal ing ones.
Ano he commen we can make on “almos su e” i ha such a p ope y may
depend on how exac ly he ansi ion p obabili y is deﬁned. The easies way is o
assign equal p obabili ies o all ansi ions om a gi en conﬁgu a ion. Al e na-
i ely, o a ansi ion ia a mul ise o ules n1
1· · · nm
mwe may assign he weigh
o a mul inomial coeﬃcien (n1+···+nm
n1,··· ,nm)=(n1+···+nm)!
n1!···nm!, which will make he co ne
cases less p obable han he a e age ones. The e can be o he ways o deﬁne an-
si ion p obabili ies, bu we would like o discuss he p ope ies o in e es wi hou
ﬁxing a speciﬁc way. We assume he ansi ion p obabili ies in an independen
subsys em a e he same as i i we e he en i e sys em.
An impo an assump ion we impose on he p obabili y dis ibu ion is ha he
p obabili y o each ansi ion is uniquely de e mined by he associa ed mul ise o
ules and by he se o all applicable mul ise s o ules, ye i does no depend on
he objec s ha canno eac , o by he p e ious his o y o he compu a ion.
2.3 Regis e machines
In wha ollows we will need o simula e egis e machines; he e we b ieﬂy ecall
hei deﬁni ion and some o hei compu a ional p ope ies. A egis e machine is
a uple M= (m, B, l0, lh, P ), whe e mis he numbe o egis e s, Pis he se o
ins uc ions bijec i ely labeled by elemen s o B,l0∈Bis he ini ial label, and
lh∈Bis he ﬁnal label. The ins uc ions o Mcan be o he ollowing o ms:
•l1: (ADD (j), l2, l3), wi h l1∈B {lh},l2, l3∈B, 1 ≤j≤m
Inc ease he alue o egis e jby one, and non-de e minis ically jump o in-
s uc ion l2o l3. This ins uc ion is usually called inc emen .

Sel -S abiliza ion in Memb ane Sys ems 5
•l1: (SUB (j), l2, l3), wi h l1∈B {lh},l2, l3∈B, 1 ≤j≤m
I he alue o egis e jis ze o hen jump o ins uc ion l3, o he wise dec ease
he alue o egis e jby one and jump o ins uc ion l2. The wo cases o his
ins uc ion a e usually called ze o- es and dec emen , espec i ely.
•lh:HALT . S op he execu ion o he egis e machine.
A egis e machine is de e minis ic i l2=l3in all i s ADD ins uc ions. A
conﬁgu a ion o a egis e machine is desc ibed by he con en s o each egis e
and by he alue o he p og am coun e , which indica es he nex ins uc ion o
be execu ed. Compu a ions s a by execu ing he ﬁ s ins uc ion o P(labeled
wi h l0), and e mina e wi h eaching a HALT -ins uc ion.
Regis e machines p o ide a simple uni e sal compu a ional model [3]. Regis e
machines can be used as accep ing o as gene a ing as well as as decision de ices.
In accep ing egis e machines, a ec o o non-nega i e in ege s is accep ed i
and only i he egis e machine hal s ha ing i as inpu . Usually, wi hou loss o
gene ali y, we may assume ha he ins uc ion lh:HALT always appea s exac ly
once in P, wi h label lh. In he gene a i e case, we s a wi h emp y egis e s
and ake as esul s o all possible hal ing compu a ions. Being used as decision
de ices, egis e machines may hal in an accep ing s a e wi h label lyes o in a
ejec ing s a e lno, espec i ely In he ollowing, we shall call a speciﬁc model o
P sys ems compu a ionally comple e i and only i o any egis e machine Mwe
can eﬀec i ely cons uc an equi alen P sys em Πo ha ype simula ing each
s ep o Min a bounded numbe o s eps and yielding he same esul s.
3 Resul s
3.1 Accep ing sys ems
Fo he ollowing heo em we conside any compu a ionally comple e model o P
sys ems as deﬁned abo e, e.g., a model wi h maximally pa allel mul ise ew i ing
o wi h con olled sequen ial mul ise ew i ing.
Theo em 1. I a model o P sys ems yields a compu a ionally comple e class, hen
weakly sel -s abilizing subclass accep s exac ly NREC.
P oo . Fo any ecu si e numbe se he e is a egis e machine Mwi h one ac-
cep ing s a e qyes and one ejec ing s a e qno, deciding i . We modi y he egis e
machine in o de o ob ain a egis e machine M′which, once he decision is made,
i.e., qyes o qno has been eached, e ases he wo kspace and hen en e s q′
yes o q′
no
espec i ely, he eby hal ing in q′
yes i and only i he inpu is accep ed o pe -
o ming an inﬁni e loop wi h q′
no : (SUB (1) , q′
no, q′
no) i and only i he inpu x
is ejec ed. This egis e machine M′now can be simula ed wi h a P sys em Π,
which by cons uc ion s a s wi h a conﬁgu a ion ep esen ing he inpu xand will
ei he end wi h hal ing in a conﬁgu a ion ep esen ing he s a e q′
yes o else looping
in a conﬁgu a ion ep esen ing he s a e q′
no, i.e., Πis weakly sel -s abilizing.
6 A. Alhazo e al.
Con e sely, conside a sel -s abilizing P sys em Π, i.e., o each inpu x,Π
pe o ms a compu a ion ha ends up in a conﬁgu a ion om a ﬁni e se Sand
hen canno each any o he conﬁgu a ion ou side S. Now conside he de i a ion
g aph o all possible compu a ions o Πon he inpu x, i.e., he nodes o his
di ec ed g aph ep esen he conﬁgu a ions and he edges indica e he de i a ion
s eps om one conﬁgu a ion o he nex one du ing one o hese compu a ions.
As he numbe o conﬁgu a ions di ec ly de i able om any conﬁgu a ion in Πis
ﬁni e, his de i a ion g aph is a connec ed di ec ed g aph wi h ﬁni e deg ee ( om
each node, only a ﬁni e numbe o edges is lea ing); mo eo e , his g aph canno
ha e a simple pa h (a pa h isi ing each node a mos once) which is inﬁni e,
as e e y compu a ion in Πhas o each a conﬁgu a ion (node) om Sand hen
canno lea e he se o conﬁgu a ions Sany mo e. Due o K¨onig’s lemma4, he o al
numbe o nodes (conﬁgu a ions) in he de i a ion g aph mus be ﬁni e. Hence,
e en wi hou knowing he se S, he b u e o ce algo i hm compu ing all possible
ansi ions om he ini ial conﬁgu a ion, bu hal ing as soon as he sys em hal s
o a conﬁgu a ion al eady passed p e iously is eached, yields a decision p ocedu e
o he se accep ed by Π.
S eng hening his esul by emo ing “weakly” is p oblema ic, e en i mo e
powe ul P sys ems a e used. Indeed, sel -s abiliza ion also om un eachable con-
ﬁgu a ions would need o handle no only he conﬁgu a ions wi hou any s a e
o wi h mul iple s a es (which could be handled wi h he join powe o maximal
pa allelism and p io i ies), bu also conﬁgu a ions ep esen ing a si ua ion wi h
only one s a e which is no he ini ial s a e o he unde lying egis e machine. We
ha e o lea e his ques ion open.
Theo em 2. I a model o P sys ems yields a compu a ionally comple e class, hen
he weakly almos su ely sel -s abilizing P sys ems o his class accep exac ly NRE.
P oo . We s a wi h he cons uc ion om Theo em 1. We wan o show ha
elaxing he p ope y “weakly sel -s abilizing” o “almos su ely” leads om e-
cu si eness o compu a ional comple eness. I suﬃces o handle he case when he
sys em ejec s he inpu by ne e hal ing. We modi y he unde lying egis e ma-
chine as ollows: add a non-de e minis ic ansi ion om e e y s a e p∈Q o a
new s a e e5, om ee ase he con en s o all egis e s and hen jump back o e.
This will no aﬀec he accep ing powe , bu i will p o ide a sel -s abilizing pa h
om any eachable non-hal ing conﬁgu a ion.
4K¨onig’s lemma: Le Gbe a connec ed g aph wi h ﬁni e deg ee. I Gcon ains an inﬁni e
numbe o nodes, hen i con ains an inﬁni e simple pa h.
5The ansi ion om p o ecan be done by p: (ADD(j), e, e), since he egis e s
hen a e emp ied anyway. Fu he mo e, he basic model o egis e machines does
no allow non-de e minism o he han p: (ADD(j), q, ). The b anching a ADD in-
s uc ions migh be done by assuming he o iginal compu a ion o be de e minis ic
and eplacing p: (ADD(j), q, q) by p: (ADD(j), q, e). The b anching a a SUB in-
s uc ion p: (SUB(j), q, ) may be done by he sequence o ules p: (ADD(j), e, p′),
p′: (SUB(j), p′′ , p′′ ), p′′ : (SUB(j), q, ).
Sel -S abiliza ion in Memb ane Sys ems 7
The p obabili y ha he compu a ion does no sel -s abilize o mo e han k
s eps dec eases exponen ially wi h espec o k. Indeed, he simula ion o a egis-
e machine by P sys em has bounded pa allelism, each ins uc ion is simula ed
in a bounded numbe o s eps, and a leas one pa h leads o sel -s abiliza ion.
Mo eo e , he e only exis s a ﬁni e numbe o diﬀe en se s o applicable mul i-
se s con aining a b anching om he simula ion in o he sel -s abiliza ion pa h, so
he minimum p obabili y o his sel -s abiliza ion pa h is s ic ly posi i e. These
obse a ions conclude he p oo . 
Theo em 3. I a model o P sys ems yields a compu a ionally comple e class,
hen he class o all almos su ely sel -s abilizing maximally pa allel/sequen ial P
sys ems wi h p io i ies accep s exac ly NRE.
P oo . Gi en a se L om NRE, we ﬁ s cons uc a P sys em Πsimula ing a
egis e machine Maccep ing Land hen ex end Π o a P sys em Π′e en ulﬁlling
he condi ion o being almos su ely sel -s abilizing.
Le M= (m, B, l0, lh, P ) a de e minis ic egis e machine accep ing L. We now
cons uc he P sys em Π= (O, l0, R, >) wi h p io i ies accep ing L:
O=B∪ {ai|1≤i≤m},
R={l1→ajl2|l1: (ADD (j), l2)∈P}
∪ {ajl1→l2, l1→l3|l1: (SUB (j), l2, l3)∈P}
>={ajl1→l2> l1→l3|l1: (SUB (j), l2, l3)∈P}.
The con en s o a egis e i, 1 ≤i≤m, is ep esen ed by he numbe o symbols
aiin Π. The s a e lo he egis e machine is ep esen ed by he co esponding
symbol lin Π, oo. When Mhal s in lhwi h all egis e s being emp y, Πalso
hal s wi h he conﬁgu a ion {lh}. Ob iously, Πaccep s L, bo h in he sequen ial
as well as in he maximally pa allel mode.
To s eng hen he esul o e en non-weak almos su e sel -s abiliza ion, we
ha e o ake in o accoun he non- eachable conﬁgu a ions, oo. The almos su ely
sel -s abilizing P sys em Π′= (O′, l0, R′, >′) wi h p io i ies accep ing Lis con-
s uc ed as ollows:
O′=B∪ {ai|1≤i≤m}∪{e},
R′={l1→ajl2|l1: (ADD (j), l2)∈P}
∪ {ajl1→l2, l1→l3|l1: (SUB (j), l2, l3)∈P}
∪ {ai→e|1≤i≤m}∪{ex →e|x∈O′}∪{e→e}
∪ {l→e|l∈B {lh}} ∪ {ll′→e|l, l′∈B},
>′={ajl1→l2> l1→l3|l1: (SUB (j), l2, l3)∈P}
∪ {ex →e > , ll′→e > |l, l′∈B, x ∈O′, ∈R}
∪ {l→e > ai→e|l∈B {lh},1≤i≤m}
∪ { > e →e| ∈R′ {e→e}} .
In addi ion o he idea o he cons uc ion gi en in he p oo o Theo em 2
using he exi eby applying a ule l→e,l∈B {lh}, i suﬃces o sel -s abilize
8 A. Alhazo e al.
om he conﬁgu a ions wi h no s a e and om he conﬁgu a ions wi h mul iple
s a es o he egis e machine. Mul iple s a es can be educed by he ules ll′→e,
l, l′∈B. I no s a e symbol is p esen , hen we may exi wi h one o he ules
ai→e, 1 ≤i≤m. All emaining cases can be cap u ed by he ules ex →e,
x∈O′. By cons uc ion, he sel -s abilizing se Sequals {{lh},{e}}. The whole
cons uc ion again is alid o he sequen ial as well as he maximally pa allel
mode. 
An open ques ion is whe he p io i ies in Theo em 3 can be eplaced by p o-
mo e s o inhibi o s.
3.2 Gene a ing sys ems
Theo em 4. Any ﬁni e se Mo numbe s can be gene a ed by some sel -s abilizing
memb ane sys em wi hou con ol.
P oo . Conside a P sys em Π= ({s, a}, s, R), whe e
R={s→an|n∈M} ∪ {amax(M)+1 →λ, ss →s}.
I is no diﬃcul o see ha Πgene a es Mand ( aking S={an|n≤max(M)}∪
{s}) i is sel -s abilizing. 
Since sel -s abiliza ion implies se -con e gence and closu e, and elaxing ei he
p ope y ( o possibly, almos su ely and/o weakly) does no comp omise he con-
s uc ion o he P sys em descibed in he p oo o Theo em 4, he lowe bound
on he gene a i e powe o associa ed sys ems es ic ed o any p ope y we ha e
deﬁned, is a leas NF IN .
Lemma 1. A possibly ﬁni e se -con e ging sys em only gene a es ﬁni e se s.
P oo . I ollows om Deﬁni ion 1 ha o a sys em possibly con e ging o a se
S,Scon ains all hal ing conﬁgu a ions. Since Sis ﬁni e, so is he se o all he
hal ing conﬁgu a ions. Hence, a mos NF IN can be gene a ed. 
Theo em 5. Any o he ollowing classes o P sys ems dpOP m(c)gene a e ex-
ac ly NF IN :
•dis possibly/almos su ely/ -
•pis sel -s abilizing/ﬁni e se -con e ging
•mis maximally pa allel/sequen ial
•cis uncon olled/wi h p omo e s/wi h inhibi o s/wi h p io i ies.
P oo . The claims o he heo em di ec ly ollow om Theo ems 4 and 5.
We now p oceed o weak p ope ies o gene a i e sys ems.
Theo em 6. Weakly almos su ely sel -s abilizing P sys ems gene a e exac ly
NF IN .
Sel -S abiliza ion in Memb ane Sys ems 9
P oo . The lowe bound is shown by Theo em 4. Now ake a weakly sel -s abilizing
P sys em Π, and i s associa ed se S om he deﬁni ion o he p ope y. Conside
an a bi a y hal ing compu a ion o Π. Le Cbe he conﬁgu a ion o Πone s ep
be o e he hal ing. In e p e ing ﬁni e se -con e gence o Cimplies ha he hal ing
conﬁgu a ion belongs o S. Since he hal ing compu a ion has been a bi a ily
chosen, he se o all hal ing conﬁgu a ions is a subse o S, and hence i is ﬁni e.
The e o e, he se gene a ed by Πis ﬁni e, oo. 
Theo em 7. I a model o P sys em yields a compu a ionally comple e class, hen
weakly possibly sel -s abilizing subclass gene a es NRE.
P oo . Conside he cons uc ion om Theo em 2, bu o a gene a i e P sys em.
The simula ion o he unde lying egis e machine is ca ied ou un il some poin .
Unless he P sys em has al eady hal ed, i always has a choice o sel -s abilize and
loop. 
4 Conclusions
We ha e p esen ed some esul s conce ning he concep o sel -s abiliza ion, e-
cen ly p oposed o memb ane compu ing. I s essence is in eachabili y and closu e
o a ﬁni e se .
Some o he ob ained esul s can be summa ized in he ollowing able:
P ope y compu a ionally comple e (sequ/maxpa )+p i Thm
sel s abilizing acc. ?/gen. NF IN -/5
almos su ely s.s. acc. ?/gen. NF IN acc. NRE/gen. NF IN 3/5
possibly s.s. acc. ?/gen. NF IN acc. NRE/gen. NF IN 3/5
weakly s.s. acc. NREC/gen. NF IN 1/6
weakly almos su ely s.s. acc. NRE/gen. NF IN 2/6
weakly possibly s.s. acc. NRE/gen. NRE 2/7
One o he ques ions we p oposed is whe he p io i ies may be eplaced by p o-
mo e s o inhibi o s in Theo em 2. Ano he open ques ion is he powe o accep ing
wi h un es ic ed sel -s abiliza ion, e en i maximal pa allelism is combined wi h
p io i ies (a commen a e Theo em 1 and he ﬁ s ques ion ma k in he able
abo e). The o he open ques ions a e also ma ked wi h ques ion ma ks in he a-
ble abo e. Any sys em in he co esponding classes mus (besides doing he ac ual
compu a ion) con e ge (deﬁni ely, in p obabili y o possibly) o some ﬁni e se
om anywhe e, wi hou using he join powe o maximal pa allelism and con ol.
We men ion wo opics ha we do no deal wi h he e. One is conside ing he
ﬁni e se as a pa ame e , possibly leading o a discussion in model checking. The
o he one conce ns eachabili y ques ions in dynamic memb ane s uc u es.

10 A. Alhazo e al.
Acknowledgemen s
The ﬁ s au ho g a e ully acknowledges he p ojec Re oNe by he Lomba dy
Region o I aly unde he ASTIL P og am ( egional dec ee 6119, 20100618). The
wo k o he second, he ou h and he ﬁ h au ho was pa ially suppo ed by
Uni e si `a degli S udi di Milano-Bicocca, Fondo di A eneo pe la Rice ca (FAR)
2011.
Re e ences
1. A. Alhazo : P ope ies o Memb ane Sys ems. Memb ane Compu ing, 12 h In e na-
ional Con e ence, CMC 2011, Fon ainebleau, Re ised Selec ed Pape s (M. Gheo ghe,
Gh. P˘aun, G. Rozenbe g, A. Salomaa, S. Ve lan, Eds.), Lec u e No es in Compu e
Science 7184, 2012, 1–13.
2. E.A. Eme son: Tempo al and Modal Logic. In: Handbook o Theo e ical Compu e
Science, Chap e 16, he MIT P ess, 1990.
3. M.L. Minsky: Fini e and In ini e Machines. P en ice Hall, Englewood Cliﬀs, New
Je sey, 1967.
4. Gh. P˘aun: Memb ane Compu ing. An In oduc ion. Sp inge , 2002.
5. Gh. P˘aun, G. Rozenbe g, A. Salomaa: The Ox o d Handbook o Memb ane Compu ing,
Ox o d Uni e si y P ess, 2010.
6. G. Rozenbe g, A. Salomaa: Handbook o Fo mal Languages, 3 ol., Sp inge , 1997.
7. P sys ems webpage: h p://ppage.psys ems.eu
8. h p://en.wikipedia.o g/wiki/Sel -s abiliza ion
Cha ac e izing he Compu a ional Powe o
Ene gy-Based P Sys ems
A iom Alhazo 1,2, Ma co An onio i1, Albe o Lepo a i1
1Uni e si `a degli S udi di Milano-Bicocca
Dipa imen o di In o ma ica, Sis emis ica e Comunicazione
Viale Sa ca 336, 20126 Milano, I aly
{a iom.alhazo ,ma co.an onio i,albe o.lepo a i}@unimib.i
2Ins i u e o Ma hema ics and Compu e Science
Academy o Sciences o Moldo a
Academiei 5, Chi¸sin˘au MD-2028 Moldo a
[email p o ec ed]
Summa y. We in es iga e he compu a ional powe o ene gy-based P sys ems, a model
o memb ane sys ems whe e a ﬁxed amoun o ene gy is associa ed wi h each objec and
he ules ans o m single objec s by adding o emo ing ene gy om hem. We answe
ecen ly p oposed open ques ions abou he powe o such sys ems wi hou p io i ies asso-
cia ed o he ules, o bo h sequen ial and maximally pa allel modes. We also conjec u e
ha de e minis ic ene gy-based P sys ems a e no compu a ionally comple e.
1 In oduc ion
Memb ane sys ems (also called P sys ems) ha e been in oduced in [11] as a class
o dis ibu ed and pa allel compu ing de ices, inspi ed by he s uc u e and unc-
ioning o li ing cells. Since hen, many a ian s o P sys ems ha e been deﬁned
in he li e a u e. In wha ollows we assume he eade is amilia wi h he basic
no ions and he e minology unde lying P sys ems. A sys ema ic in oduc ion o
he a ea can be ound in [12]; a ecen o e iew o he de elopmen s is p esen ed
in [13], whe eas he la es in o ma ion can be ound in [15].
In his pape we conside ene gy-based P sys ems [7, 8, 6], a model o compu a-
ion in he amewo k o Memb ane Compu ing in which a gi en amoun o ene gy
is associa ed o each objec , and he ene gy manipula ed du ing compu a ions is
aken in o accoun by means o conse a i e ules.
Le us no e in passing ha he e has been o he a emp s in he li e a u e
o inco po a e ce ain conse a ion laws in memb ane compu ing. One is pu ely
communica i e models, o which he mos ho oughly s udied is P sys ems wi h
sympo /an ipo [10]. In hese sys ems he compu a ion is ca ied ou by mo ing
objec s be ween he egions in g oups. To each compu a ional comple eness, he
12 A. Alhazo , M. An onio i, A. Lepo a i
wo kspace is inc eased by b inging (some ypes o ) objec s om he en i onmen ,
whe e hey can be ound in an unbounded supply. Ano he model is con o mon P
sys ems [5], whe e compu a ions a e pe o med by edis ibu ing ene gy be ween
objec s, han can also be enamed and mo ed. A ea u e o hese sys ems is ha a
diﬀe en amoun o ene gy may be embedded in he same objec a diﬀe en ime
s eps. Ye ano he app oach is o assign ene gy o memb anes, as in P sys em wi h
Uni Rules and Ene gy assigned o Memb anes (UREM P sys ems, o sho ) [1].
He e he compu a ions a e pe o med by ules enaming and mo ing an objec
ac oss a memb ane, possibly modi ying he ene gy assigned o ha memb ane.
I has been p o ed in [1] ha UREM P sys ems wo king in he sequen ial mode
cha ac e ize P sMAT, he amily o Pa ikh se s gene a ed by ma ix g amma s
wi hou appea ance checking (and wi h e asing ules), and ha hei powe is
inc eased o PsRE ( he amily o ecu si ely enume able Pa ikh se s) i p io i ies
a e assigned o he ules o he mode o applying he ules is changed o maximally
pa allel.
As s a ed abo e, in his pape we conside ene gy-based P sys ems, in which
ene gy is assigned o objec s in a way ha each objec om he alphabe is assigned
a speciﬁc alue. Ins ances o a special symbol a e used o deno e ee ene gy uni s
occu ing inside he egions o he sys em. The compu a ions a e ca ied ou by
ules enaming and possibly mo ing objec s, which may consume o elease ee
ene gy in he egion, espec ing he ene gy conse a ion law ( ha is, he o al
amoun o ene gy associa ed wi h he objec s ha appea in he le hand side o
a ule is he same as he ene gy occu ing in he igh hand side). The esul o
a compu a ion may be in e p e ed in many ways: o example, as he amoun o
ee ene gy uni s in a designa ed ou pu egion. Also o his model, o gi e he
possibili y o each compu a ional comple eness i is necessa y (bu no suﬃcien ,
as we will see) ha he e may be an unbounded amoun o ee ene gy in (a leas
one) speciﬁed egion o he sys em. In [6] i is p o ed ha ene gy-based P sys ems
wo king in he sequen ial way and using a o m o local p io i ies associa ed o
he ules a e compu a ionally comple e. Wi hou p io i ies, hei beha io can be
simula ed by ec o addi ion sys ems, and hence a e no uni e sal. Howe e , in [6]
he p ecise cha ac e iza ion o he compu a ional powe o ene gy-based P sys ems
wi hou p io i ies is le as an open p oblem. A ela ed open ques ion was whe he
ene gy-based P sys ems can each compu a ional comple eness by wo king in he
maximally pa allel mode, wi hou p io i ies, as i happens wi h UREM P sys ems
[1].
In his pape we answe hese ques ions, by showing ha he powe o ene gy-
based P sys ems con aining an inﬁni e amoun o ee ene gy and wi hou p io i ies
is exac ly PsMAT when wo king in he sequen ial mode, and PsRE when wo k-
ing in he maximally pa allel mode. None heless we will end wi h ano he open
ques ion: wha is he powe o ene gy-based P sys ems unde he es ic ion o
de e minism? We conjec u e non-uni e sali y o his case.
The es o he pape is s uc u ed as ollows. The nex sec ion con ains some
ma hema ical p elimina ies, o ﬁx he no ions, deﬁni ions and no a ions wi h which
Cha ac e izing he Compu a ional Powe o Ene gy-Based P Sys ems 13
we will wo k. Sec ion 3 con ains ou esul s conce ning he cha ac e iza ion o
he compu a ional powe o ene gy-based P sys ems wo king wi hou p io i ies,
ei he in he sequen ial o in he maximally pa allel mode. Sec ion 4 con ains he
conclusions and some discussion on he abo e men ioned open p oblem, conce ning
he compu a ional powe o de e minis ic ene gy-based P sys ems.
2 P elimina ies
We assume he eade o be amilia wi h he basics o o mal languages; on his
subjec one may e e o, e.g., [14].
We deno e by N he se o non-nega i e in ege s. An alphabe Vis a ﬁni e non-
emp y se o abs ac symbols. Gi en V, he ee monoid gene a ed by Vunde
he ope a ion o conca ena ion is deno ed by V∗; he emp y s ing is deno ed by
λ, and V∗− {λ}is deno ed by V+. By |x|we deno e he leng h o he wo d x
o e V. Le {a1, . . . , an}be an a bi a y alphabe ; he numbe o occu ences o
a symbol aiin xis deno ed by |x|ai; he Pa ikh ec o associa ed wi h xwi h
espec o a1, . . . , anis (|x|a1, . . . , |x|an).The Pa ikh image o a language L
o e {a1, . . . , an}is he se o all Pa ikh ec o s o s ings in L. Fo a amily o
languages FL, he amily o Pa ikh images o languages in FL is deno ed by PsFL.
A ﬁni e mul ise ⟨m1, a1⟩. . . ⟨mn, an⟩wi h mi∈N,1≤i≤n, is ep esen ed as
any s ing x he Pa ikh ec o o which wi h espec o a1, . . . , anis (m1, . . . , mn).
The amily o ecu si ely enume able languages is deno ed by RE, and he amily
o con ex - ee languages by CF. The amily o all ecu si ely enume able se s o
k-dimensional ec o s o non-nega i e in ege s can hus be deno ed by Ps(k)RE.
Since numbe s can be seen as one-dimensional ec o s, we can eplace P s(1) by N
in he no a ion, hus ob aining NRE.
2.1 Ma ix G amma s
A con ex - ee ma ix g amma (wi hou appea ance checking) is a cons uc
G= (N, T, S, M) whe e Nand Ta e se s o non- e minal and e minal sym-
bols, espec i ely, wi h N∩T=∅,S∈Nis he s a symbol,Mis a ﬁni e se o
ma ices,M={mi|1≤i≤n}, whe e he ma ices mia e sequences o he o m
mi= [mi,1, . . . , mi,ni], ni≥1, 1 ≤i≤n, and he mi,j , 1 ≤j≤ni, 1 ≤i≤n,
a e con ex - ee p oduc ions o e (N, T). Fo mi= [mi,1, . . . , mi,ni] and , w ∈
(N∪T)∗we deﬁne =⇒miwi and only i he e a e w0, w1, . . . , wni∈(N∪T)∗
such ha w0= , wni=w, and o each j, 1≤j≤ni,wjis he esul o he
applica ion o mi,j o wj−1. The language gene a ed by Gis
L(G) = {w∈T∗|S=⇒mi1w1. . . =⇒mikwk, wk=w,
wj∈(N∪T)∗, mij∈M o 1 ≤j≤k, k ≥1}.
Acco ding o he deﬁni ions gi en in [2], he las ma ix can al eady ﬁnish wi h
a e minal wo d wi hou ha ing applied he whole sequence o p oduc ions. The
20 A. Alhazo , M. An onio i, A. Lepo a i
e
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Fig. 1. Simula ion o he ze o- es o l1:SUB(j, l2, l3). The case when he egis e is
no ze o is shown by do ed lines and symbols in pa en heses, and he compu a ion s ops
be o e he dashed line.
P oo . We s a by no icing ha he cons uc ion om Theo em 1 p oduces he
same esul when he P sys em wo ks in he maximally pa allel mode. Thus, i
suﬃces o only add a simula ion o ze o- es ins uc ions wi hou dis up ing he
exis ing machine y. The new sys em nonde e minis ically chooses be ween dec e-
men and ze o- es , and blocks he simula ion p ocess i he ze o- es ails.
We hus add memb anes [ ]1′,[ ]2′, . . . , [ ]m′and he ollowing se s o
ules o he ene gy-based P sys em ΠMmen ioned in he p oo o Theo em 1:
R0
s={1 : l1e→(l′
1, in(j′)),3a:l′′′
1→(l1, in(j)),
5a:l′
1→(l1, in(j′)) e, 5b:zje→(Zj, in(j)) e,
7b:z′
je→(Z′
j, in(j′)),12a:l(i )
1→l3e2
|l1:SUB(j, l2, l3)∈P},
R0
j={4a:l1e→(l′
1, ou ),6b:Zj→(z′
j, ou )e∈R′′
j
|l1:SUB(j, l2, l3)∈P},
R0
j′={2 : l′
1→(l′′′
1, ou )e, 3b:zje→Zj,4b:Zj→(zj, ou ),
8b:Z′
je→Z′′
j,9b:Z′′
j→zje2,10a:l1e→l′′′
1,
11a:l′′′
1e→(l(i )
1, ou )|l1:SUB(j, l2, l3)∈P}.
The case o co ec simula ion o a ze o- es is illus a ed in Figu e 1. Indeed,
i egion jdoes no con ain any objec e, hen he ollowing sequence o mul ise s
o ules is applied: 1,2,(3a, 3b),(4a, 4b),(5a, 5b),6b, 7b, 8b, 9b, 10a, 11a, 12a. In his
way, l1is ans o med o l3while he o he objec s used (ene gy and zj) a e
ep oduced. On he o he hand, i egion jcon ains some objec e, co esponding
o a non-ze o alue o he co esponding egis e , hen he sequence o mul ise s
o ules applied is 1,2,(3a, 3b),(4a, 4b),(5a, 5b),(6b, 10a),7b, and he simula ion
p ocess is blocked. We ecall ha blocking he simula ion p ocess leads o an

Cha ac e izing he Compu a ional Powe o Ene gy-Based P Sys ems 21
inﬁni e compu a ion due o he T↔ e loop in egion . In wo ds, because o ee
ene gy in egion j, objec l1le ha egion h ee s eps ea lie . As a esul , ins ead
o objec Z′
jconsuming 1 uni o ene gy and hen eleasing 2 uni s needed o
l1, he exis ing uni o ene gy has been consumed by l1, lea ing he compu a ion
unﬁnished.
The ull P sys em, deﬁned using componen s o he cons uc ion om Theo em
1, is gi en below:
Π′′ = (A′′, ε′′ , µ′′ , e, w′′
s, . . . , w′′
, R′′
s, . . . , R′′
),whe e
A′′ =A∪ {l′′′ |l∈B}∪{zj, z′
j, Zj, Z′
j, Z′′
j|1≤j≤m},
ε′′(x) = ε(x),∀x∈A, ε(l′′′ ) = 1, ε(l(i )) = 2,∀l∈B,
ε′′(zj) = ε′′ (z′
j) = 0, ε′′(Zj) = ε′′ (Z′
j) = 1, ε′′(Z′′
j) = 2,1≤j≤m,
µ= [ [ ]1· · · [ ]m[ ]1′· · · [ ]m′[ ] ]s,
w′′
s=ws, w′′
j=wj,1≤j≤m, w′′
=w ,
w′′
j′=zj,1≤j≤m,
R′′
s=Rs∪R0
s, R′′
j=Rj∪R0
j,1≤j≤m,
R′′
j′=R0
j′,1≤j≤m, R′′
=R .
As we can see, sys em Π′′ uses he skin memb ane, one memb ane o con ol he
hal ing, and wo memb anes o each o he m egis e s, o a o al o 2m+ 2
memb anes.
Since i is known (see P oposi ion 3) ha m=β+2 egis e s suﬃce o gene a e
any ecu si ely enume able se L⊆Ps(β)RE o ec o s o non-nega i e in ege s
by nonde e minis ic egis e machines, we would ob ain 2β+ 6 memb anes. How-
e e , as ecalled abo e, when using egis e machines as gene a ing de ices we can
assume wi hou loss o gene ali y ha only ADD ins uc ions a e applied o he
ou pu egis e s. So he numbe o memb anes needed o simula e M educes o
β+ 2(m−β) + 2 = β+ 6. ⊓⊔
By pu ing β= 1 in he abo e heo em we ob ain a cha ac e iza ion o NRE:
Co olla y 2. NOP7(ene gy∗) = NRE.
whe eas i we make he union o all classes Ps(β)OPβ+6(ene gy∗) o β anging
h ough he se o non-nega i e in ege s we ob ain a cha ac e iza ion o PsRE:
Co olla y 3. PsOP∗(ene gy∗) = PsRE.
As s a ed abo e, hese esul s can be easily gene alized o he cases in which
ene gy-based P sys ems a e used as accep ing de ices o as de ices compu ing
pa ial ecu si e unc ions. Fi s o all no e ha he ene gy-based P sys ems buil
in he p oo s o Theo ems 1 and 3 can be easily modiﬁed o simula e de e minis ic
egis e machines. Conside ing he compu ing case, we know om P oposi ion 1
ha m= max{α, β}+2 egis e s suﬃce o compu e any pa ial ecu si e unc ion
:Nα→Nβ. To simula e such a egis e machine we would ob ain 2 max{α, β}+6
22 A. Alhazo , M. An onio i, A. Lepo a i
memb anes o he sys em Π′′ buil in he p oo o Theo em 3. Howe e , his
numbe can be educed o α+ max{α, β}+ 6 by conside ing ha :
•as s a ed abo e, we can assume ha only ADD ins uc ions a e applied o
he ou pu egis e s. This means ha only one memb ane (ins ead o wo) is
needed o simula e he beha iou o each ou pu egis e ;
•in gene al some inpu egis e s may also be used as ou pu egis e s. Howe e ,
any “p imed” memb ane j′associa ed wi h an inpu egis e , 1 ≤j′≤α,
canno be used also as a memb ane associa ed o an ou pu egis e , due o
he objec zj esiding in he memb ane. Hence, wi h αinpu s and βou pu s we
need αp imed memb anes plus max{α, β}non-p imed memb anes. By adding
wo memb anes o each o he 2 addi ional egis e s o M, plus memb anes
and s, we ob ain α+ max{α, β}+ 6 memb anes.
As pa icula cases, we need 2α+ 6 memb anes o he accep ing case and β+ 6
memb anes o he gene a ing case.
4 Conclusions and Fu u e Wo k
In his pape we ha e conside ed ene gy-based P sys ems, a model o memb ane sys-
ems wi h ene gy assigned o objec s. We ha e answe ed wo ques ions abou hei
compu a ional powe , and we ha e hus p o ed ha i ma ches Pa ikh mapping
o ma ix languages when he ules o he P sys ems a e applied in he sequen ial
mode, whe eas he e is compu a ional comple eness in he maximally pa allel case.
As a di ec ion o u u e esea ch, we p opose he ollowing p oblem: Wha is
he compu a ional powe o de e minis ic ene gy-based P sys ems? We conjec u e
ha hey a e no uni e sal. The ques ion o igina es om he ac ha in [7, 8]
ene gy-based P sys ems a e used o simula e F edkin ga es and F edkin ci cui s,
espec i ely; howe e , he simula ion is pe o med in a nonde e minis ic way, ely-
ing on he ac ha soone o la e he simula ion will choose he co ec sequence
o ules. No e ha i he w ong ules a e chosen he simula ion is no abo ed;
he s a e o he sys em is “ olled back” so ha a new nonde e minis ic choice can
be made, hope ully he co ec one. Clea ly his si ua ion could p oduce inﬁni e
loops; his is why one would like ins ead o ha e a de e minis ic simula ion.
He e we can only gi e an in o mal jus iﬁca ion o ou conjec u e. No ice ha
objec s only in e ac indi ec ly, ia eleasing ee ene gy uni s in a egion o con-
suming hem. Conside a dependency g aph whose nodes a e iden iﬁed by objec -
egion pai s. Two nodes a e connec ed i he co esponding objec s a e p esen
in he associa ed egions in some ule. A sys em is de e minis ic i no b anching
can be eﬀec i ely used in i s compu a ions, so emo ing unusable ules would lead
o a dependency g aph whe e each node has ou -deg ee a mos one. Hence, any
objec occu ing in he ini ial conﬁgu a ion o he sys em has some p ede e mined
e olu ion pa h, and one o he ollowing cases mus happen:
• he pa h is ﬁni e, and he objec e ol es un il he e a e no associa ed ules,
Cha ac e izing he Compu a ional Powe o Ene gy-Based P Sys ems 23
• he pa h leads o a cycle, and he objec e ol es o e e ( he compu a ion yields
no esul ),
• he e olu ion is “ ozen” because he e is no enough ene gy o he associa ed
ule.
In ene gy-based P sys ems, he only way one objec can inﬂuence he beha io
o ano he objec is by manipula ing ene gy, leading o eezing o un eezing
he compu a ional pa h o ano he objec . The e is no de e minis ic way o se
an objec o wo diﬀe en pa hs. I a “ ozen” objec ecei es enough ene gy o
con inue i s e olu ion, hen i s compu a ional pa h is he same as i i was ne e
ozen.
So he in o ma ion ha can be passed om an objec o ano he one is qui e
limi ed: gi ing he la e ene gy, as opposed o le ing i eeze o e e . Howe e ,
e e y ime his happens, some objec mus s op e ol ing o e e . Since he ini ial
numbe o objec s is ﬁxed and canno inc ease, he communica ion complexi y is
bounded and his should imply non-uni e sali y.
Howe e , e en i de e minis ic ene gy-based P sys ems we e no uni e sal, hey
could none heless be able o simula e F edkin ga es. This should be doable i
lea ing some “ga bage” in o he sys em a he end o he compu a ion is allowed.
Indeed, he ac i e objec s could un eeze he desi ed ones, p oducing he needed
esul . Mo e diﬃcul would be designing an ene gy-based P sys em ha can be
eused o simula e a F edkin ga e as many imes as desi ed. We expec he eusable
cons uc ion o be impossible, o he same easons as exposed abo e.
Acknowledgemen s
The ideas exposed in his pape eme ged du ing and a e he Ten h B ains o ming
Week on Memb ane Compu ing (BWMC 10), held in Se ille in conjunc ion wi h
he In e na ional Con e ence on De elopmen s in Memb ane Compu ing, om Jan-
ua y 30 h o Feb ua y 3 d, 2012.
The ﬁ s au ho g a e ully acknowledges he p ojec Re oNe by he Lomba dy
Region o I aly unde he ASTIL P og am ( egional dec ee 6119, 20100618). The
wo k o he o he wo au ho s was pa ially suppo ed by Uni e si `a degli S udi
di Milano-Bicocca, Fondo di A eneo pe la Rice ca (FAR) 2011.
Re e ences
1. A. Alhazo , R. F eund, A. Lepo a i, M. Oswald, C. Zand on: (Tissue) P Sys ems wi h
Uni Rules and Ene gy Assigned o Memb anes. Fundamen a In o ma icae 74(4),
2006, 391–408.
2. J. Dassow, Gh. P˘aun: Regula ed Rew i ing in Fo mal Language Theo y, Sp inge -
Ve lag, Be lin, 1989.
3. R. F eund, O.H. Iba a, Gh. P˘aun, H.-C. Yen: Ma ix Languages, Regis-
e Machines, Vec o Addi ion Sys ems. In: P oceedings o he Thi d B ain-
s o ming Week on Memb ane Compu ing, Se illa, 2005, 155–168. A ailable a :
h p://www.gcn.us.es/3BWMC/b a olpd /b a ol155.pd
24 A. Alhazo , M. An onio i, A. Lepo a i
4. R. F eund, M. Oswald: GP Sys ems wi h Fo bidding Con ex . Fundamen a In o -
ma icae 49(1-3), 2002, 81–102.
5. P. F isco: The Con o mon-P sys em: A Molecula and Cell Biology-Inspi ed Com-
pu abili y Model. Theo e ical Compu e Science 312(2-3), 2004, 295–319.
6. A. Lepo a i, D. Besozzi, P. Cazzaniga, D. Pescini, C. Fe e i: Compu ing wi h En-
e gy and Chemical Reac ions. Na u al Compu ing 9, 2010, 493–512.
7. A. Lepo a i, C. Zand on, G. Mau i: Simula ing he F edkin Ga e wi h Ene gy-Based
P Sys ems. J Uni e s Compu Sci 10(5), 2004, 600-619.
8. A. Lepo a i, C. Zand on, G. Mau i: Re e sible P Sys ems o Simula e F edkin Ci -
cui s. Fundamen a In o ma icae 74, 2006, 529–548.
9. M.L. Minsky: Fini e and In ini e Machines, P en ice Hall, Englewood Cliﬀs, New
Je sey, 1967.
10. A. P˘aun, Gh. P˘aun: The Powe o Communica ion: P Sys ems wi h Sym-
po /An ipo . New Gene a ion Compu ing 20(3), 2002, 295–306.
11. Gh. P˘aun: Compu ing wi h Memb anes. J Compu Sys Sci 1(61), 2000, 108-143.
See also Tu ku Cen e o Compu e Science, TUCS Repo No. 208, 1998.
12. Gh. P˘aun: Memb ane Compu ing. An In oduc ion, Sp inge -Ve lag, Be lin, 2002.
13. Gh. P˘aun, G. Rozenbe g, A. Salomaa: The Ox o d Handbook o Memb ane Compu -
ing, Ox o d Uni e si y P ess, 2010.
14. G. Rozenbe g, A. Salomaa: Handbook o Fo mal Languages, 3 ol., Sp inge , 1997.
15. P Sys ems Webpage. h p://www.ppage.psys ems.eu/
Asynch onous and Maximally Pa allel
De e minis ic Con olled Non-Coope a i e
P Sys ems Cha ac e ize N F IN ∪coN F IN
A iom Alhazo 1,2and Rudol F eund3
1Uni e si `a degli S udi di Milano-Bicocca
Dipa imen o di In o ma ica, Sis emis ica e Comunicazione
Viale Sa ca 336, 20126 Milano, I aly
[email p o ec ed]
2Ins i u e o Ma hema ics and Compu e Science
Academy o Sciences o Moldo a
Academiei 5, Chi¸sin˘au MD-2028 Moldo a
[email p o ec ed]
3Facul y o In o ma ics, Vienna Uni e si y o Technology
Fa o i ens . 9, 1040 Vienna, Aus ia
E-mail: [email p o ec ed]
Summa y. Memb ane sys ems (wi h symbol objec s) a e dis ibu ed con olled mul ise
p ocessing sys ems. Non-coope a i e P sys ems wi h ei he p omo e s o inhibi o s (o
weigh no es ic ed o one) a e known o be compu a ionally comple e. In his pape
we show ha he powe o he de e minis ic subclass o such sys ems is compu a ionally
comple e in he sequen ial mode, bu only sub egula in he asynch onous mode and in
he maximally pa allel mode.
1 In oduc ion
The mos amous memb ane compu ing model whe e de e minism is a c i e ion o
uni e sali y e sus decidabili y is he model o ca aly ic P sys ems, see [2] and [4].
I is also known ha non-coope a i e ew i ing P sys ems wi h ei he p omo e s
o inhibi o s a e compu a ionally comple e, [1]. Mo eo e , he p oo sa isﬁes some
addi ional p ope ies:
•Ei he p omo e s o weigh 2 o inhibi o s o weigh 2 a e enough.
•The sys em is non-de e minis ic, bu i es o es he p e ious conﬁgu a ion i
he guess is w ong, which leads o co ec simula ions wi h p obabili y 1.
The pu pose o his pape is o o mally p o e ha compu a ional comple eness
canno be achie ed by de e minis ic sys ems when wo king in he asynch onous
o in he maximally pa allel mode.

26 A. Alhazo , R. F eund
2 De ini ions
An alphabe is a ﬁni e non-emp y se Vo abs ac symbols. The ee monoid
gene a ed by Vunde he ope a ion o conca ena ion is deno ed by V∗; he emp y
s ing is deno ed by λ, and V∗ {λ}is deno ed by V+. The se o non-nega i e
in ege s is deno ed by N; a se So non-nega i e in ege s is called co- ini e i N S
is ﬁni e. The amily o all ﬁni e (co-ﬁni e) se s o non-nega i e in ege s is deno ed
by NFIN (coNFIN, espec i ely). The amily o all ecu si ely enume able se s
o non-nega i e in ege s is deno ed by NRE. In he ollowing, we will use ⊆bo h
o he subse as well as he submul ise ela ion.
Since ﬂa ening he memb ane s uc u e o a memb ane sys em p ese es bo h
de e minism and he model, in he ollowing we es ic ou sel es o conside mem-
b ane sys ems as one- egion mul ise ew i ing sys ems.
A(one- egion) memb ane sys em (P sys em) is a uple
Π= (O, Σ, w, R′),
whe e Ois a ﬁni e alphabe , Σ⊆Ois he inpu sub-alphabe , w∈O∗is a s ing
ep esen ing he ini ial mul ise , and R′is a se o ules o he o m :u→ ,
u∈O+, ∈O∗.
A conﬁgu a ion o he sys em Πis ep esen ed by a mul ise o objec s om O
con ained in he egion, he se o all conﬁgu a ions o e Ois deno ed by C(O).
A ule :u→ is applicable i he cu en conﬁgu a ion con ains he mul ise
speciﬁed by u. Fu he mo e, applicabili y may be con olled by con ex condi ions,
speciﬁed by pai s o se s o mul ise s.
De ini ion 1. A ule wi h con ex condi ions ( , (P1, Q1),· · · ,(Pm, Qm)) is appli-
cable o a con igu a ion Ci is applicable, and he e exis s some j∈ {1,· · · , m}
o which
• he e exis s some p∈Pjsuch ha p⊆Cand
•q⊆ C o all q∈Qj.
In wo ds, con ex condi ions a e sa isﬁed i he e exis s a pai o se s o mul ise s
(called p omo e se and inhibi o se , espec i ely), such ha a leas one mul ise
in he p omo e se is a submul ise o he cu en conﬁgu a ion, and no mul ise
in he inhibi o se is a submul ise o he cu en conﬁgu a ion.
De ini ion 2. AP sys em wi h con ex condi ions and p io i ies on he ules is a
cons uc
Π= (O, Σ, w, R′, R, >)
whe e (O, Σ, w, R′)is a (one- egion) P sys em as de ined abo e, Ris a se o ules
wi h con ex condi ions and >is a p io i y ela ion on he ules in R; i ule ′has
p io i y o e ule , deno ed by ′> , hen canno be applied i ′is applicable.
A Cha ac e iza ion o N F I N ∪coN F I N 27
Th oughou he pape , we will use he wo d con ol o mean ha a leas one
o hese ea u es is allowed (con ex condi ions o p omo e s o inhibi o s only and
e en ually p io i ies).
In he sequen ial mode (sequ), a compu a ion s ep consis s in he non-
de e minis ic applica ion o one applicable ule , eplacing i s le -hand side
(lhs ( )) wi h i s igh -hand side ( hs ( )). In he maximally pa allel mode
(maxpa ), mul iple applicable ules may be chosen non-de e minis ically o be ap-
plied in pa allel o he unde lying conﬁgu a ion o disjoin submul ise s, possibly
lea ing some objec s idle, unde he condi ion ha no u he ule is applicable o
hem. In he asynch onous mode (asyn), any posi i e numbe o applicable ules
may be chosen non-de e minis ically o be applied in pa allel o he unde lying
conﬁgu a ion o disjoin submul ise s. The compu a ion s ep be ween wo con-
ﬁgu a ions Cand C′is deno ed by C⇒C′, hus yielding he bina y ela ion
⇒:C(O)×C(O). A compu a ion hal s when he e a e no ules applicable o he
cu en conﬁgu a ion (hal ing con igu a ion) in he co esponding mode.
The compu a ion o a gene a ing P sys em s a s wi h w, and i s esul is |x|
i i hal s, an accep ing sys em s a s wi h wx,x∈Σ∗, and we say ha |x|is
i s esul s – is accep ed – i i hal s. The se o numbe s gene a ed/accep ed by a
P sys em wo king in he mode αis he se o esul s o i s compu a ions o all
x∈Σ∗and deno ed by Nα
g(Π) and Nα
a(Π), espec i ely. The amily o se s o
numbe s gene a ed/accep ed by a amily o (one- egion) P sys ems wi h con ex
condi ions and p io i ies on he ules wi h ules o ype βwo king in he mode
αis deno ed by NδOPα
1(β, (p ok,l, inhk′,l′)d, p i)wi h δ=g o he gene a ing
and δ=a o he accep ing case; ddeno es he maximal numbe min he ules
wi h con ex condi ions ( , (P1, Q1),· · · ,(Pm, Qm)); kand k′deno e he maximum
numbe o p omo e s/inhibi o s in he Piand Qi, espec i ely; land l′indica e
he maximum o weigh s o p omo e s and inhibi o s, espec i ely. I any o hese
numbe s k,k′,l,l′is no bounded, we eplace i by ∗. As ypes o ules we a e
going o dis inguish be ween coope a i e (β=coo) and non-coope a i e (i.e., he
le -hand side o each ule is a single objec ; β=ncoo) ones.
In he case o accep ing sys ems, we also conside he idea o de e minism,
which means ha in each s ep o any compu a ion a mos one (mul ise o )
ule(s) is applicable; in his case, we w i e de a o δ.
In he li e a u e, we ﬁnd a lo o es ic ed a ian s o P sys ems wi h con-
ex condi ions and p io i ies on he ules, e.g., we may omi he p io i ies o
he con ex condi ions comple ely. I in a ule ( , (P1, Q1),· · · ,(Pm, Qm)) we ha e
m= 1, we say ha ( , (P1, Q1)) is a ule wi h a simple con ex condi ion, and
we omi he inne pa en heses in he no a ion. Mo eo e , con ex condi ions only
using p omo e s a e deno ed by |p1,··· ,pn, meaning ( , {p1,· · · , pn},∅), o , equi a-
len ly, ( , (p1,∅),· · · ,(pn,∅)); con ex condi ions only using inhibi o s a e deno ed
by |¬q1,··· ,¬qn, meaning ( , λ, {q1,· · · , qn}), o |¬{q1,··· ,qn}. Likewise, a ule wi h
bo h p omo e s and inhibi o s can be speciﬁed as a ule wi h a simple con ex con-
di ion, i.e., |p1,··· ,pn,¬q1,··· ,¬qns ands o ( , {p1,· · · , pn},{q1,··· , qn}). Finally,
p omo e s and inhibi o s o weigh one a e called a omic.
28 A. Alhazo , R. F eund
Rema k 1. I we do no conside de e minism, hen ( he eﬀec o ) he ule
( , (P1, Q1),· · · ,(Pm, Qm)) is equi alen o ( he eﬀec o ) he collec ion o ules
{( , Pj, Qj)|1≤j≤m}, no ma e in which mode he P sys em is wo king (ob-
iously, he p io i y ela ion has o be adap ed acco dingly, oo).
Rema k 2. Le ( , {p1,· · · , pn}, Q) be a ule wi h a simple con ex condi ion; hen
we claim ha ( he eﬀec o ) his ule is equi alen o ( he eﬀec o ) he collec ion
o ules
{( , {pj}, Q ∪ {pk|1≤k < j})|1≤j≤m}
e en in he he case o a de e minis ic P sys em: I he ﬁ s p omo e is chosen
o make he ule applicable, we do no ca e abou he o he p omo e s; i he
second p omo e is chosen o make he ule applicable, we do no allow p1 o
appea in he conﬁgu a ion, bu do no ca e abou he o he p omo e s p3 o pm;
in gene al, when p omo e pjis chosen o make he ule applicable, we do no
allow p1 o pj−1 o appea in he conﬁgu a ion, bu do no ca e abou he o he
p omo e s pj+1 o pm; ﬁnally, we ha e he ule {( , {pm}, Q ∪ {pk|1≤k < m})}.
I adding {pk|1≤k < j} o Qhas he eﬀec o p ohibi ing he p omo o pj om
enabling he ule o be applied, his makes no ha m as in his case one o he
p omo e s pk, 1 ≤k < j, mus ha e he possibili y o enabling o be applied.
By cons uc ion, he domains o he new con ex condi ions now a e disjoin , so
his ans o ma ion does no c ea e (new) non-de e minism. In a simila way, his
ans o ma ion may be pe o med on con ex condi ions which a e no simple.
The e o e, wi hou es ic ing gene ali y, he se o p omo e s may be assumed o
be a single on. In his case, we may omi he b aces o he mul ise no a ion o
he p omo e mul ise and w i e ( , p, Q).
Example 1. Conside an a bi a y ﬁni e se Ho numbe s. Choose K= max (H)+
1; hen we cons uc he ollowing de e minis ic accep ing P sys em wi h p omo e s
and inhibi o s:
Π= (O, {a}, s0 0· · · K, R′, R),
O={a}∪{si, i|0≤i≤K},
R′={si→si+1 |0≤i≤K−1}∪{ i→ i|0≤i≤K},
R={si→si+1|ai+1 ,|0≤i≤K−1}
∪{ i→ i|si,¬ai+1 ,|0≤i < K, i /∈H}∪ { K→ K|sK}.
The sys em s ep by s ep, by he applica ion o he ule si→si+1|ai+1 , 0 ≤i < K,
checks i (a leas ) i+ 1 copies o he symbol aa e p esen . I he compu a ion
s ops a e is eps, i.e., i he inpu has consis ed o exac ly icopies o a, hen
his inpu is accep ed i and only i i∈H, as exac ly in his case he sys em does
no s a an inﬁni e loop wi h using i→ i|si,¬ai+1 . I he inpu has con ained
mo e han max (H) copies o a, hen he sys em a i es in he s a e sKand will
loop o e e wi h K→ K|sK. The e o e, exac ly His accep ed. To accep he
complemen o Hins ead, we simply change i /∈H o i∈Hand as well omi he
ule K→ K|sK. I is easy o see ha o he maximally pa allel mode, we can
A Cha ac e iza ion o N F I N ∪coN F I N 29
eplace each ule i→ i|si,¬ai+1 by he co esponding ule i→ i|si; in his case,
his ule may be applied wi h s ill some abeing p esen while he sys em passes
h ough he s a e si, bu i will no ge in o an inﬁni e loop in ha case.
In sum, we ha e shown ha
Nde aOPasyn
1(ncoo, (p o1,∗, inh1,∗)1)⊇FIN ∪coNFIN
and
Nde aOPmaxpa
1(ncoo, p o1,∗)⊇FIN ∪coNFIN.
Example 2. Fo P sys ems wo king in he maximally pa allel way we can e en
cons uc a sys em wi h inhibi o s only:
Π= (O, {a}, sK, R),
O={a, }∪{si|0≤i≤K},
R′={si→ si−1, si→si|1≤i≤K}∪{ →λ, s0→s0},
R={si→ si−1|¬ai|1≤i≤K}
∪ { →λ}∪{si→si|¬ |0≤i≤K, i /∈H}.
This cons uc ion does no ca y o e o he case o he asynch onous mode, as
he ule →λis applied in pa allel o he ules si→ si−1|¬aiun il he inpu ai
is eached. In his case, he sys em cano change he s a e sianymo e, and hen i
s a s o loop i and only i i /∈H. To accep he complemen o Hins ead, change
i∈H o i /∈H, i.e., in sum, we ha e p o ed ha
Nde aOPmaxpa
1(ncoo, inh1,∗)⊇FIN ∪coNF IN.
As we shall show la e , all he inclusions s a ed in Example 1 and Example 2
a e equali ies.
Rema k 3. As in a P sys em (O, Σ, w, R′, R, >) he se o ules R′can easily be
deduced om he se o ules wi h con ex condi ions R, we omi R′in he de-
sc ip ion o he P sys em. Mo eo e , o sys ems ha ing only ules wi h a simple
con ex condi ion, we omi din he desc ip ion o he amilies o se s o numbe s
and simply w i e
NδOPα
1(β, p ok,l, inhk′,l′, p i).
Mo eo e , each con ol mechanism no used can be omi ed, e.g., i no p io i ies
and only p omo e s a e used, we only w i e NδOPα
1(β, p ok,l).
2.1 Regis e machines
In wha ollows we will need o simula e egis e machines; he e we b ieﬂy ecall
hei deﬁni ion and some o hei compu a ional p ope ies. A egis e machine is
a uple M= (m, B, l0, lh, P), whe e mis he numbe o egis e s, Pis he se o
ins uc ions bijec i ely labeled by elemen s o B,l0∈Bis he ini ial label, and
lh∈Bis he ﬁnal label. The ins uc ions o Mcan be o he ollowing o ms:
36 A. Alhazo e al.
allowed hen he e iciency appa en ly dec eases [12]: no NP-comple e p oblem can
be sol ed in polynomial ime wi hou using di ision ules unless P=NP holds.
A measu e o space complexi y o P sys ems has been in oduced [6] in o de
o analyze he ime-space ade-o exploi ed when P sys ems a e used o e i-
cien ly sol e compu a ionally ha d p oblems. The space equi ed by a P sys em
is he maximal size i can each du ing any compu a ion, de ined as he sum o
he numbe o memb anes and he numbe o objec s. A uni o m amily Πo
ecognize P sys ems is said o sol e a p oblem in space :N→Ni no P sys em
in Πassocia ed o an inpu s ing o leng h n equi es mo e han (n) space.
Unde his no ion o space complexi y, in [8] i has been p o ed ha he class
o p oblems sol able in polynomial space by P sys ems wi h ac i e memb anes,
deno ed by PMCSPACEAM, coincides wi h PSPACE. This esul is p o ed by
mu ual simula ion o P sys ems and Tu ing machines.
The echniques used up o now o simula e a polynomial-space Tu ing machine
ia a polynomial-space amily o P sys ems [7] do no seem o apply when he space
bound is less s ic , i.e., exponen ial o e en supe -exponen ial. Indeed, we would
need P sys ems wi h an exponen ial numbe o memb anes wi h dis inc labels, and
such sys ems canno be buil in a polynomial numbe o s eps by a de e minis ic
Tu ing machine (as equi ed by he no ion o polynomial- ime uni o mi y usually
employed in he li e a u e [5]).
He e we show ha , by using di e en echniques, exponen ial-space Tu ing
machines can be simula ed by exponen ial-space P sys ems; hence, he classes o
p oblems sol able by P sys ems wi h ac i e memb anes and by Tu ing machines
in exponen ial space coincide; in symbols, EXPMCSPACEAM =EXPSPACE.
The es o he pape is o ganized as ollows. In sec ion 2 we ecall some de ini-
ions conce ning P sys ems wi h ac i e memb anes and hei space complexi y. In
sec ion 3 we desc ibe how P sys ems wi h es ic ed elemen a y memb anes can be
used o simula e Tu ing machines; an analysis on he esou ces ( ime and space)
needed o pe o m his simula ion is also gi en. Sec ion 4 con ains he s a emen
o ou cha ac e iza ion o EXPSPACE, while sec ion 5 p o ides he conclusions
as well as some di ec ions o u he esea ch.
2 De ini ions
We assume he eade o be amilia wi h he basic e minology and esul s con-
ce ning P sys ems wi h ac i e memb anes (see [10], chap e s 11–12 o a su ey).
He e we jus ecall some de ini ions ha a e ele an o he esul s p esen ed in
his pape .
De ini ion 1. AP sys em wi h ac i e memb anes o ini ial deg ee d≥1is a uple
Π= (Γ, Λ, µ, w1, . . . , wd, R), whe e:
•Γis an alphabe , i.e., a ini e non-emp y se o symbols, usually called objec s;
•Λis a ini e se o labels o he memb anes;

Exponen ial-Space P Sys ems wi h Ac i e Memb anes 37
•µis a memb ane s uc u e (i.e., a oo ed uno de ed ee, usually ep esen ed
by nes ed b acke s) consis ing o dmemb anes enume a ed by 1, . . . , d; u -
he mo e, each memb ane is labeled by an elemen o Λ, no necessa ily in a
one- o-one way;
•w1, . . . , wda e s ings o e Γ, desc ibing he ini ial mul ise s o objec s placed
in he d egions o µ;
•Ris a ini e se o ules.
Each memb ane possesses, besides i s label and posi ion in µ, ano he a ibu e
called elec ical cha ge (o pola iza ion), which can be ei he neu al (0), posi i e
(+) o nega i e (−) and is always neu al be o e he beginning o he compu a ion.
The ules a e o he ollowing kinds:
•Objec e olu ion ules, o he o m [a→w]α
h
They can be applied inside a memb ane labeled by h, ha ing cha ge αand
con aining an occu ence o he objec a; he objec ais ew i en in o he
mul ise w(i.e., ais emo ed om he mul ise in hand eplaced by e e y
objec in w).
•Send-in communica ion ules, o he o m a[ ]α
h→[b]β
h
They can be applied o a memb ane labeled by h, ha ing cha ge αand such
ha he ex e nal egion con ains an occu ence o he objec a; he objec ais
sen in o hbecoming band, simul aneously, he cha ge o his changed o β.
•Send-ou communica ion ules, o he o m [a]α
h→[ ]β
hb
They can be applied o a memb ane labeled by h, ha ing cha ge αand con-
aining an occu ence o he objec a; he objec ais sen ou om h o he
ou side egion becoming band, simul aneously, he cha ge o his changed o β.
•Dissolu ion ules, o he o m [a]α
h→b
They can be applied o a memb ane labeled by h, ha ing cha ge αand con ain-
ing an occu ence o he objec a; he memb ane his dissol ed and i s con en s
a e le in he su ounding egion unal e ed, excep ha an occu ence o a
becomes b.
•Elemen a y di ision ules, o he o m [a]α
h→[b]β
h[c]γ
h
They can be applied o a memb ane labeled by h, ha ing cha ge α, con ain-
ing an occu ence o he objec abu ha ing no o he memb ane inside (an
elemen a y memb ane); he memb ane is di ided in o wo memb anes ha ing
label hand cha ge βand γ; he objec ais eplaced, espec i ely, by band c
while he o he objec s in he ini ial mul ise a e copied o bo h memb anes.
•Non-elemen a y di ision ules, o he o m
[ ]+
h1· · · [ ]+
hk[ ]−
hk+1 · · · [ ]−
hnα
h→[ ]δ
h1· · · [ ]δ
hkβ
h[ ]
hk+1 · · · [ ]
hnγ
h
They can be applied o a memb ane labeled by h, ha ing cha ge α, con ain-
ing he posi i ely cha ged memb anes h1, . . . , hk, he nega i ely cha ged mem-
b anes hk+1, . . . , hn, and possibly some neu al memb anes. The memb ane h
is di ided in o wo copies ha ing cha ge βand γ, espec i ely; he posi i e
38 A. Alhazo e al.
child en a e placed inside he o me memb ane, hei cha ge changed o δ,
while he nega i e ones a e placed inside he la e memb ane, hei cha ges
changed o . Any neu al memb ane inside his duplica ed and placed inside
bo h copies.
Each ins an aneous con igu a ion o a P sys em wi h ac i e memb anes is de-
sc ibed by he cu en memb ane s uc u e, including he elec ical cha ges, o-
ge he wi h he mul ise s loca ed in he co esponding egions. A compu a ion
s ep changes he cu en con igu a ion acco ding o he ollowing se o p inciples:
•Each objec and memb ane can be subjec o a mos one ule pe s ep, excep
o objec e olu ion ules (inside each memb ane any numbe o e olu ion ules
can be applied simul aneously).
•The applica ion o ules is maximally pa allel: each objec appea ing on he
le -hand side o e olu ion, communica ion, dissolu ion o elemen a y di ision
ules mus be subjec o exac ly one o hem (unless he cu en cha ge o he
memb ane p ohibi s i ). The same easoning applies o each memb ane ha
can be in ol ed o communica ion, dissolu ion, elemen a y o non-elemen a y
di ision ules. In o he wo ds, he only objec s and memb anes ha do no
e ol e a e hose associa ed wi h no ule, o only o ules ha a e no applicable
due o he elec ical cha ges.
•When se e al con lic ing ules can be applied a he same ime, a nonde e -
minis ic choice is pe o med; his implies ha , in gene al, mul iple possible
con igu a ions can be eached a e a compu a ion s ep.
•While all he chosen ules a e conside ed o be applied simul aneously du ing
each compu a ion s ep, hey a e logically applied in a bo om-up ashion: i s ,
all e olu ion ules a e applied o he elemen a y memb anes, hen all commu-
nica ion, dissolu ion and di ision ules; hen he applica ion p oceeds owa ds
he oo o he memb ane s uc u e. In o he wo ds, each memb ane e ol es
only a e i s in e nal con igu a ion has been upda ed.
•The ou e mos memb ane canno be di ided o dissol ed, and any objec sen
ou om i canno e-en e he sys em again.
The p ecise a ian o P sys ems we use in his pape does no use dissolu ion
o non-elemen a y di ision ules.
De ini ion 2. AP sys em wi h es ic ed elemen a y ac i e memb anes is a
P sys em wi h ac i e memb anes whe e only objec e olu ion, send-in, send-ou ,
and elemen a y di ision ules a e used. This kind o P sys ems is deno ed by
AM(−d,−n).
Ahal ing compu a ion o he P sys em Πis a ini e sequence o con igu a ions
C= (C0,...,Ck), whe e C0is he ini ial con igu a ion, e e y Ci+1 is eachable by Ci
ia a single compu a ion s ep, and no ules can be applied anymo e in Ck. A non-
hal ing compu a ion C= (Ci:i∈N) consis s o in ini ely many con igu a ions,
again s a ing om he ini ial one and gene a ed by successi e compu a ion s eps,
whe e he applicable ules a e ne e exhaus ed.
Exponen ial-Space P Sys ems wi h Ac i e Memb anes 39
P sys ems can be used as ecognize s by employing wo dis inguished objec s
yes and no; exac ly one o hese mus be sen ou om he ou e mos memb ane
du ing each compu a ion, in o de o signal accep ance o ejec ion espec i ely; we
also assume ha all compu a ions a e hal ing. I all compu a ions s a ing om
he same ini ial con igu a ion a e accep ing, o all a e ejec ing, he P sys em
is said o be con luen . I his is no necessa ily he case, hen we ha e a non-
con luen P sys em, and he o e all esul is es ablished as o nonde e minis ic
Tu ing machines: i is accep ance i an accep ing compu a ion exis s. All P sys ems
in his pape a e con luen .
In o de o sol e decision p oblems (i.e., decide languages), we use amilies
o ecognize P sys ems Π={Πx:x∈Σ?}. Each inpu xis associa ed wi h
a P sys em Πx ha decides he membe ship o xin he language L⊆Σ?by
accep ing o ejec ing. The mapping x7→ Πxmus be e icien ly compu able o
each inpu leng h [3].
De ini ion 3. A amily o P sys ems Π={Πx:x∈Σ?}is said o be
(polynomial- ime) uni o m i he mapping x7→ Πxcan be compu ed by wo de e -
minis ic polynomial- ime Tu ing machines F( o “ amily”) and E( o “encod-
ing”) as ollows:
•The machine F, aking as inpu he leng h no xin una y no a ion, cons uc s
a P sys em Πn, which is common o all inpu s o leng h n, wi h a dis inguished
inpu memb ane.
•The machine E, on inpu x, ou pu s a mul ise wx(an encoding o he speci ic
inpu x).
•Finally, Πxis simply Πnwi h wxadded o he mul ise placed inside i s inpu
memb ane.3
De ini ion 4. I he mapping x7→ Πxis compu ed by a single polynomial- ime
Tu ing machine, he amily Πis said o be semi-uni o m. In his case, inpu s o
he same size may be associa ed wi h P sys ems ha ing possibly di e en memb ane
s uc u es and ules.
Any explici encoding o Πxis allowed as ou pu o he cons uc ion, as long
as he numbe o memb anes and objec s ep esen ed by i does no exceed he
leng h o he whole desc ip ion, and he ules a e lis ed one by one. This es ic-
ion is en o ced in o de o mimic a (hypo he ical) ealis ic p ocess o cons uc ion
o he P sys ems, whe e memb anes and objec s a e p esumably placed in a con-
s an amoun du ing each cons uc ion s ep, and equi e ac ual physical space
p opo ional o hei numbe ; see also [3] o u he de ails on he encoding o
P sys ems.
3No ice ha his de ini ion o uni o mi y is (possibly) weake han he o he one com-
monly used in memb ane compu ing [5], whe e he Tu ing machine Fmaps each inpu
x o a P sys em Πs(x), whe e s:Σ?→Nis a measu e o he size o he inpu ; in ou
case, s(x) is always |x|.
40 A. Alhazo e al.
Finally, we desc ibe how space complexi y o amilies o ecognize P sys ems
is measu ed, and he ela ed complexi y classes [6].
De ini ion 5. Le Cbe a con igu a ion o a P sys em Π. The size |C| o Cis de ined
as he sum o he numbe o memb anes in he cu en memb ane s uc u e and he
o al numbe o objec s hey con ain. I C= (C0,...,Ck)is a hal ing compu a ion
o Π, hen he space equi ed by Cis de ined as
|C|= max{|C0|,...,|Ck|}
o , in he case o a non-hal ing compu a ion C= (Ci:i∈N),
|C|= sup{|Ci|:i∈N}.
Non-hal ing compu a ions migh equi e an in ini e amoun o space (in symbols
|C|=∞): o example, i he numbe o objec s s ic ly inc eases a each compu-
a ion s ep.
The space equi ed by Πi sel is hen
|Π|= sup{|C|:Cis a compu a ion o Π}.
No ice ha |Π|=∞migh occu i ei he Πhas a non-hal ing compu a ion e-
qui ing in ini e space (as desc ibed abo e), o Πhas an in ini e se o hal ing
compu a ions, such ha o each bound b∈N he e exis s a compu a ion equi ing
space la ge han b.
Finally, le Π={Πx:x∈Σ?}be a amily o ecognize P sys ems, and le
:N→N. We say ha Πope a es wi hin space bound i |Πx| ≤ (|x|) o
each x∈Σ?.
By MCSPACED( (n)) we deno e he class o languages which can be decided
by uni o m amilies o con luen P sys ems o ype Dwhe e each Πx∈Πope a es
wi hin space bound (|x|). The class o languages decidable in exponen ial space
by uni o m amilies o P sys ems o ype Dis deno ed by EXPMCSPACED, while
he co esponding class o semi-uni o m amilies is EXPMCSPACE?
D. The classes
de ined in e ms o non-con luen P sys ems a e deno ed by NEXPMCSPACED
and NEXPMCSPACE?
D, espec i ely.
Fo he p ecise de ini ions and p ope ies o Tu ing machines and, in pa icula ,
he space complexi y classes PSPACE and EXPSPACE, we e e he eade o [4].
3 Simula ing a Tu ing machine
In his sec ion we show ha exponen ial-space de e minis ic Tu ing machines can
be simula ed by P sys ems wi h es ic ed elemen a y ac i e memb anes wi h a
polynomial slowdown and a polynomial g ow h in space.
Exponen ial-Space P Sys ems wi h Ac i e Memb anes 41
Theo em 1. Le Mbe a single- ape de e minis ic Tu ing machine wo king in ime
(n)and space s(n), whe e s(n)≤n+ 2p(n) o some polynomial p. Then he e
exis s a uni o m amily o con luen P sys ems wi h es ic ed elemen a y ac i e
memb anes Π={Πx:x∈Σ?}ope a ing in ime O (n)s(n) log s(n)and space
Os(n) log s(n)such ha L(Π) = L(M).
We desc ibe how he simula ion is ca ied ou by examining a speci ic example,
and gene alizing om he e. Le Mbe a Tu ing machine ha ing ape alphabe
Γ={a, b, }, whe e deno es a blank ape cell, and using space n+ 2n(i.e., we
choose p(n) = n). Also le Qbe he se o non- inal s a es o M, and
δ:Q×Γ→Q×Γ× {/, .}
i s ansi ion unc ion. Assume ha Mp ocesses he inpu x=ba o leng h 2:
hen Muses a o al o 2+22= 6 ape cells. Suppose ha , a e a ew compu a ion
s eps, M eaches he ollowing con igu a ion:
ab a b
q
010 011
100 101
110 111
ha is, he s a e o Mis q, he ape con ains he s ing baab ollowed by wo
blank cells, and he ape head is loca ed on he i h cell. The pic u e also shows
(in bina y) he non-s anda d numbe ing scheme o ape cells ha we employ:
•The i s ncells, ha ini ially con ain he inpu (highligh ed by a hick bo de ),
a e deno ed by 2p(n)−n, . . . , 2p(n)−1 (e.g., 010 and 011 in ou example). These
numbe s, when w i en in bina y o e p(n) + 1 bi s, all ha e 0 as hei mos
signi ican bi .
•The emaining 2p(n)cells a e deno ed by 2p(n),...,2×2p(n)−1 (e.g., 100 o
111 in ou example). These numbe s, when w i en in bina y o e p(n)+1 bi s,
all ha e 1 as hei mos signi ican bi .
3.1 Rep esen ing he con igu a ion o he Tu ing machine
The con igu a ion o Mdesc ibed abo e is encoded in he ollowing con igu a ion
C1o he P sys em Πxsimula ing i (how his con igu a ion o Πxis eached om
i s ini ial con igu a ion will be desc ibed la e ):

42 A. Alhazo e al.
0 1
ba210 e
s
00
b
002
a
0120100
a
0120110
b
0121100
0121110
0
q000
++0
0
10
11
0211
0
Inside he ou e mos memb ane, labeled by s, we ha e n+ 2p(n)memb anes (6 in
ou example) ep esen ing he ape cells o M;no hem a e labeled 0, . . . , n−1,
and he emaining ones (which a e gene a ed by memb ane di ision, as desc ibed
below) by . We e e o hese memb anes as ape-memb anes. Each ape-memb ane
con ains wo pieces o in o ma ion: a se o p(n)+1 (3 in ou example) subsc ip ed
bi s, he bi -objec s, encoding he numbe o he ape cell o Mi ep esen s ( he
subsc ip a e used o p ese e he o de o he bi s), and an objec aken om he
alphabe o M, deno ing he symbol w i en in ha ape cell ( he symbol-objec ).
Fo ins ance, he memb ane [120110b]0
co esponds o ape cell 101, which con ains
he symbol b.
The s a e o Mis ep esen ed by a s a e-objec (qin he example), which will
egula e he simula ion o each compu a ion s ep o M. A he beginning o he
simula ion o each compu a ion s ep o M, he s a e-objec esides in memb ane
s.
On he lowe -le side o he pic u e we ha e p(n) + 1 memb anes, called
posi ion-memb anes and labeled by p(n),...,0, whose elec ical cha ge encodes
in bina y he cu en posi ion o he ape head o M; he e a posi i e cha ge ep-
esen s a 1 bi , while a neu al cha ge deno es 0. Fo ins ance, in he pic u e we
ha e [ ]+
2[ ]+
1[ ]0
0 ep esen ing posi ion 110.
The auxilia y memb ane labeled by e, he e o -memb ane, will ha e i s cha ge
se o posi i e whene e Πxnonde e minis ically chooses a “w ong” compu a ion
pa h while simula ing a compu a ion s ep o M(see below).
Finally, on he lowe - igh side o he pic u e, we ha e memb anes labeled by
symbols om he alphabe o M( he symbol-memb anes). These will be used,
once again by se ing hei cha ge, o ead he symbol cu en ly unde he ape
head o M.
We shall now desc ibe how o simula e a compu a ion s ep o Ms a ing om
i s cu en con igu a ion, as encoded by Πx. La e on we will desc ibe how he
con igu a ion o Πx ep esen ing he ini ial con igu a ion o Mcan be ob ained.
3.2 Simula ing a compu a ion s ep o M
In o de o simula e a compu a ion s ep o M, we need o iden i y which symbol
is loca ed unde i s ape head; no e ha he s a e qis al eady s o ed in he s a e-
objec . Since mos o he ape-memb anes o Πxha e he same label (and hose
labeled by 0, . . . , n−1beha e he same way in his phase, i.e., ha e he same
Exponen ial-Space P Sys ems wi h Ac i e Memb anes 43
associa ed se o ules) he e is no way o iden i y he co ec ape-memb ane
om he ou side. Hence, we shall guess he ape-memb ane co esponding o he
cell unde he head, hen check i selec ed he igh one.
This “guessing” is pe o med by he s a e-objec , which nonde e minis ically
en e s one o he ape memb anes using one o he ollowing ules:
q[ ]0
h→[q1]0
h o q∈Qand h∈ { 0, . . . , n−1, }.
Fi s , suppose qen e s he w ong memb ane, e.g., 011 ins ead o 110, p oducing
he ollowing con igu a ion:
0 1
ba210 e
s
00
b
002
a
0120100
a
0120110
b
0121100
0121110
0
000
++0
0
10
11
0211
0
q1
The s a e-objec q1is immedia ely sen back ou , changing he cha ge o he mem-
b ane o posi i e using one o he ules
[q1]0
h→[ ]+
hq2 o q∈Qand h∈ { 0, . . . , n−1, }.
No e ha he e will always be a mos one posi i e ape-memb ane, i.e., he mem-
b ane being checked a he cu en ime.
0 1
ba210 e
s
00
b
002
a
+120100
a
0120110
b
0121100
0121110
0
000
++0
0
10
11
0211
0q2
When a ape-memb ane is posi i e, he symbol-objec i con ains (ain he ex-
ample) is sen ou , while he bi -objec s a e eplica ed in a p imed and a double-
p imed e sions. A he same ime, he s a e-objec wai s by inc easing i s subsc ip
(such wai ing s eps will be implici om now on). The co esponding ules a e
[γ]+
h→[ ]+
hγ o γ∈Γand h∈ { 0, . . . , n−1, }
[0i→00
i000
i]+
h o 0 ≤i≤p(n) and h∈ { 0, . . . , n−1, }
[1i→10
i100
i]+
h o 0 ≤i≤p(n) and h∈ { 0, . . . , n−1, }
[q2→q3]0
s o q∈Q.
In ou example, we ob ain he ollowing con igu a ion:
44 A. Alhazo e al.
0 1
ba210 e
s
00
b
0000
2
a
+120100
a
0120110
b
0121100
0121110
0
000
++0
0
100
0
100
1
0211
0q3
00
210
0
10
1
Now, he symbol-objec is sen o he co esponding symbol-memb ane, se ing
i s cha ge o posi i e ( hus allowing he s a e-objec o iden i y he symbol unde
he ape head). A he same ime, he p imed bi -objec s inside he posi i e ape-
memb ane will be sen (in nonde e minis ic o de ) o he co esponding posi ion-
memb anes and compa ed wi h hei cha ge. In ou example we ha e 00
210
110
0and
[ ]+
2[ ]+
1[ ]0
0(whe e he mos signi ican and he leas signi ican bi s di e ). I he e
is a misma ch on a ce ain bi , he co esponding bi -objec will p oduce an e o -
objec e, o he wise i will be dele ed. The e o -objec s will se he cha ge o he
e o -memb ane o posi i e, so ha he s a e-objec may iden i y he e o when
all compa isons ha e been made, and will be in u n dele ed. This phase, whose
du a ion is p(n) + 4 s eps, in ol es he ollowing ules:
γ[ ]0
γ→[γ]+
γ o γ∈Γ
[00
i]+
h→[ ]+
h00
i o 0 ≤i≤p(n) and h∈ { 0, . . . , n−1, }
[10
i]+
h→[ ]+
h10
i o 0 ≤i≤p(n) and h∈ { 0, . . . , n−1, }
00
i[ ]α
i→[00
i]α
i o 0 ≤i≤p(n) and α∈ {0,+}
10
i[ ]α
i→[10
i]α
i o 0 ≤i≤p(n) and α∈ {0,+}
[00
i→λ]0
i o 0 ≤i≤p(n)
[10
i→λ]+
i o 0 ≤i≤p(n)
[00
i]+
i→[ ]+
ie o 0 ≤i≤p(n)
[10
i]0
i→[ ]0
ie o 0 ≤i≤p(n)
e[ ]α
e→[e]+
e o α∈ {0,+}
[e→λ]+
e
[qj→qj+1]0
s o 3 ≤j≤p(n) + 6 and q∈Q.
In ou example, he compu a ion may p oceed as ollows ( o some conc e e non-
de e minis ic choices in he o de he bi -objec s a e sen ou ).
Exponen ial-Space P Sys ems wi h Ac i e Memb anes 45
0 1
ba210 e
s
00
b
0000
2
a
+120100
a
0120110
b
0121100
0121110
0
00
+++0
0
100
0
100
1
0211
0q4
00
210
0
10
1
0 1
ba210 e
s
00
b
0000
2
a
+120100
a
0120110
b
0121100
0121110
0
00
+++0
0
100
0
100
1
0211
0q5
00
2
10
0
10
1
0 1
ba210 e
s
00
b
0000
2
a
+120100
a
0120110
b
0121100
0121110
0
00
+++0
0
100
0
100
1
0211
0q6
00
2
10
0
0 1
ba210 e
s
00
b
0000
2
a
+120100
a
0120110
b
0121100
0121110
0
00
+++0
0
100
0
100
1
0211
0q7
e
10
0
52 A. Alhazo e al.
0 1
ba210 e
s
00
b a
−120100
a
0120110
b
0
b
−121110
000
+++
0
0211
0
0210
11
q
1
121100
− −
0 1
ba210 e
s
00
b a
−120100
a
0120110
b
0
b
−121110
000
+0+
0
0211
0
0210
11
q
3
121100
− −
When he subsc ip o qγ eaches p(n) + 1 (i.e., when he posi ion upda ing has
been comple ed) he sys em begins p epa ing he encoding o nex con igu a ion
o M. This equi es ese ing all cha ges o ape-memb anes o neu al: his is
accomplished by c ea ing nobjec s 0, . . . , n−1( ese ing he memb anes ha ing
he same name) and 2p(n)copies o objec c0( ese ing he memb anes ha ing
label ). The la e objec s a e c ea ed by an ini ial objec cp(n), which is ew i en
as wo copies o cp(n)−1, each o hem ew i en as wo copies o cp(n)−2, and so
on. The s a e-objec wai s o his p ocess o e mina e, and hen inally becomes
he new s a e o M, as desc ibed by he ansi ion unc ion.
[qγ
p(n)+1 →qγ
p(n)+2 0· · · n−1cp(n)]0
s o q∈Qand γ∈Γ
[qγ
p(n)+k→qγ
p(n)+k+1]0
s o q∈Q,γ∈Γand 2 ≤k≤p(n)+2
j[ ]α
j→[ j]0
j o 0 ≤j≤n−1 and α∈ {0,−}
[ j→λ]0
j o 0 ≤j≤n−1
[ci→ci−1ci−1]0
s o 1 ≤i≤p(n)
c0[ ]α
→[c0]0
o α∈ {0,−}
[c0→λ]0
[qγ
2p(n)+3 → ]0
s o q, ∈Qand γ∈Γ, i δ(q, γ)=( , σ, d)
o some σ∈Γand d∈ {/, .}.
In ou example, he compu a ion e ol es as ollows:

Exponen ial-Space P Sys ems wi h Ac i e Memb anes 53
0 1
ba210 e
s
00
b a
−120100
a
0120110
b
0
b
−121110
000
+0+
0
0211
0
0210
11
q
4
121100
0 1c2
− −
0 1
ba210 e
s
00
b a
0120100
a
0120110
b
0
b
−121110
000
+0+
0
0211
0
0210
11
q
5
121100
0 1
c1c1
0−
0 1
ba210 e
s
00
b a
0120100
a
0120110
b
0
b
−121110
000
+0+
0
0211
0
0210
11
q
6
121100
c0c0c0c0
0−
0 1
ba210 e
s
00
b a
0120100
a
0120110
b
0
b
0121110
000
+0+
0
0211
0
0210
11
q
7
121100
c0c0c0c0
00
0 1
ba210 e
s
00
b a
0120100
a
0120110
b
0
b
0121110
000
+0+
0
0211
0
0210
11
121100
0
We ha e inally eached he con igu a ion o Πxco esponding o he con igu a ion
o Ma e i has pe o med i s compu a ion s ep, and we a e eady o s a
simula ing a new s ep o M.
54 A. Alhazo e al.
3.3 C ea ing he ini ial con igu a ion
In he p e ious sec ion we desc ibed how o simula e a compu a ion s ep o M
s a ing om an a bi a y con igu a ion o he Tu ing machine. We s ill need o
desc ibe how o encode he ini ial con igu a ion o M( ep esen ed in he ollowing
pic u e) in he P sys em Πxsimula ing i .
ab
qini
010 011
100 101
110 111
We use he ollowing as he ini ial con igu a ion o he P sys em:
0 1
ba210 e
s
00
b0
002
a1
012d1d00
z20000 0 0
0
10
11
0211
0
+
This ini ial con igu a ion consis s o a memb ane scon aining:
•Memb anes 0, . . . , n−1, each con aining p(n) + 1 bi -objec s encoding he po-
si ion numbe s o he inpu cells (as desc ibed abo e).
•One single copy o memb ane , con aining he bi -objec 1p(n)( ecall ha
he mos signi ican bi o he non-inpu ape cells is always 1) and he “bi
a iables” d0, . . . , dp(n)−1.
•The posi ion-memb anes, labeled by 0, . . . , p(n) + 1, whose ini ial cha ge is 0
by de ini ion. Those which ha e o be se o posi i e in o de o se up he
ini ial head posi ion (i.e., 2p(n)−n) con ain a + objec .
•The e o -memb ane e.
•The symbol-memb anes, labeled by he elemen s o Γ.
•The objec zp(n).
All hese i ems only depend on he size o he inpu o he Tu ing machine M.
The inpu i sel is encoded by a se o objec s deno ing he symbols, subsc ip ed
by an index indica ing hei posi ion in he s ing (coun ing om 0), and placed
in o he inpu memb ane s. In ou example, he inpu ba o Mis encoded in Πx
as b0a1.
Du ing he ini ializa ion phase o Πx, se e al ope a ions a e ca ied ou . Fi s
o all, he inpu -objec s a e sen o he co esponding ape-memb anes (indica ed
in hei subsc ip s). This is accomplished by using he ollowing ules:
Exponen ial-Space P Sys ems wi h Ac i e Memb anes 55
γi[ ]0
i→[γ]0
i o γ∈Γand 0 ≤i≤n−1.
The posi ion-memb anes ha e hei cha ges se o + by sending ou he + objec s,
which a e hen dele ed using
[+]0
i→[ ]+
i+ o 0 ≤i≤p(n)
[+ →λ]0
s.
The ape-memb anes co esponding o he wo king po ion o he ape, o size
2p(n), a e c ea ed by i e a ed elemen a y memb ane di ision, s a ing om he
single ini ial ape-memb ane . The objec s dia e ew i en as 0ion one side, and
as 1ion he o he , whene e he memb ane is di ided. This p ocess c ea es all he
2p(n)cell numbe s in bina y. The co esponding ules a e
[di]0
→[0i]0
[1i]0
o 0 ≤i≤p(n)−1.
This la e ope a ion equi es p(n) s eps. The objec zp(n)has i s subsc ip dec e-
men ed o 1, and hen inally becomes he s a e-objec co esponding o he ini ial
s a e o M:
[zi→zi−1]0
s o 2 ≤i≤p(n)
[z1→qini ]0
s.
In ou example, he ini ializa ion phase p oceeds as ollows.
0 1
ba210 e
s
00
b
002
a
012d1000
z10000 +0
0
10
11
0211
0
+
12d1100
0 1
ba210 e
s
00
b
002
a
01201000
qini 0000 +0
0
10
11
0211
0
1201100
1211000
1211100
A e ha ing ini ialized he P sys em Πxacco ding o he ini ial con igu a ion
o Mon inpu x, he simula ion is ca ied ou s ep-by-s ep as desc ibed in he
p e ious sec ion.
56 A. Alhazo e al.
3.4 Hal ing and ou pu
The only missing pa o ou simula ion conce ns he ope a ions o ca y ou when
he simula ed machine hal s by accep ing o ejec ing. Assuming he ansi ion
unc ion δo Mis unde ined on i s accep ing s a e qyes and i s ejec ing s a e qno,
we can simply p oceed as ollows: i he machine en e s qyes, hen in Πx he s a e-
objec qyes appea s inside he ou e mos memb ane s; we can hen send ha objec
ou o he en i onmen as yes o make Πxaccep . The beha io is analogous o
he ejec ing s a e qno.
[qyes]0
s→[ ]0
syes [qno]0
s→[ ]0
sno.
3.5 Comple ing he p oo
The ini ializa ion phase o Πx, simula ing Mon an inpu xo leng h n, equi es
O(p(n)) ime in o de o c ea e 2p(n)copies o memb ane by a sequence o
elemen a y di isions.
Then, he (n) s eps pe o med by Ma e simula ed. Each s ep in ol es guessing
he ape-memb ane co esponding o he cell cu en ly unde he ape head; each
simula ed s ep may equi e up o O(s(n)) guesses in he wo s case. Fo each guess,
we need o check i he co ec ape-memb ane was selec ed, and his equi es
ime p opo ional o he numbe o bi posi ions, i.e., Olog s(n)s eps. I he
memb ane is inco ec , hen O(1) s eps a e equi ed o se i s cha ge o nega i e
and p epa e he sys em o a u he guess. I he memb ane was he igh one, he
s a e, head posi ion and ape symbol ha e o be upda ed, and his equi es u he
Olog s(n)s eps. Hence, each simula ed s ep o M equi es Os(n) log s(n)s eps
o Πx, o a o al o O (n)s(n) log s(n)s eps.
As he ou pu s ep only equi es cons an ime, he whole simula ion can be
ca ied ou in O (n)s(n) log s(n) ime. Since s(n) is O( (n)) o a Tu ing machine
(assuming i a leas eads i s whole inpu ), he simula ion ime can be exp essed
as a unc ion o (n) as O (n)2log (n). Hence, his is an “e icien ” simula ion:
i Mwo ks in polynomial ime, hen he amily Π={Πx:x∈Σ?}simula ing i
also wo ks in polynomial ime; i M uns in exponen ial ( esp., doubly-exponen ial)
ime, hen Πalso uns in exponen ial ( esp., doubly-exponen ial ime).
No ice ha he ac ual unning ime o he simula ion depends on he sequence
o nonde e minis ic choices pe o med when he sys em has o guess he co ec
ape-memb ane. In he bes case, when he co ec guess is always he i s one,
he ime educes o O (n) log s(n)ins ead o O (n)s(n) log s(n)as in he wo s
case.
The space equi ed by Πxis asymp o ically due o he ape-memb anes. These
a e s(n) in numbe , and each o hem con ains O(log s(n)) bi -objec s deno ing
i s posi ion on he ape. Hence, he simula ion equi es Os(n) log s(n)space:
a polynomial-space Tu ing machine is simula ed in polynomial space, and an
exponen ial-space one in exponen ial space.
Exponen ial-Space P Sys ems wi h Ac i e Memb anes 57
In o de o comple e he p oo o Theo em 1, we only need o check ha he
amily Πis polynomial- ime uni o m. I is easy o e i y ha all he ules and he
ini ial con igu a ion o Πxac ually depend only on he leng h o x(excep o he
inpu objec s). The e is a cons an numbe o di e en kinds o ules pa ame ic
wi h espec o no p(n); he la ge se s o ules a e (1) and (2) on page 47,
consis ing o On×p(n) ules each.
4 Cha ac e izing exponen ial space
In he p e ious sec ion we desc ibed a simula ion o de e minis ic Tu ing machines
wo king in exponen ial space by means o P sys ems. Combining his esul wi h
he con e se simula ion illus a ed in [8], we can show ha he compu a ional
powe o Tu ing machines and o P sys ems wi h ac i e memb anes coincide when
hese de ices ope a e wi hin an exponen ial space limi :
Co olla y 1. The ollowing inclusions hold:
EXPMCSPACEAM(−d,−n) ⊆EXPMCSPACE[?]
AM
⊆
⊆
EXPSPACE ⊇NEXPMCSPACE?
AM
whe e [?]deno es op ional semi-uni o mi y (ins ead o uni o mi y). Hence, all
classes shown in he diag am coincide.
P oo . The chain o inclusions
EXPMCSPACEAM(−d,−n) ⊆EXPMCSPACE[?]
AM ⊆NEXPMCSPACE?
AM
holds by de ini ion. Tha NEXPMCSPACE?
AM ⊆EXPSPACE is an imme-
dia e co olla y o Theo em 5 in [8]. Finally, he inclusion o EXPSPACE in
EXPMCSPACEAM(−d,−n) di ec ly ollows om Theo em 1. u
Le us ema k ha he powe o he complexi y class EXPMCSPACEAM is
mos ly due o he amilies o P sys ems hemsel es, as opposed o he Tu ing
machines p o iding he uni o mi y condi ion; indeed, hese would only be able o
sol e he s ic ly smalle [4] class Po decision p oblems.
5 Conclusions
We showed ha he class o p oblems sol able by P sys ems wi h ac i e memb anes
in exponen ial space coincides wi h he class o p oblems sol ed by Tu ing machines
in exponen ial space, ha is, EXPMCSPACEAM =EXPSPACE.

58 A. Alhazo e al.
Again, he echniques used o p o e his esul canno be applied immedia ely
when he space bound is less s ic , i.e., supe -exponen ial. In ac , in his case
we would need sys ems using indexed bi s, whe e he index anges o e a supe -
polynomial se o alues; as a consequence, such sys ems canno be gene a ed in a
uni o m way in a polynomial numbe o s eps, as eques ed by De ini ion 3. Thus,
i emains open o his case he ques ion whe he hese kinds o P sys ems wi h
ac i e memb anes ha e he same compu ing powe as Tu ing machines wo king
unde he same space cons ain s.
Le us no e ha i memb ane c ea ion [1] is used ins ead o memb ane di ision,
hen he simula ion may be s aigh o wa d and as e ( he slowdown would be
by a cons an ac o only). The simula ion would also be de e minis ic, ins ead
o equi ing “wild” nonde e minism as in ou esul . Tu ing machine cells may
be ep esen ed by nes ed memb anes, c ea ed when needed; his is a cons uc ion
ha gene alizes e en o supe -exponen ial space. Howe e , wi h memb ane di ision
only, he dep h o memb ane hie a chy canno inc ease du ing he compu a ion,
and i is o iginally polynomial unde ou cu en de ini ion.
As a di ec ion o u u e esea ch, i migh also be in e es ing o analyze he
beha io o amilies o P sys ems wi h ac i e memb anes wo king in loga i hmic
space. Howe e , he e a e wo majo issues o be conside ed in his case: i s , we
should sligh ly change he no ion o space complexi y, in o de o allow o a “ ead-
only” inpu mul ise ha is no coun ed when he space equi ed by he P sys em
is measu ed (simila ly o he inpu ape o a logspace Tu ing machine). Fu he -
mo e, he no ion o uni o mi y used o de ine he amilies o P sys ems should be
weakened, since polynomial- ime Tu ing machines cons uc ing he amilies migh
be able o sol e he p oblems al oge he by hemsel es. Mo e gene al o ms o
uni o mi y ha e al eady been in es iga ed [3], and ha wo k is going o be use ul
when a acking his p oblem.
Acknowledgemen s
A iom Alhazo g a e ully acknowledges he p ojec Re oNe by he Lomba dy
Region o I aly unde he ASTIL P og am ( egional dec ee 6119, 20100618). The
wo k o he o he au ho s was pa ially suppo ed by Uni e si `a degli S udi di
Milano-Bicocca, Fondo di A eneo pe la Rice ca (FAR) 2011.
Re e ences
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es ic ed memb ane c ea ion and objec complexi y in P sys ems. In e na ional
Jou nal o Compu e Ma hema ics, 83(7):529–547, 2006.
2. A iom Alhazo , Ca los Ma ´ın-Vide, and Linqiang Pan. Sol ing a PSPACE-
comple e p oblem by ecognizing P sys ems wi h es ic ed ac i e memb anes. Fun-
damen a In o ma icae, 58(2):67–77, 2003.
Exponen ial-Space P Sys ems wi h Ac i e Memb anes 59
3. Niall Mu phy and Damien Woods. The compu a ional powe o memb ane sys ems
unde igh uni o mi y condi ions. Na u al Compu ing, 10(1):613–632, 2011.
4. Ch is os H. Papadimi iou. Compu a ional Complexi y. Addison-Wesley, 1993.
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Al a o Rome o-Jim´enez, and Fe nando Sancho-Capa ini.
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pu ing, 2(3):265–284, 2003.
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The Powe o Sympo -3 wi h Few Ex a Symbols
A iom Alhazo 1,2and Yu ii Rogozhin2
1Uni e si `a degli S udi di Milano-Bicocca
Dipa imen o di In o ma ica, Sis emis ica e Comunicazione
Viale Sa ca 336, 20126 Milano, I aly
E-mail: [email p o ec ed]
2Ins i u e o Ma hema ics and Compu e Science
Academy o Sciences o Moldo a
Academiei 5, Chi¸sin˘au MD-2028 Moldo a
E-mail: {a iom, ogozhin}@ma h.md
Summa y. Memb ane sys ems (wi h symbol objec s) a e o mal models o dis ibu ed
pa allel mul ise p ocessing. Sympo ules mo e mul iple objec s o a neighbo ing egion.
I is known ha P sys ems wi h sympo ules o weigh a mos 3 and a single memb ane
a e compu a ionally comple e wi h 7 supe ﬂuous symbols. I is also known ha wi hou
any supe ﬂuous symbols such sys ems only gene a e ﬁni e se s.
We imp o e he lowe bounds on he gene a i e powe o P sys ems wi h ew su-
pe ﬂuous objec s as ollows. 0: emp y se and all single ons; k: all se s wi h a mos k
elemen s and all se s o numbe s k+ egula wi h up o ks a es, 1 ≤k≤5; 6: all egula
se s o non-nega i e in ege s. All esul s excep he las one a e also alid o diﬀe en
modes, e.g., sequen ial one, also o highe alues o k.
1 In oduc ion
Memb ane sys ems (wi h symbol objec s) a e o mal models o dis ibu ed pa allel
mul ise p ocessing. Sympo ules mo e p edeﬁned g oups objec s o a neighbo -
ing egion [4]. In maximally pa allel mode ( ypical o memb ane compu ing), his
alone is suﬃcien o cons uc a compu a ionally uni e sal de ice, as long as he
en i onmen may con ain an unbounded supply o some objec s. The numbe o
symbols speciﬁed in a sympo ule is called i s weigh . The esul o a compu a-
ion is he o al numbe o objec s when he sys em hal s. In some cases, howe e ,
o echnical easons he desi ed esul may only be ob ained alongside a small
numbe o supe ﬂuous objec s in he ou pu egion.
The e we e mul iple pape s imp o ing he esul s on P sys ems wi h sym-
po /an ipo o small weigh (an an ipo ule mo es objec s be ween 2 egions
in bo h di ec ions, and i s weigh is he maximum o objec s pe di ec ion), see [2]
o a su ey o esul s. Compu a ional comple eness is achie ed e en o minimal
coope a ion: ei he sympo /an ipo o weigh 1, o sympo o weigh a mos

Coun ing Cells wi h Tissue-like P Sys ems
Ioan A delean1,2, Daniel D´ıaz-Pe nil3, Miguel A. Gu i´e ez-Na anjo4,
F ancisco Pe˜na-Can illana4, Ra´ul Reina-Molina3, I is Sa chizian1
1O idius Uni e si y, Cons an ¸a 90052, Romania
2Ins i u e o Biology Bucha es , Romanian Academy, 060031 Bucha es , Romania
[email p o ec ed], [email p o ec ed]
3Resea ch G oup on Compu a ional Topology and Applied Ma hema ics
Depa men o Applied Ma hema ics - Uni e si y o Se illa, 41012, Spain
[email p o ec ed], [email p o ec ed]
4Resea ch G oup on Na u al Compu ing
Depa men o Compu e Science and A iﬁcial In elligence
Uni e si y o Se illa, 41012, Spain
[email p o ec ed], [email p o ec ed]
Summa y. Coun ing he numbe o cells ob ained in an expe imen is c ucial in many
a eas in Biology. None heless, his is usually pe o med by hand by he esea che due he
in insic diﬃcul y o he ask. In his pape , we p esen a se o echniques o coun ing
cells inspi ed in he ea men o Digi al Images ia issue-like P sys ems wi h p omo e s.
1 In oduc ion
Due o he inc easing amoun o in o ma ion s o ed as isual da a, he de elopmen
o new so wa e o dealing eﬃcien ly wi h digi al images becomes a necessi y. The
numbe o applica ion a eas is g owing and he p og ess o new echnology needs
he design o new so wa e o handling such in o ma ion. Among he classical
a eas, we can ci e biome ics [1], su eillance [15] o medical imaging [3], bu he e
a e many o he s.
Recen ly, a new esea ch line has been open by applying well-known memb ane
compu ing echniques o sol ing p oblems om digi al image y. Fo example, he
segmen a ion p oblem, [13, 14, 16, 17, 39], h esholding [12] o smoo hing [33].
Special a en ion dese es [20], whe e he symme ic dynamic p og amming s e eo
(SDPS) algo i hm [21] o s e eo ma ching was implemen ed by using simple P
modules wi h duplex channels.
We ocus he e on a p oblem om Mic obiology. Au oma ed image analysis is
inc easingly used in Mic obiology o quan i y impo an pa ame e s o esea ch
and applica ion. The mos s udied so a a e he ollowing: cell numbe s, cell
olumes, equencies o di iding cells, in si u classiﬁca ion o bac e ia, enume a ion
70 I. A delean e al.
o ac i ely espi ing bac e ia, cha ac e iza ion o bac e ial g ow h on solid medium,
iabili y and physiological ac i i y in bioﬁlms (e.g. [4, 5, 19, 23, 26, 30, 11, 34,
35, 37, 38, 40, 41, 43]). In his pape we epo ou s udy o he applica ion o
Memb ane Compu ing echniques o he p oblem o coun ing cells and show some
p elimina y esul s. The whole p ocess is a combina ion o diﬀe en echniques o
p ocessing images (bina iza ion, segmen a ion, noise educ ion . . . ) which can be
pe o med by diﬀe en amilies o P sys ems. The ﬁnal algo i hm is a sequence o
pa ial p ocesses which can be pe o med by Memb ane Compu ing echniques,
and he applica ion o such p ocesses can be seen as a global machine which akes
as inpu a digi al image showing a biological en i y (usually, a pho og aph aken
wi h a mic oscopy in a we lab) and he ou pu is he numbe o cells in he pic u e.
The diﬀe en amilies o P sys ems used in he s ages o he p ocess ha e in-
spi ed pa allel so wa e p og ams which ha e been de eloped by using a de ice a -
chi ec u e called CUDATM, (Compu e Uniﬁed De ice A chi ec u e). CUDATM is a
gene al pu pose pa allel compu ing a chi ec u e ha allows he pa allel NVIDIA1
G aphics P ocesso s Uni s (GPUs) o sol e many complex compu a ional p ob-
lems in a mo e eﬃcien way han on a CPU. GPUs cons i u e nowadays a solid
al e na i e o high pe o mance compu ing, and he ad en o CUDA allows p o-
g amme s a iendly model o accele a e a b oad ange o applica ions. This no el
a chi ec u e has been p e iously used o implemen pa allel so wa e ha simu-
la es he beha io o P sys ems [6, 8, 9, 10, 32, 33], and, in a simila way han in
o he implemen a ions, he ob ained esul s in he p oblem o coun ing cells a e
qui e p omising.
The pape is o ganized as ollows: Nex , we ecall he compu a ional model
used o design he diﬀe en amilies o P sys ems ha pe o ms he s ages o
he algo i hm. In sec ion 3, we ou line he s eps o he p ocess ha akes as an
inpu a digi al image aken in a we lab and ou pu s he numbe o cells in he
image. Sec ion 4 shows an illus a i e example and some de ails o he CUDA
implemen a ion. The pape ﬁnishes wi h some conclusions and open lines o a
u u e esea ch.
2 Fo mal F amewo k
Nex , we ecall some basics on he P sys em model chosen o implemen ing he
solu ion desc ibed below. The model is issue-like P sys ems wi h p omo e s. P o-
mo e s a e usually deﬁned on cell-like models [24] and i s ex ension o issue-like
is qui e na u al. Nex , we ecall he o mal deﬁni ion.
De ini ion 1. A issue-like P sys em wi h p omo e s o deg ee q≥1is a uple o
he o m
Π= (Γ, Σ, E, w1, . . . , wq,R, iin, iou )
whe e
1h p://www.n idia.com.
Coun ing Cells wi h Tissue-like P Sys ems 71
1. Γis a ini e alphabe , whose symbols will be called objec s;
2. Σ⊆Γis he inpu alphabe ;
3. E ⊆ Γis a ini e alphabe ep esen ing he se o he objec s in he en i onmen
a ailable in an a bi a y la ge amoun o copies;
4. w1, . . . , wqa e s ings o e Γ ep esen ing he mul ise s o objec s associa ed
wi h he cells in he ini ial con igu a ion;
5. Ris a ini e se o ules o he ollowing o m:
(p o |i, u/ , j), o 0≤i=j≤q, p o, u, ∈Γ∗
In hese ules, he labels 1, . . . , q co espond o he qcells and he label 0co -
esponds o he en i onmen ;
6. iin ∈ {1,2, . . . , q}deno es he inpu egion;
7. iou ∈ {1,2, . . . , q}deno es he ou pu egion.
The ule (p o |i, u/ , j) can be applied o e wo cells (o a cell and he en i-
onmen ) iand jsuch ha u(con ained in cell i) is aded agains (con ained in
cell j). The ule is applied i in i he objec s o he p omo e p o a e p esen . The
p omo e is no modiﬁed by he applica ion o he ule. I he p omo e is emp y,
we will w i e (i, u/ , j) ins ead o (∅ | i, u/ , j).
Rules a e used as usual in he amewo k o memb ane compu ing, ha is,
in a maximally pa allel way (a uni e sal clock is conside ed). In one s ep, each
objec in a memb ane can only be used o one ule (non-de e minis ically chosen
when he e a e se e al possibili ies), bu any objec which can pa icipa e in a
ule o any o m mus do i , i.e. in each s ep we apply a maximal mul ise o
ules. A con igu a ion is an ins an aneous desc ip ion o he sys em Π, and i
is ep esen ed as a uple (w0, w1, . . . , wq), whe e w1, . . . , wq, whe e ep esen he
mul ise o objec s con ained in he qcells and w0 ep esen he mul ise o objec s
om Γ−E placed in he en i onmen (ini ially w0=∅). Gi en a conﬁgu a ion, we
can pe o m a compu a ion s ep and ob ain a new conﬁgu a ion by applying he
ules in a pa allel manne as i is shown abo e. A sequence o compu a ion s eps
is called a compu a ion. A conﬁgu a ion is hal ing when no ules can be applied
o i .
3 Coun ing Cells
Coun ing cells in a pic u e aken by a mic oscopy in a we lab is a ha d ask.
The s udy o he cells is usually made in such condi ions (ligh ﬁl e s, noise da a,
aqueous media, . . . ) whe e i is diﬃcul o he human expe o decide whe he a
spo in he image co espond o a cell o no . In such condi ions, he de elopmen
o a so wa e ha p o ides he exac numbe o spo s in he image ha co espond
o cells is impossible, since wo diﬀe en expe s ha dly ag ee in his issue.
F om his s a ing poin , and bea ing in mind ha he esea ch in Mic obiology
needs compu e aid, we p opose a Memb ane Compu ing p o ocol o ob aining
72 I. A delean e al.
Fig. 1. (Le ) Image wi h wo ﬁlamen s o he e ocys ous cyanobac e ium labeled wi h
0.02% c ys al iole ( he a ows indica es he p esence o he e ocys s, specialized cells,
no labeled by c ys al iole in ou expe imen al labeling condi ions: 0.02% c ys al iole
concen a ion and 5 minu es o colo a ion). (Righ ) Bina iza ion o he image.
he numbe o cells in a image, o mo e exac ly, he numbe o spo s in he image
ha p obably co espond o a cell. Since his is a empi ical p oblem, ins ead o
a o mal deﬁni ion o he p oblem, we will base ou desc ip ion in a case s udy.
We s a by conside ing he image o Fig. 1. Such image co espond o ﬁlamen-
ous cyanobac e ium and has been aken by using he numbe o cells wi hin each
ﬁlamen . I is e y impo an o calcula e he a io o he numbe o he e ocys s
o non-diﬀe en ia e cells, also called ege a i e cell, as well as o calcula e he
dis ibu ion o ﬁlamen s leng hs (co ela ed o he o al numbe o cells wi hin
each ﬁlamen ) wi hin a popula ion (composed o hund eds o ﬁlamen s pe each
millili e ) o his cyanobac e ium. In mic obiological, s udies o he de e mina ion
o numbe o di iding cells ela ed o he numbe o o al cells a e also e y use-
ul o calcula e he g ow h a e in na u al popula ion o bac e ia/cyanobac e ia,
he e ocys ous o no ([2, 7, 22, 25, 27, 29, 42]).
The a ge is o ake he image as an inpu and p o ide a he numbe o cells
as ou pu and pe o ming he diﬀe en s ages o he p ocess by using Memb ane
Compu ing echniques. The s ages a e he ollowing:
•S age 1: Bina iza ion. This ﬁs s age consis on ge ing a new image om
he o iginal wi h only wo colo s (a bina y image). In his s age we use a amily
Coun ing Cells wi h Tissue-like P Sys ems 73
Fig. 2. The ﬁ s image ( op-le ) is a de ail o Figu e 1 (le ). The second and hi d
ones ( op- igh and bo om-le ) show de ails o he hinning p ocess. Finally, he las
image (bo om- igh ) shows a se o pixels inside each indi idual cells. Diﬀe en cells a e
ma ked wi h diﬀe en colo s. The numbe o such diﬀe en colo s p o ides he numbe o
cells.
o issue-like P sys ems simila o he desc ibed one in [12, 32] o his aim. A
esul o his p ocess can be seen in Figu e 1 ( igh ).
•S age 2: Segmen a ion. This is he p ocess ha spli he image in se e al
meaning ul egions. I is basic o he ea men o digi al images and i is widely
used in medical images o iden i ying he egion o in e es . We also pe o m
his s age by using Memb ane Compu ing echniques, namely, he algo i hm
desc ibed in [13].
•S age 3: Noise Reduc ion. The images aken in a we lab a e usually a
om ha ing homogeneous egions. Due o he in insic na u e o he biological
esea ch, in he image i is common ho ﬁnd li le spo s ha in a mechanic
p ocess o coun ing cells, can be easily aken as li le cells when hey a e me ely
spo s due o he noise. Ob iously, emo ing such noise imply o ake diﬃcul
decision and should be done ca e ully. We p opose h ee diﬀe en s eps o
elimina e noise om an image:

74 I. A delean e al.
–S age 3.1: Labeling black connec ed componen s. The ﬁ s s ep con-
sis s on de e mining connec ed componen s in he image. The se o pixels
o be conside ed as a cell is a connec ed se o pixels. The connec i i y
p oblem among pixels and he s udy o connec ed componen s is a p oblem
linking Digi al Image y and Algeb aic Topology and i has been ecen ly
s udied in he amewo k o Memb ane Compu ing. Fo his s ep, we p o-
pose a pa ial applica ion o he Memb ane Compu ing algo i hm p oposed
in [18].
–S age 3.2: Calcula ing A eas. This s ep s a s a e he iden iﬁca ion o
he connec ed componen s o he image. Ou p oposal is o use Memb ane
Compu ing echniques in o de o de e mina e he size o he connec ed
componen . In ou app oach, he size o he spo is i s numbe o pixels.
This p ocess is also pe o med by using he sympo /an ipo ules used
in issue like P sys ems and basically consis s on coun ing he numbe o
objec s in he P sys em which ep esen he pixels o a conc e e spo .
–S age 3.3: Elimina ing small componen s. F om he p e ious s ep, we
ha e a numbe associa ed o each spo , ep esen ing i s size. In his s age
we will conside a h eshold in o de o decide i a spo is la ge enough o
be conside ed meaning ul o i he spo should be conside ed as noise, and
hence, o be igno ed. Ob iously, he h eshold depends on he expe imen
and i mus be p o ided by he expe s.
•S age 4: Coun ing Cells. The h ee p e ious s ages can be conside ed as
p ep ocessing o he image. The algo i hm o coun ing cells s a s now. The
key poin in his s age is o conside he geome y ha a cell usually shows
in an image aken o m a mic oscopy in a we lab. Such images usually show
he cells as con ex spo s and he image o mul icellula beings usually shows
a wa y bo de . An app op ia e p ocess o hinning, inspi ed in o he hinning
p ocess2wi h Memb ane Compu ing echniques, p oduces li le isola ed se o
poin o each cell (see Figu e 2 (bo om- igh )).
•S age 5: Ou pu . In he las s age, he isola ed se o pixels ha ep esen
he cells a e codiﬁed as an app op ia e se o objec s o he alphabe in he
ou pu memb ane o he co esponding P sys em. Coun ing hese se s o objec s
p o ides he numbe o cells in he image
3.1 Implemen a ion
Inspi ed in he amilies o issue-like P sys ems ha pe o m he s ages o he p o-
cess o coun ing cells, a so wa e ool has been implemen ed by using CUDATM ,
(Compu e Uniﬁed De ice A chi ec u e) [28, 31]. CUDATM is a gene al pu pose
pa allel compu ing a chi ec u e ha allows he pa allel NVIDIA G aphics P o-
cesso s Uni s (GPUs) o sol e many complex compu a ional p oblems in a mo e
eﬃcien way han on a CPU.
2See hese p oceedings
Coun ing Cells wi h Tissue-like P Sys ems 75
The expe imen s ha e been pe o med on a compu e wi h a CPU AMD A hlon
II x4 645, which allows o wo k wi h ou co es o 64 bi s o 3.1 GHz. The compu e
has ou blocks o 512KB o L2 cache memo y and 4 GB DDR3 o 1600 MHz o
main memo y. The used g aphical ca d (GPU) is an NVIDIA Ge o ce GT240
composed by 12 S eam P ocesso s wi h a o al o 96 co es o 1340 MHz. I has
1 GB DDR3 main memo y in a 128 bi s bus o 700 MHz. So, he ans e a e
ob ained is by 54.4 Gbps. The used Cons an Memo y is 64 KB and he Sha ed
Memo y is 16 KB. I s Compu e Capabili y le el is 1.2 ( om 1.0 o 2.1). The
implemen a ion deals wi h Nblocks o h eads o he comple e image in ou
GPU o 96 co es.
3.2 Example
Figu es 1 and 2 shows se e al de ails o an illus a i e example o he p ocess. The
o iginal image in Fig. 1 shows a ﬁlamen ous cyanobac e ium able o diﬀe en ia e
he e ocys s ( he a ows indica es he p esence o he e ocys s, no labeled by c ys al
iole in ou expe imen al condi ions ) when g own on mine al medium (so called
BG0) in he absence o combined ni ogen (biological iden iﬁca ion a he le el o
genus is unde wo k). In hese condi ions he biological specimen ( he ﬁlamen ous
cyanobac e ium) is able o u ilize a mosphe ic ni ogen as a sou ce o ni ogen
o syn he izise i s own biochemical componen s (amino acids, p o eins e c); he
ﬁ s majo s eps in his u iliza ion o a mosphe ic ni ogen occu s in he eocys s,
diﬀe en ia e cells wi hin he ﬁlamen [36]. The numbe o cells ound in he image
om Fig. 2 is 13.
4 Conclusions
The de elopmen o expe imen al sciences whe e a big amoun o da a is s o ed
as digi al images needs o powe ul so wa e which helps he esea che o un-
de s and he s udied p ocesses. In pa icula , Mic obiology signiﬁcan ly beneﬁ s i
au oma ed image analysis is used by mic obiologis s in hei p o essional ac i i ies;
u he mo e he in e play be ween mic obiologis s, ma hema icians and enginee s
in his ﬁeld could be help ul in de eloping new oppo uni ies wi hin old so wa e,
o ,e en, o gene a e new so wa e mo e app op ia e o diﬀe en mic obiological
ask.
In his espec , Memb ane Compu ing de ices ha e ea u es ha make hem
sui able o he s udy o digi al images, as he encapsula ion o he in o ma ion
and i s ea men in pa allel. This pape epo s how an app op ia e combina ion
o amilies o issue-like P sys ems can sol e he p oblem o coun ing cells. The
s udy o how hese o o he amilies can be combined o sol ing mo e p oblems
om Mic obiology (o om o he expe imen al sciences) is a challenge o he nex
yea s.
76 I. A delean e al.
Acknowledgemen s
DDP and MAGN acknowledge he suppo o he p ojec TIN-2009-13192 o he
Minis e io de Ciencia e Inno aci´on o Spain and he suppo o he P ojec o
Excellence wi h In es igado de Reconocida Val´ıa o he Jun a de Andaluc´ıa, g an
P08-TIC-04200. IS and IA a e g a e ul o D . Tech. Jy ki Selinummi (Depa men
o Signal P ocessing, Tampe e Uni e si y o Technology, Finland) o e y use ul
and kind discussion conce ning digi al image analysis.
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84 E. Csuhaj-Va ju, M. Gheo ghe, M. S anne
Despi e he in insically local na u e o con olled compu a ion, he locus o
compu a ion can mig a e om one compa men o ano he one ia non-emp y
o e lap egions, as he ollowing example illus a es. Figu e 3 shows he disjoin
pa s, B4,1–B4,7, o egion 4’s co e . These a e all o he o m U Vwhe e Uand V
a e open; o example B4,3= (A4,1∩A4,2∩A4,3) ∅, and B4,1=A4,1 (A4,2∪A4,3).
A4,2
A4,3 A4,1
bc
b
c
c
(a) Open co e
B4,5
B4,6
B4,7
B4,4
B4,3
B4,2 B4,1
(b) O e lap egions
A4,1 A4,2 A4,3
(c) Key o bounda ies
Fig. 3. The ini e co e ing o egion 4 and i s disjoin o e lap egions.
Suppose, hen, ha egion 4 has he ollowing ules associa ed wi h i :
1:bc →b; 2:bcc →c; 3:cc →c.
I we conside he sys em as a P sys em wi h no opological con ol in place, he
ollowing compu a ions can ake place:
1. bc c cb 1, 1
===⇒bcb 1
==⇒bb
2. bc cc b 1, 3
===⇒bcb 1
==⇒bb
3. bc ccb 1, 2
===⇒bc 1
==⇒b
Bu when he open se s a e in place compu a ion pa h 3 is blocked, because none
o he open se s e e con ains bcc, whence 2canno be igge ed. We ha e he
ollowing wo cases ins ead:
1’ 1is applied in bo h A4,1and A4,3 esul ing in a copy o bin each o hese
open se s; i b∈A4,3is no in A4,2∩A4,3 he compu a ion s ops he e wi h bbc
sca e ed ac oss di e en open se s. I , on he o he hand, b∈A4,2∩A4,3 hen
he compu a ion can con inue; a e applying 1in A4,2a copy o bis ob ained
in each o A4,1and A4,2. In his case he esul is he same as ha ob ained in
(1);

Gene al Topologies and P Sys ems 85
2’. 3, 1 esul in copies o b∈A4,1and c∈A4,2; as in (1’) his ccan eside ei he
in he in e sec ion o ou side i ; in he i s case 1can be applied again and b
is compu ed (so ha he esul om (2) is ob ained). O he wise bbc will emain
in he memb ane unchanged.
Conside in pa icula he second case o he i s s ep o (1’). A e 1is applied
in A4,3 he esul bcan be conside ed o lie in A4,2, whence (as sugges ed abo e)
he locus o compu a ion can mig a e om A4,3 o A4,2 ia hei in e sec ion. A
simila si ua ion occu s in (2’) as well.
Mo e gene ally, suppose ha egion ihas a ule whose a ge is egion j. We
will allow he ule o be igge ed only when he wo egions a e su icien ly close
o one ano he ( hei bounda ies mus in e sec : ∂i ∩∂j 6=∅). In his case, and
p o ided all o he equi ed componen s a e a ailable wi hin a single membe o
i’s co e , he ule can i e wi h he esul ing mul ise jbeing injec ed in o an
a bi a ily selec ed membe o j’s co e . In u u e wo k we plan o in es iga e
wha happens when his es ic ion is weakened, so ha in e ac ions can occu
be ween non-neighbou ing egions.
De ini ion 1 (Ou pu o a con olled compu a ion). Fo a P sys em Πand
associa ed opology T he se o numbe s compu ed by Πwhen con olled by Twill
be deno ed NT(Π).⊓⊔
Ha ing now de ined con olled compu a ion, we will add ess he ollowing p ob-
lems. In Sec . 4 we discuss he ole o a con ol mechanism based on an associa ed
opology and show how a gene al opology in luences he compu a ion o a basic
class o P sys ems. In Sec . 5 we summa ise ou indings and discuss u u e esea ch
opics ela ed o a ious opologies associa ed wi h classes o P sys ems.
4 Basic Resul s
We will i s conside P sys ems wi h non-coope a i e ules.
Lemma 1. Fo any P sys em wi h non-coope a i e ules and ei he a bi a y a -
ge s o selec ed a ge s, Π, and any associa ed opology T,N(Π) = NT(Π).
P oo . In a P sys em wi h non-coope a i e ules he le hand side o any ule has
only one single objec , hence no in e ac ions a e in ol ed. In his case i is ob ious
he he opology Tdoes no in luence he compu a ion o ei he P sys ems wi h
a bi a y a ge s o selec ed a ge s, hence he esul s a ed holds. ⊓⊔
Fo P sys ems wi h coope a i e ules he si ua ion is o ally di e en and he
opologies associa ed wi h hem may lead o di e en compu a ions and dis inc
esul s.
Lemma 2. The e is a P sys em wi h coope a i e ules and ei he a bi a y o
selec ed a ge s, Π, such ha o any associa ed opology Twhe e o a leas one
egion no all he objec s belong o he same open se , i ollows ha N(Π)6=
NT(Π).
86 E. Csuhaj-Va ju, M. Gheo ghe, M. S anne
P oo . Le us conside Π= (O, µ, w1, w2, R1, R2, i0), whe e O={a, b, c},µ=
[[]2]1,w1=ab,w2=λ,R1={ab →c, c →(c, in)},R2=∅,i0= 2. This sys em
uses an a bi a y a ge , which, in his case, is he same as selec ed a ge , in2.
This P sys em compu es cin wo s eps in he ou pu egion, 2. Any opology, T,
associa ed wi h Π ha p o ides a co e o egion 1 wi h mo e han an open se ,
mus ha e an open se o aand ano he one o band hei in e sec ion does no
con ain any o hese wo objec s; o he wise, aand bwill s ay in he same open
se . In his case he ule ab →ccan no be applied and consequen ly cis ne e
ob ained in he ou pu egion, hence N(Π)6=NT(Π). ⊓⊔
Theo em 1. Fo any P sys em wi h ei he a bi a y o selec ed a ge s, he com-
pu a ion and he opologically con olled compu a ion a e he same when non-
coope a i e ules a e used and a e no in gene al he same o coope a i e ules.
P oo . The p oo is an immedia e consequence o Lemmas 1 and 2. ⊓⊔
The e a e P sys ems wi h coope a i e ules whe e he con en o he egions can
be ma ched agains he open se s in such a way ha he compu a ion is equi alen
o he compu a ion o he o iginal sys em. Indeed le us conside he p oblem o
checking ha a posi i e in ege mis di ided by ano he posi i e in ege k. We
p opose a P sys em below which is an adap a ion o he P sys em p esen ed in [9].
Example 1. Le us conside Π= (O, µ, w1, w2, R1, R2, i0), whe e O={a, b, c, y, n},
µ= [[]2]1,w1=ambk,w2=y,R1={ 1:ab →c, 2:ac →b, 3:bc →(n, in)},
R2={yn →n},i0= 2.
In he i s s ep a mos kobjec s ab a e eplaced by he same numbe o objec s
c(using 1a mos k imes) and hen objec s ac a e eplaced by objec s b(using
2). I kdi ides m hen he p ocess will s op a e hs eps, whe e m=kh, and in
memb ane 2 will emain y; o he wise in memb ane 1 he p ocess o al e na i ely
applying ules 1and 2will s op wi h some objec s band objec s cand he ule
3can be used. In his case nis sen in o egion 2 and inally nis ob ained in his
egion.
Now, i we aim o ob ain he same esul s in egion 2, i.e., y, when kdi ides
m, o no he wise, hen we ha e o build he opology, T, associa ed wi h Πin a
ce ain way which is subsequen ly desc ibed. Region 2 is co e ed by only one single
open se and egion 1 will ha e an a bi a y numbe o open se s, q > 1, associa ed
wi h. Any wo such open se s a e disjoin . The objec s will be dis ibu ed as ollows:
he k b′s will be andomly dis ibu ed in q−1 o he qopen se s, bk1,...,bkq−1,
ki≥0 and k1+· · · +kq−1=k. I m=kh + , hen in each o he q−1 open
se s con aining kib′s, he numbe o a′s is hkia′s. I > 0 hen one mo e awill
be conside in one o he q−1 open se s wi h b′s and he es will be associa ed
wi h he q h open se . Clea ly, in each o he q−1 open se s he compu a ion will
go o hs eps. In q−2 o hem i will be ob ained ei he only b′s o only c′s; he
open se wi h an addi ional ain i will end up a e one mo e s ep wi h a mix u e
o b′s and c′s and he ule 3will push an nin o memb ane 2 and inally will ge
nin his memb ane. Objec s a′s occu ing in he q h open se will emain he e
o e e . I ollows ha N(Π) = NT(Π). ⊓⊔
Gene al Topologies and P Sys ems 87
The ques ion o whe he he con ol s uc u e in oduced by a opology can be
igno ed, pe haps by using a mo e complex P sys em, is answe ed by he ollowing
esul . This akes in o accoun he in e p e a ion o he ou come o he compu a-
ion as being he numbe o objec s, gi en by he size o he mul ise , p esen in
he ou pu egion.
Theo em 2. Fo any P sys em, Π, and any associa ed opology, T, he e is a P
sys em, Π′, o he same deg ee wi h Π, such ha NT(Π) = N(Π′).
P oo . The idea o he p oo is o cons uc a new P sys em such ha objec s be-
longing o a egion adequa ely e e o objec s o he open se s in he co esponding
egions o he ini ial P sys em.
Le Πbe a P sys em o deg ee n,Π= (O, µ, w1,...,wn, R1, . . . , Rn, i0), and
Ta opology associa ed wi h i . In o de o build a new P sys em, Π′, o deg ee n,
a ew p elimina y no a ions a e made. Fi s , please obse e ha o each egion i,
1≤i≤n, he e exis s a amily o open se s Ai,1,...,Ai,kico e ing i . In gene al
hese open se s a e no disjoin and we desc ibe he ines disjoin pa s o he co e
by conside ing ei he some in e sec ions o open se s o he complemen o an open
se wi h espec o he es o he open se s; i ollows ha he e exis s a ini e
se , deno ed Bi, con aining he se s Bi,1,...,Bi,mi, such ha Bi,j deno es ei he
Ai,l1∩· · ·∩Ai,lj, 1 ≤lj≤kio Ai,j (Ai,1∪ · · ·∪ Ai,j−1∪Ai,j+1 ∪ · · ·∪ Ai,ki). The
se o indexes o he abo e se s Bi,j is deno ed by Ci, i.e., Ci={(i, j)|Bi,j ∈Bi}.
Each objec , a∈O, o he mul ise om egion ibelongs o a ce ain Bi,j . Fo
each a om Bi,j , he ollowing objec s a e conside ed, aα, α ∈Ci.
The P sys em Π′, o deg ee n, is buil as ollows:
Π′= (O′, µ, w′
1,...,w′
n, R′
1,...,R′
ni0),
whe e:
1. O′={aα|a∈O, α ∈Ci,1≤i≤n};
2. µis he memb ane s uc u e o Π;
3. w′
i=a(i, 1)
i,1. . . a(i, pi)
i,pi, whe e ai,j ∈Bi, j, 1 ≤j≤pi, o wi=ai,1. . . ai,pi,
ini ial mul ise o Π;
4. o each ule ai,1. . . ai,qi→bi,1. . . bi,si∈Ri,R′
icon ains a(i, 1)
i,1. . . a(i, qi)
i,qi→
b(i,s1)
i,1. . . b(i,spi)
i,pi, (i, j)∈Ci,1≤j≤qi, (i, sj)∈Ci, 1 ≤j≤piwhen a a ge ,
, appea s on he igh hand side o he ule om Ri, associa ed wi h an objec
bi,j, hen he a ge will poin o any o he open se s A ,j o he a ge egion
;
5. Πand Π′ha e he same ou pu memb ane, i0.
The codi ica ion p o ided by Π′alloca es, in a unique way, in e e y egion,
i, each objec , a, o a speci ic open se , by “s amping” i wi h he co esponding
index, (i, j)∈Ci, o he se Bi,j. Whene e a ule is applied, he esul ed mul ise
is also composed o objec s uniquely associa ed wi h ce ain open se s, ei he om
he cu en egion o om he a ge ones.
88 E. Csuhaj-Va ju, M. Gheo ghe, M. S anne
Mo e p ecisely, when in egion io Π he cu en mul ise is
u=a1. . . aq1aq1+1 . . . aq2. . . aqe−1aqe−1+1 . . . aqez
and he e a e ules ρ1,...,ρe∈R′
i, whe e ρj:aqj−1+1 . . . aqj→bpj−1+1 . . . bpj,
q0= 0, hen ρ1, . . . , ρea e applied in a compu a ion s ep, acco ding o maximal
pa allelism seman ics, o uwi h espec o opology T.
I u⇒ρ1,...ρe , wi h =b1. . . bp1. . . bpe−1. . . bpez hen each ah,qj−1+ 1 ≤
h≤qj, belongs o a ce ain Bi, hincluded in he same open se Ai,j whe e ρjis
applied. Each o he objec s bh,pj−1+ 1 ≤h≤pj, belongs o some Bi,shincluded
in he same Ai,j se .
In he P sys em Π′, in egion i, he e is
u′=a(i, 1)
1. . . a(i, q1)
q1a(i, q1+1)
q1+1 . . . a(i, q2)
q2. . . a(i, qe−1)
qe−1a(i, qe−1+1)
qe−1+1 . . . a(i, qe)
qez′,
whe e (i, j)∈Ci,j∈ { 1,..., qe}. The mul ise z′consis s o objec s (aα)c o ac
occu ing in zand α∈Ci. The e a e ules ρ′
1,...,ρ′
e, whe e
ρ′
j:a(i, qj−1+1)
qj−1+1 . . . a(i, qj)
qj→b(i,spj−1+1)
pj−1+1 . . . b(i,spj)
pj, which a e applied in a maximal
pa allel manne o u′. I u′⇒ρ′
1,...ρ′
e ′, hen
′=b(i,s1)
1. . . b(i,sp1)
p1. . . b(i,spe−1)
pe−1. . . b(i,spe)
pez′,
whe e (i, j)∈Ci,j∈ {s1,...,spe}.
The abo e cons uc ion p o es ha u⇒ρ1,...ρe in Πi and only i u′⇒ρ′
1,...ρ′
e
′in Π′. This shows ha he same numbe o symbols a e engaged in any com-
pu a ion s ep in Πand Π′, hence hese P sys ems compu e he same numbe o
symbols in i0.⊓⊔
F om he abo e p oo i is clea ha he numbe s o objec s and ules used
by he P sys em Π′a e bo h signi ican compa ed o hose o Π. The nex esul
p o ides lowe and uppe bound limi s o hese wo pa ame e s. We need a ew
mo e no a ions o desc ibe he esul .
Fo a ini e se X, le us deno e by ca d(X), he numbe o elemen s o X.
Wi h espec o he p oo o Theo em 2, he ollowing no a ions a e in oduced:
Kis he numbe o elemen s o he se O,nis he deg ee o he wo P sys ems, Π
and Π′; gi en ha o each egion i, 1 ≤i≤n, he numbe o se s Bi,j is mi, le
us deno e m=min{mi|1≤i≤n},M=max{mi|1≤i≤n},p=min{|x|,|y| |
all x→y∈Ri,1≤i≤n}and P=max{|x|,|y| | all x→y∈Ri,1≤i≤n};
i giis he maximum numbe o neighbou s ha appea in he ules o Ri, hen
g=min{gi|1≤i≤n}; inally we ha e Q=ca d(R1∪ · · · ∪ Rn). Wi h hese
no a ions we can o mula e he ollowing esul .
Co olla y 1. Fo any P sys em Πand any associa ed opology T, de ine Π′and
he associa ed no a ion as abo e. Then
(i) Kmn ≤ca d(O′)≤KMn;
Gene al Topologies and P Sys ems 89
(ii)Qmp(min{m, g})p≤ca d(R′
1∪ · · · ∪ R′
n)≤QMP(M+n−1)P.
P oo . Pa (i) ollows om he ac ha o each objec a∈O, dis inc ins ances
a e c ea ed o each o he nmemb anes and in e e y egion i(1 ≤i≤n), and
each se Bi,j (1 ≤j≤mi). Hence, ca d(O′) is bounded be ween Kmn and KMn.
To p o e (ii), we obse e ha o each ule x→y∈Ri, he ollowing ules
a e added o R′
i,xα→yβ,α∈Ci,β∈Ci∪Cj1∪ · · · ∪ Cji, whe e j1,...,jia e
neighbou s o iwhe e objec s o yβcan go o. The le hand side, xα, will ha e
elemen s om any o he mise s, Bi,j, hence he lowe and uppe bounds a e mp
and MP, espec i ely. Each o he igh hand side elemen s o yβshould belong
o ei he one o he Bi,j se s o o one o he neighbou s o i, maximum n−1, so
he lowe and uppe bounds a e (min{m, g})pand (M+n−1)P, espec i ely. We
can hen ge he wo bounda ies o ca d(R′
1∪ · · · ∪ R′
n). ⊓⊔
5 Summa y and Conclusions
In his pape we ha e in es iga ed he use o gene al opological spaces o con ol
local in e ac ions in basic memb ane sys ems. This app oach p oduces a ine g ain
desc ip ion o local ope a ions occu ing in each compa men by es ic ing he
in e ac ions be ween objec s o hose om a ce ain icini y. In ou u u e wo k we
aim o in es iga e he ole o mo e speci ic opologies, hei impac on o he ypes
o memb ane sys ems and hei applica ions o a ious p oblems. In pa icula :
1. By cons uc ion, P sys ems ha e a ee-like nes ed memb ane s uc u e. Gi en
he opological embeddings used in his pape , i is no longe clea whe he his
s uc u e is ele an ; he same p oo s appea o wo k o di e en unde lying
g aph s uc u es wi h some adjus men s.
2. I would be in e es ing o s udy he obus ness o P sys ems wi h espec o
di e en opologies. How much we can change he opology while s ill ob aining
he same o almos he same compu ed se o numbe s? To wha ex en can
locali y be e ined s a ing om a gi en opology and changing i ?
3. I we es ic a en ion o classes o con ol space (Tychono spaces, compac
Hausdo spaces, me ic spaces, e c) o which a wide ange o opological
esul s a e a ailable, can hese esul s be applied o p oduce associa ed cha -
ac e isa ions o con olled compu abili y?
Acknowledgemen s. MPS and MG we e pa ially suppo ed unde he Royal
Socie y In e na ional Exchanges Scheme ( e . IE110369); MG was also pa ially
suppo ed by he p ojec MuVe , Romanian Na ional Au ho i y o Scien i ic Re-
sea ch (CNCS – UEFISCDI) g an numbe PN-II-ID-PCE-2011-3-0688.
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Skele onizing Images by Using
Spiking Neu al P Sys ems
Daniel D´ıaz-Pe nil1, F ancisco Pe˜na-Can illana2,
Miguel A. Gu i´e ez-Na anjo2
1Resea ch G oup on Compu a ional Topology and Applied Ma hema ics
Depa men o Applied Ma hema ics
Uni e si y o Se illa
[email p o ec ed]
2Resea ch G oup on Na u al Compu ing
Depa men o Compu e Science and A iﬁcial In elligence
Uni e si y o Se illa
[email p o ec ed], [email p o ec ed]
Summa y. Skele onizing an image is ep esen ing a shape wi h a small amoun o in o -
ma ion by con e ing he ini ial image in o a mo e compac ep esen a ion and keeping
he meaning ea u es. In his pape we use spiking neu al P sys ems o sol e his p oblem.
Based on such de ices, a pa allel so wa e has been implemen ed on he GPU a chi ec-
u e. Some eal-wo ld applica ions and open lines o u u e esea ch a e also p esen ed.
1 In oduc ion
Compu e ision [32] is p obably one o he challenges o compu e scien is s
in he nex yea s. This ﬂou ishing esea ch a ea needs con ibu ions om many
o he scien iﬁc a eas as a iﬁcial in elligence, pa e n ecogni ion, signal p ocess-
ing, neu obiology, psychology o image p ocessing among o he s. I conce ns wi h
he au oma ed p ocessing o images om he eal wo ld o ex ac and in e p e
in o ma ion on a eal ime basis. F om a compu a ional poin o iew, a digi al
image is a unc ion om a wo dimensional su ace which maps each poin in he
su ace o a se o ea u es as b igh o colo . The diﬀe en ea men s o such
mappings (digi al images) p o ide a big amoun o cu en applica ions in com-
pu e ision as op ical cha ac e ecogni ion (OCR), biome ics, au omo i e sa e y,
su eillance o medical imaging.
In his pape we ocus on he p oblem o skele onizing an image. Skele oniza-
ion is one o he app oaches o ep esen ing a shape wi h a small amoun o
in o ma ion by con e ing he ini ial image in o a mo e compac ep esen a ion
and keeping he meaning ea u es. The con e sion should emo e edundan in-
o ma ion, bu i should also keep he basic s uc u e. Skele oniza ion is usually
92 D. D´ıaz-Pe nil e al.
conside ed as a p e-p ocess in pa e n ecogni ion algo i hms, bu i s s udy is also
in e es ing by i sel o he analysis o line-based images as ex s, line d awings,
human ﬁnge p in s o ca og aphy.
Many p oblems in he p ocessing o digi al images ha e ea u es which make i
sui able o echniques inspi ed by na u e. One o hem is ha he ea men o he
image can be pa allelized and locally sol ed. Rega dless how la ge is he pic u e,
he p ocess can be pe o med in pa allel in diﬀe en local a eas o i . Ano he
in e es ing ea u e is ha he local in o ma ion needed o a pixel ans o ma ion
can also be easily encoded in he da a s uc u es used in Na u al Compu ing. In he
li e a u e, we can ﬁnd many examples o he use o Na u al Compu ing echniques
o dealing wi h p oblems associa ed o he ea men o digi al images. One o he
classic examples is he use o cellula au oma a [28, 31]. O he eﬀo s a e ela ed
o a iﬁcial neu al ne wo ks as in [9, 35]. In his pape , we use spiking neu al P
sys ems.
Spiking neu al P sys ems (SN P sys ems, o sho ) we e in oduced in [16] as
a new class o dis ibu ed and pa allel compu ing de ices, inspi ed by he neu o-
physiological beha io o neu ons sending elec ical impulses (spikes) along axons
o o he neu ons. SN P sys ems a e he hi d model o compu a ion in he ame-
wo k o Memb ane Compu ing1, oge he wi h he cell-like model [25] inspi ed by
he compa men al s uc u e and unc ioning o a li ing cell and he issue-like
model [17], based on in e cellula communica ion and coope a ion be ween cells
in a issue.
Recen ly, Memb ane Compu ing echniques ha e been used o sol ing p ob-
lems om Digi al Image. Diﬀe en P sys ems models ha e been used o dealing
wi h images, as in [3] whe e cell-like P sys ems a e used o compu ing he h esh-
olding o 2D images; [4, 5, 23, 24] whe e issue-like P sys ems a e used, o e en
[10], whe e he symme ic dynamic p og amming s e eo (SDPS) algo i hm [11] o
s e eo ma ching was implemen ed by using simple P modules wi h duplex chan-
nels. To he bes o ou knowledge, his is he ﬁ s ime in which SN P sys ems
a e used o dealing wi h images.
In a simila way ha o he applica ions o P sys ems, he heo e ical ad an-
ages o he Memb ane Compu ing echniques o compu e ision need a pow-
e ul so wa e and ha dwa e o an eﬀec i e implemen a ion. In his pape , we
also p esen a pa allel so wa e de eloped by using a de ice a chi ec u e called
CUDATM , (Compu e Uniﬁed De ice A chi ec u e). CUDATM is a gene al pu pose
pa allel compu ing a chi ec u e ha allows he pa allel NVIDIA2G aphics P o-
cesso s Uni s (GPUs) o sol e many complex compu a ional p oblems in a mo e
eﬃcien way han on a CPU. GPUs cons i u e nowadays a solid al e na i e o
high pe o mance compu ing, and he ad en o CUDA allows p og amme s a
iendly model o accele a e a b oad ange o applica ions. The way GPUs ex-
1We e e o [26] o basic in o ma ion in his a ea, o [27] o a comp ehensi e p e-
sen a ion and he P sys em web page h p://ppage.psys ems.eu, o he up- o-da e
in o ma ion.
2h p://www.n idia.com.
Skele onizing Images by Using SN P Sys ems 93
ploi pa allelism diﬀe s om mul i-co e CPUs, which aises new challenges o ake
ad an age o i s emendous compu ing powe . GPU is especially well-sui ed o
add ess p oblems ha can be exp essed as da a-pa allel compu a ions.
The pape is o ganized as ollows: Fi s ly, we p esen he es ic ed model o SN
P sys ems used in he pape and ecall he Guo & Hall algo i hm o skele onizing
images. In Sec ion 4, he design o he SN P sys em o skele onizing images is
p esen ed. Nex , we show illus a i e examples o he use o ou implemen a ion.
Finally, Sec ion 6 is dedica ed o conclusions and u u e wo k.
2 Spiking Neu al P Sys ems
SN P sys ems can be iewed as an e olu ion in Memb ane Compu ing co e-
sponding o a shi om cell-like o neu al-like a chi ec u es. In SN P sys ems he
p ocessing elemen s a e called neu ons and a e placed in he nodes o a di ec ed
g aph, called he synapse g aph. The compu a ion is pe o med by sending elec i-
cal impulses among he neu ons h ough he synapses. Such elec ical impulses a e
encoded ia a single objec ype, namely he spike, which is placed in he neu ons.
The numbe o copies o such objec de e mines he elec ical cha ge o he neu on.
Each neu on may also con ain ules which allow o send spikes (possibly wi h a
delay) o o he neu ons, o o emo e a gi en numbe o spikes om i ( i ing and
o ge ing ules).
Fi ing ules allow a neu on o send in o ma ion o o he neu ons in he o m
o elec ical impulses which a e accumula ed a he a ge cell. Fo ge ing ules
emo e om he neu on a p edeﬁned numbe o spikes. The applica ion o e e y
ule is de e mined by checking he con en s o he neu on agains a egula se
associa ed wi h he ule. In each ime uni , i a neu on can use one o i s ules,
hen one o such ules mus be used. I wo o mo e ules could be applied, hen
only one o hem is nonde e minis ically chosen. Thus, he ules a e used in he
sequen ial manne in each neu on, bu neu ons wo k in pa allel wi h each o he .
A global clock is assumed, ma king he ime o he whole sys em, and hence he
unc ioning o he sys em is synch onized.
F om he seminal pape [16], o he biological ea u es ha e been explo ed in
he amewo k o SN P sys ems. One o such ex ensions (wi h ma hema ical mo-
i a ion) was in oduced in [2], whe e a neu on can emi mo e han one spike, i
he numbe o emi ed ones is no g ea e han he consumed ones. O he a ian s
including as ocy es [20], weigh s, which modi y he numbe o spikes ha a i es
o a neu on acco ding o he quali y o he link be ween neu ons [14, 21, 34], an i-
spikes [22], o neu on di ision [33] ha e also been conside ed. In his pape , we will
conside SN P sys ems wi h weigh s in he synapses and ules wi hou delay3.
In his way, he es ic ed model o SN P sys ems used in his pape can be
o mally desc ibed as ollows. A spiking neu al P sys em o deg ee m≥1 is a
cons uc o he o m
3Fo a mo e gene al desc ip ion, see [16].
100 D. D´ıaz-Pe nil e al.
espec i ely. We will add o he 512 spikes in σodd as many spikes as indica ed as
inpu o s a ing he compu a ion.
The neu on σodd has only one ule a513a∗/a513 →a512 which is applied i he
neu on has a leas 513 spikes. The neu on σe en has also one ule a512/a512 →a512.
The egula neu ons ha e wo ypes o ules: Fi ing and o ge ing ones, which will
be applied i he numbe o spikes is exac ly b= +512 wi h ∈DEL o ∈DEL,
espec i ely.
Wi h espec o he synapses, a egula neu on co esponding o a pixel Pis
linked o he eigh neu ons co esponding o he eigh neighbo pixels o P, wi h
he weigh s 1/2iwhe e i∈ {0,...,7} ollows an an i-clockwise enume a ion o he
pixels s a ing in he eas pixel (see Fig. 3 (Righ )). The neu ons σodd and σe en
a e linked each o he . The neu on σodd is also linked o all he egula neu ons σij
wi h i+jodd and, analogously, σe en is linked o all he egula neu ons σij wi h
i+je en.
4.1 How i wo ks
In o de o unde s and how he SN P sys em wo ks, ﬁ s ly we obse e ha a he
ini ial conﬁgu a ion, he se o egula neu ons σij encodes he image which will be
skele onized. We will show ha , a any ime, hese neu ons encode he successi e
images ob ained in he i e a i e p ocess o dele ing black pixels acco ding o he
Guo & Hall algo i hm. Le us ema k ha i he numbe o spikes in σij is e en,
hen he co esponding pixel is black; o he wise, i he numbe o spikes is odd,
hen he co esponding pixel is whi e. This is easily de i ed om he deﬁni ion o
cod(i, j).
Ano he obse a ion o be conside ed is ha he pa i y o he numbe o spikes
in a neu on ne e changes, since he numbe o spikes ecei ed o emo ed is
always an e en amoun , excep by he applica ion o he ule ab/a511 →a256. As
we will see below, he applica ion o his ule is in e p e ed as he dele ion o he
co esponding black pixel in he Guo & Hall algo i hm and i is applied once a
mos in each neu on.
Be o e explaining he diﬀe en s eps o he p ocess, le us conside a pixel (i, j)
in he image and a black pixel adjacen o (i, j). We iden i y his black pixel o
∈ {P1, . . . , P8}acco ding o he clockwise enume a ion desc ibed abo e. Le us
suppose ha he black pixel in belongs o he selec ed subsec ion in he cu en
s ep o he Guo & Hall algo i hm and i sa isﬁes he condi ions o be dele ed.
Le us conside now he neu ons σij and σ co esponding o he pixels in (i, j)
and . As we will show below, he h ee condi ions o (i is black, i belongs
o a selec ed subsec ion and i sa isﬁes he Guo & Hall condi ions o be dele ed)
indica es ha he numbe o spikes in σ is b= + 512 wi h ∈DEL. In his
case he ule ab/a511 →a256 is applied. As poin ed ou abo e, he applica ion o
his ule changes he pa i y o he numbe o spikes σ ( om e en, since he pixel
is black, o odd) and his change is in e p e ed as a dele ion o he pixel in .
We ocus on he inﬂuence o he dele ion o he black pixel in (o , equi alen ly,
he applica ion o he ule ab/a511 →a256 in σ ) on he neu on σij. Since he

Skele onizing Images by Using SN P Sys ems 101
numbe o spikes in σij a he ini ial conﬁgu a ion is cod(i, j) = ∑8
i=0 Hi×2iand,
in his conﬁgu a ion, he pixel is black, acco ding o he encoding, his means
ha H is ze o, o , in o he wo ds, 2 does no appea in he encoding o he
en i onmen as an addi ion o powe s o 2.
The dele ion o he pixel in changes he en i onmen o (i, j) and hen, since
he numbe o spikes in σij ep esen s such en i onmen , he numbe o spikes
mus change. In pa icula , he change co esponds o u n H o 1, o , in o he
wo ds, o add 2 o he numbe o spikes in σij.
In o de o check ha his happens, i is suﬃces o seen ha he ule ab/a511 →
a256 sends 256 = 28 om neu on σ o σij , bu hese 28mus be mul iplied by
he co esponding weigh in {1/20,...,1/27}, so only 2, 22, 23,. . . , 27o 28spikes
a i e o σij , depending on he alue o . A simple inspec ion shows ha he
numbe o spikes ha a i es o σij is exac ly 2 when he black pixel is dele ed.
Bea ing in mind hese conside a ions, we show ha o an inpu N∈
{1,...,513}, he compu a ion s eps o he SN P sys em co espond o he i -
e a i e p ocess o he Guo & Hall algo i hm whe e he ﬁ s selec ed subsec ion
co esponds o pixels wi h i+jodd. In such way we will show he ollowing
s a emen s:
•S a emen 1: The se o egula neu ons is spli in o wo sub-sec ions. One o
he sec ions is composed by he neu ons σij such ha i+jis e en. Al e na i ely,
he second sub-sec ion co esponds o he neu ons σij such ha i+jis odd.
Bo h subsec ions a e al e na i ely selec ed, s a ing wi h he odd subsec ion.
•S a emen 2: In each compu a ion s ep, only neu ons co esponding o he
selec ed subsec ion a e e alua ed. The e alua ion consis s on de e mining i he
neighbo hood o he pixel associa ed o he neu on sa isﬁes he condi ions o
he Guo & Hall algo i hm o be dele ed.
•S a emen 3: I an e alua ed black pixel sa isﬁes he Guo & Hall condi ions
o be dele ed (see Sec ion 3), hen he numbe o spikes in he co esponding
neu on changes om e en o odd.
•S a emen 4: In each conﬁgu a ion, he numbe o spikes in he egula neu-
ons is he codiﬁca ion o an image, acco ding o he encoding desc ibed abo e.
The ﬁ s key poin o he algo i hm is ha he image is spli in o wo subsec-
ions which will be explo ed al e na i ely. One black pixel will be conside ed o
i s dele ion only i i belongs o he subsec ion selec ed in he cu en s ep. We
conside ha a egula neu on σij is selec ed a he s ep i i s numbe o spikes
in he conﬁgu a ion C is g ea e han o equal o 512. O he wise, i i s numbe
o spikes is lowe han 512, hen he neu on is no selec ed.
Nex we show ha he egula neu ons wi h i+jodd and e en a e al e na i ely
selec ed. The selec ion o subsec ions is pe o med by he neu ons σodd and σe en
which send, al e na i ely, 512 spikes o he egula neu ons σij wi h i+jodd and
e en, espec i ely.
Lemma 1. Le σij be a egula neu on and N∈ {1, . . . , 513} he inpu o he SN
P sys em. Fo ∈ {0, . . . , N −1}
102 D. D´ıaz-Pe nil e al.
•I i+jis odd, hen he numbe o spikes in σij is g ea e han o equal o 512
in he conﬁgu a ion C2 +1 and i is lowe han 512 in he conﬁgu a ion C2 .
•I i+jis e en, hen he numbe o spikes in σij is g ea e han o equal
o belongs o 512 in he conﬁgu a ion C2 +2 and i is lowe han 511 in he
conﬁgu a ion C2 +1.
P oo . Le us obse e ha in he ini ial conﬁgu a ion, he numbe o spikes in
a egula neu on σij is cod(i, j)<512; he numbe o spikes in σodd is 512 + N
and he e is ze o spikes in he neu on σe en. F om his ini ial conﬁgu a ion, he
unique applicable ule is a513a∗/a513 →a512 in he neu on σodd (since N≥1).
A e applying his ule, in he conﬁgu a ion C1, he numbe o spikes in σodd is
N−1; he numbe o spikes in σe en is 512; and he numbe o spikes in σij is
cod(i, j) + 512 i i+jis odd and cod(i, j) i i+jis e en.
Le us ocus now on σodd and σe en. The unique neu on ha sends spikes o σodd
is σe en and, analogously, he unique neu on ha sends spikes o σe en is σodd. In
he conﬁgu a ion C1, he spikes in σodd and σe en a e N−1 and 512, espec i ely.
Since N≤513, he ule in σodd canno be applied in his conﬁgu a ion, bu he
ule in σe en can be applied, so he spikes in σodd and σe en in he conﬁgu a ion C2
a e 512+N−1 and 0, which is simila o he si ua ion in he ini ial conﬁgu a ion,
so we ha e ha o ∈ {1, . . . , N}, he numbe o spikes in σodd and σe en and in
he conﬁgu a ion C2 a e 512 + N− and 0.
No ice ha a he conﬁgu a ion C2N, he numbe o spikes in σodd and σe en
a e 512 and 0, espec i ely, and no mo e ules a e applied in hese neu ons.
Acco ding o he numbe o spikes in σodd and σe en in he odd and e en
conﬁgu a ions, and aken in o accoun hei synapses, hen we ha e ha , o ∈
{0, . . . , N −1}, a he conﬁgu a ion C2 +1, he egula neu ons wi h i+jodd has
a leas 512 spikes; and a C2 +2, he egula neu ons σij wi h i+je en has a
leas 512 spikes.
In o de o comple e he p oo , i is necessa y o p o e ha o ∈ {0, . . . , N −
1}, a he conﬁgu a ion C2 +1, he egula neu ons wi h i+je en has a mos
511 spikes; and a C2 +2, he egula neu ons σij wi h i+jodd has a mos 511
spikes.
Le us s a by conside ing a egula neu on σij wi h i+jodd a he con-
ﬁgu a ion C1. As we show abo e, i s numbe o spikes is b= 512 + wi h
=cod(i, j)≤511. Depending on ∈DEL o ∈DEL, one o he ules
ab/a511 →a256 o ab/a512 →λis applied.
•Le us suppose ha ∈DEL. In pa icula , his means ha he co esponding
pixel is black and he applied ule is ab/a511 →a256. The numbe o spikes in
he conﬁgu a ion C2is equal o he spikes in he conﬁgu a ion C1(512 + ),
minus he consumed ones 511 plus he con ibu ion o o he neu ons. Since σodd
does no send any spike in his s ep, he unique con ibu ion o he numbe
o spikes comes om o he egula neu ons. Each con ibu ion is he addi ion
o 2ispikes o he spikes in σij, bu , bea ing in mind ha he pixel is black,
hen 20does no appea s in he decomposi ion o cod(i, j) as sum o powe s
Skele onizing Images by Using SN P Sys ems 103
o 2, and i canno be added as a con ibu ion o o he neu on, so plus he
con ibu ion o o he neu ons is a mos 510 and hen, he numbe o spikes in
σij in C2is lowe han 512.
•I ∈ DEL, hen he ule ab/a512 →λis applied. Since 512 spikes a e con-
sumed and plus he con ibu ions o he o he egula neu ons is a mos 511,
hen he numbe o spikes in σij is lowe han 512, also in his case.
This easoning can be also applied o show ha he numbe o spikes o he
neu ons σij wi h i+je en is lowe han 512 in he conﬁgu a ion C3and in gene al
we ha e ha o ∈ {0, . . . , N −1}, a he conﬁgu a ion C2 +1, he egula neu ons
wi h i+je en has a mos 511 spikes; and a C2 +2, he egula neu ons σij wi h
i+jodd has a mos 511 spikes. 
The p e ious lemma shows ha he p ope y o ha e a leas 512 spikes changes
al e na i ely om neu ons σij wi h i+jodd and e en. F om his esul , i is easy
o check he second s a emen , since he ules in he egula neu ons can only be
applied i he numbe o spikes is a leas 512. Tha means ha only in such cases
he neu on is conside ed o e alua ion.
E alua ing a neu on consis s on deciding i he ule ab/a511 →a256 o
ab/a512 →λis applied, bu such decision depends on he se DEL which a e he
se o encodings o he neighbo hood such ha he cen al pixel mus be dele ed.
The nex key poin o he algo i hm is he dele ion o pixels. By deﬁni ion
o cod(i, j), a egula neu on σij has an odd numbe o spikes i and only i i
ep esen s a whi e pixel. Analogously, a egula neu on σij has an odd numbe o
spikes i and only i i ep esen s a whi e pixel. Bea ing in mind his coding o
black pixels, dele ing a black pixel in a compu a ion s ep consis s on emo ing an
odd amoun o spikes om a neu on wi h an e en amoun o spikes.
Lemma 2. Le us conside a black pixel and a s ep o he Guo & Hall algo i hm,
such ha he pixel belongs o he selec ed subsec ion and i sa isﬁes he condi ions
o he algo i hm o be dele ed. Then, in he co esponding s ep o he SN P sys em
compu a ion, he co esponding egula neu on σij will pass om an odd amoun
o spikes o an e en amoun .
P oo . Acco ding o he p e ious cons uc ion, a black pixel which belongs o he
selec ed subsec ion in he Guo & Hall algo i hm and e iﬁes he condi ions o
be dele ed has associa ed a egula neu on wi h 512 + spikes, ∈DEL. In
his case, he ule ab/a511 →a256 is applied. Bea ing in mind ha is e en, he
con ibu ions o o he neu ons is e en and an odd numbe o spikes is consumed,
in he nex conﬁgu a ion, he numbe spikes in he neu on is odd. 
Since all he ∈DEL a e e en and he con ibu ion o o he neu ons is always
e en, hen he ule ab/a511 →a256 wi h b= 512 + is applied a mos once in
each neu on. This means ha i a pixel is dele ed (changed om black o whi e)
i ne e becomes black o whi e, and he i e a i e p ocess o hinning he image is
also ca ied ou in he SN P sys em.
Finally, o sum up hese s a emen s. We claim he ollowing esul .
104 D. D´ıaz-Pe nil e al.
Theo em. The se o egula neu ons o he SN P sys em encodes in each conﬁg-
u a ion he successi e images ob ained in he i e a i e p ocess o hinning o he
Guo & Hall algo i hm, by aking as black he pixels wi h an e en amoun o spikes
and whi e he neu ons wi h an odd amoun o spikes.
5 Expe imen al Simula ion
Simula ion o diﬀe en a ian s o P sys ems ha e been widely s udied in he las
yea s. Since he e do no exis implemen a ions o P sys ems in i o no in i o,
he na u al way o explo e he beha io o designed P sys ems is o simula e i
in con en ional compu e s. A sho desc ip ion o some o hese simula o s can be
ound in [7, 15]. Cu en ly, a big eﬀo is being de eloped in he P-lingua p ojec [8],
by combining an eﬃcien simula ion engine wi h an ad-hoc p og amming language.
In his pape , a so wa e ools based on he design o he SN P sys em has
been implemen ed by using CUDATM , (Compu e Uniﬁed De ice A chi ec u e)
[18, 19]. CUDATM is a gene al pu pose pa allel compu ing a chi ec u e ha allows
he pa allel NVIDIA G aphics P ocesso s Uni s (GPUs) o sol e many complex
compu a ional p oblems in a mo e eﬃcien way han on a CPU.
The expe imen s ha e been pe o med on a compu e wi h a CPU AMD A hlon
II x4 645, which allows o wo k wi h ou co es o 64 bi s o 3.1 GHz. The compu e
has ou blocks o 512KB o L2 cache memo y and 4 GB DDR3 o 1600 MHz o
main memo y.
The used g aphical ca d (GPU) is an NVIDIA Ge o ce GT240 composed by 12
S eam P ocesso s wi h a o al o 96 co es o 1340 MHz. I has 1 GB DDR3 main
memo y in a 128 bi s bus o 700 MHz. So, he ans e a e ob ained is by 54.4
Fig. 4. Scheme o he h eads
Skele onizing Images by Using SN P Sys ems 105
Fig. 5. Example o image wi h aﬃc signals
Gbps. The used Cons an Memo y is 64 KB and he Sha ed Memo y is 16 KB. I s
Compu e Capabili y le el is 1.2 ( om 1.0 o 2.1). The implemen a ion deals wi h
Nblocks o h eads o he comple e image in ou GPU o 96 co es, as we can see
in Fig. 4. We need mo e h eads han pixels i he heigh and wid h o he image
a e no mul iples o 16; i.e., we can ha e useless h eads (see Figu e 4).
5.1 Examples
A ﬁ s example is shown in Fig 2. Skele onizing hand-w i en ex s is one o he
challenges o skele onizing, since he skele on keeps he opological s uc u e and
meaning o he o iginal and he ex can be easily s o ed.
Nex , we p o ide se e al examples o a ealis ic ecognizing p oblem wi h ap-
plica ions in he au omo i e indus y. In Fig. 5, we can see a pho og aph aken in
a oad. I has been bina ized by using a h eshold me hod by using a h eshold
100 on a g ay scale 0, . . . , 255. We can see ha he skele onized images keep he
in o ma ion o he aﬃc signals and hey can be used in a u he pa e n ecog-
ni ion p oblem (see Fig. 6). In Fig. 7, wo mo e examples o skele ionizing eal
images a e shown.
We ﬁnish his sec ion by showing he esul s o some expe imen s pe o med
wi h ou implemen a ion. We ha e aken 36 o ally black images o n×npixels8,
om n= 125 o n= 4500 wi h a egula inc emen o 125 pixels o side. Figu e 8
( op) shows he ime in milliseconds o ou so wa e ool inspi ed in he designed
SN P sys em o implemen ing he Guo & Hall algo i hm o 1, 30, 60 and 90
s eps in he skele onizing p ocess. Figu e 8 (bo om) shows he same s udy o a
sequen ial implemen a ion o he algo i hm.
8Theo e ically, his is he wo s case, since he ime in e ed by he algo i hm depends
on he size o he bigges black connec ed componen o he o iginal image.

106 D. D´ıaz-Pe nil e al.
Fig. 6. (Top le ) The bina iza ion o he image om Figu e 5 (Top igh ) In e se bina-
iza ion o he image. (Bo om le ) I s skele onizing. (Bo om igh ) The skele onizing
o he in e se h esholding.
6 Conclusions
The de elopmen o new bioinspi ed pa allel echniques p o ides a chance o e-
isi ing classical sequen ial algo i hms. In his pape , we ha e conside a classical
algo i hm o skele onizing images, bu many o he algo i hms can be conside ed.
In pa icula , he bio-inspi ed compu ing echniques ha e ea u es as he encapsu-
la ion o he in o ma ion, a simple ep esen a ion o he knowledge and pa allelism,
which a e app op ia e wi h dealing wi h digi al images.
None heless, he use o new compu a ional pa adigms o de eloping he bio-
inspi ed ideas needs, on he one hand, he con ibu ion o heo e ical esea ch
ha allows us o design new bio-inspi ed eﬃcien algo i hms, and, on he o he
hand, he use o he mos ecen pa allel compu e a chi ec u es o a eal pa allel
implemen a ion o he algo i hms.
In his pape we p o ide a new s ep in bo h di ec ions, since we s udy he
skele oniza ion o images by using Spiking Neu al P sys ems and show he esul s
o a new so wa e based on he SN P sys em by using he GPU a chi ec u e. This
esea ch line can be ollowed by conside ing mo e classical p oblems and s udying
he possible imp o emen s om a bio-inspi ed pe spec i e, o , by s udying he
same skele oniza ion p oblem in o he P sys em models.
Skele onizing Images by Using SN P Sys ems 107
Fig. 7. O iginal images and hei skele ons
Acknowledgemen s
DDP and MAGN acknowledge he suppo o he p ojec s TIN2008-04487-E and
TIN-2009-13192 o he Minis e io de Ciencia e Inno aci´on o Spain and he suppo
o he P ojec o Excellence wi h In es igado de Reconocida Val´ıa o he Jun a
de Andaluc´ıa, g an P08-TIC-04200.
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116 R. F eund e al.
Fo a mul ise o ules R∈R◦and a conﬁgu a ion C ∈ Cwe deﬁne he se
o mul ise s Applicable(R, C)⊆(R×N∗)◦gi ing he se o mul ise s o ins an i-
a ed ules ha can be compu ed based on Rand he conﬁgu a ion C. This se is
compu ed as ollows.
Le R={ 1, . . . , n}( he ules a e no necessa ily diﬀe en ) and le IC( i) =
( i,1, . . . , iki), 1 ≤i≤n. Conside an a bi a y ec o o ule ins ances =
( 1,j1, . . . , n,jn), 1 ≤ji≤ki, 1 ≤i≤n. The mul ise {( 1, 1,j1), . . . , ( n, n,jn)}
is added o Applicable(R, C) i
•Fo all p∈P e m( i)∪DP e m( i), p⟨ ⟩ ⊆ ¯
Cm, 1 ≤i≤n.
•Fo all q∈F o ( i), q⟨ ⟩ ⊆ ¯
Cm, 1 ≤i≤n.
•∪n
i=1 Bound( i)⟨ ⟩ ⊆ ¯
Cm.
•The consecu i e applica ion o g aph ansduce s Change–Rela ion( i) and
Change–Rela ion( j) yields he same esul ega dless o he o de o he ap-
plica ion, 1 ≤i, j ≤n.
I is clea ha he e is a bound on he size o he mul ise o ules R o which
Applicable(R, C) is no emp y. We deno e by Applicable(C)⊆(R×N∗)◦ he union
o co esponding mul ise s:
Applicable(C) = ∪
Applicable(R,C)=∅
Applicable(R, C).
Fo a P sys em Πha ing a se o ules Rwe deﬁne Applicable(Π, C) =
Applicable(R, C). Following [3] i is possible o deﬁne now he ansi ion modes as
a es ic ion o his se . Howe e , i should be no ed ha since he co esponding
mul ise s con ain ins an ia ed ules, addi ional es ic ions based on ins ances can
be placed.
Now we a e eady o deﬁne he applica ion o a mul ise o ules R.
Le C= (L, ρ) be he cu en conﬁgu a ion and le RI ∈Applicable(R, C),
RI ={( 1, 1), . . . , ( n, n)}be a mul ise o ins an ia ed ules. We now deﬁne he
ope a ion Apply(RI, C)∈Cwhich is he esul o he applica ion o RI o C.
Be o e gi ing he algo i hm we ema k ha a ule is composed om h ee pa s:
he ew i ing o objec s and he label change (R), he memb ane dele ion (D) and
he memb ane c ea ion (G). The o de o he applica ion o hese pa s is ex emely
impo an , e.g. doing he ew i ing be o e he memb ane c ea ion pe mi s o copy
he esul o he ew i ing o he new memb anes. In his a icle we conside
ha he applica ion o de is RGD, i.e. ew i ing, c ea ion and hen dele ion. This
o de co esponds o he ac ual s a e o a in he a ea o P sys ems wi h ac i e
memb anes. O he o de s a e also possible and his can be an in e es ing opic o
a u he esea ch.
The algo i hm o he compu a ion o Apply(RI, C) is he sequence consis ing
o he ollowing s eps.
1. ( ew i ing applica ion): L1={(i1, l1, w′
1)...(in, ln, w′
n)}whe e:

A Fo mal F amewo k o P Sys ems wi h Dynamic S uc u e 117
w′
j=wj+∪
( k, k)∈RI
(s, u → )∈Rew i e( k)
k|s=j
(−u+ ).
2. (label change): L2={(i1, l′
1, w′
1). . . (in, l′
n, w′
n)}whe e:
l′
j=


es, he e is ( k, k)∈RI such ha (s, es)∈Label–Rename( k)
and k|s=j,
lj,o he wise.
3. (memb ane c ea ion): (m1. . . m +sa e new ids). We deﬁne he lis s o newly
c ea ed cells Lcand L′
c:
Lc( k) =(m1, h1, u1). . . (m , h , u ),( k, k)∈RI and
Gene a e( k) = {(1′, h1, u1). . . ( ′, h , u )}.
Lc=
n
∏
k=1
Lc( k).
L′
c( k) = (m +1, h +1, w′
n1−u1+ 1)...(m +s, h +s, w′
ns−us+ s),whe e
( k, k)∈RI and
Gene a e–and–Copy( ) ={(( + 1)′, h +1, n +1, u +1 → +1). . .
(( +s)′, h +s, n +s, u +s→ +s)},
(ij, l′
j, w′
j)∈L2,1≤j≤n
L′
c=
n
∏
k=1
L′
c( k).
By deﬁni ion, we pu k|q′=mq, 1 ≤q≤ +s.
We also conside a g aph ansduce CREAT E–NODES ha c ea es nodes
m1, . . . , m +s.
We pu L3=L2·Lc·L′
c( his is he Cmpa o he esul o he applica ion o
R).
4. (memb ane dele ion):
Conside a ec o P= (p1, . . . , pn) deﬁned as ollows:
pj=















∗,i he e exis s ( k, k)∈RI such ha s∈Dele e( k)
and k|s=j,
k|m,i he e exis s ( k, k)∈RI such ha
(s, m)∈Dele e–and–Mo e( k) and k|s=j,
j, o he wise.
118 R. F eund e al.
The ﬁ s wo cases co espond o hose ids j o which he co esponding cells
should be dele ed. We ema k ha o any pksuch ha pk=ik, he e is a
alue z∈N∪ {∗} such ha he e is a sequence x1, . . . , xmwi h x1=pk,
xm=z, and xj=pxj−1, 2 ≤j≤m. We deno e his by z=las (x). The
abo e aﬃ ma ion ollows om ha ac ha he Dele e–and–Mo e ela ion
(conside ed as a pa en ela ion) induces a o es on he ids o he cells ha
should be dele ed. The oo s o he ob ained ees a e gi en by he unc ion
las and hey will collec he objec s om all he cells in he ee (i hey a e
diﬀe en om ∗).
Nex we desc ibe how he con en s is mo ed:
L4={(i1, l′
1, w′′
1). . . (in, l′
n, w′′
n)}whe e (ik, l′
k, w′
k)∈L3, 1 ≤k≤n, and
w′′
j=w′
j+∪
las (k)=j
w′
k.
The dele ion o cells induces changes o he ela ion ρ. We collec hese modi-
ﬁca ions as a g aph ansduce DELET E–NODES ha will be un a e he
Change–Rela ion ansduce . This ansduce dele es all e ices jsuch ha
pj=jas well as all edges ha a e incoming o hese dele ed nodes.
We also emo e he co esponding cells om L4:
L5= (i1, l′
1, w′′
1). . . (in1, l′
n1, w′′
n1) whe e (ij, l′
j, w′′
j)∈L4and pj=ij.
5. ( ela ion change) The new ela ion C′
ρis compu ed by unning he g aph ans-
duce s CREAT E–NODES,Change–Rela ion( ⟨ k⟩) and
DELET E–NODES o all ( k, k)∈Ron Cρ.
3 Taxonomy
In o de o simpli y he no a ion we conside se e al a ian s o ule no a ion:
Simple ew i ing ule (R- ule)
An R- ule is deﬁned only by he ollowing componen s:
= (Labels( ), ρ( ), Rew i e( ))
Simple ew i ing ule wi h label ename (LR- ule)
An LR- ule is deﬁned only by he ollowing componen s:
= (Labels( ), ρ( ), Rew i e( ), Label–Rename( ))
Simple c ea ion ule (C- ule)
AC- ule is deﬁned by he ollowing componen s:
= (Labels( ), ρ( ), Gene a e( ), Gene a e–and–Copy( ),
Change–Rela ion( ))
A Fo mal F amewo k o P Sys ems wi h Dynamic S uc u e 119
Simple c ea ion ule wi h label ename (CL- ule)
ACL- ule is deﬁned by he ollowing componen s:
= (Labels( ), ρ( ), Label–Rename( ), Gene a e( ),
Gene a e–and–Copy( ), Change–Rela ion( ))
Simple dissolu ion ule (D- ule)
AD- ule is deﬁned by he ollowing componen s:
= (Labels( ), ρ( ), Dele e( ), Dele e–and–Mo e( ),
Change–Rela ion( ))
In he case o he pa en ela ion ( ee case), we can simpli y he ules and omi
ρ( ) by supposing ha Labels( ) is o size 2. In his case we implici ly assume ha
(1,2) ∈ρ( ). The ype o co esponding ules wi h pa en ela ion will addi ionally
con ain he le e P(e.g., P C- ule).
In a mo e gene al way we can combine se e al componen s: L– label ename,
R– ew i ing, C– memb ane c ea ion, D– memb ane dele ion (and ge RD ules
o example).
Rules ha ing a non-emp y Dele e–and–Mo e( ) componen can be simpli-
ﬁed by educing hei Change–Rela ion( ) componen in he case o he pa en
ela ion.
In he abo e case we will assume ha Change–Rela ion( ) con ains he ans-
duce MOV E −CONNECTIONS desc ibed below. This ansduce adds he
ollowing edges o ρ:{(ax, by)|(x, y)∈ Cρand ax, by=∗}, whe e (ax, by) is de-
ﬁned as ollows (imis he id o memb ane m):
(ax, by) = {(las (x), y),(x, y)∈ρand px=ix,
(y, las (x)),(y, x)∈ρand px=ix.
The abo e ans o ma ions co espond o he dele ion o cells and o he mo e-
men o hei con en s acco ding o Dele e–and–Mo e ela ion.
4 Some examples
4.1 Ac i e memb anes
Le us s a wi h he example o adi ional ac i e memb ane ules (e.g., see Sec ion
11.2 om handbook).
Pola iza ion can be ea ed in wo ways – as a special objec inside a mem-
b ane o like a special label; we he e conside he la e case, i.e., he couple
(label,pola iza ion) will be a new ype o label.
Thus, a ule : [a→ ]e
hwill be ea ed as : [a→ ]⟨e,h⟩and i can be
ansla ed as he ollowing P R- ule:
120 R. F eund e al.
:Labels( ) = (⟨e, h⟩),
Rew i e( ) = {(1, a → )}.
In he u u e we indica e eins ead o ⟨e, h⟩.
A ule a[]e1→[b]e2can be ansla ed as he ollowing g oup o PLR- ules
(∀p∈Lab):
:Labels( ) = (e1, p),
Rew i e( ) = (2, a)→(1, b),
Label–Rename( ) = {(1, e2)}.
A ule [a]e1→[]e2bcan be ansla ed as he ollowing g oup o PLR- ules
(∀p∈Lab):
:Labels( ) = (e1, p),
Rew i e( ) = (1, a)→(2, b),
Label–Rename( ) = {(1, e2)}.
A ule [a]e→bcan be ansla ed as he ollowing g oup o PD- ules (∀p∈Lab):
:Labels( ) = (e, p),
Rew i e( ) = {(1, a)→(1, b)},
Dele e–and–Mo e( ) = {(1,2)}.
A ule [a]e1→[b]e2[c]e3can be ansla ed as he ollowing g oup o PCLR- ules
(∀p∈Lab):
:Labels( ) = (e, p),
Rew i e( ) = (1, a)→(1, b),
Label–Rename( ) = {(1, e2)},
Gene a e–and–Copy( ) = {(1′, e3,1, b →c)},
Change–Rela ion( ) = INSERT −EDGE(1′,2).
4.2 Rules wi hou pola iza ions
(Acco ding o Sec ion 11.4 om he handbook). Since in ou case he label is a
couple ⟨e, h⟩, he e is no dis inc ion wi h espec o he p e ious case.
4.3 C ea ion ules
Conside c ea ion ules like on p. 326 in handbook.
A ule [a→[u]h1]h2can be ansla ed as ollowing PCR- ule:
:Labels( ) = (h2),
Rew i e( ) = (1, a)→(1, λ),
Gene a e( ) = {(1′, h1, u)},
Change–Rela ion( ) = INSERT −EDGE(1′,1).
A Fo mal F amewo k o P Sys ems wi h Dynamic S uc u e 121
4.4 S ong di ision
A ule [[]h1. . . []hk[]hk+1 . . . []hn]h→[[]h1. . . []hk]h[[]hk+1 . . . []hn]hcan be deﬁned as
he ollowing C- ule:
:Labels( )=(h1,...,hn, h),
ρ( ) = {(i, n + 1) |1≤i≤n},
Rew i e( ) = ∅
Gene a e( ) = {(1′, h, λ)},
Gene a e–and–Copy( ) = ∅,
Change–Rela ion( ) = DELET E −EDGE(k, n + 1), i + 1 ≤k≤n, and
INSERT −EDGE(k, 1′).
4.5 Di ision based on pola iza ions
Conside a ule o ype []h→[+]h[−]h2[0]h3 ha eg oups all memb anes wi h
he same pola iza ion in h ee new memb anes. This can be simula ed wi h he
ollowing C- ule:
:Labels( ) = (h),
ρ( ) = ∅,
Rew i e( ) = ∅,
Gene a e( ) = {(1′, h1, λ),(2′, h2, λ)},
Gene a e–and–Copy( ) = ∅,
Change–Rela ion( ) =
DELET E–EDGE(k, 1),and INSERT –EDGE(k, 1′),
o all ksuch ha lab(k) = −
DELET E–EDGE(k, 1),and INSERT –EDGE(k, 2′),
o all ksuch ha lab(k)=0
5 Conclusions
In his pape we p esen ed a amewo k o P sys ems wi h dynamic s uc u e.
The ob ained me a-language has a p ecise seman ics cen e ed a ound 2 no ions:
(1) he e olu ion o he objec s and memb ane labels and (2) he e olu ion o he
memb ane s uc u e (c ea ion and dele ion o nodes and edges). As a consequence
i pe mi s o easily desc ibe diﬀe en ea u es o exis ing P sys ems wi h dynam-
ical s uc u e, which pe mi s o p o ide an in e es ing ool o he compa ison o
diﬀe en a ian s o P sys ems. Mo eo e , he ansla ion o he amewo k allows

122 R. F eund e al.
o a be e unde s anding o he co esponding P sys em and p o ides ways o
ex end i s deﬁni ion by new ea u es. We ema k ha in he case o he sys ems
wi h a s a ic s uc u e a simila app oach using he amewo k om [3] pe mi ed
o deﬁne new a ian s o P sys ems and o be e exp ess some exis ing ones [1, 4].
The in oduced model wo ks wi h an a bi a y (bina y) ela ion be ween mem-
b anes, so i could be in e es ing o conside ela ions diﬀe en om he pa en
ela ion widely used in P sys ems. As an in e es ing candida e we sugges he
b o he /sis e ela ion on a ee. I could also be in e es ing o conside a gene al-
iza ion o he amewo k o an a bi a y n-a y ela ion. In his case he ela ion ρ
induces a hype g aph, so he componen s changing he s uc u e o ρha e o be
adap ed o wo k on hype g aphs.
Ano he di ec ion o he de elopmen o he amewo k is o conside ha
o a mul ise o ules he o de o he applica ion o Change–Rela ion p oduces
diﬀe en esul s. This implies ha he o de o ules is impo an , by consequence
he se Applicable(Π, C, δ) will con ain ec o s (o lis s) o ules. This in e es ing
idea was no ye conside ed in he amewo k o P sys ems and we hink ha i
can lead o in e es ing esul s.
Re e ences
1. A. Alhazo , M. Oswald, R. F eund, S. Ve lan, Pa ial Hal ing and Minimal Pa allelism
Based on A bi a y Rule Pa i ions, Fundamen a In o ma icae 91(1), 2009, 17–34.
2. R. F eund, B. Habe s oh, A ibu ed Elemen a y P og ammed G aph G amma s,
P oceedings 17 h In e n. Wo kshop on G aph-Theo e ic Concep s in Compu e Sci-
ence, Lec u e No es in Compu e Science 570, Sp inge , 1991, 75–84.
3. R. F eund, S. Ve lan, A Fo mal F amewo k o S a ic (Tissue) P Sys ems, Memb ane
Compu ing, 8 h In e na ional Wo kshop, WMC 2007, Thessaloniki, G eece, June 25-
28, 2007 Re ised Selec ed and In i ed Pape s, Lec u e No es in Compu e Science
4860 , 271–284, Sp inge , 2007.
4. R. F eund, S. Ve lan, (Tissue) P sys ems wo king in he k- es ic ed minimally o
maximally pa allel ansi ion mode, Na u al Compu ing 10(2), 2011, 821–833.
5. Gh. P˘aun, Memb ane Compu ing. An In oduc ion. Sp inge –Ve lag, 2002.
6. G. P˘aun, G. Rozenbe g, A. Salomaa, The Ox o d Handbook O Memb ane Compu ing.
Ox o d Uni e si y P ess, 2009.
7. The Memb ane Compu ing Web Page:h p://ppage.psys ems.eu
8. G. Rozenbe g, A. Salomaa, eds., Handbook o Fo mal Languages. Sp inge –Ve lag,
Be lin, 1997.
P Sys ems wi h Minimal Le and Righ
Inse ion and Dele ion
Rudol F eund1, Yu ii Rogozhin2, and Se gey Ve lan3
1Facul y o In o ma ics, Vienna Uni e si y o Technology
Fa o i ens . 9, 1040 Vienna, Aus ia
Email: [email p o ec ed]
2Ins i u e o Ma hema ics and Compu e Science
Academy o Sciences o Moldo a
S . Academiei 5, Chi¸sin˘au, MD-2028, Moldo a
Email: [email p o ec ed]
3LACL, D´epa emen In o ma ique, Uni e si ´e Pa is Es
61, a . G´en´e al de Gaulle, 94010 C ´e eil, F ance
Email: [email p o ec ed]
Summa y. In his a icle we in es iga e he ope a ions o inse ion and dele ion pe -
o med a he ends o a s ing. We show ha using hese ope a ions in a P sys ems
amewo k (which co esponds o using speciﬁc a ian s o g aph con ol), compu a ional
comple eness can e en be achie ed wi h he ope a ions o le and igh inse ion and
dele ion o only one symbol.
1 In oduc ion
The ope a ions o le and igh inse ion and dele ion ha we conside in his
a icle co espond o he ope a ions o le and igh conca ena ion and quo ien
wi h a ﬁni e language. While hese ope a ions a e known o a long ime, hei
join in es iga ion in a dis ibu ed amewo k o igina es om he a ea o na u-
al compu ing, whe e hey we e used in he con ex o ne wo ks o e olu iona y
p ocesso s (NEP) [7]. Such ne wo ks a e a special ype o ne wo ks o language
p ocesso s [6] ha ea u e a se o ( ew i ing) nodes ew i ing languages and a -
e ha edis ibu ing some egula subse s be ween he nodes. In ne wo ks o
e olu iona y p ocesso s, he ew i ing ope a ions a e eplaced by h ee ypes o
ope a ions ha ing a biological mo i a ion: inse ion, dele ion, and mu a ion (sub-
s i u ion). The co esponding sys ems a e qui e powe ul and we e e o [8] o
mo e de ails. The edis ibu ion o he node con en s based on a egula condi ion
is a e y powe ul ope a ion. Accep ing hyb id ne wo ks o e olu iona y p ocesso s
(AHNEP) eplace his condi ion by andom con ex condi ions, howe e , he se
124 R. F eund, Y. Rogozhin, S. Ve lan
o ope a ions is changed and now includes he inse ion and dele ion ope a ions a
he ex emi ies o he s ings; we e e o [21, 9] o mo e de ails on AHNEP.
The ope a ions o inse ion and dele ion on he ex emi ies o a s ing can also
be seen as a pa icula case o a mo e gene al a ian , whe e inse ion and dele ion
can be pe o med anywhe e in he s ing. The inse ion ope a ion deﬁned in such
a way was ﬁ s conside ed in [14, 15] and a e ha ela ed inse ion and dele ion
ope a ions we e in es iga ed in [17, 18]. Ano he gene aliza ion o he inse ion
and dele ion ope a ions ha in ol es he checking o con ex s o he inse ion and
dele ion was conside ed wi h a linguis ic mo i a ion in [13, 20] and wi h a biological
mo i a ion in [4, 5, 18, 26]. Gene ally, i he leng h o he con ex s and/o o he
inse ed and dele ed s ings a e big enough, hen he inse ion-dele ion closu e o
a ﬁni e language leads o compu a ional comple eness. The e a e nume ous esul s
es ablishing he desc ip ional complexi y pa ame e s suﬃcien o achie e his goal,
we e e o [31, 30] o an o e iew o his a ea.
Some desc ip ional complexi y pa ame e s lead o a ian s ha a e no com-
pu a ionally comple e. An in es iga ion o inse ion and dele ion ope a ions com-
bined wi h egula ing mechanisms was done o hese cases, mo e p ecisely, wi h
he g aph-con olled, he ma ix, and he andom-con ex con ols [11, 28, 16]. As
i was shown in hese a icles, in mos o he cases he addi ional con ol leads
o compu a ional comple eness. The g aph-con olled egula ion is o pa icula
in e es , as i can be ela ed o he no ion o P sys ems. Such sys ems o mal-
ize he unc ioning o a li ing cell ha opologically delimi s p ocessing uni s by
memb anes, hus leading o a ee (o g aph) s uc u e o p ocessing nodes. The
elemen s p ocessed in some node (memb ane) hen a e dis ibu ed among he
neighbo s in he s uc u e. We e e o [24, 25] and o he web page [29] o mo e
de ails on P sys ems. In he case o he ope a ions o inse ion and dele ion ac ing
on s ings his di ec ly co esponds o a g aph con ol whe e he con ol nodes
co espond o he memb anes.
The esea ch on con ex - ee inse ion and dele ion (i.e., wi hou con ex ual
dependency) shows ha i he leng hs o he inse ed and dele ed s ings a e 2 and
3 (o 3 and 2), espec i ely, hen he inse ion-dele ion closu e o ﬁni e languages
is compu a ionally comple e [22]. When one o hese pa ame e s is dec eased, his
esul is no ue anymo e [32]; mo eo e , e en he g aph-con olled a ian canno
achie e compu a ional comple eness [19]. This changes when a g aph con ol wi h
appea ance checking is used [1] o in he case o a andom con ex con ol [16]. In
bo h a ian s, minimal ope a ions (in ol ing only one symbol) we e conside ed,
leading o RE ( he amily o ecu si ey enume able languages) in he case o se -
con olled andom con ex condi ions and o PsRE ( he amily o Pa ikh se s o
RE) in he case o g aph con ol wi h appea ance checking.
We no e ha he ope a ions o le and igh inse ion and dele ion a e incom-
pa able wi h no mal inse ion and dele ion: because o he posi ional in o ma ion,
he egula language a+b+can be ob ained e en wi h le and igh inse ions o
only one symbol, ye no when inse ions a e possible a a bi a y posi ions in he
s ing. On he o he hand, he Dyck language canno be ob ained when inse ion
P Sys ems wi h Minimal Le and Righ Inse ion and Dele ion 125
is only possible a he ends o he s ings, while wi h no mal inse ion his can be
done easily. In [1, 3], le and igh inse ion and dele ion ope a ions (unde he
name o exo-inse ion and -dele ion) we e conside ed in he P sys ems amewo k
(i.e., wi h a g aph con ol) and i was shown ha sys ems wi h inse ion o s ings
o leng h 2 ( espec i ely 1) and dele ion o s ings o leng h 1 ( espec i ely 2)
lead o compu a ional comple eness. In he case o minimal inse ion and dele ion
(i.e., o only one symbol), a p io i y o dele ion o e inse ion (co esponding o
an appea ance check) was used o show compu a ional comple eness.
In his a icle we con inue hese in es iga ions and we conside P sys ems wi h
minimal le and igh inse ion and dele ion and p o e ha compu a ional com-
ple eness can be achie ed e en in his case, wi h he s uc u e o he P sys em we
need being ma ix-like. We also di ec ly show ha ma ix g amma s using minimal
le inse ion and minimal igh dele ion ules a e compu a ionally comple e (wi h
ma ices o leng h a mos 3). Mo eo e , we also p o e ha using an addi ional
minimal mu a ion ope a ion (subs i u ion o one symbol by ano he one) allows
o educing he heigh o he ee s uc u e o he P sys em o he minimum size
1.
2 P elimina ies
A e some p elimina ies om o mal language heo y, we deﬁne he s ing ew i -
ing ules o be used in his pape . As s ing ew i ing sys ems, we will conside
Pos sys ems, ma ix g amma s, and sequen ial P sys ems. Mo eo e , we will gi e
some examples and p elimina y esul s o illus a e ou deﬁni ions.
The se o non-nega i e in ege s is deno ed by N. An alphabe Vis a ﬁni e non-
emp y se o abs ac symbols. Gi en V, he ee monoid gene a ed by Vunde
he ope a ion o conca ena ion is deno ed by V∗; he elemen s o V∗a e called
s ings, and he emp y s ing is deno ed by λ;V∗ {λ}is deno ed by V+. Le
{a1, ..., an}be an a bi a y alphabe ; he numbe o occu ences o a symbol ai
in xis deno ed by |x|ai; he numbe o occu ences o all symbols om Vin xis
deno ed by |x|. The amily o ecu si ely enume able s ing languages is deno ed
by RE. Fo mo e de ails o o mal language heo y he eade is e e ed o he
monog aphs and handbooks in his a ea as [10, 27].
We he e conside s ing ew i ing ules only wo king a he ends o a s ing:
Pos ew i ing ule P[x/y] wi h x, y ∈V∗:P[x/y] (wx) = yw o w∈V∗.
Le subs i u ion SL[x/y] wi h x, y ∈V∗:SL[x/y] (xw) = yw o w∈V∗.
Righ subs i u ion SR[x/y] wi h x, y ∈V∗:SR[x/y] (wx) = wy o w∈V∗.
I in a (le o igh ) subs i u ion SL[x/y] o SR[x/y]xis emp y, hen we
call i an inse ion and w i e IL[y] and IR[y], espec i ely; i in a (le o igh )
subs i u ion SL[x/y] o SR[x/y]yis emp y, hen we call i a dele ion and w i e
DL[x] and DR[x], espec i ely. I we only inse one symbol a, hen we will also
w i e +a,a+, −a, and a− o IL[a], IR[a], DL[a], and DR[a], espec i ely.
132 R. F eund, Y. Rogozhin, S. Ve lan
(h, 8) ScwDH H−,F+
(h, 7) +S cwDH ScwD E+, F+
(h, 6) +c wDH ScwDE M+, F+
(h, 5) H+wD ScwDEM −S,F+
(h, 4) D+w cwDEM M−,F+
(h, 3) a−wa cwDE J+, F+
(h, 2) b−wab cwDEJ J−,F+
(h, 1) K−wabK cwDE E−,F+
1K+wab cwD D−
cw
Simula ion o h:P[ab/c]
(h, 8) SbcwDH H−,F+
(h, 7) +S bcwDH SbcwD E+, F+
(h, 6) +b cwDH SbcwDE M+, F+
(h, 5) +c wDH SbcwDEM −S,F+
(h, 4) H+wD bcwDEM M−,F+
(h, 3) D+w bcwDE J+, F+
(h, 2) a−wa bcwDEJ J−,F+
(h, 1) K−waK bcwDE E−,F+
1K+wa bcwD D−
bcw
Simula ion o h:P[ab/c]
( , 6) SwDH H−,F+
( , 5) +S wDH SwD E+, F+
( , 4) H+wD SwDE S−,F+
( , 3) D+w wDE J+, F+
( , 2) a−wa wDEJ J−,F+
( , 1) K−waK wDE E−,F+
1K+wa wD D−
w
Simula ion o h:P[a/λ], a=Z
(h, 6) SbwD D−,F+
(h, 5) +S wD Sbw E+, F+
(h, 4) +b wD SbwE −S,F+
(h, 3) D+w bwE J+, F+
(h, 2) a−wa bwEJ J−,F+
(h, 1) K−waK bwE E−,F+
1K+wa bw
Simula ion o h:P[a/b]
F om he desc ip ions gi en in he ables abo e, i is easy o see how a suc-
cess ul simula ion o a ule h:P[xh/yh]∈Pwo ks. I we en e a memb ane
(h, 1) wi h a s ing no being o he o m uxh, hen a some momen he only

P Sys ems wi h Minimal Le and Righ Inse ion and Dele ion 133
chance will be o use F+, in oducing he ap symbol Fwhich canno be e ased
anymo e and deﬁni ely leads o a non-hal ing compu a ion. The addi ional sym-
bols D, E, H, J, M in e media ely in oduced on he igh -hand side o he s ing
gua an ee ha loops inside he linea memb ane s uc u e o he simula ion o a
ule h:P[xh/yh]∈Pcanno lead o success ul compu a ions as well. In sum, we
conclude L(Π) = L(G).
Due o he ma ix-like memb ane s uc u e o he simple P sys ems cons uc ed
in he p eceding p oo , we could ob ain he compu a ional comple eness o ma ix
g amma s o ype D1I1as an ob ious consequence o Theo em 5, ye he di ec
ans o ma ion o he cons uc ion gi en in he p oo o his heo em would yield
a lo o ma ices wi h leng hs mo e han 3, whe eas he di ec p oo gi en in
Theo em 3 only needed ma ices o leng h a mos 3.
4 Conclusion
In his pape we ha e conside ed s ing ew i ing sys ems using he ope a ions o
minimal le and igh inse ion and dele ion. Using e en only he ope a ions o
minimal le inse ion and minimal igh dele ion, ma ix g amma s each com-
pu a ional comple eness wi h ma ices o leng h a mos 3; ou conjec u e is ha
his equi ed leng h canno be educed o 2. As ou main esul , we ha e shown
ha sequen ial P sys ems using he ope a ions o minimal le and igh inse ion
and dele ion a e compu a ionally comple e, hus sol ing an open p oblem om
[2]. The simple P sys em cons uc ed in he p oo o Theo em 5 had a he la ge
ee heigh ; i emains an open ques ion o educe his complexi y pa ame e . On
he o he hand, in Theo em 4 we ha e shown ha using minimal le inse ion,
minimal igh dele ion, and, in addi ion, minimal igh mu a ion (subs i u ion o
one symbol by ano he one on he igh -hand side o a s ing) we can educe he
heigh o he ee s uc u e o he P sys em o he minimum 1 and e en a oid he
use o he a ge he e. Mo eo e , we would also like o a oid he a ge he e in
he case o simple P sys ems using minimal le and igh inse ion and dele ion,
as wi h a oiding he a ge he e, he applica ions o he ules could be in e p e ed
as being ca ied ou when passing a memb ane, in he sense o a molecule passing
a speciﬁc meb ane channel om one egion o ano he one. We shall e u n o his
qes ion and ela ed ones in an ex ended e sion o his pape .
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Simula ing La ge-Scale ENPS Models
by Means o GPU
Manuel Ga c´ıa–Quismondo1, Ana B ˆandu¸sa Pa el2, and Ma io J. P´e ez–Jim´enez1
1Resea 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,ma pe }@us.es
2Depa men o Au oma ic Con ol and Sys ems Enginee ing, Poli ehnica Uni e si y
o Bucha es
Splaiul Independen ¸ei, N . 313, sec o 6, 090042, Bucha es , Romania
E-mail: [email p o ec ed]
Summa y. Enzyma ic Nume ical P Sys ems (ENPS), an ex ension o Nume ical P
Sys ems, ha e been success ully applied o model obo con olle s. GPGPU is an
inno a i e echnological pa adigm which applies he pa allel a chi ec u e o g aphic ca ds
o sol e pa allel, gene al–pu pose p oblems. In p e ious wo k, a GPU simula o o ENPS
was in oduced. In his pape , a pe o mance analysis on he simula o is pe o med in
o de o expe imen ally measu e he speed-up ac o s esul ing om he simula ions.
Keywo ds: Enzyma ic Nume ical P Sys ems, GPU, simula ion
1 Enzyma ic Nume ical P Sys ems
Memb ane compu ing is an in e disciplina y ield which s udies compu a ional models
inspi ed by he compa men al s uc u e o biological cells. The e exis many ypes o
memb ane sys ems [12], also known as P sys ems a e ma hema ician Gh. P˘aun, who
in oduced hem. Nume ical P sys ems (NPS) a e a ype o P sys ems in which nume ical
a iables e ol e inside he compa men s by means o p og ams; a p og am (o ule) is
composed o a p oduc ion unc ion and a epa i ion p o ocol [11]. The a iables ha e a
gi en ini ial alue and he p oduc ion unc ion is a mul i a ia e polynomial. The alue
o he p oduc ion unc ion o he cu en alues o he a iables is dis ibu ed among
a iables in ce ain compa men s acco ding o a epa i ion p o ocol. A o mal de ini ion
o NPS can be ound in [11], whe e his ype o P sys em is in oduced wi h possible
applica ions in economics.
Enzyma ic nume ical P sys ems (ENPS) ep esen an ex ension o NPS, p oposed and
used in he con ex o modeling obo con olle s [13], [14]. ENPS is a mo e powe ul
modelling ool han NPS, as i is p o en in se e al a icles [13], [3], [17]. ENPS models
allow he exis ence o mo e han one ule pe memb ane han NPS while keeping he
de e minis ic beha io . By using a special ype o a iables e e ed as enzymes, ENPS
models p o ide a selec ion mechanism o he ac i e ules du ing he compu a ional

138 M. Ga c´ıa–Quismondo e al.
p ocess. The e o e, ENPS a e a lexible and e icien modelling amewo k ha can
be success ully used o modeling obo beha io s like obs acle a oidance, localiza ion,
ollowe , e c. [13], [4]. An ENPS model o deg ee m, m ≥1 is o mally de ined as ollows:
Π= (H, µ, (V a 1, E1, P 1, V a 1(0)),...,(V a m, Em, P m, V a m(0))) (1)
whe e:
•His an alphabe ha con ains msymbols ( he labels o he memb anes);
•µis a memb ane s uc u e;
•V a iis he se o a iables om compa men i, and he ini ial alues o hese
a iables a e V a i(0);
•Eiis a se o enzyme a iables om compa men i,Ei⊆V a i
•P iis he se o p og ams ( ules) om compa men i. P og ams p ocess a iables
and ha e wo componen s; a p oduc ion unc ion and a epa i ion p o ocol. In ENPS
models, p og ams can ha e one o he wo ollowing o ms:
1. Non-enzyma ic o m, which is exac ly like he one om s anda d NPS:
P j,i = (Fj,i(x1,i,...,xki,i), cj,1| 1+... +cj,ni| ni) (2)
2. Enzyma ic o m
P j,i = (Fj,i(x1,i,...,xki,i), ej,i, cj,1| 1+... +cj,ni| ni) (3)
whe e:
–Fj,i(x1,i, ..., xki,i) is he p oduc ion unc ion;
–ej,i ∈Ei, is he enzyme–like a iable associa ed o ej,i;
–ki ep esen s he numbe o a iables in memb ane i;
–cj,1| 1+. . . +cj,ni| niis he epa i ion p o ocol;
–ni ep esen s he numbe o a iables con ained in memb ane i, plus he numbe
o a iables con ained in he pa en memb ane o i, plus he numbe o a iables
con ained in he child en memb anes o i.
In ENPS models, all ac i e ules a e execu ed in pa allel on each compu a ional s ep. A
ule is always ac i e i i is in he non-enzyma ic o m. O he wise, i i is in he enzyma ic
o m, a ule is ac i e only i he associa ed enzyme–like a iable has a g ea e alue han
he minimum o he absolu e alues o he a iables in ol ed in he p oduc ion unc ion.
The epa i ion p o ocol wo ks like in classical NPS [11]
Bo h NPS and ENPS models ha e been success ully used o modelling obo con olle s
[4], [13], [14]. The ad an ages o using ENPS o his kind o applica ions a e poin ed
ou in [3]. Fo es ing he obo con olle s, a Ja a implemen a ion o a nume ical P
sys ems simula o was implemen ed and used. This Ja a simula o , SimP, simula es
ENPS models. Since ENPS a e an ex ension o NPS, SimP can be used o simula e
classical NPS as well. SimP was p oposed in [15] and i is a ailable as a ee execu able
e sion ( o a ee execu able e sion o SimP, please con ac [email p o ec ed]).
SimP allows he compu a ion o a ional p oduc ion unc ions and no only polynomials.
This is use ul o implemen ing mo e complex models equi ed o obo ics applica ions.
An example o model o a a ional p oduc ion unc ion is p esen ed in subsec ion 4.3.
In his pape , he pe o mance o a pa allel and dis ibu ed simula o which compu es
ENPS s uc u es is analysed. The simula o was i s p oposed in [7]. In o de o es he
Simula ing La ge-Scale ENPS Models by Means o GPU 139
pa allel simula o and analyse i s pe o mance, a eplica ion mechanism o memb anes
was used. Fo ins ance, in swa m obo ics applica ions, he obo s in a g oup may ha e
simila beha io s which need o un in pa allel. Fo example, each obo needs o un an
obs acle a oidance beha io . The e o e, he ENPS s uc u e o obs acle a oidance [13]
mus be execu ed in pa allel o all he obo s in he swa m ( igu e 1). Those applica ions
in which each obo needs o un one o e en mo e memb ane con olle s in pa allel and
all obo s mus wo k in pa allel as well ake eal ad an age o he pa allel and dis ibu ed
ENPS simula o , which will be u he discussed.
Fig. 1. Replica ion o ENPS models can be used o swa m obo ics applica ions
2 An in oduc ion o GPU Compu ing
G aphic ca ds, also known as G aphic P ocessing Uni s (GPUs) a e de ices whose main
ask is o sol e image ende ing p oblems. These p oblems a e usually massi e pa allel
p oblems, as hey can commonly be educed o ende pixels and e ices in a pa allel
ashion. As a esul , he indus y has u ned hese de ices in o powe ul highly–pa allel
compu e s wi h a la ge numbe o p ocesso s. Howe e , hey ha e been o limi ed use o
scien is o qui e a long ime. The eason is ha GPUs we e only sui able o image–
ela ed p oblems, wi h li le applica ion in he scien i ic wo ld ou o image p ocessing
i sel . The e o e, he amoun o e o equi ed o ansla e gene al–pu pose p oblems
in o hei g aphical in e p e a ions made scien is s use o he pa allel a chi ec u es such
as compu e clus e s and FPGAs [1] [9] [16].
Howe e , NVIDIA changed his landscape by p o iding a oolki o gene al–pu pose
GPU compu ing named Compu e Uni ied De ice A chi ec u e (CUDA) [19]. This API
pe mi ed de elope s o sol e scien i ic, non–g aphical p oblems on GPUs. F om hen
on, he adop ion o GPU compu ing by he scien i ic communi y has gone widesp ead.
As a esul nume ous scien i ic pape s ha e shown mode a e o imp essi e pe o mance
imp o emen s on a GPU o e a CPU [2].
140 M. Ga c´ıa–Quismondo e al.
2.1 The CUDA p og amming model
Nowadays, he numbe o p ocesso s in a GPU can each 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
p ocessing uni s [19]. Thus, GPUs a e s uc u ed as a g id o mul ip ocesso co es o
s eaming mul ip ocesso s. Each one o hese mul ip ocesso s ( igu e 2) is composed o
se e al simple p ocessing uni s known as s eaming p ocesso s. This hie a chy allows
p og amme s o s uc u e hei code in a dis ibu ed way, so hey can couple p ocesses
wi h a high communica ion bandwid h wi h each o he , hus clus e ing and alloca ing
hem wi hin he same mul ip ocesso in he GPU. All p ocessing uni s execu e he
same code a he same ime, hence applying a compu a ional pa adigm known as Single
Ins uc ion Mul iple P og am (SIMP) [16]. CUDA p o ides an abs ac model o GPU
Fig. 2. Simpli ied ha dwa e block diag am o an N idia G aphic Ca d
a chi ec u e. This model is known as he CUDA p og amming model. The idea is o make
use o an abs ac ion o he speci ic g aphic ca d on which he code uns, so as no o
o ce he code depend on pa icula de ices. This model is composed o a g id o elemen s
known as blocks, which a e abs ac ions o he s eaming mul ip ocesso s men ioned
abo e. Each one o hese blocks is composed o compu ing elemen s known as h eads [5].
Thus, he CUDA p og amming model is a mul idimensional h ee–le elled a chi ec u e
, as he dimension o g ids and blocks can be 1, 2 o 3, depending on he con igu a ion
op ions se by he de elope . Figu e 3 desc ibes g aphically his p og amming model.
2.2 P og amming on CUDA–C
The i s s s eps owa ds ob aining languages o gene al–pu pose GPU compu ing
in ol ed w apping cu en ly exis ing languages o g aphics p ocessing wi h mains eam
gene al–pu pose languages, such as C and Fo an. Howe e , hese app oaches we e
s ill hinde ed by he limi a ions o g aphic ca ds on ea ing gene al–pu pose da a
s uc u es and da a lows as pixels and e ices. Howe e , he CUDA p og amming
Simula ing La ge-Scale ENPS Models by Means o GPU 141
Fig. 3. The CUDA p og amming model
model, combined wi h mo e ad anced g aphic ha dwa e wi h suppo o gene al–pu pose
pa allel p og amming allowed o ex end mains eam languages wi h speci ic p imi i es
o GPU compu ing [16]. In his di ec ion, N idia issued he i s e sion o he CUDA–C
language in 2006 [19]. This language was an ex ension o he C language wi h p imi i es
and ope a ions o gene al–pu pose pa allel p oblems [16].
A CUDA–C p og am is composed o wo di e en pa s: he hos code and he de ice
code. The hos code is he pa o he code execu ed on he CPU. This code con ains
calls o pieces o he de ice code, which is execu ed on he GPU. The de ice code is
composed o CUDA–C unc ions which a e execu ed on he GPU, known as ke nels. The
SIMP pa adigm de ines ha each ke nel is execu ed on all h eads a he same ime.
When i comes o execu ion, h eads a e bundled in packages known as wa ps. Th eads
in he same wa p communica e ia a as on–chip memo y. Howe e , he communica ion
wi h h eads in di e en wa ps is pe o med by using a slow o –chip memo y. Tha is
he eason why h eads in he same wa p should ha e a high communica ion bandwi h
wi h each o he , as well as a low communica ion bandwi h wi h h eads ou o he wa p
[5].
In p ac ice, he CUDA p og amming model claims ha each block is assigned a sequence
o wa ps. The e o e, each block akes a wa p om i s sequence and execu es i , assigning
each wa p h ead o a di e en block h ead [16]. The o de o his sequence, as well as
he block o which each wa p is associa ed, is no con olled by he de elope . This en ails
ha he p oblem should no depend on he o de in which he wa ps a e execu ed, as his
o de canno be gua an eed [5]. A mo e ho ough desc ip ion o he CUDA p og amming
model can be ound in [10].
244 M. Gheo ghe, Gh. P˘aun, M.J. P´e ez-Jim´enez, eds.
links be ween algeb aic opology and MC ha e s a ed o be explo ed ia homology
heo y [6, 7, 12]. Homology heo y is a b anch o algeb aic opology ha a emp s
o dis inguish be ween spaces by cons uc ing algeb aic in a ian s ha eﬂec he
connec i i y p ope ies o he space. Homology g oups ( ela ed o he diﬀe en
n-dimensional holes, connec ed componen s, unnels, ca i ies, e c., o a geome ic
objec ) a e in a ian s om algeb aic opology which a e equen ly used in digi al
image analysis and s uc u al pa e n ecogni ion.
In a simila way wi h o he applica ions o P sys ems, he heo e ical ad an-
ages o he MC echniques o compu e ision need a powe ul so wa e and
ha dwa e o an eﬀec i e implemen a ion. The use o hese new echnologies o
he pa allel implemen a ion o P sys ems echniques applied o compu e ision
ha e s a ed o be explo ed wi h p omising expe imen al esul s [1, 17, 18].
An app op ia e combina ion o MC echniques oge he wi h an eﬃcien pa -
allel implemen a ion on he new ha dwa e a chi ec u es can p o ide compe i i e
algo i hms o diﬀe en p oblems om compu e ision. Among hem, we can ci e
dealing wi h ex u es, colo s and/o 3D objec s (o e en 4D objec s, whe e he
e olu ion o objec s in ime is also conside ed). F om algeb aic opology, he cal-
culus o complex opological in a ian s o 2D and 3D objec s can be a sou ce o
new open p oblems o MC.
Re e ences
1. J. Ca ne o, D. D´ıaz-Pe nil, M.A. Gu i´e ez-Na anjo: Designing issue-like P sys-
ems o image segmen a ion on pa allel a chi ec u es. Nin h B ains o ming Week
on Memb ane Compu ing (M.A. Ma ´ınez del Amo e al., eds.), F´enix Edi o a,
Se illa, 2011, 43–62
2. R. Ce e chi, R. G ama o ici, N. Jonoska, K.G. Sub amanian: Tissue-like P sys ems
wi h ac i e memb anes o pic u e gene a ion. Fundamen a In o ma icae, 56 (2003),
311–328.
3. R. Ce e chi, M. Mu yam, Gh. P˘aun, K.G. Sub amanian: A ay- ew i ing P sys ems.
Na u al Compu ing, 2 (2003), 229–249.
4. H.A. Ch is inal, D. D´ıaz-Pe nil, M.A. Gu i´e ez-Na anjo, M.J. P´e ez-Jim´enez:
Th esholding o 2D images wi h cell-like P sys ems. Romanian Jou nal o In o -
ma ion Science and Technology, 13 (2010), 131–140.
5. H.A. Ch is inal, D. D´ıaz-Pe nil, P. Real: Segmen a ion in 2D and 3D image using
issue-like P sys em. LNCS 5856, Sp inge , 2009, 169–176.
6. H.A. Ch is inal, D. D´ıaz-Pe nil, P. Real: Using memb ane compu ing o ob aining
homology g oups o bina y 2D digi al images. LNCS 5852, Sp inge , Be lin, 2009,
383–396.
7. H.A. Ch is inal, D. D´ıaz-Pe nil, P. Real: P sys ems and compu a ional algeb aic
opology. Ma hema ical and Compu e Modelling, 52 (2010), 1982–1996.
8. H.A. Ch is inal, D. D´ıaz-Pe nil, P. Real: Region-based segmen a ion o 2D and 3D
images wi h issue-like P sys ems. Pa e n Recogni ion Le e s, 32 (2011), 2206–2212.
9. K.S. De sanambika, K. K i hi asan: Con ex ual a ay P sys ems. In e n. Jou nal o
Compu e Ma hema ics, 81 (2004), 955–969.

F on ie s o Memb ane Compu ing 245
10. D. D´ıaz-Pe nil, M.A. Gu i´e ez-Na anjo, H. Molina-Ab il, P. Real: A bio-inspi ed
so wa e o segmen ing digi al images. P oc. Fi h In e na ional Con e ence on Bio-
Inspi ed Compu ing. Theo ies and Applica ions BIC-TA (A.K. Naga e al., eds.),
ol. 2 (2010), 1377–1381.
11. D. D´ıaz-Pe nil, M.A. Gu i´e ez-Na anjo, H. Molina-Ab il, P. Real: Designing a new
so wa e ool o digi al image y based on P sys ems. Na u al Compu ing, 2011
h p://dx.doi.o g/10.1007/s11047-011-9287-4.
12. D. D´ıaz-Pe nil, M.A. Gu i´e ez-Na anjo, P. Real, V. S´anchez-Canales,: Compu ing
homology g oups in bina y 2D image y by issue-like P sys ems. Romanian Jou nal
o In o ma ion Science and Technology, 13 (2010), 141–152.
13. C. Fe e i, G. Mau i, C. Zand on: P sys ems wi h s ing objec s. Chap e 7 o The
Ox o d Handbook o Memb ane Compu ing (Gh. P˘aun, G. Rozenbe g, A. Salomaa,
eds.), Ox o d Uni e si y P ess, 2010, 168–197.
14. G. Gimel’ a b, R. Nicolescu, S. Raga an: P sys ems in s e eo ma ching. LNCS 6855,
Sp inge , Be lin, 2011, 285–292.
15. G. Gimel’ a b: P obabilis ic egula isa ion and symme y in binocula dynamic p o-
g amming s e eo. Pa e n Recogni ion Le e s, 23 (2002), 431–442.
16. S.N. K ishna, R. Rama, K. K i hi asan: P sys ems wi h pic u e objec s. Ac a Cy-
be ne ica, 15 (2001), 53–74.
17. F. Pe˜na-Can illana, D. D´ıaz-Pe nil, A. Be ciano, M.A. Gu i´e ez-Na anjo: A pa allel
implemen a ion o he h esholding p oblem by using issue-like P sys ems. LNCS
6855, Sp inge , 2011, 277–284.
18. F. Pe˜na-Can illana, D. D´ıaz-Pe nil, H.A. Ch is inal, M.A. Gu i´e ez-Na anjo: Imple-
men a ion on CUDA o he smoo hing p oblem wi h issue-like P sys ems. In e n.
Jou nal o Na u al Compu ing Resea ch, 2 (2011), 25–34.
26 Open P oblems on Simula ion o Memb ane Compu ing
Models
Manuel Ga c´ıa-Quismondo, Luis F. Mac´ıas-Ramos,
Miguel A. Ma ´ınez-del-Amo , Ignacio P´e ez-Hu ado,
Luis Valencia-Cab e a
Resea ch G oup on Na u al Compu ing
Dp . o Compu e Science and A iﬁcial In elligence, Uni e si y o Se illa, Spain
{mga ciaquismondo,l macias ,mdelamo ,pe ezh,l alencia}@us.es
Resea ch ideas ela ed o he ex ension o he P-lingua dedica ed languages
and on he implemen a ion o P sys ems on econﬁgu able o pa allel ha dwa e
(e.g., NVIDIA a chi ec u es) a e men ioned.
Requi ed No ions: P sys em models, GPU compu ing, P-Lingua, MeCoSim
The de elopmen o P sys em simula o s, and o o he ela ed so wa e ools,
becomes a c i ical poin in he p ocesses o model alida ion and i ual expe i-
men a ion. Fo his pu pose, a so wa e amewo k o speci ying and simula ing
246 M. Gheo ghe, Gh. P˘aun, M.J. P´e ez-Jim´enez, eds.
P sys ems, called P-Lingua, was de eloped [7]. Mo eo e , a gene ic so wa e o
gene a e g aphical applica ions based on P-Lingua, called MeCoSim, was also de-
eloped. Finally, in o de o accele a e he simula ion by implemen ing P sys ems
pa allelism on high pe o mance pla o ms, some simula o s we e de eloped by
using GPU compu ing [5]. Resea ch in all hese di ec ions a e unde de elopmen .
(A) Simula ion F amewo k: P-Lingua and PLinguaCo e
P–Lingua has been success ully applied o ecosys em modeling p oblems [6],
o mal model checking and o sol e compu a ionally ha d p oblems [7]. I suppo s
se e al P sys em models, such as ac i e memb ane models [7], Tissue P sys em
models and Spiking Neu al P Sys ems (SN P) [9] sys ems.
Ne e heless, he e is s ill plen y o do in o de o ex end he capabili ies o
P-Lingua. Bo h exp essi i y and unc ionali y issues should be imp o ed o enew
he P-Lingua menu and a ac new use s. Fi s , inclusion o pa sing di ec i es
should be implemen ed in o de o, say, modi y he beha io o exis ing models. A
as lea ne example would be he ’asynch onous’ beha io , ha is, c acking he
uni e sal clock ha eigns he compu a ion p ocess in mos o s anda d models
( his is al eady included o he case o SN P Sys ems). Then, in eg a ion o new
models, such as nume ical P sys ems and some speciﬁc ypes o SN P Sys ems,
inco po a ing weigh s and as ocy es wi h hei diﬀe en ﬂa o s, emains unex-
plo ed. Finally, e- ac o ing o he wo k done o b ing some exo ic elemen s o he
eac ion sys ems.
(B) Gene ic end-use g aphical applica ions: MeCoSim
In he las ew yea s, he e ha e been some in e es ing, use - iendly, success ul
so wa e applica ions o modeling and simula ing P sys ems, mainly ocused on
biological sys ems: Me aPlab [4] o in e nal mechanisms o biological sys ems by
means o MP Sys em; BioSimWa e [1] and In obio ics [2] o P sys em based mul i-
compa men al s ochas ic simula ions o complex biological sys ems, he las one
including Syn he ic Biology; and EcoSim [6], a amily o p obabilis ic simula o s
o diﬀe en ecosys ems.
Howe e , a gene al applica ion o he s udy, analysis, modeling, isual simu-
la ion, model checking, op imiza ion o as many as possible a ian s o P sys ems
has no been p o ided. A ﬁ s app oach has been de eloped wi h MeCoSim [10].
Some plugins ha e been de eloped o p o ide some analysis and model checking
capabili ies. I has been success ully applied as an assis an ool o he i e a i e
design o ecosys em models, and o sol e compu a ionally ha d p oblems by using
issue and SN P sys em models.
Ne e heless, he e a e many challenges o sol e. The co e o he isual appli-
ca ion should include mo e analysis and modeling ools o ease he wo k o he P
sys ems designe . Also, some in e aces o communica e wi h diﬀe en simula ion
engines should be de eloped o un simula ion agains simula o s implemen ed in
diﬀe en local o emo e pla o ms and a chi ec u es. I should in eg a e wi h di -
e en applica ions o o mal model checking, enabling he use o ex ac and/o
alida e p ope ies o he s udied models. E en ually, new P sys ems models should
F on ie s o Memb ane Compu ing 247
be added o MeCoSim, p o iding he gene al unc ionali ies and new possible plu-
gins o many po en ial P sys ems designe s.
(C) Simula ion on High Pe o mance Pla o ms: GPU compu ing
So a , he e ha e been many eﬀo s on he de elopmen o GPGPU based
simula o s o P sys ems. In ac , he ollowing P sys ems models ha e been suc-
cess ully simula ed by means o GPUs: P sys ems wi h ac i e memb anes and
di ision ules [5], a amily o P sys ems wi h ac i e memb anes sol ing SAT in
linea ime [5], SN P sys ems (SNP) wi h and wi hou delays [3], and ENPSs [8].
On he o he hand, he e exis many o he models which, o he bes o ou
knowledge, a e ye o be simula ed by means o GPGPU. These models include
issue P sys ems, popula ion dynamics P sys ems, s ochas ic P sys ems, hype dag
P sys ems, nume ical P sys ems and s ing P sys ems, o name jus a ew.
Ano he challenge is he in eg a ion o GPU simula o s on end-use MC so -
wa e amewo ks. Al hough some s eps ha e been aken in his di ec ion wi h he
P-Lingua au oma ic gene a ion [7] o P sys em ﬁles o be pa sed by GPU simula-
o s [5], he e is s ill a long way o walk o an eﬃcien in e ac ion be ween hese
wo kinds o echnological ools.
Las bu no leas , a ho ough pe o mance compa ison be ween GPU simula-
o s and o he HPC app oaches is ye o be de eloped. These app oaches include
econﬁgu able ha dwa e (FGPA, DSP), compu e clus e s wi h OpenMP and MPI,
e c. The need o some wo ks on his di ec ion has been p e iously no iced [5, 8].
Finally, i is also in e es ing o po cu en de elopmen s o GPU simula o s o
he las GPU pla o ms, based on bo h NVIDIA and AMD ATI a chi ec u es, and
on GPU based clus e s.
Acknowledgemen s. The au ho s a e suppo ed by he p ojec TIN 2009-13192
om “Minis e io de Econom´ıa y Compe i i idad” o Spain, and by “P oyec o de
Excelencia con In es igado de Reconocida Val´ıa” P08-TIC-04200 om Jun a de
Andaluc´ıa, bo h co-ﬁnanced by FEDER unds. Manuel Ga c´ıa-Quismondo is also
suppo ed by he Na ional FPU G an P og amme om he Spanish Minis y o
Educa ion.
Re e ences
1. D. Besozzi, P. Cazzaniga, G. Mau i, D. Pescini: BioSimWa e: A so wa e o he
modeling, simula ion and analysis o biological sys ems. LNCS 6501, Sp inge , 2011,
119–143.
2. J. Blakes, J. Twyc oss, F.J. Rome o–Campe o, N. K asnogo : The in obio ics wo k-
bench: an in eg a ed in silico modelling pla o m o Sys ems and Syn he ic Biology.
Bioin o ma ics, 27 (2011), 3323–3324.
3. F.C. Caba le, H. Ado na, M.A. Ma ´ınez-del-Amo : A spiking neu al P sys em sim-
ula o based on CUDA. LNCS 7184, Sp inge , Be lin, 2012, 87–103.
4. A. Cas ellini, V. Manca: Me aPlab: A compu a ional amewo k o me abolic P
sys ems. P e-p oceedings o WMC’08, 2008, Edinbu gh, Sco land.
248 M. Gheo ghe, Gh. P˘aun, M.J. P´e ez-Jim´enez, eds.
5. J.M. Cecilia, J.M. Ga c´ıa, G.D. Gue e o, M.A. Ma ´ınez-del-Amo , I. P´e ez-
Hu ado, M.J. P´e ez-Jim´enez: Simula ion o P sys ems wi h ac i e memb anes on
CUDA. B ie ings in Bioin o ma ics, 11 (2010), 313–322.
6. M.A. Colome , A. Ma galida, D. Sanuy, M.J. P´e ez–Jim´enez: A bio–inspi ed com-
pu ing model as a new ool o modeling ecosys ems: The a ian sca enge s as a case
s udy. Ecological modelling, 222 (2011), 33–47.
7. M. Ga c´ıa-Quismondo, R. Gu i´e ez-Escude o, M.A. Ma ´ınez-del-Amo , E.
O ejuela-Pinedo, I. P´e ez-Hu ado: P-Lingua 2.0: A so wa e amewo k o cell-like
P sys ems. In e na ional Jou nal o Compu e s, Communica ions and Con ol, 4
(2009), 234–243.
8. M. Ga c´ıa-Quismondo, M.J. P´e ez–Jim´enez, L.F. Mac´ıas–Ramos: Implemen ing
ENPS by means o GPUs o AI applica ions. Beyond AI. In e disciplina y Aspec s o
A i icial In elligence (BAI 2011), 08/12/2011–09/12/2011, Pilsen, Czech Republic.
9. L.F. Mac´ıas-Ramos, I. P´e ez-Hu ado, M. Ga c´ıa-Quismondo, L. Valencia-Cab e a,
M.J. P´e ez-Jim´enez, A. Riscos-N´u˜nez: A P-Lingua based simula o o spiking neu al
P sys ems. LNCS 7184, Sp inge , 2012, 257–281.
10. I. P´e ez-Hu ado, L. Valencia-Cab e a, M.J. P´e ez-Jim´enez, M.A. Colome , A.
Riscos-N´u˜nez: MeCoSim: A gene al pu pose so wa e ool o simula ing biological
phenomena by means o P Sys ems. IEEE P oceedings o BIC-TA 2010, I, 637–643.
Closing Rema ks
As also said in he In oduc ion, his collec ion o open p oblems and esea ch
opics in MC was ini ially mean o be a wo king ma e ial, o 10 h BWMC, and
i was upda ed and comple ed se e al imes. Howe e , no such lis can be com-
ple e, nei he uni o m, in wha conce ns he ype o p oblems, hei echnicali y,
diﬃcul y, ange o in e es . As expec ed, some p oblems a e local, o he s a e e y
gene al, while he sec ions a e no a all uni o m in s yle (we ha e p ese ed in a
g ea ex en he con ibu o s w i ing). Mo eo e , many u he esea ch ideas wai
o be add essed in MC, o ins ance, in he P and dP au oma a a ea, he SN P
sys ems a ea, complexi y, dynamical sys ems app oach – no o speak abou ap-
plica ions ( om biology and bio-medicine, o ecology, obo con ol, app oxima e
op imiza ion). S ill, we belie e ha such a lis is use ul, on he one hand, because
i can en ail coope a ion abou he co-au ho s o he pape and he eade s, and,
on he o he hand, because i poin s ou ac i e esea ch a eas o MC, indica ing
i s “ on ie s”. Ac ually, his “mega-pape ” p o ed al eady o be use ul du ing he
10 h BWMC, whe e se e al o he p oposed esea ch opics we e add essed – he
pape inco po a es some changes due o hese ecen p og esses.
Acknowledgemen s
The wo k o Gh. P˘aun and M.J. P´e ez-Jim´enez was suppo ed by P oyec o de
Excelencia con In es igado de Reconocida Val´ıa, de la Jun a de Andaluc´ıa, g an
P08 – TIC 04200. The wo k o M. Gheo ghe was pa ially suppo ed by p ojec
F on ie s o Memb ane Compu ing 249
MuVe , Romanian Na ional Au ho i y o Scien iﬁc Resea ch (CNCS, UEFISCDI)
g an numbe PN-II-ID-PCE-2011-3-0688.
The edi o s a e much indeb ed o all MC esea che s who ha e con ibu ed
o his “mega-pape ”, o G zego z Rozenbe g o many sugges ions du ing he
p epa a ion o he pape , as well o wo anonymous e e ees o ca e ully eading
a p e ious e sion o he ex .

A Fo mal F amewo k o Clock- ee
Ne wo ks o Cells
Se giu I ano
Ins i u e o Ma hema ics and Compu e Science
Academy o Sciences o Moldo a
Academiei 5, Chi¸sin˘au MD-2028 Moldo a
[email p o ec ed]
Uni e si y o he Academy o Science o Moldo a,
Facul y o Real Sciences,
3/2, Academiei s ., MD-2028, Chi¸sin˘au, Republic o Moldo a
1 In oduc ion
Clock- ee P sys ems we e in oduced in [7] as a na u al ex ension o ansi ional
memb ane sys ems. The idea spa ks om he obse a ion o he undamen al di -
e ence be ween how ansi ional P sys ems e ol e and how p ocesses ake place
in biological cells: ansi ional P sys ems e ol e in a se ies o c isp e olu ion s eps,
unde he con ol o a global clock, while hei biological p o o ypes ha e no hing
simila o such a de ice.
In clock- ee P sys ems, he a emp is made o b idge his di e ence by dis-
ca ding any global s ep synch onisa ion mechanism. Any ule applica ion las s
di e en ly and he e is no way o knowing when exac ly he igh -hand side o a
ule will be added o he sys em.
The clock- ee model p oduces wo in ui i e imp essions. Fi s ly, i seems o be
much close o he eal-wo ld p ocesses in he cell: he du a ion o clock- ee ules
is no egula ed by any ex e nal mechanism and is exp essed as a eal numbe .
Secondly, he absence o any buil -in global s ep synch onisa ion seems o be e y
speci ic and qui e unwieldy o manage. In ac , one o he bes -wo king app oaches
o p oducing meaning ul esul s wi h clock- ee P sys ems is cu ing down on
pa allelism as much as possible: his is how he compu a ional comple eness o
hese de ices is shown in [7].
The e a e o he ways o in oduce ime in o P sys ems, an example could be
imed P sys ems (see [1]). In his model, howe e , he global clock is s ill p esen .
In his pape we p o ide a o mal desc ip ion o he seman ics o clock- ee P
sys ems, o a he he mo e gene al concep o clock- ee ne wo ks o cells, and
show how hese de ices can be modelled in ansi ional P sys ems.
252 S. I ano
This pape is hea ily based on [3]. While we ied o in oduce all he el-
e an concep s in his pape as well, ge ing acquain ed wi h [3] would s ill be
ecommended.
2 P elimina ies
2.1 Mul ise s
Gi en a ini e se A, by |A|we unde s and he numbe o elemen s in A.
Le Vbe a ini e alphabe ; hen V∗is he se o all ini e s ings o a V, and
V+=V∗− {λ}, whe e λis he emp y s ing. By Nwe deno e he se o all
non-nega i e in ege s, by Nk– he se o all ec o s o non-nega i e in ege s.
Le Vbe a ini e se , V={a1, . . . , ak},k∈N. A ini e mul ise Mo e
Vis a mapping M:V→N. Fo each a∈V,M(a) indica es he numbe o
“occu ences” o ain M. The alue M(a) is called he mul iplici y o ain M. The
size o he mul ise Mis |M|=Pa∈VM(a), i.e., he o al coun o he en ies o
he mul ise . A mul ise Mo e Vcan also be ep esen ed by any s ing xwhich
con ains exac ly M(ai) ins ances o ai, 1 ≤i≤k. The suppo o Mis he se
supp(M) = {a∈V|M(a)≥1}, which is he se which con ains all elemen s
o he mul ise . Fo example, he mul ise o e {a, b, c}de ined by he mapping
{a→3, b →1, c →0}can be w i en as a3b. The suppo o his mul ise is {a, b}.
The class o all ini e mul ise s o e Vis deno ed by hV, Ni. One may also
conside mappings Mo he o m M:V→N∞, whe e N∞=N∪ {∞}, i.e., he
elemen s may ha e in ini e mul iplici y. A mul ise Mis in ini e i ∃i∈N1≤
i≤kM(ai) = ∞, i.e., a leas one elemen is o in ini e mul iplici y. The
class o mul ise s Mo e Vwi h M:V→N∞is deno ed by hV, N∞i. Fo
W⊆V,W∞is he mul ise in which e e y elemen is o in ini e mul iplici y:
∀a∈WW∞(a) = ∞.
Le x, y ∈ hV, N∞ibe wo (possibly in ini e) mul ise s o e V. Then xis called
asubmul ise o y, w i en as x≤y, i and only i ∀a∈Vx(a)≤y(a). I
∀a∈Vx(a)< y(a) hen xis called a s ic submul ise o y. The sum o x
and y, deno ed by x+yis de ined in he ollowing way: ∀a∈V(x+y)(a) =
x(a) + y(a). The di e ence o xand y, deno ed by x−y, is de ined simila ly:
∀a∈V(x−y)(a) = x(a)−y(a). The seman ics o he symbol ∞obey he
usual ules: ∀n∈Nn≤ ∞ ∧ ∞ +n=∞ − n=∞. When alking abou x−y,
we assume ha y∈ hV, Ni, i.e., he ha sub ac ed mul ise is ini e.
I X= (x1, . . . , xm) and Y= (y1, . . . , ym) a e ec o s o mul ise s o e V, hen
he ela ion X≤Yis de ined as ollows X≤Y⇔∀i∈N1≤i≤mxi≤
yi, i.e., X≤Yi and only i each componen o Xis a submul ise o Y. Simila ly,
we de ine X+Yand X−Yin a componen -wise way.
Fo u he de ails on hese opics see [2] and [6].
A Fo mal F amewo k o Clock- ee Ne wo ks o Cells 253
2.2 Clock- ee P Sys ems
Clock- ee P sys ems we e o iginally in oduced in [7]; an in ui i e app oach o
wo king wi h he clock- ee seman ics was explo ed in [4].
Aclock- ee memb ane sys em is de ined by a uple
Π= (O, C, µ, w1, w2,· · · , wm, R1, R2, . . . , Rm, i0),whe e
Ois a ini e se o objec s,
Cis a ini e se o ca alys s, C∈O,
µis a hie a chical s uc u e o mmemb anes, bijec i ely labeled
by 1, . . . , m; he in e io o each memb ane de ines a egion;
he en i onmen is e e ed o as egion 0,
wiis he ini ial mul ise in egion i, 1≤i≤m,
Riis he se o ules o egion i, 1≤i≤m,
i0is he ou pu egion.
The ules o a clock- ee memb ane sys em ha e he o m u→ , whe e u∈O+
and ∈(O×Ta )∗. In he case o non-coope a i e ules, u∈O. The a ge
indica ions om Ta ={he e, ou }∪{inj|1≤j≤m}a e w i en in he
ollowing way: (a, ), a∈O, ∈T a and he a ge he e is ypically omi ed.
A ule associa ed wi h memb ane imus only speci y a label o he immedia ely
inne memb ane in a a ge indica ion inj.
The ules a e applied in a maximally pa allel way: no u he ule should be
applicable o he idle objec s. In he case o non-coope a i e sys ems, all objec s
e ol e by he associa ed ules in he co esponding egions (excep objec s ain
egions isuch ha Ridoes no con ain any ule a→u, bu hese objec s do no
con ibu e o he esul ). Rules a e non-de e minis ically chosen a each momen
in ime when a change occu s in he con igu a ion o he P sys em. The p ocess o
choosing which ules should be applied does no ake any ime.
In ui i ely, clock- ee ule applica ions wo k in he ollowing way. A he s a
o applica ion, he mul ise in he le -hand side o he ule is sub ac ed om
he con en o he co esponding egion. When a ule applica ion is comple e, he
mul ise in he igh -hand side o he ule is added o he co esponding egion.
The ime be ween he s a and he end o a ule applica ion is a eal alue, may
be di e en o di e en applica ions o he same ule, and he e is impossible o
know in ad ance.
Fo u he de ini ions and de ails, see [7].
2.3 Ne wo ks o Cells
Ne wo ks o cells a e a gene al amewo k o desc ibing memb ane sys ems wi h a
s a ic memb ane s uc u e. Ne wo ks o cells a e o mally desc ibed in dep h in [3].
In ui i ely, wi h his app oach, memb ane sys ems a e conside ed as collec ions o
260 S. I ano
De ini ion 2. A clock- ee con igu a ion C∗= (C, H)o a ne wo k o cells Π
e ol ing unde he e olu ion mode ϑis hal ing when all ules ha e inalised and
he e a e no mo e applicable ules:
H=∅^Appl(Π, C, ϑ) = ∅.
This hal ing condi ion co esponds o he clock- ee seman ics as desc ibed
in [7]. Ob iously, jus as wi h o he a ian s o ne wo ks o cells, he p edica e
de ining he hal ing condi ion can be de ined in a di e en way.
In wha ollows, we will e e o ne wo ks o cells wi h clock- ee con igu a ions,
ope a ing acco ding o Algo i h 2, as o clock- ee ne wo ks o cells.
3.4 Example o Clock- ee E olu ion (Algo i hm 2)
We will now u n back o he example o a compu a ion o he clock- ee P sys em
Π0shown in Figu e 1. The compu a ion s a s wi h he ini ial con igu a ion:
C∗
0=a3,∅.
In his con igu a ion he mul ise o ules o apply is selec ed o be {(a→b)→
1,(a→u)→1,(a→x)→1}; he h ee ins ances o aa e co espondingly
emo ed. The ule (a→u) is immedia ely picked o be inalised, so he nex
con igu a ion is
C∗
1=u,(a→b)→1,(a→x)→1.
Since he e a e no ules which consume only u, no hing is applicable in C∗
2, so
A=∅once again. This ime F={(a→b)→1}, so he sys em ansi ions in o
C∗
2=bu,(a→x)→1}.
In his con igu a ion he ule b→cbecomes applicable, so i s applica ion mus be
s a ed: he only bis emo ed and H0is co espondingly modi ied. The ule a→x
is inalised (F={(a→x)→1}):
C∗
3=ux,(b→c)→1.
The e is he ule xu →x, so i mus be s a ed in his con igu a ion. The algo i hm
hen picks bo h ules which a e “s ill being applied” (including he xu →x, which
has jus been s a ed) and inalises hem:
C∗
4=cx,∅.
Since no ules a e applicable and H4=∅, he sys em has a i ed a a hal ing
con igu a ion.

A Fo mal F amewo k o Clock- ee Ne wo ks o Cells 261
3.5 Sui abili y o he Fo malisa ion
In his sec ion we will discuss whe he he o malism in oduced and desc ibed in
he p e ious sec ion is compa ible wi h he in ui i e desc ip ion o clock- eeness
p o ided in [7]. In his pape we de ine clock- eeness on he ounda ion o ne wo ks
o cells, while [7] s a s om ansi ional P sys ems. To b idge he ob ious gap,
we will conside ne wo ks o cells wi h ules wo king in clock- ee mode. This will
b ing a common g ound o he (ex ended) de ini ions om [7] and he o mal
de ini ions sugges ed in his pape .
De ini ion 3. A*-ne wo k o cells is a ne wo k o cells wi h ules ope a ing in
clock- ee mode, in he sense o [7].
Acco ding o he de ini ion, ule applica ions in a *-ne wo k o cells las o
a di e en eal- alued ime in e al each. In his sec ion we will only conside
ϑ=max o bo h *-ne wo ks o cells and clock- ee ne wo ks o cells, because clock-
ee P sys ems e ol e unde maximal de i a ion mode. The hal ing condi ion o
*-ne wo ks o cells will be he condi ion ha all ule applica ions ha e inalised and
no mo e ules a e applicable, while ne wo ks o cells wi h clock- ee con igu a ions
will ha e he hal ing condi ion in oduced in he p e ious sec ions. Since he goal
o he compu a ion and he way o in e p e he esul do no di ec ly pe ain
o he subjec o his sec ion, we will conside hese wo pa ame e s as ha ing a
ce ain well-de ined alue, he same o bo h kinds o analysed ne wo ks o cells.
Ob iously, clock- ee P sys ems as de ined in [7] a e a pa icula case o *-
ne wo ks o cells.
Theo em 1 (Sui abili y o he o malism). Conside a ne wo k o cells Π=
(n, V, w, In , R)and a ini e sequence o clock- ee con igu a ions o Π:C∗=
(C∗
i)m
i=0, wi h C∗
0= (C0,∅)being an ini ial con igu a ion and C∗
m= (Cm,∅)
being a hal ing con igu a ion. Le ∗Π= (n, V, w, In , R)be a *-ne wo k o cells.
C∗is a clock- ee compu a ion i and only i he sequence o he i s componen s
C= (Ci)m
i=0 is a alid compu a ion in ∗Π.
P oo . Acco ding o he co esponding de ini ions, C∗
0and C∗
na e alid clock- ee
ini ial and hal ing con igu a ions in Πco espondingly i and only i C0and Cn
a e alid ini ial and hal ing con igu a ions in ∗Πco espondingly. Ob iously, he
s a emen o he heo em holds o C∗
0=C∗
m, he e o e we will ocus on he cases
when m > 0.
Conside C∗
0and C∗
1and suppose ha C∗
0⇒(Π,max)C∗
1. Then, in ∗Π, we can
conside he ansi ion om C0 o C1cons uc ed in he ollowing way: s a he
applica ions o he ules belonging he mul ise Achosen in Algo i hm 2, hen
conside ha he ules collec ed in o Fas cons uc ed in Algo i hm 2 inalise hei
applica ion a one and he same ime. The momen hese ules comple e is he
momen when C1will occu . The e o e, C0⇒(∗Π,max)C1.
Now suppose ha C0⇒(∗Π,max)C1. This means ha , in ∗Π, some ule appli-
ca ions s a ed in C0, and some o hese ules inished o esul in con igu a ion
262 S. I ano
C1. We can he e o e conside ha in con igu a ion C∗
0, Algo i hm 2 chose o s a
he same ules as he ones which s a ed in in ∗Πand hen immedia ely inalised
hose which led o he occu ence o C1in ∗Π. Fo C∗
1cons uc ed in his way,
he ollowing is ue: C∗
0⇒(Π,max)C∗
1.
We ha e he e o e p o ed ha
C∗
0⇒(Π,max)C∗
1⇔C0⇒(∗Π,max)C1.
Mo eo e , we ha e p o ed ha H1con ains hose and only hose ules which could
ha e been s a ed in ∗Πin con igu a ion C0and migh ha e no been inalised in
ansi ion o C1. By epea ing he same easoning o any pai C∗
iand C∗
i+1, i is
now possible o p o e he s a emen o he heo em by induc ion.
4 Simula ions o Clock- eeness
Now ha we ha e o mally de ined clock- eeness, i is ime o pose one o he
mos impo an ques ions: how “ a away” a e clock- ee ne wo ks o cells om
he adi ional ne wo ks o cells? I u ns ou ha i is e y easy o simula e
clock- eeness wi h adi ional, s a ic ne wo ks o cells, as hey a e o malised in
[3].
Indeed, conside a clock- ee ne wo k o cells Π= (n, V, w, In , R) and a ule
i= (X→Y, P, Q)∈R0. Acco ding o clock- ee seman ics, an applica ion o iis
s a ed in a con igu a ion C∗
io Πby emo ing i s le -hand side om he sys em,
and is inalised in a con igu a ion C∗
jby adding i s igh -hand side o he sys em.
Conside now an o dina y ne wo k o cells Π0= (n, V 0, w, In , R0), wi h R0and
V0cons uc ed in he ollowing way:
V0=V∪ξi
 i= (X→Y, P, Q)∈R,
R0=nX→(ξi, k∗), P, Q,(ξi, k∗)→(ξi, k∗),¯
λ, ¯
λ,
(ξi, k∗)→Y, ¯
λ, ¯
λ
 i= (X→Y, P, Q)∈R},
1≤k∗≤ |R|,
whe e ¯
λis he ec o o size no emp y mul ise s. The alphabe o Π0includes all
symbols om he alphabe o Π, bu also a ξipe each ule wi h index i.
Fo each ule in R, h ee ules a e added o R0. When he ule iis applicable,
ins ead o Xbeing di ec ly ans o med in o Y,Xis ini ially ew i en in o ξi,
which is placed in o he cell wi h index k∗. The choice o he index k∗is o ally
a bi a y, i may e en be di e en o di e en ules. The symbol ξican ei he
ep oduce i sel o add Y o he sys em; ei he o hese will uncondi ionally happen;
he choice be ween he wo op ions is nonde e minis ic.
We claim ha Π0accu a ely simula es he e olu ion o Π. Indeed, a ule (X→
(ξiq, k∗), P, Q)∈R0is applicable in Π0i and only i he co esponding ule i=
(X→Y, P, Q)∈Ris applicable. Rew i ing Xin o ξi hus co esponds o emo ing
he le -hand side o he ule om he sys em and adding i o H0( he ope a ions
A Fo mal F amewo k o Clock- ee Ne wo ks o Cells 263
pe o med in Algo i hm 2). I , in a ce ain con igu a ion, ξiis ew i en in o i sel ,
hen, a his s ep, he ule iwas no picked by Algo i hm 2 o be inalised. The
case when ξiis ew i en in o Yco esponds o he scena io when Algo i hm 2
picked ule i o inalise.
Obse e now ha i , in a ce ain con igu a ion o Π0, only he ules which
ew i e ξi, 1 ≤i≤ |R|, we e chosen o be applied, hen he sys em will a i e a
exac ly he same con igu a ion a he nex s ep. I is possible o de ec such si u-
a ions and cu o such compu a ions. This howe e , does no make he ollowing
esul signi ican ly di e en on he o e all.
Theo em 2 (Simula ion o clock- eeness). Conside a clock- ee ne wo k
o cells Π= (n, V, w, In , R)and a sequence o clock- ee con igu a ions C∗=
(C∗
i)m
i=0, wi h C∗
0being an ini ial con igu a ion and C∗
mbeing a hal ing con ig-
u a ion. I C∗is a compu a ion hen i is possible o cons uc a compu a ion
K= (Ki)l
i=0 o an o dina y ne wo k o cells Π0so ha he e is a mapping
:K → C∗wi h he p ope ies:
1. (Ki) = C∗
j= (Cj, Hj)⇔Ci≤Ki( (Ki) = C∗
ii and only i Kicon ains a
leas all he symbols in Ki),
2. Ki⇒∗
(Π0,max)Kj⇔ (Ki)⇒∗
(Π,max) (Kj)( maps he bina y ela ion
⇒∗
(Π0,max)in o ⇒∗
(Π,max)and ice e sa),
3. ∀i∃j (Kj) = C∗
i( is su jec i e),
whe e 1≤i≤m,1≤j≤l.
P oo . Acco ding o he cons uc ions in he p e ious pa ag aphs, o a clock- ee
ne wo k o cells Π, one can associa e an o dina y ne wo k o cells Π0which simu-
la es Π. The mapping can hus be de ined as ollows: om he egion i∗o Ki
emo e all ins ances o ξi, 1 ≤i≤ |R|; his will be he i s componen o (Ki).
The second componen o (Ki) is ob ained by s a ing wi h an emp y mul ise
and adding an ins ance o ule ipe each ins ance o symbol ξi, 1 ≤i≤ |R|. The
equi ed p ope ies o ollow om he cons uc ions shown in his sec ion.
5 Conclusion
In his pape we ha e o n apa he concep o clock- ee P sys ems as de ined
in [7] and ha e sepa a ed clock- eeness as a s and-alone ing edien . We ha e o -
mally de ined his ing edien wi hin he amewo k o ne wo ks o cells [3] and
shown ha his de ini ion is consis en wi h he o iginal concep . We ha e also
shown ha clock- eeness can be simula ed wi h usual ne wo ks o cells in a qui e
s aigh o wa d way.
The ac ha clock- eeness can be simula ed so easily goes agains he in ui i e
imp ession p oduced by clock- ee P sys ems and poses he impo an ques ion o
how aluable his ing edien is. Indeed, i seems ha almos any p oblem in clock-
ee sys ems can be equi alen ly o mula ed o he co esponding clocked sys ems.
264 S. I ano
We belie e ha clock- eeness is ai ly impo an , hough, because i is (in ui i ely)
much close o how p ocesses ake place in biological cells. The ac ha clock-
eeness is easy o simula e is hus bene icial and shows how i is possible o mo e
close o eal li e wi hou sac i icing oo much.
We ema k, o cou se, ha he chemical eac ions aking place in he cell ha e
been s udied well enough o app oxima ely p edic hei du a ions o , a leas , com-
pa e hem o o he cellula p ocesses in e ms o speed. Clock- eeness, howe e ,
allows us o abs ac away hese de ails. Me apho ically pu , an implemen a ion
o an ope a ion in a clock- ee ne wo k o cells (o clock- ee P sys em) can su i e
changes in he physical implemen a ion, because i does no depend on he du a-
ions o unde lying chemical ela ions. This easoning is o cou se hypo he ical a
he momen , bu i may become p ac ical qui e soon.
In his pape we ha e used he e m “clock- eeness” o deno e he mode o
e olu ion o a ne wo k o cells in which ule applica ions may las o an a bi a y
long o sho amoun o ime. The wo d “ ee” in he e m “clock- ee”, he e o e,
e e s o a di e en concep han he same wo d in he e m “ ime- ee” [1]. How-
e e , he majo i y o clock- ee P sys ems conside ed in [7] and, o example, [4],
a e in ac independen o he wha he du a ions o ules a e. I hus is possible
o conside ha he e ms “clock- eeness” and “ ime- eeness” do ha e some hing
in common. Obse e ha Theo em 2 allows ansla ing he p oblem o indepen-
dence o he du a ions o ules in clock- ee ne wo ks o cells in o he p oblem o
con luence in egula ne wo ks o cells.
Acknowledgemen s
The au ho g a e ully acknowledges A iom Alhazo , PhD o he ema kably ui -
ul discussions du ing he wo k on [4], wi hou which he p esen pape would no
ha e happened in he i s place.
Re e ences
1. A. Alhazo , M. Ca alie e: E olu ion-communica ion P Sys ems: Time-F eeness. P oc.
Thi d B ains o ming Week on Memb ane Compu ing (M.A. Gu i´e ez-Na anjo e al.
eds.), 2005, 11–18.
2. J. Dassow, Gh. P˘aun: On he powe o memb ane compu ing. Jou nal o Uni e sal
Compu e Science, 5 (1999), 33–49.
3. R. F eund, S. Ve lan: A Fo mal F amewo k o S a ic (Tissue) P Sys ems. 8 h In e -
na ional Wo kshop on Memb ane Compu ing, WMC2007 (G. Ele he akis e al., eds.),
LNCS 4860, Sp inge , 2007.
4. S. I ano : Basic Concu ency Resolu ion in Clock-F ee P Sys ems. P oc. 12 h In e -
na ional Con e ence on Memb ane Compu ing (M. Gheo ghe e al., eds.), LNCS 7184,
Sp inge , 2012, 226–242.
5. Gh. P˘aun: Memb ane Compu ing. An In oduc ion. Sp inge , Be lin, 2002.
6. G. Rozenbe g, A. Salomaa, eds.: Handbook o Fo mal Languages, 3 ols., Sp inge ,
Be lin, 1997.
A Fo mal F amewo k o Clock- ee Ne wo ks o Cells 265
7. D. Sbu lan: Clock- ee P Sys ems. P e-p oc. Fi h Wo kshop in Memb ane Compu ing
(G. Mau i, Gh. Paun, C. Zand on, eds.), 2004, 372–383.
8. F. Be na dini, M. Gheo ghe, M. Ma gens e n, S. Ve lan: Ne wo ks o Cells and
Pe i Ne s. P oc. 5 h B ains o ming Week on Memb ane Compu ing (M.A. Gu i´e ez-
Na anjo e al., eds.), 3 2007, 3–62.
9. The P Sys ems Web Page: h p://ppage.psys ems.eu.

On he Simula ions o E olu ion-Communica ion
P Sys ems wi h Ene gy wi hou An ipo Rules
o GPUs
Richelle Ann B. Juayong1, F ancis Geo ge C. Caba le1, Hen y N. Ado na1,
Miguel A. Ma ´ınez–del–Amo 2
1Algo i hms & Complexi y Lab
Depa men o Compu e Science
Uni e si y o he Philippines Diliman
Diliman 1101 Quezon Ci y, Philippines
E-mail: [email p o ec ed],[email p o ec ed],[email p o ec ed]
2Resea ch G oup on Na u al Compu ing
Depa men o Compu e Science and A i icial In elligence
Uni e si y o Se ille
A da. Reina Me cedes s/n, 41012 Se illa, Spain
E-mail: [email p o ec ed]
Summa y. In his epo , we p esen ou ini ial p oposal on simula ing compu a ions on
a es ic ed a ian o E olu ion-Communica ion P sys em wi h ene gy (ECPe sys em)
which will hen be implemen ed in G aphics P ocessing Uni s (GPUs). This ECPe sys-
ems a ian p ohibi s he use o an ipo ules o communica ion. Se e al possible le els
o pa alleliza ions o simula ing ECPe sys ems compu a ions on GPUs a e emphasized.
Ou wo k is based on a localized ma ix ep esen a ion o he men ioned a ian gi en
in a p e ious li e a u e. Ou p oposal employs a me hodology o o wa d compu ing also
discussed in he said li e a u e.
Key wo ds: Memb ane compu ing, Pa allel compu ing, GPU compu ing
1 In oduc ion
E olu ion-Communica ion P sys ems wi h ene gy (ECPe sys ems) [1] is a a i-
an o P sys ems in oduced in 2009 o ini ia e a amewo k o communica ion
complexi y. I o igina es om E olu ion-Communica ion P (ECP) sys ems [10], a
hyb id o wo well-in es iga ed a ian s, T ansi ion P sys ems [9] and P sys ems
wi h Sympo and An ipo ules [11]. The di e ence be ween ECPe and ECP
sys ems is he p esence o a special objec called ‘ene gy’ in he o me , which can
be p oduced h ough e olu ion ules and consumed in communica ion ules. One
268 R.A.B. Juayong e al.
c ucial es ic ion o ECPe sys ems includes he use o ene gy o each communi-
ca ion ule. Thus, no objec can be communica ed wi hou using some ‘quan a’ o
ene gy. Mo eo e , upon being deli e ed o a ecei ing egion, he ene gy used in
a communica ion does no pass h ough any memb ane. In his manne , i is said
ha he ene gy used in he p ocess o communica ion is being ’los ’.
Se e al ecen wo ks ha e in oduced he concep o ep esen ing ce ain P
sys em a ian s and hei compu a ions as ma ices and ec o -ma ix ope a ions,
espec i ely. In pa icula , a ian s known as Spiking Neu al P (SNP) sys ems and
hei ma ix ep esen a ions we e in oduced in [5] whe eas ma ix ep esen a ions
o ECPe sys ems we e gi en in [6]. Aside om c ea ing a ‘con enien ’ and ela-
i ely compac way o desc ibing he sys ems and hei compu a ions, he ma ix
ep esen a ions add addi ional ease o hei simula ion and implemen a ion in pa -
allel ha dwa e. Vec o and ma ix ope a ions a e highly pa allelizable and can be
e icien ly implemen ed in pa allel ha dwa e, including G aphics P ocessing Uni s
(GPUs). GPUs a e massi ely pa allel ha dwa e no like cu en gene a ion CPUs.
Using hei espec i e ma ix ep esen a ions, SNP sys ems ha e been success ully
implemen ed in GPUs in [2] and mo e ecen ly in [3]. The in en ion o his cu -
en wo k is o con inue such end i.e. o p o ide ou me hodology on how o
implemen ECPe sys ems compu a ions (using he ec o -ma ix ep esen a ions)
on pa allel ha dwa e, in pa icula GPUs.
2 E olu ion-Communica ion P Sys ems wi h Ene gy
2.1 Fo mal De ini ion o ECPe sys ems
Be o e we p oceed, we no e ha he eade s a e assumed o be amilia wi h he
undamen als o o mal language heo y and memb ane compu ing [9].
A ela i ely new a ian o E olu ion-Communica ion P sys ems [10] has been
in oduced in [1] o e alua e communica ion ha is dependen on some ene gy
p oduced om e olu ion ules. A special objec eis in oduced o he sys em o
ep esen a quan um o ene gy. We use he de ini ion o EC P sys em wi h ene gy
(ECPe sys em) om [1].
De ini ion 1. An EC P sys em wi h ene gy is a cons uc o he o m
Π= (O, e, µ, w1, . . . , wm, R1, R0
1, . . . , Rm, R0
m, iou )
whe e:
(i) mpe ains o he o al numbe o memb anes;
(ii) Ois he alphabe o objec s;
(iii) µis he memb ane s uc u e which can be deno ed by a se o pai ed squa e
b acke s wi h labels. We say ha memb ane iis he pa en memb ane o a
memb ane j, deno ed pa en (j), i he pai ed squa e b acke s ep esen ing
On he Simula ions o ECPe Sys ems wi h Ene gy o GPUs 269
memb ane jis loca ed inside he pai ed squa e b acke s ep esen ing mem-
b ane i, i.e. [i. . . [j]j]i. Re e sely, we say ha memb ane jis a child mem-
b ane o memb ane i, deno ed j∈child en(i) whe e child en(i) e e s o he
se o memb anes con ained in memb ane i. The ela ion o pa en and child
memb ane becomes mo e appa en when we ep esen he memb ane s uc-
u e as a ee. Since o de does no ma e in ou model, he e can be mul iple
ees (isomo phic wi h espec o child en o a node), each co esponding o
he same memb ane s uc u e ep esen a ion.
(i ) w1, . . . , wma e s ings o e O∗whe e wideno es he mul ise o objec p esen
in he egion bounded by memb ane i.
( ) R1, . . . , Rma e se s o e olu ion ules, each associa ed wi h a egion delimi ed
by a memb ane in µ;
•An e olu ion ule is o he o m a→ whe e a∈O, ∈(O∪ {e})∗. In
he e en ha his ype o ule is applied, he objec a ans o ms in o a
mul ise o objec s in he nex ime s ep. Th ough e olu ion ules, objec
ecan be p oduced, bu eshould ne e be in he ini ial con igu a ion and
objec eis no allowed o e ol e.
( i) R0
1, . . . , R0
ma e se s o communica ion ules, each associa ed wi h a memb ane
in µ; A communica ion ule can ei he be a sympo o an an ipo ule:
•A sympo ule can be o he o m (aei, in) o (aei, ou ), whe e a∈O,
i≥1. By using his ule, icopy o eobjec s a e consumed o anspo
objec ainside (deno ed by in) o ou side (deno ed by ou ) he memb ane
whe e he ule is de ined. To consume copies o objec emeans ha upon
comple ion o he anspo a ion o objec a, he occu ences o ea e los ,
hey do no pass om a egion o ano he one.
•An an ipo ule is o he o m (aei, ou ;bej, in) whe e a, b ∈Oand i, j ≥1.
By using his ule, we know ha he e exis s an objec ain he egion im-
media ely ou side he memb ane whe e he ule is decla ed, and an objec
binside he egion bounded by he memb ane. In he applica ion o his
ule, objec aand objec ba e swapped using iand jcopies o objec ein
he di e en egions, espec i ely. As in sympo ules, he copies o objec
ea e los a e he applica ion.
We say ha a communica ion ule has a sending and ecei ing egion. Fo
a ule ∈R0
iassocia ed wi h an in label, i s ecei ing egion is egion i
and i s sending egion is he pa en (i). On he o he hand, he sending and
ecei ing egions a e e e sed o a ule ∈R0
iassocia ed wi h an ou label.
Fo an an ipo ule ∈R0
i, egion iand pa en (i) a e bo h sending and
ecei ing egion. Also, no e ha no communica ion can be applied wi hou
he u iliza ion o objec e.
( ii) iou ∈ {0,1, . . . , m}is he ou pu memb ane. I iou = 0, his means ha he
en i onmen shall be he placeholde o he ou pu .
Rules a e applied in a nonde e minis ic, maximally pa allel manne . Nonde e -
minism, in his case, has he ollowing meaning: when he e a e mo e han wo
e olu ion ules ha can be applied o an objec , he sys em will andomly choose
276 R.A.B. Juayong e al.
S ep 3 and 4
The igge ma ix o bo h egion 1 and 2 a e shown below
TMΠECP e ,1=








100
100
101
011
010
001








TMΠECP e ,2=0 1
1 0 
The ex ended applica ion ec o s a0
i,1and a0
i,2 ep esen ing he ec o o un-
knowns has he same index as ha o he ows o hei co esponding e ec ma ix.
S ep 5
The esul ing sys em o linea equa ions achie ed om Equa ion (2) o egion 1
and 2 is gi en below:
a0
i−1,1( 11) + a0
i−1,1( 12) + a0
i−1,1( 0
21)=2
a0
i−1,1( 0
22) + a0
i−1,1(Add11)=1
a0
i−1,1( 0
21) + a0
i−1,1( 0
22) + a0
i−1,1(Add12)=2
a0
i−1,2( 21)=0
a0
i−1,2(Add21)=1
As can be aced, he e a e 11 possible ex ended applica ion ec o s o egion 1
while he e exis s a unique ex ended applica ion ec o o egion 2. Shown below
is ex ended applica ion ec o o egion 2:
a0
i−1,2=0 1 
S ep 6
The addi ional inequali y in egion 1 equi es ha :
a0
i,1(Add12)<1
o cases whe e he ap objec # emain. Thus, hese solu ions a e possible:
ai,1=002010ai,1=101100
ai,1=011100
Fo cases whe e he ap objec does no emain, he ollowing solu ions a e also
possible:
ai,1=110101ai,1=020101
ai,1=200101
A e s ep 6, he 11 solu ions in s ep 5 we e educed o six.

On he Simula ions o ECPe Sys ems wi h Ene gy o GPUs 277
S ep 7
Pe o ming Equa ion (4) and Equa ion (5), below a e he possible applica ion
ec o combina ions:
Solu ion 1 : ai,1=0020ai,2=020
Solu ion 2 : ai,1=1011ai,2=011
Solu ion 3 : ai,1=0111ai,2=011
Solu ion 4 : ai,1=1101ai,2=001
Solu ion 5 : ai,1=0201ai,2=001
Solu ion 6 : ai,1=2001ai,2=001
The co esponding con igu a ion ec o s o each solu ion is as ollows:
Solu ion 1 : Ci,1=010Ci,2=3 0 
Solu ion 2 : Ci,1=200Ci,2=2 1 
Solu ion 3 : Ci,1=002Ci,2=2 1 
Solu ion 4 : Ci,1=203Ci,2=1 1 
Solu ion 5 : Ci,1=005Ci,2=1 1 
Solu ion 6 : Ci,1=401Ci,2=1 1 
2.4 A Sequen ial Implemen a ion o Compu a ion on ECPe Sys ems
wi hou An ipo using Ini ial Ma ix Rep esen a ion
Gi en he ep esen a ion and algo i hm o o wa d compu ing p esen ed in Sec ion
2.3, we we e able o do a sequen ial implemen a ion o compu a ion on ECPe
sys ems wi hou an ipo using he C p og amming language.
The sys em s a s wi h eading wo inpu iles con aining in o ma ion o ind-
ing alid applica ion ec o s and de e mining he nex con igu a ion ec o gi en a
cu en ly examined con igu a ion ec o . The ollowing a e he names o he inpu
iles:
•File ans ile. x which con ains he in o ma ion needed o ansi ioning om
one con igu a ion o he nex .
•File o wComp ile. x which con ains he in o ma ion needed o ind alid con-
igu a ion ec o /s gi en a cu en con igu a ion ec o .
A discussion abou he o ma o he speci ied iles is gi en in he appendix. The
iles will be used o ini ialize he necessa y a iables and poin e s needed o he
simula ion. The sys em has wo ou pu iles ep esen ing he ee-s uc u e o he
con igu a ion his o y. The ollowing a e he name o he ou pu iles:
278 R.A.B. Juayong e al.
•File con . x which con ains a lis o con igu a ion.
•File con index. x which con ains he index o he con igu a ion in he ee
s uc u e.
P esen ed in Figu e 2 is he lowcha o how he p og am wo ks. Upon eading
inpu iles and loading a iables and poin e s needed o compu ing, he ini ial
con igu a ion is placed in he con . x and he associa ed index 1 is placed in
con index. x . A e wa ds, he sys em will en e a loop o de e mining he con-
igu a ions gene a ed by a cu en ly examined con igu a ion. The examina ion o
con igu a ion will be execu ed in o de o hei posi ion in he ile. Gi en wo con-
igu a ion Cand C0whe e he posi ion o Cp ecedes C0, hen con igu a ion C
will be examined i s be o e con igu a ion C0. Examining con igu a ion shall hal
only when he sys em eaches wo s opping c i e ion:
•Upon achie ing a p e-speci ied uppe bound on he numbe o i e a ions
•Upon eaching a s a e whe e he e a e no mo e con igu a ions o examine.
Fig. 2. An o e iew o ou sequen ial implemen a ion o ECPe sys ems wi hou an ipo
in he C p og amming language.
Fo e e y loop, a con igu a ion in he ile is examined by i s de e mining all alid
applica ion ec o s which is applicable o he cu en ly examined con igu a ion. In
inding a alid applica ion ec o , we use he concep o localized ep esen a ion
and ex ended applica ion ec o s, and ollow he s eps in Sec ion 2.3 o o wa d
On he Simula ions o ECPe Sys ems wi h Ene gy o GPUs 279
compu ing. I he only applica ion ec o applicable is he ze o ec o which means
no mo e ule combina ion can be applied o he cu en con igu a ion, i will p o-
ceed o he nex con igu a ion o examine. Each non-ze o applica ion ec o is
used o gene a e he nex se o con igu a ions. A e wa ds, ou pu iles con . x
and con index. x will be upda ed o accoun all he newly gene a ed nex con-
igu a ion ec o s. No e ha in his sys em, we ha e no ye implemen ed he
me hodology o de ec epea ing con igu a ion.
On Gene a ing Ex ended Valid Applica ion Vec o s and Fil e ing
The goal o s ep 5 in Sec ion 2.3 is he gene a ion o all possible ex ended ap-
plica ion ec o s a0
i,h sa is ying equa ion (2). In ou sequen ial implemen a ion,
we achie e his by examining each equa ion esul ing om he co esponding and
equi alen sys em o linea equa ion. We now s udy he cha ac e is ics o he e-
sul ing sys em by using he example p esen ed o o wa d compu ing.
Shown below a e he equa ions yield om Equa ion (2) o egion 1 o he
example in Sec ion 2.2, also shown in Sec ion 2.3.
β1:a0
i−1,1( 11) + a0
i−1,1( 12) + a0
i−1,1( 0
21)=2
β2:a0
i−1,1( 0
22) + a0
i−1,1(Add11) = 1
β3:a0
i−1,1( 0
21) + a0
i−1,1( 0
22) + a0
i−1,1(Add12) = 2
I can be obse ed ha each equa ion in he esul ing sys em ep esen s an objec
condi ion; he objec e e ing o possible objec s ha may en e an examined
egion. Also, o sending egions, an equa ion o ene gy condi ion (β3) mus also
be p esen in a esul ing sys em. In he gene al case, each a iable ( ep esen ing
ule applica ion o a ce ain communica ion ule) in he ene gy equa ion is p esen
in exac ly one o he objec condi ion. This objec is he communica ion igge
ha will be communica ed upon ac i a ion o he ule ep esen ed by he said
a iable. Fo example, he a iable a0
i−1,1( 0
21) is p esen in bo h β1and β3. The
same goes o a iable a0
i−1,1( 0
22) which is p esen in bo h β2and β3.
O he han such ype o a iables, no mo e a iables can be p esen in mo e
han one equa ion. Mo eo e , while he coe icien s o he e ms in he ene gy
equa ion can con ain any posi i e in ege , he coe icien s o he e ms o non-
ene gy condi ion will always be one (due o he es ic ion o noncoope a i e ule
o ma ). Mo eo e , he se o ules ∈T R(h) a e all ep esen ed by he union o
all ules ( a iables) ep esen ed in he non-ene gy condi ions wi hou he iden i y
ules.
Gi en such obse a ion, possible ec o s a0
i,h a e de e mined by i s sol ing he
condi ion posed o ene gy. Since he e can be mul iple solu ion o ules in ol ing
ene gy ( he ules include he iden i y ule o ene gy since i is o Ca ego y 2),
we shall de e mine he possible ex ended applica ions ec o s esul ing om a
alid ene gy solu ion. In he linea equa ion shown abo e, we i s on de e mining
solu ions o β3. The possible ene gy solu ions a e
(1,1,0),(1,0,1),(0,1,1),(2,0,0),(0,2,0),(0,0,2),
280 R.A.B. Juayong e al.
whe e in each ec o , he i s elemen co esponds o alue o a0
i−1,1( 0
21), he sec-
ond co esponds o he alue o a0
i−1,1( 0
22) and he hi d elemen is o he ene gy
iden i y ule a0
i−1,1(Add12). Fo each solu ion, we hen copy he ule applica ion
o he associa ed objec o communica e. A e wa ds, he ule applica ion is ans-
e ed o he igh -hand side o he equa ion, i.e. sub ac ed om he cu en coun
o he co esponding communica ed objec . As an example, he esul ing modi ied
objec condi ion β0
1caused by he alue o communica ion ulea0
i−1,1( 0
21) in he
i s ene gy solu ion will be
β0
1:a0
i−1,1( 11) + a0
i−1,1( 12) = 1
I he esul ing coun is nega i e, hen, he esul ing ene gy solu ion canno be
applied. The e o e, no ec o a0
i,h can be gene a ed gi en such nega i e esul .
This il e ing on ene gy solu ion is e iden in applying ene gy solu ion (0,2,0) on
β2. Upon sub ac ing he esul ing ule applica ion om he igh -hand side o he
co esponding communica ed objec condi ion, he esul ing objec condi ions can
be analyzed one a a ime.
Upon ealizing solu ions o each objec condi ion, s ep 6 o he o wa d com-
pu ing me hodology can al eady be execu ed pe objec condi ion. Iden i y ules
o ca ego y 2 objec s can be u he checked o execu e he il e ing pa , done
in S ep 6 o he me hodology in Sec ion 2.3 o see i he numbe o ca ego y 2
objec s emaining in he egion can be allowed o emain ( ha is, he case doesn’
alida e he ule ha all objec s ha can e ol e o be communica ed mus do so).
O he wise, he objec solu ion will be d opped. No e ha while s ep 6 in Sec ion
2.3 e alua es i s all ex ended applica ion ec o s be o e his s ep, we pe o m his
s ep pe objec condi ion since he iden i y ules o any ca ego y 2 objec s can
only be p esen in he co esponding ca ego y 2 objec equa ion. P e e ably, he
o de o analyzing ca ego y 2 objec s ollow he so ed lis Lis ca 2so ha i an
objec solu ion is no alid, i can e mina e immedia ely a he i s unsa is ied
ca ego y 2 objec .
The esul ing alue o each il e ed a iable se pe solu ion can be combined,
one solu ion om each objec , and each combined lis cons i u es one ex ended
applica ion ec o . To illus a e his, we examine he possible ex ended applica ion
ec o s ha can be yield om ene gy solu ion (1,1,0). Fo β1condi ion, he objec
solu ions a e (0,1) and (1,0) whe e he i s elemen o he said ec o s co espond
o a0
i−1,1( 11) and he second, o a0
i−1,1( 12). Fo β2condi ion, he objec solu ion
only assigns he alue 1 o a0
i−1,1(Add11). The e o e, o ene gy solu ion (1,1,0),
he co esponding ex ended applica ion ec o s yielded a e
ai,1=101100ai,1=011100
3 Simula o design and implemen a ion
In his sec ion, we elay how we can employ GPUs o pa allelize he ask o inding
all possible objec solu ions. NVIDIA in oduced he Compu e Uni ied De ice
On he Simula ions o ECPe Sys ems wi h Ene gy o GPUs 281
A chi ec u e (CUDA) in 2007 [7]. CUDA is a so wa e and ha dwa e a chi ec u e
o gene al pu pose compu a ions in NVIDIA’s GPUs [7]. CUDA ex ends high-
le el languages such as C o allow p og amme s o easily c ea e so wa e ha will
be execu ed in pa allel, a oiding low-le el g aphics and ha dwa e p imi i es [12].
GPUs in oduce inc eased pe o mance speedups o e CPU only implemen-
a ions wi h linea algeb a compu a ions (among o he ypes o compu a ions)
because o he GPU a chi ec u e. The common CPU a chi ec u es a e composed
o ansis o s which a e di ided in o di e en blocks o pe o m he basic asks o
CPUs (gene al compu a ion): con ol, caching, DRAM, and ALU (a i hme ic and
logic). In con as , only a ac ion o he CPU’s ansis o s alloca ed o con ol
and caching a e used by GPUs, since a mo e ansis o s a e used o ALU [7]
(see Figu e 3 o an illus a ion). This a chi ec u al di e ence is a e y dis inc
and signi ican eason why GPUs o e la ge pe o mance inc eases o e CPU only
implemen a ion o pa allel code wo king on la ge amoun s o inpu da a. Howe e
i he p oblem o be sol ed canno be o ganized in a da a pa allel o m (a ask
pe o ming compu a ions on da a need no depend hea ily on o he ask’s esul s)
hen he pe o mance o GPUs o e CPUs will no be ully u ilized.
Code w i en o CUDA can be spli up in o mul iple h eads wi hin mul iple
h ead blocks, each con ained wi hin a g id o ( h ead) blocks. These g ids belong
o a single de ice o GPU. Each de ice has mul iple co es, each capable o unning
i s own h eads. Each co e in he de ice is able o un a se o h eads.A h ead
block is assigned o each mul ip ocesso , whe e each p ocesso is made up o se -
e al co es [7, 12]. A unc ion known as a ke nel unc ion is one ha is called om
he hos o CPU bu execu ed in he de ice. Using ke nel unc ions, he p og am-
me can speci y he GPU esou ces: he layou o he h eads ( om one o h ee
dimensions) in a h ead block, and he h ead blocks ( om one o wo dimensions)
in a g id. Table 1 shows he esou ces o cu en CUDA enabled NVIDIA GPUs.
GPU esou ces Values
Global memo y Up o 4GB
Max numbe o h eads pe dimension (x, y, z) (1024,1024,64)
Max numbe o h ead blocks pe g id (x, y, z) (65535,65535,65535)
Table 1. Typical esou ces o CUDA enabled Fe mi a chi ec u e GPUs ( om [7, 12]) .
On Pa allelizing T ansi ions
Ano he appa en possibili y in o de o simula e he pa allel compu a ions o
ECPe sys ems (as well as capi alize on hei ep esen a ions as ma ices) on GPUs,
we can ha e ini ially a leas wo le els o pa allelism: he i s le el is he compu a-
ion o Equa ion (1) in pa allel by h eads in a block; he second le el in ol es he
compu a ion o all he possible nex con igu a ions gi en a cu en con igu a ion,
so ha each block in a g id o h ead blocks pe o ms his le el.

282 R.A.B. Juayong e al.
Fig. 3. (a) Common ansis o alloca ion o CPUs and GPUs (b) Compu ing uni hie -
a chy o GPUs, om [12].
The i s le el is highly pa allelizable since ec o -ma ix mul iplica ion and
ec o addi ion a e highly da a independen . Each h ead can mul iply a ec o
o one column o he ma ix, hus pe o ming ai,h ·MΠ,h. Each h ead sums he
p oduc s hen adds hese o ano he ec o , pe o ming he addi ion o Ci−1,h.
Fo he second le el, i he e a e qnumbe o ai,h’s and hence qnumbe o nex
con igu a ions, hen qblocks will pe o m Equa ion (1) q imes.
Because o he physical limi a ions o cu en NVIDIA GPUs, no mo e han
1024 h eads pe block a e allowed o Fe mi a chi ec u e GPUs so ha a mos
ma ices o a mos 1024 columns can be simula ed in a block. In Fe mi GPUs, he
maximum numbe o allowable h ead blocks in a g id is 65535 (See Table 1) so q
is cu en ly uppe bounded by his alue. Ano he simula ion conside a ion, aside
om he compu ing uni s ( h eads, h ead blocks) is he ela i ely mo e limi ed
memo y o cu en GPUs compa ed o CPUs. In his case, s o ing all qnumbe
o applica ion ec o s (each o which a e o leng h |R(h)|) and he qnumbe o
nex con igu a ions (each o which a e o leng h |PO(h)|) esul ing om hose
applica ion ec o s mus i in o he GPU’s global memo y.
On Gene a ing Objec Solu ions
Gi en an examined ene gy solu ion, we check each objec condi ion whe e each
a iable co esponding o a communica ion ule ha e al eady been de e mined ( ia
On he Simula ions o ECPe Sys ems wi h Ene gy o GPUs 283
he examined ene gy solu ion) and he alue a he igh -hand side o a communi-
ca ed objec has al eady been upda ed. As can obse ed, his p oblem is educed o
an in ege pa i ion p oblem whe e, gi en an objec equa ion α1+α2+...+αk=n,
we need o ind ec o con aining αi∈N’s, i.e. (α1, α2, ...., αk).
To gene a e solu ions o non-ene gy objec equa ion, we i s ob ain a lexi-
cog aphic o de o pa i ion o he alue n. In [13], gi en X= (x1, x2, ..., xk0)
and Y= (y1, y2, ..., yk00 ), Xp ecedes Ylexicog aphically i and only i o some
j≥1, xi≥yiwhen i < j, and xjp ecedes yj. As an example, pa i ions o 5 in
lexicog aphic o de a e: 11111,2111,311,221,311,32,41,5.
Each esul ing pa i ion will be padded wi h ze oes acco dingly so ha he
pa i ion can be ep esen ed in a k-dimensional ec o . Each pa i ion will be
assigned o a h ead. Each h ead will be esponsible o he gene a ion o dis inc
pe mu a ion o he k-dimensional ec o ep esen ing he pa i ion. The union o
solu ions gene a ed by each pa i ion co esponds o he se o objec solu ions o
a ce ain examined objec . Since he il e ing s ep o he o wa d me hodology, as
explained in Sec ion 2.4, can be done pe objec condi ion, his s ep can also be
pe o med wi hin he cu en h eads.
The idea o pa alleliza ion in Sec ion 3 may also be used o gene a ing ene gy
solu ions o he o m c1α1+c2α2+... +ckαk=n, whe e a lexicog aphic o de on
he pa i ions o nis i s accomplished. Again, each esul ing pa i ion will hen
be padded wi h ze oes acco dingly. Upon assigning each pa i ion o a h ead,
and gene a ing dis inc pe mu a ion o a pa i ion, each ec o ep esen a ing a
pe mu a ion can be equa ed wi h he ec o (c1α1, c2α2, ..., ckαk), a e which he
co esponding alue o he α0
iscan be ob ained.
4 Conclusions and u u e wo k
In his epo , we we e able o desc ibe a sequen ial implemen a ion o a o wa d
compu ing me hodology o ECPe sys ems wi hou an ipo ules. We also we e
able o show how we ex end his sequen ial wo k o employ GPUs o pa allelizing
some key a eas in he implemen a ion p ocedu es. We we e also able o show ou
p oposed ideas o pa allelizing o he pa s o he code.
As u u e wo k, we would like o implemen ou p oposed ideas o pa alleliza-
ion and es he e iciency o he esul ing implemen a ion. Mo eo e , we hope o
imp o e hese ideas o be e capi alize he pa allelizabili y o ec o -ma ix ep-
esen a ions o he compu a ions on GPUs. As pa o ou u u e wo ks, we also
would like o ex end he me hodology o o wa d compu ing o apply o a gene al
ECPe sys ems whe e an ipo ules a e allowed. The di icul y in allowing such
communica ion ules a e in luenced by he implica ion ha a egion can be bo h
a sende and ecei e egion. Thus, an ipo ules need o main ain ela ionship
be ween adjacen egions a c ucial pa s o he o wa d compu ing me hodology.
The mo e icky pa is he ac ion done when conside ing objec s ha a e un-
mo ed due o an ipo ule. The an ipo ule inc eases he numbe o possible
cases dic a ing why a ca ego y 2 objec can emain in a ce ain egion.
284 R.A.B. Juayong e al.
5 Acknowledgmen s
R.B. Juayong and F.G.C. Caba le a e pa ially suppo ed by he DOST-ERDT
schola ship p og am. H. Ado na is unded by he DOST-ERDT esea ch g an and
he Alexan p o esso ial chai o he UP Diliman Depa men o Compu e Science,
Uni e si y o he Philippines Diliman. M.A. Ma ´ınez–del–Amo is suppo ed by
“P oyec o de Excelencia con In es igado de Reconocida Val´ıa” o he “Jun a de
Andaluc´ıa” unde g an P08-TIC04200, and by he p ojec TIN2009–13192 o he
“Minis e io de Ciencia e Inno aci´on” o Spain, bo h co- inanced by FEDER unds.
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and Technology, Vol. 13 No. 2 pp. 113-130, 2010
2. F.G.C. Caba le, H. Ado na, M.A. Ma ´ınez-del-Amo : A Spiking Neu al P sys em
simula o based on CUDA, M. Gheo ghe e al. (Eds.), 12 h In ’l Con e ence on
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103. Sp inge -Ve lag, 2012.
3. F.G.C. Caba le, H. Ado na, M.A. Ma ´ınez-del-Amo , M.J. P´e ez-Jim´enez: Im-
p o ing GPU Simula ions o Spiking Neu al P Sys ems, ( o appea ) Romanian
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USA, 2010.
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na ional Jou nal o Compu e Ma hema ics, Vol. 70, No. 2., pp. 319-332, (1998)
On he Simula ions o ECPe Sys ems wi h Ene gy o GPUs 285
Appendix A:
On File Fo ma s o Implemen a ion o ECPe sys em in C
Fig. 4. Fo ma o ile ans ile. x o he sequen ial implemen a ion o ECPe sys em
in C.
Shown in Figu e 4 abo e is he o ma o inpu ile ans ile. x . This ile
con ains he in o ma ion equi ed o ind a succeeding con igu a ion gi en a cu en
one. Fi s , he numbe o egions mus be speci ied. I he e will be a case whe e he
en i onmen will be needed du ing he compu a ion, he alue o his pa ame e
mus be he numbe o memb anes inc emen ed by one in o de o accoun o he
en i onmen . The memb ane s uc u e is ep esen ed by pai ed squa e b acke s as
ypical ep esen a ion o a memb ane s uc u e. No e ha o ei he a closed o
open squa e b acke , he symbol mus be ollowed by a nume al o indica e he label
o he memb ane. Fo he ini ial con igu a ion, he e is a need o ollow a o al o de
as discussed in Sec ion 2.3. The e o e, he ini ial con igu a ion is simply a ini ial
con igu a ion ec o whe e each cell con ains he numbe o copies o a ce ain
objec a he s a o he compu a ion. The sepa a o o he cells will be he space
symbol. The expec ed ansi ion ma ix has dimensions ollowing he p e iously
speci ied numbe o ules in ol ing a speci ic egion and he numbe o possible
objec s ha may en e he egion. The ows will be sepa a ed by newline, whe eas,
he columns a e sepa a ed by spaces. A ule ype can ei he be an e olu ion ule in
292 R. Le ica u e al.
he e ha e also been signi ican de elopmen s in using he P sys ems pa adigm o
model, simula e and o mally e i y a ious sys ems [5, 23].
Up o now, wo main a eas conce ning P sys em o mal e i ica ion ha e been
in es iga ed: p ope y e i ica ion h ough model checking and p ope y ex ac-
ion. Howe e , he e is no app oach which in eg a es hese wo aspec s (o o he
ela ed aspec s such as simula ion). Ini ial esea ch on P sys em model check-
ing has ackled he p oblem o iden i ying decidable (o undecidable) p oblems
[7, 8]. Ve i ying p ope ies o P sys ems implies de ining and implemen ing an
ope a ional seman ics o he P sys em and using a co esponding model-checke .
Among he ools used we men ion: Maude [2], he p obabilis ic model-checke
P ism [24, 4], he symbolic model e i ie NuSMV [17], he Spin [19, 18] and P oB
model checke s [16].
P ope y ex ac ion using Daikon, a dynamic in a ian de ec o , and u he
e i ica ion o he P sys ems was ackled in [4, 15]. In [4] a simple example in ol -
ing a egula o y ne wo k is p esen ed, along wi h he p ope ies (p econdi ions,
pos condi ions, in a ian s) in e ed by Daikon. The ela ionships disco e ed e-
ga d he bounda ies o he numbe o objec s (e.g., 0 ≤p o ≤205) and ela ions
be ween objec s, such as na < o ig( ep) o ( na = 0) →(p o = 0). They ha e
been checked using he P ism p obabilis ic model checke . In [15] simple cell like
P sys ems ha e been used and in a ian s like 2∗c−d= 0, (b= 0) →(o ig(b) = 0)
ha e been ob ained and u he e i ied using NuSMV.
In his pape , we p opose an in eg a ed me hodology o modelling, simula ion,
analysis, p ope y ex ac ion (in a ian de ec ion) and e i ica ion h ough model
checking o P sys ems. The app oach in eg a es a modelling and simula ion en i-
onmen (P–Lingua and MeCoSim) wi h model checke s, p ope y ex ac ion ools
(Daikon) and ools o ma hema ical and symbolic calculus (Maple). App op ia e
in eg a ion ools (plugins) ha e also been de eloped (see Fig. 1).
S a ing om a p oblem, his p ocess in ol es: he modelling o he p oblem by
means o P sys ems, he model ansc ip ion in o a language like P–Lingua [14],
unde s andable by a machine; he de ini ion o a isual in e ace, o en e he
needed inpu s and show he desi ed ou pu s om he compu a ion; he simula-
ion o he model unde di e en ini ial pa ame e s; he da a ex ac ion om he
simula ion; he in a ian s de ec ion om he ex ac ed da a, and he analysis and
e i ica ion o he de ec ed p ope ies. A de ailed desc ip ion o he me hodology
has been p o ided in sec ion 4, applying he p ocess un il he in a ian s de ec ion
o he 3–Col p oblem in subsec ion 4.4. The app oach is illus a ed on a case
s udy in ol ing he issue P sys em model o he well-known 3-colou ing (3-Col)
p oblem [10, 12].
The pape is s uc u ed as ollows. We s a by p esen ing in Sec ion 2 he
no a ion and main concep s o be used in he pape . Sec ion 3 p esen s a se o
ini ial p ope ies o he 3-colou ing p oblem, which ha e been e i ied using Spin
and P oB model checke . In he nex wo sec ions a e p esen ed a me hodology
o p ope ies ex ac ion (in a ian s de ec ion), i s in eg a ion wi h he MeCoSim

An In eg a ed App oach o Model Simula ion o P Sys ems 293
pla o m [20] and he empi ical de ec ion and alida ion o addi ional p ope ies.
Finally, conclusions a e d awn in Sec ion 6.
{ Simula ion }
MeCoSim
P-lingua model
Simula ion
aces
{ Ve i ica ion }
Spin
{ In a ian de ec o }
Daikon
P omela
model
P ope y
ex ac ion cycle
P ope ies
Fig. 1. Me hodology O e iew
2 Backg ound
Be o e p esen ing ou app oach, le us es ablish he no a ions used and de ine he
class o P sys ems add essed in he pape .
Gi en a ini e alphabe V={a1, ..., ap}, a mul ise is ei he deno ed by a
s ing u∈V∗(in which he o de is no impo an , he s ing no a ion is only
used as a con en ion), o by an associa ed ec o o non-nega i e in ege s, ΨV(u) =
(|u|a1, ..., |u|ap), whe e |u|aideno es he numbe o aioccu ences in u, o each
1≤i≤p.
The ollowing de ini ion e e s o a model o issue P sys ems wi h cell di ision,
in oduced in [22]. This model can be seen as a ne wo k o cells, whose s uc u e
is no s a ic: i is inspi ed by he way cells a e duplica ed in a na u al way ia
mi osis.
De ini ion 1. Fo mally, a issue P sys em wi h cell di ision o deg ee q≥1is a
uple o he o m
Π= (Γ, w1, . . . , wq, ε, R, i0),
whe e:
1. q≥1is he ini ial deg ee o he sys em; he sys em con ains qcells, labelled
wi h 1,2, . . . , m;0 ep esen s he en i onmen .
294 R. Le ica u e al.
2. Γis a ini e alphabe (called wo king alphabe ), whose symbols will be called
objec s.
3. w1, . . . , wqa e s ings o e Γ, desc ibing he mul ise s o objec s placed in he
qcells o he sys em.
4. ε⊆Γis he se o objec s p esen in he en i onmen in a bi a ily many
copies each.
5. Ris a ini e se o de elopmen al ules o he ollowing o m:
a) Communica ion ules: (i, u/ , j), o i, j ∈ {0,1,2, . . . , q},i6=j,u, ∈Γ∗.
When applying a ule (i, u/ , j), he objec s o he mul ise ep esen ed by
ua e sen om egion i o egion jand simul aneously he objec s o he
mul ise a e sen om egion j o egion i.
b) Di ision ules: [a]i→[b]i[c]i, whe e i∈ {1,2, . . . , q}and a, b, c ∈Γ. The
cell wi h label iis di ided in wo cells wi h he same label; in he i s copy
he objec ais eplaced by b, in he second copy he objec ais eplaced by
c; all o he objec s a e eplica ed and copies o hem a e placed in he wo
new cells.
6. i0∈ {0,1,2, . . . , q}deno es he ou pu egion (which can be he egion inside
a memb ane o he en i onmen ).
Rules a e applied as usual in a maximally pa allel way, wi h only one es ic ion:
when a cell is di ided, he di ision ule is he only one which is applied o ha
cell in ha s ep; he objec s inside ha cell do no e ol e in ha s ep.
This class o P sys ems can be u he ex ended, o sol ing NP-comple e p ob-
lems, o ecognize P sys ems. A ecognize issue P sys em wi h cell di ision is
a uple (Γ, Σ, w1, . . . , wq, ε, R, iin, i0), which has, in addi ion o a issue P sys em
wi h cell di ision:
•Two dis inguished objec s yes, no ∈Γ, p esen in a leas one copy in
w1, w2, . . . , wq, bu no p esen in ε.
•An inpu alphabe Σs ic ly con ained in Γ.
•An inpu cell iin ∈ {1, . . . , q}.
Also, i mus sa is y he ollowings:
•The ou pu egion i0is he en i onmen .
•All compu a ions hal .
•I Cis a compu a ion o Π, hen ei he he objec yes o he objec no (bu
no bo h) mus ha e been eleased in o he en i onmen , and only in he las
s ep o he compu a ion.
3 Ve i ying a i s se o p ope ies o he 3-colou ing
p oblem
In his sec ion we will in oduce a simpli ied e sion o a issue P sys em sol ing he
3-colou ing p oblem, we will p esen some o i s p ope ies and hei e i ica ion
using he Spin and P oB model checke s.
An In eg a ed App oach o Model Simula ion o P Sys ems 295
3.1 A P sys em o he 3-colou ing p oblem
In o de o illus a e ou app oach ega ding p ope y ex ac ion and e i ica ion
o P sys ems, we ha e conside ed a simpli ied e sion o he 3-colou ing p oblem
om [9, 11].
The k-colou ing p oblem is o mula ed as ollows: gi en an undi ec ed g aph
G= (V, E), decide whe he o no Gis k-colou able; ha is, i he e exis s a alid
k-colou ing o G( o e e y edge {u, } ∈ E he colou s o uand a e di e en ).
The 3-colou ing p oblem can be sol ed in linea ime by a amily o ecognize
issue P sys ems wi h cell di ision [9]. The solu ion p oposed in [9] is using a b u e
o ce algo i hm, in he amewo k o ecognize issue P sys ems wi h cell di ision,
which consis s o 4 s ages:
1. Gene a ion S age: an ini ial cell, labelled by 2, is di ided in o wo new cells;
his p ocess is epea ed un il all possible candida e solu ions o he p oblem
a e gene a ed (one solu ion o each memb ane).
2. P echecking S age: a e ob aining all possible 3-colou ings (in cells labelled
by 2), addi ional objec s a e gene a ed in he cells, o e e y edge o he g aph.
3. Checking S age: i is e i ied i he e exis s a pai o adjacen e ices in he
g aph, wi h he same colou in he co esponding candida e solu ion.
4. Ou pu S age: he sys em sends o he en i onmen he igh answe acco ding
o he esul s o he p e ious s age (yes o no).
As we will ocus in he es o he pape only on he p ope ies om he
Gene a ion S age, we will omi om he ecognize P sys em model gi en in [9]
some ules and objec s, which would hinde he unde s anding o he mechanism.
Mo e p ecisely, we will conside only he di ision ules and a es ic ed se o
objec s, so we can de ine he model using he basic class o issue P sys ems wi h
cell di ision. Howe e , o a comple e speci ica ion in e ms o a amily o ecognize
issue P sys ems, [9] can be consul ed.
Le Π(n) = (Γ(n), w1, w2(n), ε, R(n), i0) be a amily o issue P sys ems wi h
cell di ision o deg ee 2, whe e:
1. Γ(n) = {Ai, Ri, Ti, Bi, Gi: 1 ≤i≤n}
2. w1=∅,w2(n) = {A1, . . . , An}
3. R(n) is a se o di ision ules:
• 1,i ≡[Ai]2→[Ri]2[Ti]2 o i= 1, . . . , n
• 2,i ≡[Ti]2→[Bi]2[Gi]2 o i= 1, . . . , n
In his model Aiencodes he i- h e ex o he g aph; Ri, Bi, Gi ep esen he
h ee colou s ed, blue, g een. Appendix A p esen s wo examples o compu a ion
o Π(2) and Π(3). I can be obse ed ha , a e app op ia e di isions, in he
s ep 2nwe ge exac ly 3ncells encoding all he possible 3-colou ings o he g aph
ha ing e ices A1, . . . ,An. Appendix B p esen s he numbe o cells labelled 2 a
each compu a ion s ep o 2 ≤n≤11, simula ion esul s which help us o mula e
some in e es ing p ope ies.
296 R. Le ica u e al.
Gi en he amily o P sys ems Π(n) p e iously de ined, an ini ial se o p op-
e ies ha e been iden i ied manually (wi hou using p ope y ex ac ion ools):
P1Fo each compu a ion Co he P sys em Π(n), he e a e 3ncells labelled wi h
2 a con igu a ion C2n.
P2Fo each compu a ion Co he P sys em Π(n), he con igu a ion Cn+1 has
exac ly 2n+1 −1 cells labelled 2.
P3Fo each compu a ion Co he P sys em Π(n), o each 0 ≤j≤n he con ig-
u a ion Cjhas exac ly 2jcells labelled 2.
P4Fo each combina ion (X1, X2, . . . , Xn), Xi∈ {Ri, Gi, Bi},i= 1 . . . n, he e
exis s, a con igu a ion C2n, one and only one cell labelled by 2 ha con ains
he mul ise {X1, X2, . . . , Xn}.
3.2 3-Col p ope y e i ica ion using model checking
This ini ial se o p ope ies is now e i ied using wo model checke s, Spin and
P oB.
Spin is a model checke widely used in indus ies ha build c i ical sys ems
and is conside ed one o he mos powe ul model checke s a ailable [3]. I is de-
signed o modelling and e i ying concu en and dis ibu ed sys ems speci ied
in P omela (P ocess o P o ocol Me a Language), a e i ica ion modelling lan-
guage. The p ope ies o be e i ied can be exp essed in LTL o by using asse ion
s a emen s.
P oB is an anima ion and model checking ool in eg a ed wi hin he Rodin pla -
o m, which accep s E en -B models [1]. Unlike mos model checking ools, P oB
wo ks on highe -le el o malisms and so i enables a mo e con enien modelling.
Besides e i ica ion o p ope ies (exp essed using he LTL o he CTL o malism),
i also p o ides anima ion acili ies, allowing o isualize, a any momen , he s a e
space o o execu e a gi en numbe o ope a ions.
P ope y e i ica ion using Spin
As explained in [18], he execu able speci ica ion o he Spin model checke
associa ed o a P sys em will con ain ex a s a es and a iables, co esponding
o in e media e s eps, which ha e no co espondence in he P sys em con igu a-
ions. Fo his eason, he p ope ies o be e i ied, ha e e o he P sys em,
need o be e o mula ed as equi alen LTL o mulas o he associa ed P omela
implemen a ion.
The P sys em p ope ies e i ied in his sec ion a e o he o m ‘G (φ→ψ)’,
and he equi alen LTL o mula o he P omela model is ‘[](!φ||ψ|| !pInS)’, as
o mally p o en in [19] (pInS is used o exp ess i he cu en con igu a ion in he
P omela model ep esen s also a s a e in he P sys em).
In ou expe imen s, we ha e used one P omela speci ica ion ile o each pa -
icula P sys em Π(n), o all n∈ {2,...,9}because hese iles we e (semi-)
An In eg a ed App oach o Model Simula ion o P Sys ems 297
au oma ically gene a ed om each co esponding ins ance o P–Lingua de ini ion
ile using he plinguaco e lib a y. We ha e success ully simula ed all hese models
wi h Spin, o n∈ {2,...,9}; ypical s a e explosion p oblems appea ed when we
a emp ed o e i y he p ope ies men ioned ea lie o n≥4.
Fo n∈ {2,3}we ha e e i ied all he o mulas p esen ed bellow, using also
some echniques ha Spin p o ides o educe he memo y use. S a ing wi h n= 4,
we ob ained ou o memo y o p ope ies ha in ol ed checking many s eps o
he compu a ions.
The p ope ies e i ied, exp essed as LTL o mulas o he Spin model checke ,
o n= 3, a e:
P1:[] ((!(noO S eps == 6) || (noO Cells == 27) || (!pInS))
P2:[] ((!(noO S eps == 4) || (noO Cells == 15) || (!pInS))
P3:[] ((!(noO S eps >= 0 && noO S eps <= 3)) || (noO Cells ==
pow2noO S eps)|| (!pInS)), whe e pow2noO S eps is a a iable which com-
pu es 2noO S eps
P4:This p ope y is ha d o e i y wi h Spin because o he complex ope a ions
in ol ed.
P ope y e i ica ion using P oB
The E en -B model o a P sys em wi h ac i e memb anes can be speci ied
using wo unc ions cell and cellp, ep esen ing he numbe o objec s o each ype
con ained in e e y cell and he numbe o objec s p oduced be ween wo s eps
o maximal pa allelism, espec i ely. The ules a e ep esen ed by e en s. Each
di ision ule adds a cell o he domain o hese unc ions. Addi ionally, a special
e en called upda e, enabled a e each s ep o maximal pa allelism, is used o add
each alue o cellp o cell and o ese all he alues o cellp o 0. Mo e de ails can
be ound in [6].
Fi s , we de eloped an E en -B model, using he Rodin pla o m, o each pa -
icula P sys em Π(n), wi h n∈ {2,3,4}. Then, he possibili y o use quan i ie s
in P oB allowed us o de elop a gene al E en -B model o he amily o P sys ems
Π(n), ha has been ins an ia ed o pa icula alues o n. We anima ed he mod-
els in o de o see how he sys em e ol es and we e i ied hei p ope ies using
he model checke P oB. Unlike he P omela speci ica ion, whe e he numbe o
cells wi h label 2 was inc emen ed a e each di ision ule, we could speci y he
p ope ies P1-P4neglec ing he ex a in e media y) s a es; his is because hese
p ope ies e e only o he numbe o s eps o maximal pa allelism in he e olu-
ion o Π(n) (coun ed by a a iable called noO S eps) and o he numbe o cells
wi h label wo (coun ed by a a iable called noO Cells), whose alues a e modi ied
only in he upda e e en . On he o he hand, we used an addi ional s a e, Hal , o
ma k inal con igu a ions. Howe e , we we e able o e i y all he p ope ies only
o n∈ {2,3,4}; o n= 5, due o he s a e explosion p oblem, he model checke
c ashed wi h an ou o memo y e o be o e eaching he inal con igu a ion (a e

298 R. Le ica u e al.
p oducing 196 cells labelled 2). Consequen ly, o n > 5, we could e i y only some
simple p ope ies, ha do no in ol e e minal con igu a ion (e.g. P3 o small
alues o j).
The p ope ies we e speci ied using LTL as ollows:
P1:G{s a e =Hal ⇒noO S eps = 2 ∗n&noO Cells = 3n}
P2:G{noO S eps =n+ 1 ⇒noO Cells = 2n+1 −1}
P3:G{!j.j >= 0 &j <=n&noO S eps =j⇒noO Cells = 2j}
P4:We we e able o e i y P4spli ing i in wo p ope ies, he i s one o he
exis ence and he second one o he unici y. Fo n= 2 hese p ope ies we e
o mula ed as ollows:
– Fo all x, y symbols in {Ri, Gi, Bi},i∈ {1,2}, he e exis one cell in he
inal con igu a ion ha con ains one xand one y:
G{s a e =Hal ⇒(!x, y.x :{R1, R2, G1, G2, B1, B2}&y:{R1, R2, G1, G2,
B1, B2}& (x/ =y)⇒(#c.c :dom(cell) & cell(c)(x) = 1 & cell(c)(y) = 1))}
– In he inal con igu a ion any wo di e en cells c1, c2 ha e di e en con-
en s:
G{s a e =Hal ⇒(!c1, c2.c1 : dom(cell) & c2 : dom(cell) & (c1/=c2) ⇒
(#s.s :{R1, R2, G1, G2, B1, B2}&cell(c1)(s)/=cell(c2)(s)))}
Fo highe alues o n he o mulas o hese p ope ies a e e y la ge and
o space conside a ions, we will omi hem.
He e “!”, “#” and “:” co espond o he uni e sal quan i ie “∀”, exis en ial
quan i ie “∃” and membe ship ope a o “∈”, espec i ely.
4 In eg a ing Daikon in MeCoSim and inding new ela ions
The nex s age in he p oposed me hodology is he au oma ic ex ac ion o new
p ope ies om simula ion aces. The main ools used (MeCoSim and Daikon)
and hei in eg a ion a e p esen ed nex . The p ocess o iden i ying new p ope ies,
b oken down in a numbe o indi idual s eps, is also desc ibed.
4.1 Modelling and o maliza ion
Once he 3-Col p oblem has been s udied and modelled by means o P sys ems, his
model mus be exp essed in a language ha may be unde s ood by a simula ion
machine. Fo his pu pose, he s anda d P–Lingua [14] language has been chosen
as ou modelling language. The ini ial mul ise s and ules a e exp essed as ollows:
@ms(2) += A{i} : 1<=i<=n;
/* 1 */ [A{i}]’2 --> [R{i}]’2 [T{i}]’2 : 1<=i<=n;
/* 2 */ [T{i}]’2 --> [B{i}]’2 [G{i}]’2 : 1<=i<=n;
An In eg a ed App oach o Model Simula ion o P Sys ems 299
The ull code o he model in P–Lingua is showed in Appendix C. This ile
will se e as an inpu o MeCoSim, so ha he P sys em can be simula ed, ana-
lyzed and debugged, and in a ian s can be de ec ed, o be hen e i ied by model
checking.
4.2 MeCoSim
In o de o p o ide an in eg a ed me hodology o model simula ion, p ope ies
ex ac ion and e i ica ion, we need an in eg a ed en i onmen o simpli y he
use ’s p ocess.
In his sense, a gene al pu pose memb ane compu ing simula o , MeCoSim
[20], was p o ided. I was ini ially designed o enable he use de ined cus omized
in e aces, wi h inpu s, ou pu s, cha s, e c., adap ed o each amily o P sys ems.
This pe mi s en e ing da a o di e en ini ial condi ions, ins an ia ing di e en P
sys ems o he amily.
The ini ial aim o his so wa e en i onmen has been ex ended such ha i can
co e a mo e gene al se o applica ions by p o iding lexible and powe ul me hods
o in eg a e a ious so wa e applica ions and packages as MeCoSim plugins. These
kind o plugins can be easily added o MeCoSim by se ing app op ia e pa ame e s
in a con igu a ion ile. Keeping in mind his a chi ec u e and he de eloped plugins,
MeCoSim may p o ide a pla o m o he in eg a ion o di e en ools o he
modelling, simula ion, analysis, p ope y ex ac ion and e i ica ion o P sys ems.
Some o his ools ha e al eady been de eloped and/o in eg a ed, o he s a e being
de eloped, and many o he could be added in a simila way.
To ake ad an age o his amewo k o s udying he 3-Col p oblem, we need
o de ine ou cus omized inpu s, ou pu s, ex ac ions, e c. The main s eps o his
p ocess a e illus a ed in he nex pa ag aphs.
MeCoSim pe mi s se ing he hie a chy o abs o be shown in he isual use
in e ace, including inpu and ou pu ables inside each lea ab. In ou case, we
di ide he in o ma ion in wo abs, Inpu and Ou pu (plus an addi ional ab,
Debug console, p o ided by de aul in MeCoSim, used o debugging he models).
Fo ou example, we only need one inpu pa ame e , n, so one inpu able is de ined
inside he ab Inpu , as showed in Fig. 2.
Now he simula ion could be pe o med, in such a way ha n akes he alue
om he inpu able, his pa ame e complemen s he P–Lingua ile o ins an ia e
he ini ial con igu a ion o he P sys em and he compu a ion s eps un un il
a hal ing condi ion is eached. In he debug console, we can un s ep by s ep,
looking a all he objec s o he mul ise s inside each cell and compa men , o
each compu a ion s ep.
Howe e , his p ocess could be e y slow i we a e in e es ed in se e al objec s,
memb anes o s eps, so we need a way o de ine cus omized ou pu s showing
he desi ed in o ma ion only. MeCoSim p o ides his mechanism, and we de ine
ou pu s as shown in Fig. 3 o s udy di e en issues: en i e con igu a ions, objec s
pe memb ane, objec s by ype (R, G, B), numbe o cells, e c.
300 R. Le ica u e al.
Fig. 2. MeCoSim window - Inpu ab: alue o n
Tabs
Tables
Resul s
Ex ac ions
Fig. 3. MeCoSim window - Ou pu ab: numbe o cells by s ep
E en ually, as we see in Fig. 3, we se he in o ma ion o be ex ac ed o
Daikon. This makes up he ou pu iles om he simula ion, o be used as an
inpu o he MeCoSim Daikon plugin.
4.3 MeCoSim new plugin o Daikon in eg a ion
MeCoSim p o ides an easy way o add plugins, en iching he de aul unc ionali y.
Taking ad an age o his a chi ec u e, a new plugin has been de eloped o in eg a e
Daikon wi h MeCoSim.
Daikon [13] is a ool which dynamically de ec s p og ams in a ian s, based on
hei execu ion aces. I can disco e p ope ies om C, C++, Ei el, Ja a, o
Pe l p og ams, om sp eadshee iles and o he da a sou ces. The usual ope a ion
o Daikon is he ollowing: i ecei es da a ace iles abou he alues o some
a iables ac oss a sequence o s eps om he execu ion o a p og am, and ies
o de ec p ope ies o ypes: p econdi ion, pos condi ion and in a ian . We a e
mainly in e es ed in he las one, bu he p e ious ones could be also use ul o
checking he co ec ness o he models.
An In eg a ed App oach o Model Simula ion o P Sys ems 301
As pa o he p oposed me hodology, we aim o in eg a e his ool wi h
MeCoSim, so in a ian s could be de ec ed om he desi ed ou pu s o he simu-
la ion. Fo his pu pose, a plugin has been de eloped; an o e iew o he en i e
(simula ion and p ope y ex ac ion) p ocess in ol es he ollowing s eps:
1. The model o he P sys em (w i en in P–Lingua, possibly pa ame ized) is
loaded in MeCoSim.
2. The ini ial pa ame e s (nin he case o he 3–Col p oblem) o ins an ia ing
he speci ic P sys em a e p o ided by he use in a isual way h ough he
inpu ables.
3. The simula ion uns, gene a ing ex ac ion iles o he ou pu s p e iously se .
4. The plugin can be called om a menu op ion (Plugins >“Daikon”).
When he plugin is launched om MeCoSim, a window wi h a lis ing o a ail-
able ex ac ion iles is isualized, as showed in igu e 4.
Fig. 4. MeCoSim window - Daikon plugin - File selec ion
Once one o he ex ac ion iles is selec ed, he Daikon plugin uns om his
inpu ile. I au oma ically eads he simula ion ex ac ion ile, gene a es he aces
in he app op ia e o ma and launches Daikon om his aces ile, ying o
de ec as many in a ian s as possible. They a e e en ually isualized, as showed
in igu e 5.
Fu he echnical de ails conce ning he Daikon plugin and i s in eg a ion in o
MeCoSim is p o ided in Appendix D.
4.4 Me hodology
In he p e ious sec ions, di e en ools and languages o modelling, simula ion,
e i ica ion and in a ian de ec ion ha e been explained. In o de o in eg a e he
di e en ools in a sys ema ic way, a no el me hodology has been de ised, as
ou lined in Fig. 1.
In he i s s age, we need o model he p oblem, w i ing he model in P–
Lingua, P omela and o he languages, in o de o se e as he de aul inpu o
Spin and o he possible model checke s.
308 R. Le ica u e al.
•expand(exp ): command used o dis ibu e p oduc s o e sums in he gi en
exp ession.
Fo k= 4, using g ind ecu (4,[(81 −28),(227 −29),(585 −210),(1403 −
211),(3185 −212),(6947 −213),(14729 −214),(30619 −215)], , ) we ob ained he
ecu ence: ( )=4∗ ( −1) −6∗ ( −2) + 4 ∗ ( −3) − ( −4). Sol ing his
ecu ence wi h he command sol e({ ( ) = 4∗ ( −1)−6∗ ( −2)+4∗ ( −3)−
( −4), (4) = −175, (5) = −285, (6) = −439, (7) = −645}, (n)) we ob ain
he gene al e m: −19+12(n+1) 1
2n+ 1−14n−8(n+1) 1
2n+ 11
3n+ 1,
and expanding i : −15 −32
3n−2n2−4
3n3. Adding 2n+4 o his gene al e m we
ob ain he numbe o cells wi h label 2 in he con igu a ion n+ 4. The ecu ences
om Table 2 ha e been ob ained simila ly.
Con igu a ion Recu ence
n+ 2 ( ) = 2 · ( −1) − ( −2)
n+ 3 ( ) = 3 · ( −1) −3· ( −2) + ( −3)
n+ 4 ( ) = 4 · ( −1) −6· ( −2) + 4 · ( −3) − ( −4)
n+ 5 ( ) = 5 · ( −1) −10 · ( −2) + 10 · ( −3) −5· ( −4) + ( −5)
n+ 6 ( ) = 6 · ( −1) −15 · ( −2) + 20 · ( −3)
−15 · ( −4) + 6 · ( −5) − ( −6)
Table 2. Recu ence o he numbe o cells wi h label 2 in con igu a ion n+k
We no ice ha all hese ecu ences ha e, o each con igu a ion n+k, he
ollowing o m: ( ) =
k
P
i=1
(−1)i+1 ·Ci
k· ( −i). Un o una ely, he cu en limi-
a ions o ou ools (n < 20) does no allow us o con inue he calculus wi h he
nex s eps and so we canno es ablish unequi ocally i he ecu ences ha e he
same o m o la ge alues o k.
Sol ing hese ecu ences (using sol e), expanding he exp essions (wi h ex-
pand) and adding 2n+k, we ob ain he numbe o cells wi h label 2 in he con ig-
u a ion n+k, o k∈ {2,3,4,5,6}, as p esen ed in Table 3.
All hese o mulas we e e i ied using Algo i hm 2 om Appendix E o n < 20.
As we can see om Table 3 he coe icien o he polynomial ha ollows 2n+ka e
a ional bu we could no es ablish any u he ule o hem excep he ac ha ,
in each case, he ee e m is −2k+ 1.

An In eg a ed App oach o Model Simula ion o P Sys ems 309
Con igu a ion Numbe o cells
n+ 2 2n+2 −2n−3
n+ 3 2n+3 −2n2−4n−7
n+ 4 2n+4 −4
3·n3−2·n2−32
3·n−15
n+ 5 2n+5 −2
3·n4−28
3·n2−20 ·n−31
n+ 6 2n+6 −4
15 ·n5+2
3·n4−20
3·n3−32
3·n2−676
15 ·n−63
Table 3. Numbe o cells wi h label 2 in con igu a ion n+k
6 Conclusion
In his pape , we ha e ou lined an in eg a ed me hodology o P sys em o mal e -
i ica ion, comp ising modelling using P–Lingua, simula ion wi h MeCoSim, p op-
e ly ex ac ion using Daikon and model checking using ools such as Spin and
P oB. A plugin which allows Daikon o be called and used wi hin MeCoSim has
been de eloped and a (semi)-au oma ic P omela implemen a ion has been gene -
a ed om he P–Lingua model. A numbe o s eps in ol ed in p ope y ex ac ion
using Daikon ha e been iden i ied and he whole p ocess has been illus a ed wi h
an example, a issue P sys em model o he 3-colou ing p oblem; his is a complex
p oblem since, by using ac i e memb anes (cell di ision), he numbe o cells g ows
exponen ially. As some o he sough p ope ies ha e p o ed o be qui e complex
and could no be di ec ly ex ac ed using Daikon, a ool o ma hema ical and
symbolic calculus (Maple) has been used o supplemen ou me hodology.
Fu he wo k in ol es he de elopmen o comple ely in eg a ed en i onmen
o au oma ic modelling, simula ion and e i ica ion o P sys ems as well as ap-
plying he p oposed me hodology o o he , mo e complex, P sys ems.
Acknowledgmen
The wo k o MG, FI and RL was suppo ed by a g an o he Romanian Na ional
Au ho i y o Scien i ic Resea ch, CNCS–UEFISCDI, p ojec numbe PN-II-ID-
PCE-2011-3-0688. The au ho s MPJ and LVC acknowledge he suppo o he
p ojec TIN2009–13192 o he Minis e io de Ciencia e Inno aci´on o Spain, co-
inanced by FEDER unds, and he suppo o he P ojec o Excellence wi h
In es igado de Reconocida Val´ıa o he Jun a de Andaluc´ıa, g an P08–TIC–
04200. AT was suppo ed by p ojec POSDRU - “Dez ol a ea ¸scolilo doc o ale
p in aco da ea de bu se ine ilo doc o anzi cu ec en ¸˘a”- 88/1.5/S/52826.
310 R. Le ica u e al.
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Appendices
A Compu a ion examples o Π(n), n∈ {2,3}
C . No. o cells Cu en con igu a ion
s ep labelled 2 (only cells labelled wi h 2)
0 1 [A1A2]2
1 2 [R1A2]2[T1A2]2
2 4 [R1R2]2[R1T2]2[B1A2]2[G1A2]2
3 7 [R1R2]2[R1B2]2[R1G2]2[B1R2]2[B1T2]2
[G1R2]2[G1T2]2
4 9 [R1R2]2[R1B2]2[R1G2]2[B1R2]2[B1B2]2
[B1G2]2[G1R2]2[G1B2]2[G1G2]2
Table 4. Compu a ion example o Π(2)
C . No. o cells Cu en con igu a ion
s ep labelled 2 (only cells labelled wi h 2)
0 1 [A1A2A3]2
1 2 [R1A2A3]2[T1A2A3]2
2 4 [R1R2A3]2[R1T2A3]2[B1A2A3]2[G1A2A3]2
3 8 [R1R2R3]2[R1R2T3]2[R1B2A3]2[R1G2A3]2
[B1R2A3]2[B1T2A3]2[G1R2A3]2[G1T2A3]2
4 15 

[R1R2R3]2[R1R2B3]2[R1R2G3]2[R1B2R3]2[R1B2T3]2
[R1G2R3]2[R1G2T3]2[B1R2R3]2[B1R2T3]2[B1B2A3]2
[B1G2A3]2[G1R2R3]2[G1R2T3]2[G1B2A3]2[G1G2A3]2


5 23






[R1R2R3]2[R1R2B3]2[R1R2G3]2[R1B2R3]2[R1B2B3]2
[R1B2G3]2[R1G2R3]2[R1G2B3]2[R1G2G3]2[B1R2R3]2
[B1R2B3]2[B1R2G3]2[B1B2R3]2[B1B2T3]2[B1G2R3]2
[B1G2T3]2[G1R2R3]2[G1R2B3]2[G1R2G3]2[G1B2R3]2
[G1B2T3]2[G1G2R3]2[G1G2T3]2






6 27








[R1R2R3]2[R1R2B3]2[R1R2G3]2[R1B2R3]2[R1B2B3]2
[R1B2G3]2[R1G2R3]2[R1G2B3]2[R1G2G3]2[B1R2R3]2
[B1R2B3]2[B1R2G3]2[B1B2R3]2[B1B2B3]2[B1B2G3]2
[B1G2R3]2[B1G2B3]2[B1G2G3]2[G1R2R3]2[G1R2B3]2
[G1R2G3]2[G1B2R3]2[G1B2B3]2[G1B2G3]2[G1G2R3]2
[G1G2B3]2[G1G2G3]2








Table 5. Compu a ion example o Π(3)
An In eg a ed App oach o Model Simula ion o P Sys ems 313
B Numbe o cells labelled wi h 2
nNumbe o cells labelled 2 a each con igu a ion ( om 0 o 17)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
2 1 2 4 7 9
3 1 2 4 8 15 23 27
4 1 2 4 8 16 31 53 73 81
5 1 2 4 8 16 32 63 115 179 227 243
6 1 2 4 8 16 32 64 127 241 409 585 697 729
7 1 2 4 8 16 32 64 128 255 495 891 1403 1867 2123 2187
8 1 2 4 8 16 32 64 128 256 511 1005 1881 3185 4673 5857 6433 6561
9 1 2 4 8 16 32 64 128 256 512 1023 2027 3891 6947 11043 15203 18147 19427
10 1 2 4 8 16 32 64 128 256 512 1024 2047 4073 7945 14729 24937 37289 48553
11 1 2 4 8 16 32 64 128 256 512 1024 2048 4095 8167 16091 30619 54395 87163
12 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8191 16357 32425 62801 115633
13 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16383 32739 65139 127651
14 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32767 65505 130617
15 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65535 131039
16 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131071
17 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072
18 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072
19 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072
Table 6. Numbe o cells labelled 2 a each con igu a ion. S eps 0 o 17.
nNumbe o cells labelled 2 a each con igu a ion ( om 18 o 25)
18 19 20 21 22 23 24 25
2 - 8
9 19683
10 55721 58537 59049
11 123131 152827 169979 176123 177147
12 195953 297457 399089 475633 516081 529393 531441
13 241235 427219 689363 994003 1273811 1467347 1561555 1590227
14 257929 496537 909689 1543801 2372729 3261817 4014969 4496249
15 261627 519163 1012395 1902763 3363179 5460331 8007275 10538603
16 262109 523705 1042417 2050721 3927553 7168705 12186689 18927937
17 262143 524251 1047923 2089827 4135555 8028995 15023811 26524099
18 262144 524287 1048537 2096425 4185673 8315209 16300105 31081801
19 262144 524288 1048575 2097111 4193499 8378523 16686555 32930523
Table 7. Numbe o cells labelled 2 a each con igu a ion. S eps 18 o 25.

314 R. Le ica u e al.
nNumbe o cells labelled 2 a each con igu a ion
26 27 28 29 30 31 32
2 - 12
13 1594323
14 4713337 4774777 4782969
15 12526187 13705835 14201451 14332523 14348907
16 26553153 33603393 38758209 41613121 42735425 43013953 43046721
17 43301315 64409027 86709699 105964995 119129539 125961667 128484803
18 56571721 96349513 151011657 215527753 279384393 331059529 364220745
19 63682011 118735323 209492955 343710683 517551067 710439899 889828315
Table 8. Numbe o cells labelled 2 a each con igu a ion. S eps 26 o 32.
nNumbe o cells labelled 2 a each con igu a ion
33 34 35 36 37 38
2 - 16
17 129074627 129140163
18 380408137 386044233 387289417 387420489
19 1026339803 1108849627 1146860507 1159377883 1161999323 1162261467
Table 9. Numbe o cells labelled 2 a each con igu a ion. S eps 33 o 38.
C P–Lingua model ile
The con en o he ile in P–Lingua o ma , con aining he speci ica ion o he
model, is shown below.
@model< issue_psys ems>
de main()
{
/* issue P sys em skele on */
call example_ issue(n);
}
de example_ issue(n)
{
call ini _cells();
call ini _mul ise s(n);
call ini _ ules(n);
}
de ini _cells()
{
@mu = [[]’2]’0;
}
de ini _ ules(n)
{
An In eg a ed App oach o Model Simula ion o P Sys ems 315
/* 1 */ [A{i}]’2 --> [R{i}]’2 [T{i}]’2 : 1<=i<=n;
/* 2 */ [T{i}]’2 --> [B{i}]’2 [G{i}]’2 : 1<=i<=n;
}
de ini _mul ise s(n)
{
@ms(2) += A{i} : 1<=i<=n;
}
D Daikon in eg a ion in MeCoSim - plugin and con ig iles
This appendix p esen s some echnical de ails abou he de eloped plugin, and he
simple p ocess o in eg a ion wi h MeCoSim.
The main code o he p og am is con ained in DaikonIn e ace.ja p og am,
ha ecei es an inpu ile wi h a compa ible o ma o Daikon in a ian s de ec o .
Howe e , an addi ional ja ile has been de eloped, DaikonPlugin.ja , o pe mi
selec ing among he di e en ex ac ion iles gene a ed by he simula o .
To implemen he in eg a ion o he p og am, he only wo k we ha e o do is
he addi ion o a ew lines in he ile plugins-p ope ies o MeCoSim, as ollows:
plugin-daikon = daikonPlugin.Main
pluginname-daikon = Daikon
pluginme hod-daikon = pluginHook
pluginpa am-daikon-1 = use iles/daikon- iles. x
pluginja -daikon-1 = DaikonIn e ace.ja
pluginja -daikon-2 = DaikonPlugin.ja
316 R. Le ica u e al.
E Algo i hms
Algo i hm 1 Calcula ing he numbe o memb anes
unc ion T iColo (n)
numbe CellsS ep[0..2n]
o i=0 do 2n-1
numbe CellsS ep[i]←0
end o
a←n
←0
s ep ←0
s ack.Push(a, , s ep)
numbe CellsS ep[s ep]←numbe CellsS ep[s ep] + 1
while no s ack.IsEmp y() do
a, , s ep ←s ack.Pop()
s ep ←s ep + 1
i > 1 hen
← −1
s ack.Push(a, , s ep)
s ack.Push(a, , s ep)
numbe CellsS ep[s ep]←numbe CellsS ep[s ep] + 1
else i = 1 hen
i a > 0 hen
← −1
s ack.Push(a, , s ep)
s ack.Push(a, , s ep)
numbe CellsS ep[s ep]←numbe CellsS ep[s ep] + 1
else
numbe CellsS ep[s ep]←numbe CellsS ep[s ep] + 1
end i
else i a > 0 hen
s ack.Push(a−1, , s ep)
s ack.Push(a−1, + 1, s ep)
numbe CellsS ep[s ep]←numbe CellsS ep[s ep] + 1
end i
end while
o i=1 do 2n-1
numbe CellsS ep[i]←numbe CellsS ep[i−1] + numbe CellsS ep[i]
end o
e u n numbe CellsS ep
end unc ion
An In eg a ed App oach o Model Simula ion o P Sys ems 317
Algo i hm 2 Tes ing he in a ian s
unc ion Fn2(n)
e u n 2∗ ∗(n+ 2) −2∗n−3
end unc ion
unc ion Fn3(n)
e u n 2∗ ∗(n+ 3) −2∗n∗ ∗2−4∗n−7
end unc ion
unc ion Fn4(n)
m1←n∗ ∗3∗4
1←m1 mod 3
m1←m1/3
m2←2∗n∗ ∗2
m3←32 ∗n
3←m3 mod 3
m3←m3/3
←( 1 + 3)/3
e u n 2∗ ∗(n+ 4) −m1−m2−m3− −15
end unc ion
unc ion Fn5(n)
m1←n∗ ∗4∗2
1←m1 mod 3
m1←m1/3
m2←n∗ ∗2∗28
2←m2 mod 3
m2←m2/3
m3←20 ∗n
←( 1 + 2)/3
e u n 2∗ ∗(n+ 5) −m1−m2−m3− −31
end unc ion

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