No Cycles in Compartments. Starting from Conformon-P Systems

No Cycles in Compa men s. S a ing om Con o mon-P Sys ems Pie luigi F isco1, Gheo ghe P˘aun2 1School o Ma hema ical and Compu e Sciences He io -Wa Uni e si y Edinbu gh, EH14 4AS, UK E-mail: [email p o ec ed] 2Ins i u e o Ma hema ics o he Romanian Academy PO Box 1-764, 014700 Bucha es , Romania and Depa men 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-mails: [email p o ec ed], [email p o ec ed] Summa y. S a ing om p oo s o esul s abou he compu ing powe o con o mon- P sys ems, we in e se e al esul s abou he powe o ce ain classes o issue-like P sys ems wi h (coope a i e) ew i ing ules used in an asynch onous way, wi hou cycles in compa men s. This las ea u e is ela ed o an impo an es ic ion appea ing when dealing wi h lab implemen a ions o P sys ems, ha o a oiding local e olu ion loops o objec s. 1 In oduc ion This no e add esses a echnical issue which appea ed in he amewo k o he ecen a emp o implemen a P sys em in biochemical e ms, a Technion in- s i u e, Hai a, Is ael, namely o a oiding cyclical e olu ion o chemicals in any compa men o he sys em – see a mo e p ecise desc ip ion o he p oblem in [9]. He e we conside a class o issue-like P sys ems, namely as in oduced in [16], wi h ew i ing ules p esen in memb anes, and wi h a ge indica ions o he o ms he e, go associa ed wi h he “p oduc s o eac ions”: ules o he o m u→ , whe e uand a e mul ise s o objec s and he objec s in ha e asso- cia ed a ge indica ions he e, go (ac ually, he e is omi ed) indica ing ha he espec i e objec emains in he same compa men o i has o go o any o he adjacen compa men s, non-de e minis ically choosing he des ina ion. We also conside an e olu ion-communica ion (EC) e sion o hese sys ems, ollowing he ideas o [1], i.e., using e olu ion ules wi hou a ge indica ions and using sepa- a e communica ion ules (o he o m (a, go), wi h he ob ious meaning: objec 158 P. F isco, Gh. P˘aun ais communica ed o any o he adjacen memb anes). In o de o ans e in a di ec way o hese sys ems esul s om con o mon-P sys ems a ea, we add o he de ini ion in [16] se e al “non-s anda d” ing edien s: we wo k asynch onously (in any s ep, in any compa men , a ule may be used o no ), maybe wi h a p io - i y ela ion among ules, o a global ype (in each compa men , e olu ion ules ha e p io i y o e communica ion ules: i an objec can e ol e and, a he same ime, communica ed, an e olu ion ule is applied i s ), an acknowledging mem- b ane ( he compu a ion s ops when any objec is sen o his memb ane, which is emp y in he beginning o he compu a ion). The numbe o memb anes we use is a bi a y ( a he high, i we ake in o accoun he numbe o memb anes used in con o mon-P sys ems simula ing egis e machines), bu , on he good side, he e olu ion ules we need o simula e a con o mon-P sys em a e o a e y es ic i e o m: each o he mul ise s u, om a ule u→ has exac ly wo objec s. Al hough, o he sake o eadabili y, we ecall he e he de ini ions o con o mon- P sys ems and o P sys ems wi h a g aph s uc u e, we do no en e in o de ails, and we assume he eade o be amilia wi h basic elemen s o memb ane compu - ing. Howe e , we indica e a se ies o pape s ela ed o con o mons. This concep was in oduced independen ly in [10] and [17]. Following he de ini ion gi en in [10] con o mons and con o mon-like en i ies ha e been classi ied in o 10 amilies ac- co ding o hei biological unc ions [12]. To know mo e abou he Bhopala o e e o [11, 13]. The e m con o mon was adop ed in [14, 15] whe e he au ho s s a ed o de elop a quan um mechanical heo y based on his concep . Con o mon-P sys ems ha e been in oduced in [3] and la e s udied, among o he s, in [4, 6]. Con o mon-P sys ems ha e also been success ully used as a pla o m o model biological p ocess. The in e es ed eade can e e o [8, 2, 7]. 2 Basic De ini ions Le Vbe an alphabe (a ini e se o abs ac symbols), and Nbe he se o na u al numbe s, including 0. A mul ise o e Vis a unc ion M:V−→ N∪ {+∞}. The suppo o M( he se o elemen s a∈V o which M(a)>0) is deno ed by supp(M) and he ca dinali y o M( he sum o mul iplici ies o all elemen s in supp(M)) is deno ed by |M|. 2.1 Con o mon-P Sys ems In wha ollows, a con o mon is an elemen o V×N, deno ed by [a, n]. We e e o aas he name o he con o mon [a, n] and o nas i s alue. Two con o mons can in e ac acco ding o an in e ac ion ule. An in e ac ion ule is o he o m ae →b, whe e a, b ∈Vand e∈N, and i says ha a con o mon wi h name acan gi e e om i s alue o he alue o a con o mon ha ing name b. I , o ins ance, he e a e con o mons [a, 5] and [b, 9] and he ule a3 →b, one applica ion his ule leads o [a, 2] and [b, 12]. As he e we conside ha he alue No cycles in compa men s 159 o a con o mon canno be a nega i e numbe , he ule a3 →bcanno be applied o [a, 2]. Each memb ane p esen in a con o mon-P sys em has associa ed a label, di e en om he labels o o he memb anes. These memb anes a e placed in he nodes o a di ec ed g aph, hence hey a e connec ed in a unidi ec ionally way. Each connec ion has associa ed a p edica e, which is an elemen o he se p ed(N) = {≥ n, ≤n|n∈N}. I , o ins ance, he e a e wo compa men s (wi h labels) m1and m2and he e is an connec ion om m1 o m2ha ing p edica e ≥4, hen con o mons ha ing alue g ea e han o equal o 4 can pass om m1 o m2. Acon o mon-P sys em is a cons uc Π= (V, µ, ωz, ack, L1, . . . , Lm, R1, . . . , Rm), whe e: Vis a ini e alphabe ; µ= (Q, E) is a di ec ed labelled g aph unde lying Π, whe e Q={1, . . . , m}is he se o memb anes (we also say compa men s) o Π; E⊆Q×Q×p ed(N) de ines di ec ed labelled edges be ween e ices, indi- ca ed by (i, j, p ed), i, j ∈Q, i 6=j, whe e p ed ∈p ed(N) is a p edica e; ωzwi h ω∈ {in, ou }and z∈Qindica es whe he Πis an accep ing (ω=in) o gene a ing (ω=ou ) de ice; he compa men zcon ains he inpu o ou pu , espec i ely; ack ∈Qindica es he acknowledging compa men ; Li: (V×N)→N∪{+∞}, i ∈Q, a e mul ise s o con o mons ini ially associa ed wi h he e ices in Q; Ri, i ∈Q, a e ini e se s o in e ac ion ules associa ed wi h he e ices in Q, wi h supp(Lack) = ∅. Le Miand Ribe he mul ise o con o mons and he se o ules, espec i ely, associa ed wi h he compa men i∈Q. Two con o mons p esen in compa men ican in e ac acco ding o a ule in Risuch ha he mul ise o con o mons Mi changes in o M0 i. I , o ins ance, [a, p],[b, q]∈Mi, a e →b∈Riand p≥e, hen M0 i= (Mi− {[a, p],[b, q]})∪ {[a, p −e],[b, q +e]}. A con o mon [a, p] p esen in compa men ican pass o compa men ji (i, j, p ed)∈Eand p ed(p) holds. This passage changes he mul ise s o con o - mons Miand Mjin o M0 iand M0 j, espec i ely, such ha M0 i=Mi− {[a, p]}and M0 j=Mj∪ {[a, p]}. A he momen we do no assume any equi emen (such as maximal pa al- lelism, p io i ies, e c.) on he applica ion o ope a ions. I a con o mon can pass o ano he compa men o in e ac wi h ano he con o mon acco ding o an in e ac- ion ule, hen one o he wo ope a ions o none o hem is non-de e minis ically chosen. 160 P. F isco, Gh. P˘aun The possibili y o ca y ou one o he wo allowed ope a ions in a compa - men o none o hem le s con o mon-P sys ems o be non-de e minis ic. Non- de e minism can also a ise om he con igu a ions o a con o mon-P sys em i in a compa men a con o mon can in e ac wi h mo e han one con o mon and also om he g aph unde lying Πi a compa men has edges wi h he same p edica e going o di e en compa men s. Acon igu a ion o Πis an m- uple (M1, . . . , Mm) o mul ise s o e V×N. The m- uple (L1, . . . , Lm), is called ini ial con igu a ion ( emembe ha supp(Lack) = ∅, so in he ini ial con igu a ion he acknowledging compa men does no con ain any con o mon) while any con igu a ion ha ing supp(Mack)6=∅is called inal con igu a ion. In a inal con igu a ion no ope a ion is pe o med e en i i could. Fo wo con igu a ions (M1, . . . , Mm),(M0 1, . . . , M0 m) o Πwe w i e (M1, . . . , Mm)⇒(M0 1, . . . , M0 m) indica ing a ansi ion om (M1, . . . , Mm) o (M0 1, . . . , M0 m), ha is, he applica ion o one ope a ion o a leas one con o - mon. In o he wo ds, in any con igu a ion in which supp(Lack) = ∅any con o mon p esen in a compa men can ei he in e ac wi h ano he con o mon p esen in he same compa men o pass o ano he compa men o emain in he same com- pa men unchanged. I no ope a ion is applied o a mul ise Mi, hen M0 i=Mi. The e lexi e and ansi i e closu e o ⇒is indica ed by ⇒∗. Acompu a ion is a ini e sequence o ansi ions be ween con igu a ions o a sys em Πs a ing om (L1, . . . , Lm). In case Πis an accep ing de ice (ω=in), hen he inpu is gi en by he numbe o con o mons (coun ed wi h hei mul iplici y) p esen in Lz. The inpu is accep ed by Πi i eaches a con igu a ion in which any con o mon is p esen in ack, hal ing in his way he compu a ion. Fo mally: N(Π) = {|Lz| | (L1, . . . , Lm)⇒∗(M0 1, . . . , M0 m)⇒(M1, . . . , Mm), supp(M0 ack) = ∅, supp(Mack)6=∅}. In case Πis a gene a ing de ice (ω=ou ), hen supp(Lz) = ∅. The esul o a compu a ion is gi en by Mzwhen any con o mon is p esen in ack. When his happens he compu a ion is hal ed and he numbe o con o mons (coun ed wi h hei mul iplici y) p esen in Mzde ines he numbe gene a ed by Π. Fo mally: N(Π) = {|Mz| | (L1, . . . , Lm)⇒∗(M0 1, . . . , M0 m)⇒(M1, . . . , Mm), supp(M0 ack) = ∅, supp(Mack)6=∅}. In he con o mon-P sys ems a ea, in gene al one uses g aphical ep esen a ions ins ead o o mal de ini ions in o de o speci y sys ems appea ing in examples o p oo s. We ecall now some con en ions used in hese ep esen a ions – de ails can be ound in he pape s men ioned in he end o In oduc ion. Memb anes/compa men s a e ep esen ed by labelled o als, ha ing inside he associa ed con o mons and in e ac ion ules. Con o mons p esen in he ini ial No cycles in compa men s 161 con igu a ion o a sys em a e w i en in bold inside a memb ane while he ones w i en in no mal on a e p esen in ha compa men in one o he possible con igu a ions o he sys em. A slash (/) be ween alues in a con o mon indica es ha a con o mon can ha e any o he indica ed alues. The mul iplici y is indica ed only o con o mons which appea in mo e han one copy. Di ec ed edges be ween compa men s a e ep esen ed as a ows wi h hei p edica e indica ed close o hem. Se e al edges connec ing wo compa men s a e depic ed as jus one edge wi h di e en p edica es sepa a ed by a slash (/). Fo ins ance, Figu e 1 p esen s a con o mon-P sys em which accep s any posi i e e en numbe ( he inpu memb ane is he one wi h label 1 and he acknowledging one is memb ane 11). ≤11 ≥1 ≥2 ≥3 ≥5 ≥6 ≤2 ≤5 ≤1 ≤0 ≥7 ≥7 ≤0 ≥7 ≥8 ≤3 ≤6 ≥14 10 ≤8 ≤0/≥14 [B,3/14] [C,11] 2 ([A, 0], q) 15 [B, 7] 11 [B, 14] [C, 0] 14 B11 →C [A, 8] 9 [B, 1] [A, 6] 10 A6 →B [C, 5] [A, 2] 12 A2 →C [B, 7] [C, 7] 13 C7 →B 7 6 5 4 3 [B, 3] B2 →A [C, 11] C6 →A ([A,0], p) 1 [B, 1/3] [A, 2/6/8] [C, 5/11] [A, 6/8] [C, 11] [A, 6/8] [C, 5/11] [B, 3] [A, 6/8] [C, 5/11] [B, 3] [A, 2/6/8] [C, 5/11] [A, 8] [C, 11] ≥11 8 ≥3 Fig. 1. A con o mon-P sys em accep ing e en numbe s. In p oo s he e appea la ge con o mon-P sys ems, ha is why i is use ul o conside modules which a e so o sho cu s o g aphical ep esen a ions. Such modules a e explained in de ail in se e al pape s, e.g., in [3]. The basic modules a e he spli e (i selec s con o mons depending on hei alues; speci ically, when con o mons o ype [a, pi],1≤i≤h, a e p esen in a 162 P. F isco, Gh. P˘aun gi en compa men , hey can pass o speci ic di e en compa men s depending on alues pi) and he sepa a o (i selec s con o mons depending on hei name; speci ically, when con o mons o ype [ai, p],1≤i≤h, a e p esen in a compa - men , hey can pass o speci ic di e en compa men s depending on ai). In he pic o ial ep esen a ions o con o mon-P sys ems he modules a e indi- ca ed by ick o als, linked by a ows ma ked wi h p edica es, which a e o he o m =niin he case o spli e s and o he o m [a, pi] in he case o sepa a o s; usual memb anes and a ows ma ked wi h p edica es can be in e lea ed wi h modules. Fo ins ance, in Figu e 2 we gi e a e sion o he sys em ep esen ed in Figu e 1 whe e a spli e is also in ol ed. ≤11 ≤0 ≥7 ≥7 ≤0 ≥7 ≥14 ≤0/≥14 = 8 = 1/ = 6 ≥1= 3/ = 11 = 2/= 5 spl [B,3/14] [C,11] 2 ([A, 0], q) 15 [B, 7] 11 [B, 14] [C, 0] 14 B11 →C [A, 8] 9 [B, 1] [A, 6] 10 A6 →B [C, 5] [A, 2] 12 A2 →C [B, 7] [C, 7] 13 C7 →B [B, 3] B2 →A [C, 11] C6 →A ([A,0], p) 1 ≥3 [B, 1/3] [A, 2/6/8] [C, 5/11] Fig. 2. The con o mon-P sys em wi h a spli e associa ed o he sys em in Figu e 1. 2.2 Asynch onous Tissue-like P Sys ems We in oduce he P sys ems o he o m we ha e desc ibed in he In oduc ion, wi h a se ies o ing edien s as p esen ed be o e o con o mon-P sys ems. Because we wo k only wi h asynch onous sys ems, om now on we omi men ioning his ea u e. An EC issue-like P sys em o deg ee mis a uple No cycles in compa men s 163 Π= (V, µ, ωz, ack, L1, . . . , Lm, R1, . . . , Rm, P1, . . . , Pm), whe e: Vis a ini e alphabe whose elemen s a e called objec s; µ= (Q, E) is a g aph indica ing he unde lying compa men s uc u e o Π, whe e Q={1, . . . , m}is he se o memb anes/compa men s; E⊆Q×Qis he se o di ec ed edges be ween compa men s; ωzwi h ω∈ {in, ou }and z∈Qindica es i Πis an accep ing (ω=in) o gene a ing (ω=ou ) de ice; he compa men zcon ains he inpu o ou pu , espec i ely; ack ∈Qindica es he acknowledging compa men ; Li:V→N∪{+∞},1≤i≤m, a e mul ise s o objec s in V, wi h supp(Lack) = ∅; Ri,1≤i≤m, a e se s o e olu ion ules o he o m ab →cd wi h a, b, c, d ∈V; Pi,1≤i≤m, a e se s o communica ion ules o he o m (a, go) wi h a∈V. A issue-like P sys em is cycle- ee i ab →cd ∈Riimplies ha cd →ab does no belong o Ri(wi h some abuse o no a ion we ep esen mul ise s by s ings and all hei pe mu a ions). Acon igu a ion o Πis an m- uple (M1, . . . , Mm) o mul ise s o e V. The m- uple (L1, . . . , Lm), is called ini ial con igu a ion (in he ini ial con igu a ion he acknowledge compa men does no con ain any objec ) while any con igu a ion ha ing supp(Mack)6=∅is called inal con igu a ion. In a inal con igu a ion no ope a ion is pe o med e en i i could. Fo wo con igu a ions (M1, . . . , Mm),(M0 1, . . . , M0 m) o Πwe w i e (M1, . . . , Mm)⇒(M0 1, . . . , M0 m) indica ing a ansi ion om (M1, . . . , Mm) o (M0 1, . . . , M0 m), ha is, he applica ion o one ule in a compa men acco ding o he ollowing. I a, b ∈Miand ab →cd ∈Ri, hen M0 i=Mi− {a, b} ∪ {c, d}. I a∈Miand (a, go)∈Pi, hen M0 i=Mi− {a}, M0 j=Mj∪ {a}i (i, j)∈E. I no ule is applied o a mul ise Mi, hen M0 i=M0 i. The e lexi e and ansi i e closu e o ⇒is indica ed by ⇒∗. I in a con igu a ion a symbol can be subjec o mo e han one ule, hen one o hem is non-de e minis ically applied. Acompu a ion is a ini e sequence o ansi ions be ween con igu a ions o he sys em Πs a ing om (L1, . . . , Lm). In case Πis an accep ing de ice (ω=in), hen he inpu is gi en by he numbe o symbols (coun ed wi h hei mul iplici y) p esen in Lz. The inpu is accep ed by Πi i eaches a con igu a ion in which any con o mon is p esen in ack, hal ing in his was he compu a ion. Fo mally, N(Π) = {|Lz| | (L1, . . . , Lm)⇒∗(M0 1, . . . , M0 m)⇒(M1, . . . , Mm), supp(M0 ack) = ∅, supp(Mack)6=∅}. In case Πis a gene a ing de ice (ω=ou ), hen supp(Lz) = ∅. The esul o a compu a ion is gi en by Mzwhen any symbol is p esen in ack. When his 164 P. F isco, Gh. P˘aun happens he compu a ion is hal ed and he numbe o symbols (coun ed wi h hei mul iplici y) p esen in Mzde ines he numbe gene a ed by Π. Fo mally, N(Π) = {|Mz| | (L1, . . . , Lm)⇒∗(M0 1, . . . , M0 m)⇒(M1, . . . , Mm), supp(M0 ack) = ∅, supp(Mack)6=∅}. As usual in P sys ems (i.e., wi hou sepa a ing e olu ion om communica- ion), we a oid ules o he o m (a, go) and associa e a ge indica ion di ec ly o e olu ion ules: an objec which has o be communica ed will appea in he igh hand side o a ule pai ed wi h go ( he objec s wi hou such a pai emain in he same memb ane). No e he impo an de ail ha his ime he communica ion o an objec cappea ing in he o m (c, go) in a ule mus be done immedia ely, his does no mean applica ion o a ule, bu i is jus pa o using he e olu ion ule. This is a di e ence wi h espec o con o mon-P sys ems and o EC issue-like P sys ems, bu in he p oo s below we will no ha e o ake ca e o his aspec : communica ion will be done by e olu ion ules o he o m a→(a, go) which a e di ec ly associa ed wi h communica ion ules o he o m (a, go). 3 Compu ing wi h Con o mon-P Sys ems We ecall now some esul s conce ning he compu ing powe o con o mon-P sys- ems. P oo s can be ound, e.g., in [3]. A con o mon-P sys ems is called alue- es ic ed (in sho , VR) i in i s ini ial con igu a ion all con o mons p esen in an unbounded numbe o copies ha e alue 0. In his way, he o al alue o con o mons p esen in he sys em a any s ep o a compu a ion is ini e. Theo em 1. The amily o se s o numbe s gene a ed by VR con o mon-P sys ems coincides wi h he amily o se s o numbe s gene a ed by pa ially blind egis e machines. The con o mon-P sys em which can simula e a pa ially blind egis e machine is based on he cons uc ion indica ed in Figu e 3. We ecall i because la e we will poin ou some basic ea u es o his cons uc ion use ul in in e ing esul s abou (asynch onous) issue-like P sys ems. F om Theo em 2 in [5] we know ha i in he con o mon-P sys em desc ibed in he p e ious heo em ei he p io i ies, maximal concu ency, o maximal pa al- lelism a e added, hen he esul ing sys ems a e compu a ionally comple e. Theo em 2. The amily o se s o numbe s gene a ed by VR con o mon-P sys ems whe e e olu ion has p io i y on communica ion (i a con o mon can be subjec o an in e ac ion ule and i can also pass o ano he memb ane, hen he in e ac ion should be done) coincides wi h he amily o se s o numbe s gene a ed by egis e machines (hence wi h he amily o Tu ing compu able se s o numbe s). No cycles in compa men s 165 [s00 j,γ ,7] [s ,7] [s0 j,γ ,6] [s00 j,γ ,4] [si,1] [si,3] si 1 →s0 j,γ si 3 →s00 j,γ [s0 j,γ ,2] [γ, 4] ([γ, 0],kγ) γ4 →s0 j,γ s00 j,γ 3 →γ [s00 j,γ ,4] [s ,7] [si,3][si,1]/ 2 3 [s0 j,γ ,2]/ [γ, 4] 4 [s0 j,γ ,6] [s00 j,γ ,4] [si,0]/ [s0 j,γ ,7]/ [s00 j,γ ,7] [s0 j,γ ,0]/[sj,7]/ [s00 j,γ ,0] 5 ([γ, 0],+∞) [s0 j,γ ,6] s0 j,γ 4 →γ [s00 j,γ ,1] [γ, 3] γ3 →s00 j,γ [s00 j,γ ,4] [s0 j,γ ,0] [s00 j,γ ,0] [si,7] 1 si 6 →s0 j,γ si 4 →s00 j,γ [s0 j,γ ,6] [γ, 3] [s00 j,γ ,1]/ [sj,0] [s0 j,γ ,7] 6 s00 j,γ 7 →sj s0 j,γ 7 →sj Fig. 3. The con o mon-P sys em ela ed o Theo em 1. Also o his case we ecall – in Figu e 4 – he cons uc ion used in p o ing ha a con o mon-P sys em wi h p io i y as abo e can simula e a egis e machine. 4 F om Con o mon- o Tissue-like P Sys ems Fi s , le us poin ou a di ec passage om con o mon-P sys ems o EC issue-like P sys ems. Theo em 3. Gi en any VR con o mon-P sys em Π= (V, µ, ωz, ack, L1, ..., Lm, R1, . . . , Rm), we can cons uc an EC issue-like P sys em Π0= (V0, µ0, ωz, ack, L0 1, . . . , L0 m, R0 1, . . . , R0 m, P 0 1, . . . , P0 m)such ha N(Π0) = N(Π). P oo . Conside a con o mon-P sys em Πas abo e, wi h µ= (Q, E); deno e by S he sum o he alues o he con o mons in Π. We cons uc he issue-like P sys em Π0wi h: V0={ap|a∈V, 0≤p≤S}; µ0= (Q, E0) wi h (i, j)∈E0 o each (i, j, p ed)∈E; L0 i(ap) = ki Li([a, p]) = k o 1 ≤i≤m; apbq→ap−ebq+e∈R0 ii ae →b∈Ri, 0 ≤p, q ≤S, p ≥e; (ap, go)∈P0 ii (i, j, ≥ )∈E o ≤p≤So (i, j, ≤ )∈E o 0 ≤p≤ .