The P Versus NP Problem Through Cellular Computing with Membranes

The P Ve sus NP P oblem Th ough Cellula Compu ing wi h Memb anes Ma io J. P´e ez-Jim´enez, Al a o Rome o-Jim´enez, and Fe nando Sancho-Capa ini Abs ac . We s udy he P e sus NP p oblem h ough memb ane sys- ems. Language accep ing P sys ems a e in oduced as a amewo k al- lowing us o ob ain a cha ac e iza ion o he P = NP ela ion by he polynomial ime unsol abili y o an NP–comple e p oblem by means o a P sys em. 1 In oduc ion The P e sus NP p oblem [2] is he p oblem o de e mining whe he e e y language accep ed by some non-de e minis ic algo i hm in polynomial ime is also accep ed by some de e minis ic algo i hm in polynomial ime. To define he abo e p oblem p ecisely we mus ha e a o mal defini ion o he concep o an algo i hm. The heo e ical model o be used as a compu ing machine in his wo k is he Tu ing machine, in oduced by Alan Tu ing in 1936 [10], se e al yea s be o e he in en ion o mode n compu e s. A de e minis ic Tu ing machine has a ansi ion unc ion p o iding a unc- ional ela ion be ween configu a ions; so, o e e y inpu he e exis s only one compu a ion (fini e o infini e), allowing us o define in a na u al way when an inpu is accep ed ( h ough an accep ing compu a ion). In a non-de e minis ic Tu ing machine, o a gi en configu a ion se e al suc- cesso configu a ions can exis . The e o e, i could happen ha o a gi en inpu diffe en compu a ions exis . In hese machines, an inpu is accep ed i he e exis s a leas one fini e accep ing compu a ion associa ed wi h i . The class P is he class o languages accep ed by some de e minis ic Tu ing machine in a ime bounded by a polynomial on he leng h (size) o he inpu . F om an in o mal poin o iew, he languages in he class P a e iden ified wi h he p oblems ha ing an efficien algo i hm ha gi es an answe in a easible ime; he p oblems in P a e also known as ac able p oblems. The class NP is he class o languages accep ed by some non-de e minis ic Tu ing machine whe e o e e y accep ed inpu he e exis s a leas one accep ing compu a ion aking an amoun o s eps bounded by a polynomial on he leng h o he inpu . E e y de e minis ic Tu ing machine can be conside ed as a non-de e minis ic one, so we ha e P⊆NP. In e ms o he p e iously defined classes, he P e sus NP p oblem can be exp essed as ollows: is i e ified he ela ion NP ⊆P? The P? =NP ques ion is one o he ou s anding open p oblems in heo e - ical compu e science. The ele ance o his ques ion does no lie only in he inhe en pleasu e o sol ing a ma hema ical p oblem, bu in his case an an- swe o i could p o ide an in o ma ion o a high p ac ical in e es . Fo ins ance, a nega i e answe o his ques ion would confi m ha he majo i y o cu en c yp og aphic sys ems a e secu e om a p ac ical poin o iew. On he o he hand, a posi i e answe could no only en ail he ulne abili y o c yp og aphic sys ems, bu his kind o answe is expec ed o come oge he wi h a gene al p o- cedu e which will p o ide a de e minis ic algo i hm sol ing any NP-comple e p oblem in polynomial ime. Mo eo e , he p oblems known o be in he class NP bu no known o be in Pa e a ied and o highes p ac ical in e es . An NP–comple e p oblem is a ha des (in ce ain sense) p oblem in NP; ha is, any p oblem in NP could be efficien ly sol ed using an efficien algo i hm which sol es a fixed NP–comple e p oblem. These p oblems a e he sui able candida es o a ack he P e sus NP p oblem. In he las yea s se e al compu ing models using powe ul and inhe en ools inspi ed om na u e ha e been de eloped (because o his eason, hey a e known as bio-inspi ed models) and se e al solu ions in polynomial ime o p oblems om he class NP ha e been p esen ed, making use o non-de e minism o o an exponen ial amoun o space. This is he eason why a p ac ical implemen- a ion o such models (in biological, elec onic, o o he media) could p o ide a quan i a i e imp o emen o he esolu ion o NP-comple e p oblems. In his wo k we ocus on one o hese models, he cellula compu ing model wi h memb anes, specifically, on one o i s a ian s, he language accep ing P sys ems, in o de o de elop a compu a ional complexi y heo y allowing us o a ack he P e sus NP p oblem om o he poin o iew han he classical one. The pape is s uc u ed as ollows. The nex sec ion is de o ed o he de - ini ion o language accep ing P sys ems. In sec ion 3 a polynomial complexi y class o he abo e model is in oduced. Sec ions 4 and 5 p o ides simula ions o de e minis ic Tu ing machines by P sys ems and language accep ing P sys ems by de e minis ic Tu ing machines. Finally, in sec ion 6 we es ablish a cha ac e - iza ion o he P e sus NP p oblem h ough P sys ems. 2 Language Accep ing P Sys ems Un il he end o 90’s decade se e al na u al compu ing models ha e been in- oduced simula ing he way na u e compu es a he gene ic le el (gene ic al- go i hms and DNA based molecula compu ing) and a he neu al le el (neu al ne wo ks). In 1998, Gh. P˘aun [5] sugges s a new le el o compu a ion: he cellula le el. Cells can be conside ed as machines pe o ming ce ain compu ing p ocesses; in he dis ibu ed amewo k o he hie a chical a angemen o in e nal esicles, he communica ion and al e a ion o he chemical componen s o he cell a e ca ied ou . O cou se, he p ocesses aking place in he cell a e complex enough o no a emp ing o comple ely model hem. The goal is o c ea e an abs ac cell-like compu ing model allowing o ob ain al e na i e solu ions o p oblems which a e in ac able om a classical poin o iew. The fi s cha ac e is ic o poin ou om he in e nal s uc u e o he cell is he ac ha he diffe en uni s composing he cell a e delimi ed by se e al ypes o memb anes (in a b oad sense): om he memb ane ha sepa a es he cell om he en i onmen in o which he cell is placed, o hose delimi ing he inne esicles. Also, wi h ega d o he unc ionali y o hese memb anes in na u e, i has o be emphasized he ac ha hey do no gene a e isola ed compa men s, bu hey allow he chemical compounds o flow be ween hem, some imes in selec i e o ms and e en in only one di ec ion. Simila ideas we e p e iously conside ed, o ins ance, in [1] and [3]. P sys ems a e desc ibed in [4] as ollows: a memb ane s uc u e consis s o se e al memb anes a anged in a hie a chical s uc u e inside a main memb ane (called he skin) and delimi ing egions (each egion is bounded by a mem- b ane and he immedia ely lowe memb anes, i he e a e any). Regions con ain mul ise s o objec s, ha is, se s o objec s wi h mul iplici ies associa ed wi h he elemen s. The objec s a e ep esen ed by symbols om a gi en alphabe . They e ol e acco ding o gi en e olu ion ules, which a e also associa ed wi h he egions. The ules a e applied non-de e minis ically, in a maximally pa allel manne (in each s ep, all objec s which can e ol e mus do so). The objec s can also be mo ed (communica ed ) be ween egions. In his way, we ge an- si ions om one con igu a ion o he sys em o he nex one. This p ocess is synch onized: a global clock is assumed, ma king he ime uni s common o all compa men s o he sys em. A sequence (fini e o infini e) o ansi ions be- ween configu a ions cons i u es a compu a ion; a compu a ion which eaches a configu a ion whe e no ule is applicable o he exis ing objec s is a hal ing compu a ion. Wi h each hal ing compu a ion we associa e a esul ,by aking in o conside a ion he objec s collec ed in a specified ou pu memb ane o in he en i onmen . Fo an exhaus i e o e iew o ansi ion P sys ems and o hei a ian s and p ope ies, see [4]. Th oughou his pape , we will s udy he capaci y o cellula sys ems wi h memb anes o a ack he efficien sol abili y o p esumably in ac able decision p oblems. We will ocus on a specific a ian o ansi ion P sys ems: language accep ing P sys ems. These sys ems ha e an inpu memb ane,andwo kinsuch a way ha when in oducing in he inpu memb ane a p ope ly encoded s ing, a “message” is sen o he en i onmen , encoding whe he his s ing belongs o no o a specified language. Defini ion 1. Amemb ane s uc u e isa oo ed ee,whe e henodesa ecalled memb anes, he oo is called skin,and helea esa ecalledelemen a y mem- b anes. Defini ion 2. Le µ=(V(µ),E(µ)) be a memb ane s uc u e. The memb ane s uc u e wi h ex e nal en i onmen associa ed wi h µis he oo ed ee such ha : (a) he oo o he ee is a new node ha we deno e by en ; (b) he se o nodes is V(µ)∪{en }; and (c) he se o edges is E(µ)∪{{en , skin}}. The node en is called en i onmen o he s uc u e µ. So, e e y memb ane s uc u e has associa ed in a na u al way an en i onmen . Defini ion 3. Alanguage accep ing P sys em (wi h inpu memb ane and ex- e nal ou pu ) is a uple Π=(Σ,Γ,Λ,#,µ Π,M1, ..., Mp,(R1,ρ 1), ..., (Rp,ρ p),i Π) e i ying he ollowing p ope ies: –The inpu alphabe o Πis Σ. –The wo king alphabe o Πis Γ,wi hΣΓand #∈Γ−Σ. –µΠis a memb ane s uc u e consis ing o pmemb anes, wi h he memb anes (and hence he egions) injec i ely labelled wi h 1,2,...,p. –iΠis he label o he inpu memb ane. –The ou pu alphabe o Πis Λ={Yes,No}. –M1, ..., Mpa e mul ise s o e Γ−Σ, ep esen ing he ini ial con en s o he egions o 1,2,...,p o µΠ. –R1, ..., Rpa e ini e se s o e olu ion ules o e Γassocia ed wi h he egions 1,2,...,p o µΠ. –ρi,1≤i≤p, a e pa ial o de ela ions o e Rispeci ying a p io i y ela ion among ules o Ri. An e olu ion ule is a pai (u, ), usually ep esen ed u→ ,whe euis a s ing o e Γand = o = δ,wi h as ingo e Γ×{he e, ou }∪{ini|i=1,...,p}. Conside a ule u→ om a se Ri. To apply his ule in memb ane imeans o emo e he mul ise o objec s specified by u om memb ane i( he la e mus con ain, he e o e, sufficien objec s so ha he ule can be applied), and o in oduce he objec s specified by , in he memb anes indica ed by he a ge commands associa ed wi h he objec s om . Specifically, o each (a, ou )∈ an objec awill exi he memb ane iand will become an elemen o he memb ane immedia ely ou side i ( ha is, he a he memb ane o memb ane i), o will lea e he sys em and will go o he en- i onmen i he memb ane iis he skin memb ane. I con ains a pai (a, he e), hen he objec awill emain in he same memb ane iwhe e he ule is applied (when speci ying ules, pai s (a, he e) a e simply w i en a, he indica ion he e is omi ed). Fo each (a, inj)∈ an objec ashould be mo ed in he memb ane wi h label j, p o iding ha his memb ane is immedia ely inside memb ane i ( ha is, memb ane iis he a he o memb ane j); i memb ane jis no di ec ly accesible om memb ane i( ha is, i memb ane jis no a child memb ane o memb ane i), hen he ule canno be applied. Finally, i δappea s in , hen memb ane iis dissol ed; ha is, memb ane iis emo ed om he memb ane s uc u e, and all objec s and memb anes p e iously p esen in i become el- emen s o he immedia ely uppe memb ane ( he a he memb ane) while he e olu ion ules and he p io i y ela ions o he dissol ed memb ane a e emo ed. The skin memb ane is ne e dissol ed; ha is, no ule o he o m u→ δis applicable in he skin memb ane. All hese ope a ions a e done in pa allel, o all possible applicable ules u→ , o all occu ences o mul ise s uin he memb ane associa ed wi h he ules, and o all memb anes a he same ime. The ules om hese Ri,1≤i≤p, a e applied o objec s om memb ane isynch onously, in a non-de e minis ic maximally pa allel manne ; ha is, we assign objec s o ules, non-de e minis ically choosing he ules and he objec s assigned o each ule, bu in such a way ha a e his assigna ion no u he ule can be applied o he emaining objec s. The e o e, a ule can be applied in he same s ep as many imes as he numbe o copies o objec s allows i . On he o he hand, we in e p e he p io i y ela ions be ween he ules in as ong sense:a uleu→ in a se Rican be used only i no ule o a highe p io i y exis s in Riand can be applied a he same ime wi h u→ . Acon igu a ion o Πis a uple (µ, ME,M i1,...,M iq), whe e µis a memb ane s uc u e ob ained by emo ing om µΠall memb anes diffe en om i1,...,i q (o cou se, he skin memb ane canno be emo ed), MEis he mul ise o objec s con ained in he en i onmen o µ,andMijis he mul ise o objec s con ained in he egion ij. Fo e e y mul ise mo e Σ( he inpu alphabe o he P sys em), he ini ial con igu a ion o Πwi h inpu mis he uple(µΠ,∅,M1, ..., MiΠ∪m, ..., Mp). Tha is, in any ini ial configu a ion o Π he en i onmen is emp y. We will deno e by IΠ he collec ion o possible inpu s o he sys em Π. Gi en a configu a ion Co a P sys em Π, applying p ope ly he e olu ion ules as desc ibed abo e, we ob ain, in a non-de e minis ic way, a new configu- a ion C.Wedeno ebyC⇒ΠC, and we say ha we ha e a ansi ion om C o C.Ahal ing con igu a ion is a configu a ion in which no e olu ion ule can be applied. Acompu a ion Co a P sys em is a sequence o configu a ions, {Ci}i< , whe e: C0is an ini ial configu a ion o he sys em; Ci⇒ΠCi+1, o e e y i< ; and, ei he ∈N+( ha is, i is a non-ze o na u al numbe ) and C −1is a hal ing con igu a ion,o =∞, in which case i is said ha Cis no hal ing. Fo a compu a ion C={Ci}i< we will deno e by Mj E he con en o he en i onmen in he configu a ion Cj. Nex we define he ou pu o he P sys em. Defini ion 4. The ou pu o a compu a ion C={Ci}i< is: Ou pu (C)=     Yes,i Cis hal ing, Yes∈M −1 Eand No ∈ M −1 E, No,i Cis hal ing, No ∈M −1 Eand Yes∈ M −1 E, no de ined,o he wise. I Csa is ies any o he wo i s condi ions, hen we say ha i is a success ul compu a ion. Defini ion 5. A language accep ing P sys em is said o be alid i e e y hal ing compu a ion is a success ul compu a ion and e e y hal ing compu a ion, and only hem, sends ou he symbol #(and only in he las s ep). We deno e by LA he class o alid language accep ing P sys ems. Nex we define wha i means ha such P sys ems accep o decide alan- guage. Defini ion 6. Le Lbe a language o e an alphabe Ω. We say ha he sys em Π∈LAaccep s he language Li he ollowing p ope ies a e e i ied: –The e exis s a o al unc ion, cod :Ω∗→IΠ, compu able and injec i e, encoding s ings o e Ωby means o mul ise s o e he inpu alphabe o Π. –Fo e e y s ing w∈Ω∗i is e i ied ha : •I w∈L, hen he e exis s acompu a ion Co Πwi h inpu cod(w)such ha Cis hal ing and Ou pu (C)=Yes. •I he e exis s acompu a ion Co Πwi h inpu cod(w)such ha Cis hal ing and Ou pu (C)=Yes, henw∈L. Defini ion 7. Le Lbe a language o e an alphabe Ω. We say ha he sys em Π∈LAdecides he language Li he ollowing p ope ies a e e i ied: –E e y compu a ion o Πis hal ing. –The e exis s a o al unc ion, cod :Ω∗→IΠ, compu able and injec i e, encoding s ings o e Ωby means o mul ise s o e he inpu alphabe o Π. –Fo e e y s ing w∈Ω∗i is e i ied ha : •I w∈L, hen o e e y compu a ion Co Πwi h inpu cod(w)i is e i ied ha Ou pu (C)=Yes. •I w∈ L, hen o e e y compu a ion Co Πwi h inpu cod(w)i is e i ied ha Ou pu (C)=No. 3 A Polynomial Complexi y Class in Cellula Sys ems In o de o gi e a o mal defini ion o compu a ional complexi y classes in his model, we ha e o fi s speci y wha we mean by a decision p oblem. Defini ion 8. Adecision p oblem,X,isapai (IX,θ X)such ha IXis a lan- guage (o e a ini e alphabe ) whose elemen s a e called ins ances o he p oblem and θXis a o al Boolean unc ion o e IX. A decision p oblem Xis sol able by a Tu ing machine TM i IXis he se o inpu s o TM, o any w∈IX he Tu ing machine hal s o e w,andwis accep ed i and only i θX(w)=1. To sol e a p oblem by means o P sys ems, we usually cons uc a amily o such de ices so ha each elemen decides he ins ances o equi alen size,ina ce ain sense which will be specified below. Defini ion 9. Le g:N+→N+be a o al compu able unc ion. We say ha a decision p oblem Xis sol able by a amily o alid language accep ing P sys ems, in a ime bounded by g, and we deno e his by X∈MCLA(g),i he eexis sa amily o P sys ems, Π=Π(n)n∈N+, wi h he ollowing p ope ies: 1. Fo e e y n∈Ni is e i ied ha Π(n)∈LA. 2. The e exis s a Tu ing machine cons uc ing Π(n) om nin polynomial ime (we say ha Πis polynomially uni o m by Tu ing machines). 3. The e exis wo unc ions, cod :IX→n∈N+IΠ(n)and s:IX→N+, compu able in polynomial ime, such ha : –Fo e e y w∈IX,cod(w)∈IΠ(s(w)). –The amily Πis bounded, wi h ega d o (X, cod, s, g); ha is, o each w∈IXe e y compu a ion o he sys em Π(s(w)) wi h inpu cod(w)is hal ing and, mo eo e , i pe o ms a mos g(|w|)s eps. –The amily Πis sound, wi h ega d o (X, cod, s); ha is, o each w∈IX i he e exis s an accep ing compu a ion o he sys em Π(s(w)) wi h inpu cod(w), henθX(w)=1. –The amily Πis comple e, wi h ega d o (X, cod, s); ha is, o each w∈IXi θX(w)=1, hen e e y compu a ion o he sys em Π(s(w)) wi h inpu cod(w)is an accep ing compu a ion. No e ha we impose a ce ain kind o con luence o he sys ems, in he sense ha e e y compu a ion wi h he same inpu mus e u n he same ou pu . As usual, he polynomial complexi y class is ob ained using as bounds he polynomial unc ions. Defini ion 10. The class o decision p oblems sol able in polynomial ime by a amily o cellula compu ing sys ems belonging o he class LA,is PMCLA = gpoly. MCLA(g). This complexi y class is closed unde polynomial- ime educibili y. P oposi ion 1. Le Xand Ybe wo decision p oblems such ha Xis poly- nomial- ime educible o Y.I Y∈PMCLA, henX∈PMCLA. 4 Simula ing De e minis ic Tu ing Machines by P Sys ems In his sec ion we conside de e minis ic Tu ing machines as language decision de ices. Tha is, he machines hal o e any s ing on he inpu alphabe , wi h he hal ing s a e equal o he accep ing s a e, in he case ha he s ing belongs o he decided language, and wi h he hal ing s a e equal o he ejec ing s a e in he case ha he s ing does no belong o he language. I is possible o associa e wi h a Tu ing machine a decision p oblem, and his will pe mi us o define wha means ha such a machine is simula ed by a amily o P sys ems. Defini ion 11. Le TM be a Tu ing machine wi h inpu alphabe ΣTM.The decision p oblem associa ed wi h TM is he p oblem XTM =(I,θ),whe eI= Σ∗ TM, and o e e y w∈Σ∗ TM,θ(w)=1i and only i TM accep s w. Ob iously, he decision p oblem XTM is sol able by he Tu ing machine TM. Defini ion 12. We say ha a Tu ing machine TM is simula ed in polynomial ime by a amily o sys ems o he class LA,i XTM ∈PMCLA. Nex we s a e ha e e y de e minis ic Tu ing machine can be simula ed in polynomial ime by a amily o sys ems o he class LA. P oposi ion 2. Le TM be a de e minis ic Tu ing machine wo king in polyno- mial ime. Then XTM ∈PMCLA. See chap e 9 o [8], which ollows ideas om [9], o de ails o he p oo . 5 Simula ing Language Accep ing P Sys ems by De e minis ic Tu ing Machines In his sec ion we a e going o p o e ha i a decision p oblem can be sol ed in polynomial ime by a amily o language accep ing P sys ems, hen i can also be sol ed in polynomial ime by a de e minis ic Tu ing machine. Fo he design o he Tu ing machine we we e inspi ed by he wo k o C. Zand on, C. Fe e i and G. Mau i [11], wi h he diffe ence ha he men ioned pape deals wi h P sys ems wi h ac i e memb anes. P oposi ion 3. Fo e e y decision p oblem sol able in polynomial ime by a amily o alid language accep ing P sys ems, he e exis s a Tu ing machine sol ing he p oblem in polynomial ime. P oo . Le Xbe a decision p oblem such ha X∈PMCLA. Then, he e exis s a amily o alid language accep ing P sys ems Π=Π(n)n∈N+such ha : 1. The amily Πis polynomially uni o m by Tu ing machines. 2. The e exis wo unc ions cod :IX→n∈N+IΠ(n)and s:IX→N+, compu able in polynomial ime, such ha : –Fo e e y w∈IX,cod(w)∈IΠ(s(w)). –The amily Πis polynomially bounded, wi h ega d o (X, cod, s). –The amily Πis sound and comple e, wi h ega d o (X, cod, s). Gi en n∈N+,le Anbe he numbe o symbols in he inpu alphabe o Π(n), Bn he numbe o symbols in he wo king alphabe , Cn he numbe o symbols in he ou pu alphabe , Dn he numbe o memb anes, En he maximum size o he mul ise s ini ially associa ed wi h hem, Fn he o al numbe o ules o he sys em, and Gn he maximum leng h o hem. Since he amily Πis polynomially uni o m by Tu ing machines, hese numbe s a e polynomial wi h espec o n. Le mbe an inpu mul ise o he sys em Π(n). Gi en a compu a ion Co Π(n) wi h inpu m,wedeno ebyHn(m) he maximum numbe o digi s, in base 2, o he mul iplici ies o he objec s con ained in he mul ise s associa ed wi h he memb anes o he sys ems and wi h he en i onmen , in any s ep o C. Na u ally, his numbe depends on C, bu wha we a e in e es ed in, and we will p o e a he end o he p oo , is ha any compu a ion o he sys em Π(s(w)) wi h inpu cod(w) e ifies ha Hs(w)(cod(w)) is polynomial in he size o he s ing w. Nex , we associa e wi h he sys em Π(n) a de e minis ic Tu ing machine, TM(n), wi h mul iple apes, such ha , gi en an inpu mul ise mo Π(n), he machine ep oduces aspecific compu a ion o Π(n)o e m. The inpu alphabe o he machine TM(n) coincides wi h ha o he sys em Π(n). On he o he hand, he wo king alphabe con ains, besides he symbols o he inpu alphabe o Π(n) he ollowing symbols: a symbol o each label as- signed o he memb anes o Π(n); he symbols 0 and 1, ha will allow o ope a e wi h numbe s ep esen ed in base 2; h ee symbols indica ing i a memb ane has no been dissol ed, has o be dissol ed o has been dissol ed; and h ee symbols ha will indica e i a ule is awai ing, is applicable o is no applicable. Subsequen ly, we speci y he apes o his machine. –We ha e one inpu ape, ha keeps a s ing ep esen ing he inpu mul ise ecei ed. –Fo each memb ane o he sys em we ha e: •One s uc u e ape, ha keeps in he second cell he label o he a he memb ane, and in he hi d cell one o he h ee symbols ha indica e i he memb ane has no been dissol ed, i he memb ane has o dissol e, o i he memb ane has been dissol ed. •Fo each objec o he wo king alphabe o he sys em: ∗One main ape, ha keeps he mul iplici y o he objec , in base 2, in he mul ise con ained in he memb ane. ∗One auxilia y ape, ha keeps empo a y esul s, also in base 2, o applying he ules associa ed wi h he memb ane.