Rewriting in P Systems: An Algebraic Approach

Rew i ing in P Sys ems: An Algeb aic App oach Rodica Ce e chi Facul y o Ma hema ics and Compu e Science, Uni e si y o Bucha es Academiei 14, RO-010014, Bucha es , Romania [email p o ec ed] Summa y. We e o mula e in algeb aic e ms he maximal pa allel ew i ing o symbols which occu inside memb anes o a P sys em. 1 In oduc ion We ela e o he o mal de ini ion o a P sys em wi h symbol-objec s, om Sec ion 3.5 o Chap e 3 o he monog aph [2]. We gi e an equi alen o mula ion o he symbol-objec ew i ing ules which occu in one memb ane o a P sys em, e o mula ion which emphasizes he unde - lying algeb aic s uc u e o commu a i e monoid wi h a ini e se o gene a o s. We limi ou sel es o sys ems wi h only one memb ane. Thus no dissol ing ac ions a e aken in o conside a ion, and no a ge indica ions o ules. 2 An Algeb aic Re o mula ion: Symbol Rew i ing in One Memb ane Le V={a1,· · · , an}be an alphabe . Le us conside (V+,+, λ) he commu a i e monoid eely gene a ed by V. Deno e by φV:V→V+ he canonical inclusion o Vin V+,φV(ai) = ai. We ecall ha V+is he unique (up o isomo phism) commu a i e monoid which includes V, wi h he ollowing uni e sali y p ope y: o any commu a i e monoid Mwhich includes V ia ψ:V→M he e exis s a unique mo phism o monoids ρV:V+→Mwhich commu es wi h φVand ψ, i.e., o which he equali ies ρV(ai) = ψ(ai) hold in M o all 1 ≤i≤n.ρVis called he canonical mo phism. We will use addi i e no a ion o commu a i e monoids. I (M, +,0) is such a monoid and a∈M, o a na u al numbe n,na s ands o a+a+· · · +awi h n occu ences o a. The elemen s o (V+,+, λ) will hus be w i en as w=Pn i=1 miai, o mi∈N. 166 R. Ce e chi Two o he commu a i e monoids, isomo phic o (V+,+, λ) a e gi en by he ollowing: •(V∗/σ, ·, λ), whe e V∗is he (non-commu a i e) ee monoid gene a ed by V,σ is he equi alence ela ion de ined by commu a ion o le e s, ·is he ope a ion induced on classes by ca ena ion, and λis he class o he emp y wo d. •(Nn,+,0), whe e Nnis he n- h powe se o na u al numbe s, wi h + compo- nen wise addi ion. This is ela ed o se e al possible ep esen a ions o mul ise s which we ecall bellow: •S ing no a ion: w=am1 1· · · amn n, and any pe mu a ion in his s ing s ands o he same mul ise . •Func ion no a ion: wis iden i ied wi h he unc ion Mw:V−→ Nwhich associa es o each ai∈Vi s “mul iplici y” in w,mi=Mw(ai) = |w|ai. A compac ep esen a ion is as se o pai s {(a1, m1), . . . , (an, mn)}. •Vec o no a ion: wis iden i ied wi h i s Pa ikh ec o (m1, . . . , mn)∈Nn. •Addi i e no a ion: w=Pn i=1 miai. In [2] he i s h ee abo e p esen a ions o mul ise s a e used. The monoid V+is endowed wi h a na u al pa ial o de ela ion (which is he same o de as mul ise inclusion) gi en by: n X i=1 miai≤ n X i=1 m0 iaii mi≤m0 i o all 1 ≤i≤n. In he o malism o [2], Sec ion 3.5, e olu ion ules a e iples (αi, βi, δi), whe e αi∈V+s ands o he le -hand side, βis ands o he igh -hand side, and is a mul ise wi h a ge indica ions, and δi∈ {¬δ, δ}is a symbol indica ing he dissolu ion o non-dissolu ion o he memb ane. Since we ha e only one memb ane, no a ge indica ions and no dissolu ion, such a ule will be educed o a pai (αi, βi) wi h αi, βi∈V+. Le {(α1, β1),· · · ,(αn, βn)}be such a se o e olu ion ules (o symbol ew i ing ules), associa ed o ou unique memb ane. Le Λ={α1,· · · , αn} ⊂ V+be he subse composed o le -hand sides o he se o ules. Conside RΛ=Λ+ he commu a i e monoid eely gene a ed by Λin V+. Conside µΛ:RΛ−→ V+ he canonical mo phism gi en by µ(αi) = αi o all i= 1, . . . , n. I s image µ(RΛ) = MΛis he submonoid o V+gene a ed by Λ= {α1,· · · , αn}. Le w∈V+be a ixed s ing ( he “axiom” o he unique memb ane). Conside in MΛ he abo e pa ial o de and i s in e sec ion wi h he in e al [λ, w] w. . . his o de . (MΛ∩[λ, w],≤Λ) is nonemp y and ini e. Le CΛ,w deno e he se o he maximal elemen s o (MΛ∩[λ, w],≤Λ). I is nonemp y. Le RΛ,w =µΛ −1(CΛ,w). I is nonemp y and ini e. Rew i ing in P Sys ems: An Algeb aic App oach 167 De ini ion 1. Aone s ep compu a ion s a ing wi h axiom wand applying symbol e olu ion ules (α1, β1),· · · ,(αn, βn)is: (i) I RΛ,w ={λ}, hen no hing happens. (ii) I RΛ,w 6={λ}, hen an elemen m1α1+m2α2+· · · +mnαn∈RΛ,w is chosen a andom. (iii)Replace wby z, ob ained om was z=w− n X i=1 miαi+ n X i=1 miβi. Lemma 1. The one s ep compu a ion de ined abo e is he same as he one s ep ansi ion in a P sys em wi h one memb ane con aining wand symbol e olu ion ules {(α1, β1),· · · ,(αn, βn)}. P oo . Case (i) co esponds o he si ua ion ha he ules a e no applicable. The s ings zob ained by cases (ii) and (iii) o he abo e a e p ecisely he s ings ob ained in one memb ane, wi h ini ial con en w, by he maximal pa allel appli- ca ion o he se o ew i ing ules (α1, β1),· · · ,(αn, βn), a e one s ep o (non- de e minis ic) compu a ion (see [2]). u 3 The Case o S ing Rew i ing We wan o see o wha ex en , and how, he cons uc ion o he p e ious sec ion can be ex ended o co e he case o s ing ew i ing sys ems. This eopens he discussion on se e al ypes o pa allel p ocessing ea u es p esen in a memb ane sys em. Acco ding o [2], we ha e h ee le els o pa allelism; wo o hem a e sha ed by he sys ems wi h symbols and hose wi h s ings, and a hi d one, speci ic o s ing- ew i ing sys ems: •Pa allel p ocessing inside one memb ane: he ules a e applied o all symbols o s ings inside a memb ane. •Pa allel p ocessing a he le el o he sys em: p ocessing occu s simul aneously in all memb anes. •Pa allel p ocessing o each s ing: his would mean pa allel ew i ing o each s ing, and would be applicable only o s ing- ew i ing sys ems. This hi d ype o pa allelism, speci ic o Lindenmeye sys ems, is no used by P sys ems: he ules ew i e each s ing in only one place. I we wan o conside his hi d ype o pa allelism, i.e. he pa allel ew i ing o each s ing, and only one s ing is p esen , in one memb ane, hen we ha e a non-commu a i e e sion o he p e ious cons uc ion. A good candida e o he o de ela ion in e ms o which o exp ess maximali y is he sca e ed subwo ds o de . Mo e p ecisely, o V={a1,· · · , an}an alphabe , conside (V∗,·, λ) he (noncommu a i e) ee monoid gene a ed by V. We conside he pa ial o de ela ion on V∗gi en by he sca e ed subwo ds o a wo d: 168 R. Ce e chi uis a sca e ed subwo d o , and we w i e u≤ , i he e exis a decomposi ion o uas u=u1· · · unand s ings z0, z1,· · · , zn∈A∗such ha can be decomposed as =z0u1z1u2z2· · · unzn. Fo an axiom-s ing w∈V∗, and a se o s ing ew i ing ules (α1, β1),. . . , (αn, βn), αi, βi∈V∗, one can epea he cons uc ion om he p e ious sec ion, using he uni e sali y p ope y o eely gene a ed non-commu a i e monoids, and he abo e pa ial o de , and de ine, in algeb aic e ms, he one-s ep pa allel ew i - ing o one s ing. I we wan o conside he s ing ew i ing sys ems which a e p esen in he li e a u e, and which ha e only he i s wo ea u es o pa allelism abo e, i.e., o which each s ing is ew i en in only one place, hen we will ha e a di e en cons uc ion. This is a ma e o ongoing esea ch. 4 Conclusions and Open P oblems De ini ion 1 is gi en in pu ely algeb aic e ms. Some mo e in ica e no ions om Sec ion 3.5 o [2] can be ecap u ed in his o malism: o ins ance, (m1, . . . , mn) is an applicabili y ec o (see De ini ion 3.5.9 o [2]). The o malism can be ex ended in se e al di ec ions. Fi s , o deal wi h se e al memb anes, and nex , wi h he ea u es added by communica ion be ween hem. Communica ion in he o m o sympo /an ipo ules can (a a i s glance) be mo e easily adap ed o his o malism. Thus ECP sys ems in oduced in [1] seem good candida es. Second, as ou lined in Sec ion 3, o deal wi h s ing ew i ing sys ems. In he uni ying amewo k o his o malism, he gap be ween he case o symbol ew i ing and he case o s ing ew i ing could be b idged. The concep could also be use ul o conside ing uzzy ex ensions. Acknowledgemen s. The au ho hanks he colleague Paul Flondo o sug- ges ing he p esen app oach, and o many discussions on his opic. She also hanks he Resea ch G oup on Na u al Compu ing, Depa men o Compu e Science and A i icial In elligence o he Uni e si y o Se illa, o a s imula ing wo king en i onmen . This was made possible by p ojec TIN2005- 09345-C04-01 o he Minis e io de Educaci´on y Ciencia o Spain, co inanced by FEDER unds. Re e ences 1. M. Ca alie e: E olu ion-Communica ion P Sys ems. In Memb ane Compu ing, In- e na ional Wo kshop, WMC-CdeA 2002, Cu ea de A ge¸s, Romania, Augus 19-23, LNCS 2597, Sp inge , 2003. 2. Gh. P˘aun: Memb ane Compu ing. An In oduc ion. Sp inge , Be lin, 2002.