Priorities, Promoters and Inhibitors in Deterministic Non-Cooperative P Systems

P io i ies, P omo e s and Inhibi o s in De e minis ic Non-Coope a i e P Sys ems A iom Alhazo 1, Rudol F eund2 1Ins 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] 2Facul y o In o ma ics, Vienna Uni e si y o Technology Fa o i ens . 9, 1040 Vienna, Aus ia [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. Since ecen ly, i is known ha he powe o he de e minis ic subclass o such sys ems is sub egula . We p esen new esul s on he weigh o p omo e s and inhibi o s, as well as o cha ac e izing he sys ems wi h p io i ies only. 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 [3] and [6]. 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, [2]. 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. Recen ly, in [1] i was shown ha he compu a ional comple eness canno be achie ed by de e minis ic non-coope a i e sys ems wi h p omo e s, inhibi o s and p io i ies (in maximally pa allel o asynch onous mode, unlike he sequen ial mode), and cha ac e iza ions o he co esponding classes we e ob ained: 28 A. Alhazo , R. F eund NFIN ∪coNFIN =Nde aOPasyn 1(ncoo, p o1,∗, inh1,∗) =Nde aOPmaxpa 1(ncoo, p o1,∗) =Nde aOPmaxpa 1(ncoo, inh1,∗) =Nde aOPasyn 1(ncoo, (p o∗,∗, inh∗,∗)∗, p i) =Nde aOPmaxpa 1(ncoo, (p o∗,∗, inh∗,∗)∗, p i),bu NRE =Nde aOP sequ 1(ncoo, p o1,1, inh1,1). A ew in e es ing ques ions ha e been le open. Fo ins ance, wha is he powe o P sys ems, e.g., in he maximally pa allel mode, when we only use p io i ies, o when we es ic he weigh o he p omo ing/inhibi ing mul ise s. These a e he ques ions we add ess in his pape . 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 NF IN (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. Le Pi,Qibe ( ini e) se s o mul ise s o e O,1≤i≤m. A ule wi h con ex condi ions ( , (P1, Q1),· · · ,(Pm, Qm)) is applicable 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. Con ols in De e minis ic Non-Coope a i e P Sys ems 29 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. 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 ), a mul ise 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, pos- sibly lea ing some objec s idle, unde he condi ion ha no u he applicable ule can be added o ha mul ise (i.e., no supe mul ise o he chosen mul ise is applicable o he same configu a ion). Maximal pa allelism is he mos common compu a ion mode in memb ane compu ing, see also Defini ion 4.8 in [5]. In he asynch onuous mode (asyn), any posi i e numbe o applicable ules may be cho- sen 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 configu 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 o 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. 30 A. Alhazo , R. F eund 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. 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). 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 ols in De e minis ic Non-Coope a i e P Sys ems 31 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). 3 Resul s 3.1 Recen esul s We fi s ecall om [1] he bounding ope a ion o e mul ise s, wi h a pa ame e k∈Nas ollows: o u∈O∗,bk(u) = wi h | |a= min(|u|a, k) o all a∈O. The mapping bk“c ops” he mul ise s by emo ing copies o e e y objec a p esen in mo e han kcopies un il exac ly k emain. Fo wo mul ise s u, u′, bk(u) = bk(u′) i o e e y a∈O, ei he |u|a=|u′|a< k, o |u|a≥kand |u′|a≥k. Mapping bkinduces an equi alence ela ion, mapping O∗in o (k+ 1)|O| equi alence classes. Each equi alence class co esponds o speci ying, o each a∈ O∗, whe he no copy, one copy, o · · · k−1 copies, o “kcopies o mo e” a e p esen . We deno e he ange o bkby {0,· · · , k}O. Lemma 1. [1] Con ex condi ions a e equi alen o p edica es de ined on bound- ings. Theo em 1. [1] P io i ies a e subsumed by condi ional con ex s. Rema k 4. I is wo h o no e, see also [4], ha i no o he con ol is used, he p io i ies can be mapped o se s o a omic inhibi o s. Indeed, a ule is inhibi ed p ecisely by he le side o each highe p io i y ule. This is s aigh o wa d in case when he p io i y ela ion is assumed o be a pa ial o de . I i is no , hen bo h he seman ics o compu a ion in P sys ems and he educ ion o p io i ies o inhibi o s is a bi mo e complica ed, bu he claim s ill holds. Fix an a bi a y de e minis ic con olled non-coope a i e P sys em. Take kas he maximum o size o all mul ise s in all con ex condi ions. Then, he bounding does no influence applicabili y o ules, and bk(u) is hal ing i and only i uis hal ing. We ecall ha bounding induces equi alence classes p ese ed by any compu a ion. Lemma 2. [1] Assume u→xand →y. Then bk(u) = bk( )implies bk(x) = bk(y). Co olla y 1. [1] I bk(u) = bk( ), hen uis accep ed i and only i is accep ed. 32 A. Alhazo , R. F eund Finally, he “a mos NFIN ∪coNFIN” pa o cha ac e izing Nde aOPmaxpa 1(ncoo, (p o∗,∗, inh∗,∗)∗, p i) ( he main heo em o [1]) is shown wi h he ollowing a gumen : Each equi alence class induced by bounding is comple ely accep ed o comple ely ejec ed. I no infini e equi alence class is accep ed, hen he accep ed se is fini e (con aining numbe s no exceeding (k−1) · |O|). I a leas one infini e equi alence class is accep ed, hen he ejec ed se is fini e (con aining numbe s no exceeding (k−1) · |O|). 3.2 P io i ies only We s a wi h an example how o de e minis ically ew i e an objec depending on he p esence o absence o objec a. Example 1. Π= ({a, A, A′, , ′, +, −},{a}, A, R, R, >),whe e R={1 : → ′,2 : a→λ, 3 : A→A′,4 : ′→ +,5 : ′→ −,6 : A′→λ}, >={a→λ > A →A′, A →A′> ′→ −, A′→λ > ′→ +}. Indeed, objec wai s o one s ep by becoming ′, while Ahas o change o A′o wai , depending on he p esence o a. Then, objec ′becomes ei he + o −, depending on whe he Ao A′is p esen . No ice, e.g., how adding ei he ule +→ +o ule −→ −leads o a sys em accep ing {0}o N {0}. O cou se, accep ing only ze o could ins ead be done by a i ial one- ule sys em, bu his example is impo an because such a deciding subsys em can be used, wi h sui able delays, as a building block o checking combina ions o p esence/absence o mul iple symbols. We now p oceed wi h cha ac e izing sys ems wi h p io i ies only. Theo em 2. Nde aOPmaxpa 1(ncoo, p i) = {Nk,Nk∪ {0} | k≥0} ∪ {{0},∅}. P oo . We al eady know ha he p io i ies co espond o se s o a omic inhibi o s. This means ha each sys em accep s a union o some equi alence classes induced by bounding b1(i.e., checking p esence/absence). No e ha a ious combina ions o “= 0” and “≥1” yield nume ic se s {0}and Nk(whe e k > 0 is he numbe o diffe en symbols p esen ). The amily o all unions o hese se s is Fp i ={Nk,Nk∪ {0} | k≥0} ∪ {{0},∅}. I ollows ha Nde aOPmaxpa 1(ncoo, p i)⊆Fp i. We p oceed wi h he con e se inclusion. Le Π0= ({a, },{a}, , R, R, >), hen R={ → }and emp y ela ion >yields ∅. To accep {0}, we ins ead ake R={a→a}and emp y ela ion >. Con ols in De e minis ic Non-Coope a i e P Sys ems 33 Now suppose we wan o accep Nk. I would suffice o coun ha we ha e a leas one o each objec s a1,· · · , ak(we ecall ha we need o accep a leas one inpu o size j o each j≥k, o ejec he inpu i j > k). To accep Nk∪ {0} ins ead, we may fi s pe o m a simul aneous check o he absence o all inpu symbols. Using he idea om Example 1, we cons uc he sys em Π1= (O, Σ ={ai,0|1≤i≤k}, A0,0· · · Ak,0, R, R, >),whe e O={ai,j |1≤i≤k, 0≤j≤i+ 1} ∪ {Ai,j |0≤i≤k, 0≤j≤i+ 2} ∪ { , z, p}∪{ i|0≤i≤i+ 1}, R={1 : ai,j →ai,j+1 |1≤i≤k, 0≤j≤i} ∪ {2 : Ai,j →Ai,j+1 |1≤i≤k, 0≤j≤i+ 1} ∪ {3 : → 0,4 : 0→z, 5 : 0→ 1,6 : p→p} ∪ {7 : i→ i+1,8 : i→p|1≤i≤k}, >={ai,0→ai,1> A0,0→A0,1|1≤i≤k} ∪ {A0,0→A0,1> 0→z, A0,1→A0,2> 0→ 1} ∪ {ai,i →ai,i+1 > Ai,i →Ai,i+1 |1≤i≤k} ∪ {Ai,i →Ai,i+1 > i→p, Ai,i+1 →Ai,i+2 > i→ i+1}. Such sys em accep exac ly Nk∪ {0}. Indeed, a e fi s s ep, A0,0is p esen i all inpu symbols we e absen , o he wise A0,1is p esen ins ead. Fo any i, 1 ≤i≤k, a e s ep 1+i, objec Ai,i is p esen i inpu symbol ai,0was p esen in he inpu , and o he wise Ai,i+1 is p esen ins ead. These “decision symbols” a e used by i, 0≤i≤k, o build he “p esence pic u e”. We ecall ha i suffices o accep when all inpu symbols a e p esen , o when none o hem is p esen . In he fi s case, 0becomes z, and he compu a ion only con inues by ules om g oups 1 and 2, leading o hal ing. Le us assume ha he fi s so he inpu symbols a e p esen , s<k. Then, 0becomes 1, and hen ···, s, and hen he absence o s+1 will change sin o p, leading o an infini e compu a ion. Finally, i all inpu symbols a e p esen , hen he compu a ion will hal wi h k+1. I emains o no ice ha accep ing Nk,k≥1, can be done by simply adding a ule z→z. 3.3 P omo e s o inhibi o s o weigh 2 We s a om examples, illus a ing de e minis ic choice o ew i ing p, depending on whe he objec ais absen , occu s exac ly once, o occu s mul iple imes. Example 2. Symbols A,Ba e p imed i inpu is p esen (mul iple inpu symbols a e p esen ). Then p imed and unp imed symbols o m mu ually exclusi e condi- ions. 34 A. Alhazo , R. F eund Π= (O={p, p′, p′′, p>, p1, p0, A, B, a}, Σ ={a}, pAB, R′, R),whe e R′={1 : p→p′,2 : A→A′,3 : B→B′, 4 : p′→p>,5 : p′→p′′,6 : p′′ →p1,7 : p′′ →p0}, R′={1 : p→p′,2 : A→A′|a,3 : B→B′|aa, 4 : p′→p>|B,5 : p′→p′′|B′,6 : p′′ →p1|A,7 : p′′ →p0|A′}. Example 3. No ice ha i we eplace all p omo e s by inhibi o s wi h he same con ex , he effec o blocking ules will be e e sed, bu he esul will be he same. Indeed, he ole o A′and B′will swi ch om ound aand ound aa, espec i ely, o no ound aand no ound aa, espec i ely. R′={1 : p→p′,2 : A→A′|¬a,3 : B→B′|¬aa, 4 : p′→p>|¬B,5 : p′→p′′|¬B′,6 : p′′ →p1|¬A,7 : p′′ →p0|¬A′}. We now p oceed wi h cha ac e izing sys ems wi h con ex o weigh wo. No ice ha we al eady know ha hei powe does no exceed NFIN ∪coNFIN. Theo em 3. Nde aOPmaxpa 1(ncoo, p o2) = Nde aOPmaxpa 1(ncoo, inh2) = NFIN ∪coNFIN. P oo . We use he echnique om Example 2 o all inpu symbols and combine he ex ac ed in o ma ion. Conside an a bi a y fini e se M, and le max(M) = n. We will use he ollowing s a egy: o accep a numbe j∈M, we will accep an inpu mul ise wi h exac ly jsymbols appea ing once, and no hing else. To accep he complemen o M, we spli i in o se s M′′ ={j|j > n}and M′={j|j≤ n, j /∈M}. While M′is ea ed simila ly o M, i only emains o accep M′′, which is co e ed by equi alence classes when all symbols a e p esen , and a leas one is p esen mo e han once. Π= (O, Σ ={ai|1≤i≤n}, A1· · · AnB1···Bn, R′, R),whe e O={ i,j, Ti,j , ′ i,j, T ′ i,j |1≤i≤n+ 1,0≤j≤n} ∪ {Ai, A′ i, Bi, B′ i|1≤i≤n}∪{ , #}, R′={ i,j →Ti+1,j+1, Ti,j →Ti+1,j+1, i,j → ′ i,j, Ti,j →T′ i,j, ′ i,j → i+1,j+1, T′ i,j →Ti+1,j+1, ′ i,j → i+1,j, T ′ i,j →Ti+1,j, Ai→A′ i, Bi→B′ i|1≤i≤n}∪{ → 1,0#→#} ∪ {Ti,n+1 →#|1≤i≤n}∪{ i,n+1 →#|i /∈M}, R={ i,j →Ti+1,j+1|Bi, Ti,j →Ti+1,j+1|Bi, i,j → ′ i,j|B′ i, Ti,j →T′ i,j|B′ i, ′ i,j → i+1,j+1|Ai, T′ i,j →Ti+1,j+1|Ai, ′ i,j → i+1,j|A′ i, T′ i,j →Ti+1,j|A′ i, Ai→A′ i|ai, Bi→B′ i|aiai|1≤i≤n}∪{ → 1,0,#→#} ∪ {Ti,n+1 →#|1≤i≤n}∪{ i,n+1 →#|i /∈M}. Con ols in De e minis ic Non-Coope a i e P Sys ems 35 The meaning o Ti,n+1 is ha exac ly iinpu symbols a e p esen , and a leas one o hem is p esen mul iple imes. The meaning o i,n+1 is ha he inpu consis ed o exac ly idiffe en symbols. This is how an a bi a y fini e se is accep ed. To accep ins ead o Mi s complemen , eplace i /∈Mby i∈Mand emo e ule Tn,n+1 →#. The e o e, de e minis ic P sys ems wi h p omo e s o weigh wo accep exac ly NFIN ∪coNFIN. Fo he inhibi o coun e pa , no ice ha he compu a ion o he numbe o diffe en symbols p esen , as well as checking i any symbol is p esen mul iple imes, s ays co ec by simply changing p omo e s o he inhibi o s wi h he same condi ion, jus like in Example 3. Rules p ocessing objec s i,n+1 and Ti,n+1 will ha e an opposi e effec , accep ing he complemen o he se accep ed by he sys em wi h p omo e s, again yielding NFIN ∪coNFIN. I is s ill open whe he only inhibi o s in he ules o only p omo e s in he ules a e sufficien o yield NF IN ∪coNF IN wi h he asynch onuous mode, oo. 4 Conclusion We ha e shown he cha ac e iza ions o de e minis ic non-coope a i e P sys ems wi h inhibi o s o weigh 2, wi h p omo e s o weigh 2, and wi h p io i ies. The fi s wo cases did no educe he accep ing powe wi h espec o un es ic ed weigh . Re e ences 1. A. Alhazo , R. F eund: Asynch onuous 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 and coN F IN .The Ten h B ains o ming Week in Memb ane Compu ing, ol. 1, Se illa, 2012, 25–34, and Mem- b ane Compu ing - 13 h In e na ional Con e ence, CMC13, Budapes (E. Csuhaj- Va j´u, M. Gheo ghe, G. Rozenbe g, A. Salomaa, Gy. Vaszil, Eds.), Lec u e No es in Compu e Science 7762, 2013, 101-111. 2. A. Alhazo , D. Sbu lan: Ul ima ely Confluen Rew i ing Sys ems. Pa allel Mul ise - Rew i ing wi h Pe mi ing o Fo bidding Con ex s. In: G. Mau i, Gh. P˘aun, M.J. P´e ez-Jim´enez, G. Rozenbe g, A. Salomaa: Memb ane Compu ing, 5 h In e na ional Wo kshop, WMC 2004, Milano, Re ised Selec ed and In i ed Pape s, Lec u e No es in Compu e Science 3365, Sp inge , 2005, 178–189. 3. R. F eund, L. Ka i, M. Oswald, P. Sos´ık: Compu a ionally Uni e sal P Sys ems wi hou P io i ies: Two Ca alys s a e Sufficien , Theo e ical Compu e Science 330, 2, 2005, 251–266. 4. R. F eund, M. Kogle , M. Oswald, A Gene al F amewo k o Regula ed Rew i ing Based on he Applicabili y o Rules. In: J. Kelemen, A. Kelemeno ´a, Compu a ion, Coope a ion, and Li e, Sp inge , Lec u e No es in Compu e Science 6610, 2011, 35–53.