P Systems-Based Computing Polynomials With Integer Coefficients: Design and Formal Verification

P Sys ems-Based Compu ing Polynomials Wi h In ege Coe icien s: Design and Fo mal Ve i ica ion Ming Zhu, Gexiang Zhang , Membe , IEEE , Qiang Yang, Haina Rong, Wei ao Yuan, and Ma io J. Pé ez-Jiménez Abs ac —Au oma ic design o mechanical p ocedu es sol ing abs ac p oblems is a ele an scien i ic challenge. In pa icula , au oma ic design o memb anes sys ems pe - o ming some p e ixed asks is an impo an and use ul esea ch opic in he a ea o Na u al Compu ing. In his con ex , de e minis ic memb ane sys ems we e designed in o de o cap u e he alues o polynomials wi h na u al num- be s coe icien s. Following ha wo k, his pape ex ends he p e ious esul o polynomials wi h in ege numbe s coe icien s.Speci ically,a de e minis ic ansi ionP sys em using p io i ies in he weak in e p e a ion, associa ed wi h an a bi a y such kind polynomial, is p esen ed. The con- igu a ion o he unique compu a ion o he sys em will be encoded by means o wo dis inguished objec s, he alues o he polynomial o na u al numbe s. The desc ip i e com- pu a ional esou ces equi ed by he designed memb ane sys em a e also analyzed. Index Te ms —Memb ane compu ing, P sys ems, au o- ma ic design o memb ane sys ems, polynomials wi h in ege coe icien s. I. INTRODUCTION MEMBRANE compu ing is a apidly g owingb anch o na u al compu ing ini ia ed in [1], which abs ac s compu ing models om he a chi ec u e and he unc ioning o li ing cells, as well as om he o ganiza ion o cells in issues, o gans ( he b ain included), o o he highe -deg ee s uc u es. In he pas wen y yea s, se e al classes o com- pu ing models (called P sys ems) we e in oduced, inspi ed This wo k was suppo ed in pa by he Scien i ic Resea ch Fund o Sichuan P o incial Science and Technology Depa men unde G an s 2015JY0257 and 2017FZ0010, in pa by he Na ional Na -u al Science Founda ion o China unde G an s 61672437, 61702428, and 61373047, in pa by he Sichuan Science and Technology P o-g am unde G an s 2018GZ0185, 2018GZ0085, and 2017GZ0159, and in pa by he Fundamen al Resea ch Funds o he Cen al Uni e si ies unde G an A0920502051619-36. (Co esponding au ho : Gexiang Zhang.) M. Zhu and Q. Yang a e wi h he College o Con ol Enginee ing, Chengdu Uni e si y o In o ma ion Technology, Chengdu 610225, China (e-mail: [email p o ec ed]; [email p o ec ed]). G. Zhang, H. Rong, and W. Yuan a e wi h he School o Elec ical Engi- nee ing, Sou hwes Jiao ong Uni e si y, Chengdu 610031, China (e-mail: [email p o ec ed]; [email p o ec ed]; [email p o ec ed]). M. J. Pé ez-Jiménez is wi h he Depa men o Compu e Science and A i icial In elligence, Uni e si y o Se illa, 41012 Se illa, Spain (e-mail: [email p o ec ed]). Fig. 1. Schema ic g aph showing he oolbox o a i icial neu al ne wo ks. Fig. 2. Schema ic g aph showing he aim o au oma ic design o P sys ems. om biological ac s o mo i a ed om ma hema ical o com- pu e science poin s o iew [2], [3]. Many P sys em classes a eable osimula e egis e machines and he e o e hey a e compu a ionally comple e, ha is, hey a e equi alen in powe o Tu ing machines [4]–[9]. I is well known ha some P sys ems a e e icien , in he sense ha hey ha e he abili y o sol e compu a ionally ha d p oblems by making use o an exponen ial wo kspace c ea ed in a na u al way, in polynomial ime [10]–[12]. Memb ane compu ing models ha e been used in a ious applica ions like in he a eas o app oxima e op imiza ions, sys ems and syn he ic biology and eal-li e complex p oblems [13]–[19]. Like he oolbox o a i icial neu al ne wo ks (ANN) o p oducing success ul ANNs sa is ying use s’ equi emen s, which is shown in Fig. 1, he au oma ic design o P sys ems is o de elop a me hodology o gene a ing success ul P sys ems mee ing designe s’ equi emen s, as shown in Fig. 2. This is a e y complica ed and challenging ask. So a , he me hods epo ed in he li e a u e can be classi ied in o wo g oups: heu is ic and easoning echniques [20]. The i s ype o me hods ocused on he use o heu is ic algo i hms, such as gene ic algo hms (GAs) and quan um-inspi ed e olu iona y algo i hm (QIEAs), o make a popula ion o P sys ems e ol e owa d a success ul one [15]. This kind o me hods began om he selec ion o an app op ia e subse om a edundan se o e olu ion ules o design a cell-like P sys em, whe e a memb ane s uc u e and ini ial objec s we e p e-de ined and ixed in he p ocess o design [15], [21]–[24]. In [21], a gene ic algo i hm was employed o design a P sys em o calcula e 42. In [22], a bina y encoding echnique was p esen ed o deno e an e olu ion ule se o a P sys em and a QIEA was used o make a popula ion o P sys ems e ol e owa d success ul ones. This me hod success ully sol ed he design o P sys- ems o compu e 42and n2( o na u al numbe s n≥2). In [23], an e alua ion app oach conside ing non-de e minism and hal ing penal y ac o s and a gene ic algo i hm wi h he bina y encoding echnique in [22] we e in oduced o design P sys ems o 42,n2and he gene a ion o he lan- guage {a2nb3n|n>1}. In hese s udies men ioned abo e, a speci ic edundan e olu ion ule se was designed o a speci ic compu a ional ask. This was de eloped in [15], [24] by applying one p e-de ined edundan e olu ion ule se o design mul iple di e en P sys ems, each o which execu es a compu a ion ask. In [24], an au oma ic design me hod o a cell-like P sys em amewo k o pe o ming i e basic a i h- me ic ope a ions (addi ion, sub ac ion, mul iplica ion, di ision and powe ) was p esen ed. In [15], a common edundan se o e olu ion ules was applied o design success ul P sys ems o ul illing eigh compu a ional asks, i.e., eigh compu ing se s o na u al numbe s: 2(n−1),2n−1, n2,1 2[n(n−1)], n(n−1),(n−1)2+2n+2, a2nb3nand 1 2(3n−1),(n>1o 2). A signi ican de elopmen in his opic is he wo k in [25] in which a cell-like hal ing P sys em o 42was designed by uning memb ane s uc u es, ini ial objec s and e olu ion ules. In ha wo k, a gene ic algo i hm wi h a bina y encod- ing echnique was discussed o codi y he h ee ing edien s o a P sys em, he memb ane s uc u e, ini ial objec s and e olu ion ules. Following his wo k, an au oma ic design me hod, Pe mu a ion Penal y Gene ic Algo i hm (PPGA), o a de e minis ic and non-hal ing memb ane sys em by uning memb ane s uc u es, ini ial objec s and e olu ion ules was p oposed in [26]. The main ideas o PPGA a e he in oduc ion o he pe mu a ion encoding echnique o a memb ane sys em, a penal y unc ion e alua ion app oach o a candida e mem- b ane sys em and a gene ic algo i hm o making a popula ion o P sys ems e ol e owa d a success ul one ul illing a gi en compu a ional ask. A cell-like memb ane sys em o compu ing he squa e o n2( o na u al numbe s n≥1) was success ully designed. In addi ion, he au oma ic design o he minimal memb ane sys ems wi h espec o hei memb ane s uc u es, alphabe , ini ial objec s and e olu ion ules o ul ill he gi en ask we e also discussed in [26]. The second ype o me hods use easoning echniques o ul ill he design o a P sys em. In [27], a easoning me hod o design a k-deg ee (k ≥ 2) polynomial P sys em was epo ed by ana- lyzing he syn ax and seman ics o cell-like P sys ems. In he s udy o [27], de e minis ic ansi ion P sys ems o compu ing polynomials wi h na u al numbe coe icien s we e designed. The nume ical alues o such polynomials, p(n), o n ∈ N, a e always posi i e and he P sys ems compu - ing p(n) handle only posi i e numbe s h ough he mul iplic- i y o objec s in an usual manne . In his pape , he wo k in [27] is ex ended o conside he design o de e minis ic ansi ion P sys ems o compu ing polynomials wi h in ege coe icien s, whe e he nume ical alues o p(n), o n ∈ N, may be posi i e o nega i e and, consequen ly, he P sys ems compu ing p(n) mus p ocess in ege numbe s by using na u al numbe s in he mul iplici y o objec s. This ask is much mo e challenging. The aim o his pape is o ind a “minimal” such P sys em compu ing an a bi a y polynomial wi h in ege coe icien s. He e he concep “minimal” e e s o some syn ac ical ing e- dien s associa ed wi h P sys ems: he memb ane s uc u e has only one memb ane and he numbe o objec s used is e y es ic i e. The es pa s o his pape a e o ganized as ollows. Sec ion II ecalls some p elimina ies needed in he ollowing sec ions, including he speci ic a ian o memb ane sys ems conside ed in his wo k. The main concep o polynomial wi h in ege coe icien s compu ed by a de e minis ic an- si ion P sys em is de ined in Sec ion III. The design and o mal e i ica ion o a de e minis ic P sys em associa ed wi h an a bi a y polynomial whose coe icien s a e in ege num- be s, is p esen ed in Sec ion III-A. The desc ip i e compu a- ional esou ces equi ed by he designed k-deg ee polynomial P sys em is analyzed in Sec ion IV. The compa ison wi h me aheu is ic app oaches is discussed in Sec ion V. Finally, conclusions and u u e wo k a e gi en in Sec ion VI. II. PRELIMINARIES In his sec ion, some gene al concep s a e b ie ly desc ibed in o de o make he wo k sel -con ained. A. Alphabe and Mul ise s An alphabe is a non-emp y se and hei elemen s a e called symbols.As ing u o e is an o de ed ini e sequence o symbols, ha is, a mapping om a na u al numbe n∈N on o . The numbe nis called he leng h o he s ing u and i is deno ed by |u|. The emp y s ing (wi h leng h 0) is deno ed by λ.Amul ise o e an alphabe is a mapping om on o he se o na u al numbe s N. Fo each symbol a∈, he na u al numbe (a)is called he mul iplici y o symbol ain mul ise . We deno e by M() he se o all mul ise s o e . B. Roo ed T ee An undi ec ed g aph G is an o de ed pai (V,E),whe eV is a se whose elemen s a e called nodes and E={{x,y}| x,y∈V,x= y}whose elemen s a e called edges.Apa h o leng h k≥1 om x∈V o y∈Vis a sequence (x0,...,xk) such ha x0=xand xk=y.I x0=xk hen we say ha he pa h is a cycle. An undi ec ed g aph is connec ed i e e y pai o nodes is connec ed by a pa h. An undi ec ed g aph wi h no cycle is said o be acyclic.A oo ed ee is a connec ed, acyclic, undi ec ed g aph in which one o he e ices (called he oo o he ee) is dis inguished om he o he s. C. T ansi ion P Sys ems The basic model o memb ane sys ems was in oduced by Gh. P˘aun in i s seminal pape [1]. A ansi ion P sys em o deg ee q≥1 is a uple =(, μ, M1,...,Mq,(R1,ρ 1),...,(Rq,ρ q), iou ), whe e: –is a ini e alphabe . –μis a oo ed ee. –M1,...,Mqa e mul ise s o e . –Ri,1≤i≤q, is a ini e se o e olu ion ules o he ollowing o ms: (a) [u]i→ 1[ 2[ 3]j]i;and (b) [u]i→ 1[ 2[ 3]j]iδ,whe ei,j∈{1,...,q}, i= j,u, 1, 2, 3∈M() and δis a dis inguished symbol such ha δ/∈. –ρi,1≤i≤q, is an s ic pa ial o de o e Ri. –iou ∈{0,1,...,q}. A ansi ion P sys em =(, μ, M1,...,Mq, (R1,ρ 1),...,(Rq,ρ q), iou ),o deg eeq≥1 can be iewed as a se o qmemb anes injec i ely labeled by 1,...,q, a anged in a hie a chical s uc u e μgi en by a oo ed ee whose oo is called he skin memb ane o he sys em, and wi h an en i onmen labeled by 0 such ha : (a) M1,...,Mq a e mul ise s o e he wo king alphabe  ep esen ing he objec s ini ially placed in he qmemb anes o he sys em; (b) Ri,1≤i≤n, is he se o ules associa ed wi h memb ane i,andρip o ides p io i ies be ween ules in Ri, in such a manne ha i ( 1, 2)∈ρiwe say ha ule 1has a highe p io i y han 2and we deno e i by 1> 2;and (c) iou ∈{1,...,q} ep esen s a dis inguished memb ane ( he ou pu memb ane). Acon igu a ion a an ins an o a ansi ion P sys em is desc ibed by he memb ane s uc u e a ins an and all mul ise s o objec s o e associa ed wi h all he memb anes p esen in he sys em. The ini ial con igu a ion o he sys em is (μ, M1,··· ,Mq). Gi en a ansi ion P sys em ,wesay ha con igu a ion C yields con igu a ion C +1in one ansi ion s ep, i we can pass om C o C +1by applying he ules om R1,...,Rqsynch onously, in a non-de e minis ic maximally pa allel manne . This means he ollowing: he objec s o e ol e in a ansi ion s ep and he ules by which hey e ol e a e chosen in a non-de e minis ic manne , bu in such a way ha in each memb ane we ha e a maximally pa allel applica ion o ules (a each ansi ion s ep a mul ise o ules which is maximal is applied, no u he applicable ule can be added). A compu a ion o is a ( ini e o in ini e) sequence o con igu a ions such ha : (a) he i s e m o he sequence is he ini ial con igu a ion o he sys em; (b) each non- i s e m o he sequence is ob ained om he p e ious con igu a ion by applying ules o he sys em in a non-de e minis ic maximally pa allel manne ; and (c) i he sequence is ini e hen he las e m o he sequence is a con igu a ion, whe e no ule o he sys em is applicable o i . I is wo h poin ing ou ha in his pape he p io i y be ween ules is used in he weak in e p e a ion, ha is,ina ansi ion s ep a ule is used always when objec s exis , which we e no used by a ule o a highe p io i y. In his pape we deal wi h de e minis ic ansi ion P sys ems, whe e he e is only one compu a ion s a ing om an ini ial con igu a ion. Besides, only ules o he ype [u]i→[ ]iwill be used and hey a e b ie ly deno ed by u→ when he memb ane being wo ked wi h is unde s ood. Le us conside wo auxilia y unc ions +and − om he se o in ege numbe s Zin o he se o na u al numbe s N, de ined as ollows: +(x)=xi x≥0 0i x<0 −(x)=0i x≥0 −xi x<0 I is wo h poin ing ou ha o each in ege numbe x ∈ Z we ha e +(x) ≥ 0, −(x) ≥ 0, +(x) + −(x) =|x| and +(x) − −(x) = x. III. DETERMINISTIC TRANSITION PSYSTEMS COMPUTING POLYNOMIALS WITH INTEGER COEFFICIENTS In his sec ion we de ine he meaning o compu ing a polynomial p(n) whose coe icien s a e in ege numbe s, by a de e minis ic ansi ion P sys em p(n) associa ed wi h i . The idea is he ollowing: o each na u al numbe ∈ N he alue p( ) will be compu ed/encoded by he con igu a ion C +1 o he unique compu a ion o p(n). Fo ha , ou dis inguished objec s (o1, o2, p1, p2) will be conside ed in he wo king alphabe o p(n), in such a manne ha o1, o2 will be used o encode/ ep esen in ege numbe s by means o hei mul iplici ies, and p1, p2 will be used as hei co esponding ansi ion compu ing objec s. De ini ion 1: Le p(n) be a polynomial wi h in ege numbe s coe icien s. We say ha p(n) is compu ed by a de e minis ic ansi ion P sys em p(n)=(, μ, M1,...,Mq,(R1,ρ 1),...,(Rq,ρ q), iou ) i he ollowing holds: •The wo king alphabe has ou dis inguished objec s: o1,o2( he ou pu objec s) and p1,p2( ansi ion compu - ing objec s). •Fo each ∈N, a con igu a ion C +1 he con en o he ou pu memb ane labeled by iou encodes he alue p( ) h ough he mul iplici y o objec s o1and o2as ollows: (a) I p( )≥0 hen he mul iplici y o o1is p( )and he mul iplici y o o2is 0; and (b) i p( )<0 hen he mul iplici y o o2is −p( )and he mul iplici y o o1is 0. A. Design In his sec ion, a de e minis ic ansi ion P sys em p(n) o deg ee 1 ha compu es, in he sense o De ini ion 1, he polynomial p(n)=a0+a1·n···+ak·nko deg ee k≥1, wi h in ege coe icien s ai∈Z,0≤i≤k, is designed. I is easy o check ha o each na u al numbe ∈N he ollowing holds: p( +1)−p( ) =[a11 0+a22 0+···+ak−1k−1 0+akk 0]· 0 +[a22 1+···+ak−1k−1 1+akk 1]· 1 ............................................. +[ak−1k−1 k−2+akk k−2]· k−2 +[akk k−1]· k−1 Le us deno e: a0 k=a11 0+a22 0+···+ak−1k−1 0+akk 0 a1 k=a22 1+···+ak−1k−1 1+akk 1 ....................................... ak−2 k=ak−1k−1 k−2+akk k−2 ak−1 k=akk k−1 Then, p( +1)−p( )=a0 k+a1 k· +a2 k· 2+···+ak−2 k· k−2+ ak−1 k· k−1= k−1  i=0 ai k· i, ha is,p( +1)=p( )+ k−1  i=0 ai k· i. De ini ion 2: Le p(n)=a0+a1·n+ ··· + ak·nk be a polynomial o deg ee k≥1, wi h in ege coe icien s ai∈Z,0≤i≤k. We associa e p(n)wi h he de e minis ic ansi ion P sys em p(n)=(, μ, M1,(R1,ρ 1), iou )o deg ee 1, de ined as ollows: •={o1,o2,p1,p2,b1,b2,b3,···bk} •μ=[] 1 •M1=o +(a0) 1o −(a0) 2b1 •R1is he se o he ollowing e olu ion ules: 1≡b1→p +(a0 k) 1p −(a0 k) 2b(0 0) 1b(1 0) 2b(2 0) 3···b(k−2 0) k−1b(k−1 0) k 2≡b2→p +(a1 k) 1p −(a1 k) 2b(1 1) 2b(2 1) 3···b(k−2 1) k−1b(k−1 1) k 3≡b3→p +(a2 k) 1p −(a2 k) 2b(2 2) 3···b(k−2 2) k−1b(k−1 2) k . . . − k−1≡bk−1→p +(ak−2 k) 1p −(ak−2 k) 2b(k−2 k−2) k−1b(k−1 k−2) k k≡bk→p +(ak−1 k) 1p −(ak−1 k) 2b(k−1 k−1) k k+1≡p1p2→λ k+2≡p1o2→λ k+3≡p2o1→λ k+4≡p1→o1 k+5≡p2→o2 •ρ1is he se o p io i ies ela ion among ules in R1: {( k+1, k+2), ( k+1, k+3), ( k+2, k+4), ( k+2, k+5), ( k+3, k+4), ( k+3, k+5)}which can be in o mally desc ibed as: k+1>{ k+2, k+3}>{ k+4, k+5}. •iou =1. B. Fo mal Ve i ica ion We show in his subsec ion ha he memb ane sys em p(n) associa ed wi h he polynomial p(n), designed in he p e ious sec ion, compu es he alues p( )acco ding o De ini ion 1, o each ∈N. Theo em 1: Le p(n)=a0+a1·n+···+ak·nkbe a polynomial o deg ee k≥1 such ha ai∈Z,0≤i≤k. Le p(n)be he de e minis ic ansi ion P sys em conside ed in De ini ion 1. Fo each ≥0, a con igu a ion C +1 he con en o memb ane labeled by 1 is he ollowing mul ise : {o +(p( )) 1o −(p( )) 2pk−1 i=0 +(ai k)· i 1pk−1 i=0 −(ai k)· i 2 b1b( +1) 2b( +1)2 3··· b( +1)k−1 k} P oo : Le us p o e he esul by induc ion on . Le us s a wi h he base case =0. A he ini ial con igu a ion C0, he con en o memb ane labeled by 1 is he mul ise o +(a0) 1o −(a0) 2b1. Then, con igu a ion C0yields con igu a ion C1by applying ule 1once. Thus, a con igu- a ion C1 he con en o memb ane labeled by 1 is he mul- ise o +(a0) 1o −(a0) 2p +a0 k 1p −a0 k 2b1b2b3··· bk. Because o p(0)=a0= +(a0)− −(a0), he esul holds o =0. By induc ion hypo hesis, le us assume he esul holds o ≥0, ha is, a con igu a ion C +1 he con en o memb ane labeled by 1 is he mul ise {o +(p( )) 1o −(p( )) 2pk−1 i=0 +(ai k)· i 1pk−1 i=0 −(ai k)· i 2 b1b( +1) 2b( +1)2 3··· b( +1)k−1 k}. In o de o ob ain he con en o memb ane labeled by 1 a con igu a ion C +2, le us analyze all he possible cases ha may happen: Case 1: k−1  i=0 ai k· i≥ −(p( )) In his case, p( +1)−p( )≥ −(p( )) and he ollowing holds: (a) k−1  i=0 +(ai k)· i− k−1  i=0 −(ai k)· i≥ −(p( )). Indeed, i is su icien o no ice ha ai k= +(ai k)− −(ai k), o 0≤ i≤k−1. (b) k−1  i=0 +(ai k)· i≥ k−1  i=0 −(ai k)· i. Indeed, (b) ollows om (a) ecalling ha −(p( )) ≥0. (c) p( +1)≥0. Indeed, i p( )≥0 henp( +1)≥p( )+ −(p( )) ≥0, and i p( )<0 hen −(p( )) =−p( ), so p( +1)≥p( )+ −(p( )) =0. The e o e, in his case con igu a ion C +1yields con igu a- ion C +2as ollows: (1) F om (b) we deduce ha a con igu a ion C +1in mem- b ane labeled by 1 he e a e mo e copies o objec p1 han copies o objec p2,so ule k+1≡p1p2→λwill be applied k−1  i=0 −(ai k)· i imes, consuming all copies o p2and “ emaining” k−1  i=0 +(ai k)· i− k−1  i=0 −(ai k)· i copies o p1wi hou e ol ing. (2) F om (a) we deduce ha a con igu a ion C +1in memb ane labeled by 1 he e a e mo e copies o he “ emaining” objec p1 han copies o objec o2,so ule k+2≡p1o2→λwill be applied −(p( )) imes, consuming all copies o o2and “ emaining” k−1  i=0 +(ai k)· i− k−1  i=0 −(ai k)· i− −(p( )) copies o p1wi hou e ol ing. (3) Rule k+4≡p1→o1will be applied α= k−1  i=0 +(ai k)· i− k−1  i=0 −(ai k)· i− −(p( )) imes, consuming all copies o p1and p oducing α copies o o1. (4) Fo each j,1≤j≤k, ule j≡bj→p +(aj−1 k) 1 p −(aj−1 k) 2b(j−1 j−1) j...b(k−1 j−1) kwill be applied ( +1)j−1 imes. All he p e ious ules a e applied in pa allel in one ansi ion s ep. Thus, in his case, a con igu a ion C +2 he con en o memb ane labeled by 1 is he mul ise which con ains objec s o1,o2,p1,p2and bj,(1≤j≤k)wi h he ollowing mul iplici ies: –Mul iplici y o o1: +(p( +1)). Indeed, a e execu ion o ules om (2), αnew copies o o1a e p oduced. Thus, he o al numbe o copies o o1 will be: +(p( )) + k−1  i=0 +(ai k)· i− k−1  i=0 −(ai k)· i− −(p( )) =p( )+ k−1  i=0 ai k· i=p( +1)(c) = +(p( +1)). –Mul iplici y o o2:0. Indeed, a e execu ion o ules om (2), all copies o objec o2a e consumed and any new copies o objec o2 a e p oduced o any ule applied in his ansi ion s ep. Thus, a con igu a ion C +2in memb ane labeled by 1 he mul iplici y o o2is 0. –Mul iplici y o p1: k−1  i=0 +(ai k)·( +1)i. Indeed, a e execu ion o ules om (1), (2) and (3), all copies o objec p1a e consumed bu by applying ules om (4), he o al numbe o copies o p1p oduced is k  j=1 +(aj−1 k)·( +1)j−1= k−1  i=0 +(ai k)·( +1)i. –Mul iplici y o p2: k−1  i=0 −(ai k)·( +1)i. Indeed, a e execu ion o ules om (1), (2) and (3), all copies o objec p2a e consumed bu by applying ules om (4), he o al numbe o copies o p2p oduced is k  j=1 −(aj−1 k)·( +1)j−1= k−1  i=0 −(ai k)·( +1)i. –Mul iplici y o objec bj, o each j,1≤j≤k: ( +2)j−1. Indeed, om (3) he o al numbe o copies o objec bj p oduced is j−1  s=0j−1 s( +1)s=[( +1)+1]j−1=( +2)j−1 Hence, in his case he esul holds o +1. Case 2: 0≤ k−1  i=0 ai k· i< −(p( )) In his case, he ollowing holds: (a) p( )<0andp( +1)<0. Indeed, on he one hand, as −(p( )) > 0weha e −(p( )) =−p( ). On he o he hand, p( +1)=p( )+ k−1  i=0 ai k· i=− −(p( )) + k−1  i=0 ai k<0. (b) 0≤ k−1  i=0 +(ai k)· i− k−1  i=0 −(ai k)· i< −(p( )). Indeed, i su ices o no ice ha ai k= +(ai k)− −(ai k), o 0≤ i≤k−1. (c) p( +1)=k−1  i=0 +(ai k)· i− k−1  i=0 −(ai k)· i− −(p( )). Indeed, as −(p( )) > 0weha ep( )<0and −(p( )) =−p( ).So, p( +1)=p( )+ k−1  i=0 ai k· i =− −(p( )) + k−1  i=0 +(ai k)· i− k−1  i=0 −(ai k)· i. The e o e, in his case con igu a ion C +1yields con igu a ion C +2as ollows: (1) F om (b) we deduce ha a con igu a ion C +1in mem- b ane labeled by 1 he e a e mo e copies o objec p1 han copies o objec p2,so ule k+1≡p1p2→λwill be applied k−1  i=0 −(ai k)· i imes, consuming all copies o p2and “ emaining” k−1  i=0 +(ai k)· i− k−1  i=0 −(ai k)· i copies o p1wi hou e ol ing. (2) Con igu a ion C +1in memb ane labeled by 1 he e a e mo e copies o objec o2 han copies o objec p1, hen ule k+2≡p1o2→λ will be applied k−1  i=0 +(ai k)· i− k−1  i=0 −(ai k)· i imes, consuming all copies o p1and “ emaining” −(p( ))− k−1  i=0 +(ai k)· i− k−1  i=0 −(ai k)· icopies o o2wi hou e ol ing. (3) Fo each j,1≤j≤k, ule j≡bj→p +(aj−1 k) 1 p −(aj−1 k) 2b(j−1 j−1) j...b(k−1 j−1) kwill be applied ( +1)j−1 imes. All he p e ious ules a e applied in pa allel in one ansi ion s ep. Thus, in his case, a con igu a ion C +2 he con en o memb ane labeled by 1 is he mul ise which con ains objec s o1,o2,p1,p2and bj,(1≤j≤k)wi h he ollowing mul iplici ies: –Mul iplici y o objec o1: +(p( +1)). Indeed, he applied ules do no a ec o objec o1and om (a) we deduce ha +(p( +1)) =0= +(p( )). –Mul iplici y o objec o2: −(p( +1)). Indeed, a e execu ion o he ci ed ules, he mul iplici y o o2is −(p( )) −k−1  i=0 +(ai k)· i− k−1  i=0 −(ai k)· i (b) =−p( +1)(c) = −(p( +1)). –Mul iplici y o objec p1: k−1  i=0 +(ai k)·( +1)i. Indeed, a e execu ion o he ules om (1) and (2), he mul iplici y o p1is 0, bu a e he applica ion o ules om (3) i s mul iplici y becomes k  i=1 +(ai−1 k)·( +1)i−1= k−1  i=0 +(ai k)·( +1)i –Mul iplici y o objec p2: k−1  i=0 −(ai k)·( +1)i. Indeed, because a e execu ion o he ules om (1) and (2), he mul iplici y o p2is 0, bu a e he applica ion o ules om (3) i s mul iplici y becomes k  i=1 −(ai−1 k)·( +1)i−1= k−1  i=0 −(ai k)·( +1)i –Mul iplici y o objec bj, o each j,1≤j≤k: ( +2)j−1. Indeed, om (3) he o al numbe o copies o objec bj p oduced is j−1  s=0j−1 s( +1)s=[( +1)+1]j−1=( +2)j−1 Hence, in his case he esul holds o +1. Case 3: k−1  i=0 ai k· i<0∧ +(p( )) + k−1  i=0 ai k· i≤0 In his case, he ollowing holds: (a) k−1  i=0 +(ai k)− k−1  i=0 −(ai k)· i<0. Indeed, i su ices o bea in mind ha k−1  i=0 ai k· i<0, and ai k= +(ai k)− −(ai k), o 0≤i≤k−1. (b) +(p( )) ≤−k−1  i=0 +(ai k)· i− k−1  i=0 −(ai k)· i Indeed, i is enough o no ice ha +(p( )) ≤− k−1  i=0 ai k· i =− k−1  i=0 +(ai k)· i− k−1  i=0 −(ai k)· i (c) p( +1)≤0. Indeed, i p( )≥0 henp( )= +(p( )) ≤− k−1  i=0 ai k· i and p( +1)=p( )+ k−1  i=0 ai k· i;i p( )<0 henp( +1)= p( )+ k−1  i=0 ai k· i<0. The e o e, in his case con igu a ion C +1yields con igu a ion C +2as ollows: (1) F om (a) we deduce ha a con igu a ion C +1in mem- b ane labeled by 1 he e a e mo e copies o objec p2 han copies o objec p1,so ule k+1≡p1p2→λwill be applied k−1  i=0 +(ai k)· i imes, consuming all copies o p1and “ emaining” k−1  i=0 −(ai k)· i− k−1  i=0 +(ai k)· i copies o p2wi hou e ol ing. (2) F om (b) we deduce ha a con igu a ion C +1in mem- b ane labeled by 1 he e a e mo e copies o objec p2 han copies o objec o1,so ule k+3≡p2o1→λ will be applied +(p( )) imes, consuming all copies o o1and “ emaining” − +(p( )) + k−1  i=0 ai k· icopies o p2wi hou e ol e bu hese copies mus e ol e by means o ule k+5. (3) Rule k+5≡p2→o2will be applied − +(p( )) + k−1  i=0 ai k· i imes, consuming all copies o p2and p oducing − +(p( )) + k−1  i=0 ai k· inew copies o objec o2. (4) Fo each j,1 ≤j≤k, ule j≡ bj→p +(aj−1 k) 1p −(aj−1 k) 2b(j−1 j−1) j...b(k−1 j−1) kwill be applied ( +1)j−1 imes. All he p e ious ules a e applied in pa allel in one ansi ion s ep. Thus, in his case, a con igu a ion C +2 he con en o memb ane labeled by 1 is he mul ise which con ains objec s o1,o2,p1,p2and bj,(1≤j≤k)wi h he ollowing mul iplici ies: –Mul iplici y o objec o1: +(p( +1)). Indeed, om (2) all copies o objec o1a e consumed, bu +(p( +1)) (c) =0. –Mul iplici y o objec o2: −(p( +1)). Indeed, a e execu ion o he ules, i s mul iplici y will be −(p( )) − +(p( )) + k−1  i=0 ai k· i =−p( )− k−1  i=0 ai k· i =−p( +1)(c) = −(p( +1)). –Mul iplici y o objec p1: k−1  i=0 +(ai k)·( +1)i. Indeed, a e execu ion o he ules om (1) all copies o p1a e consumed bu om (4) he p oduced copies a e he ollowing: k  i=1 +(ai−1 k)·( +1)i−1= k−1  i=0 +(ai k)·( +1)i –Mul iplici y o objec p2: k−1  i=0 −(ai k)·( +1)i. Indeed, a e execu ion o he ules om (1), (2) and (3), all copies o objec p2a e consumed bu by applying ules in (4) new copies o p2a e p oduced, in o al he numbe o copies will be: k  i=1 −(ai−1 k)·( +1)i−1= k−1  i=0 −(ai k)·( +1)i –Mul iplici y o objec bj, o each j,1≤j≤k: ( +2)j−1. Indeed, om (4) he o al numbe o copies o objec bj p oduced is j−1  s=0j−1 s( +1)s=[( +1)+1]j−1=( +2)j−1 Hence, in his case he esul holds o +1. Case 4: k−1  i=0 ai k· i<0∧ +(p( )) + k−1  i=0 ai k· i>0 In his case, he ollowing holds: (a) k−1  i=0 +(ai k)− k−1  i=0 −(ai k)· i<0. Indeed, i is enough o no ice ha k−1  i=0 ai k· i<0, and ai k= +(ai k)− −(ai k), o 0≤i≤k−1. (b) +(p( )) > −k−1  i=0 +(ai k)· i− k−1  i=0 −(ai k)· i. Indeed, i su ices o bea in mind ha +(p( )) > − k−1  i=0 ai k· i=−k−1  i=0 +(ai k)· i− k−1  i=0 −(ai k)· i and ai k= +(ai k)− −(ai k), o 0≤i≤k−1. (c) p( )>0andp( +1)>0. Indeed, om he hypo hesis in his case we ha e +(p( )) > − k−1  i=0 ai k· i>0. So, p( )>0and +(p( )) =p( ). Thus, p( +1)=p( )+ k−1  i=0 ai k· i= +(p( )) + k−1  i=0 ai k· i>0 The e o e, in his case con igu a ion C +1yields con igu a ion C +2as ollows: (1) F om (a) we deduce ha a con igu a ion C +1in mem- b ane labeled by 1 he e a e mo e copies o objec p2 han copies o objec p1,so ule k+1≡p1p2→λwill be applied k−1  i=0 +(ai k)· i imes, consuming all copies o p1and “ emaining” k−1  i=0 −(ai k)· i− k−1  i=0 +(ai k)· i copies o p2wi hou e ol ing. (2) F om (b) we deduce ha a con igu a ion C +1in mem- b ane labeled by 1 he e a e mo e copies o objec o1 han copies o objec p2,so ule k+3≡p1o1→λ will be applied k−1  i=0 −(ai k)· i− k−1  i=0 +(ai k)· i imes, consuming all copies o p2and “ emaining” +(p( ))− k−1  i=0 −(ai k)· i− k−1  i=0 +(ai k)· icopies o o1wi hou e ol ing. (3) Fo each j,1 ≤j≤k, ule j≡bj→ p +(aj−1 k) 1p −(aj−1 k) 2b(j−1 j−1) j...b(k−1 j−1) kwill be applied ( + 1)j−1 imes. All he p e ious ules a e applied in pa allel in one ansi ion s ep. Thus, in his case, a con igu a ion C +2 he con en o memb ane labeled by 1 is he mul ise which con ains objec s o1,o2,p1,p2and bj,(1≤j≤k)wi h he ollowing mul iplici ies: –Mul iplici y o objec o1: +(p( +1)). Indeed, om (2) we deduce ha he numbe o copies o objec o1is: +(p( )) −k−1  i=0 −(ai k)· i− k−1  i=0 +(ai k)· i = +(p( )) + k−1  i=0 ai k· i(c) =p( )+ k−1  i=0 ai k· i =p( +1)(c) = +(p( +1)) all copies o objec o1a e consumed, bu +(p( +1)) (c) =0. –Mul iplici y o objec o2: −(p( +1)). Indeed, objec o2is no in ol ed by he applica ion o ules o each con igu a ion C +2 om con igu a ion C +1, so he mul iplici y o o2is −(p( )) (c) =0(c) = −(p( +1)). –Mul iplici y o objec p1: k−1  i=0 +(ai k)·( +1)i. Indeed, om (1) all copies o objec p1a e consumed bu om (3) he o al numbe o p oduced copies is k  j=1 +(aj−1 k)·( +1)j−1= k−1  i=0 +(ai k)·( +1)i. –Mul iplici y o objec p2: k−1  i=0 −(ai k)·( +1)i. Indeed, om (1) all copies o objec p2a e consumed bu om (3) he o al numbe o p oduced copies is k  j=1 −(aj−1 k)·( +1)j−1= k−1  i=0 −(ai k)·( +1)i. –Mul iplici y o objec bj, o each j,1≤j≤k: ( +2)j−1. Indeed, om (3) he o al numbe o copies o objec bj p oduced is j−1  s=0j−1 s( +1)s=[( +1)+1]j−1=( +2)j−1 Hence, in his case he esul holds o +1. Co olla y 1: Le p(n)=a0+a1·n+ ··· + ak·nkbe a polynomial o deg ee ksuch ha ai∈Z,i=0,1,...,k. Le p(n)be he de e minis ic ansi ion P sys em conside ed in De ini ion 2. Then, polynomial p(n)is compu ed by he sys em p(n)acco ding wi h De ini ion 1. P oo : F om Theo em 1 we deduce ha o each ∈N a con igu a ion C +1 he con en o memb ane labeled by 1 is he ollowing mul ise : {o +(p( )) 1o −(p( )) 2pk−1 i=0 +(ai k)· i 1pk−1 i=0 −(ai k)· i 2 b1b( +1) 2b( +1)2 3··· b( +1)k−1 k} In o de o know he mul iplici y o objec s o1and o2in memb ane labeled by 1 a con igu a ion C +1, wo cases a e dis inguished: •I p( )≥0 hen +(p( )) =p( )and −(p( )) =0. So, he mul iplici y o o1is +(p( )) =p( )and he mul iplici y o o2is −(p( )) =0. •I p( )<0 hen +(p( )) =0and −(p( )) =−p( ). Thus, he mul iplici y o o1is +(p( )) =0and he mul iplici y o o2is −(p( )) =−p( ). IV. DESCRIPTIVE COMPUTATIONAL RESOURCES In his sec ion, he desc ip i e compu a ional esou ces equi ed by he de e minis ic ansi ion P sys em p(n) con- side ed in De ini ion 2 which compu es polynomial p(n) wi h in ege numbe s coe icien s, is depic ed. •The size o he wo king alphabe : k+4. •The ini ial numbe o objec s: 1 +|a0|. •The numbe o ules: k+5. •The o al numbe o objec s in ol ed in he ules is 2k+k+9+ k−1 |ai k|. i=0 Hence, he o al amoun o desc ip i e compu a ional esou ces is exponen ial in he size o he polynomial. V. DISCUSSIONS Un il now, wo kinds o me hods ha e been epo ed in li - e a u e o implemen au oma ic design o memb ane sys ems. One is he easoning way p esen ed in his pape and [27], which is called REASON. The o he is he me aheu is- ic app oaches (META) al eady used in memb ane sys ems design, such as gene ic algo i hms [21], [23], [25], in pa icula Pe mu a ion Penal y Gene ic Algo i hms (PPGAs) [26], and quan um-inspi ed e olu iona y algo i hm (QIEAs) [22], [24]. REASON and META ha e he ollowing di e ences: •Concep : META uses a me aheu is ic app oach o e ol e a popula ion o candida e P sys ems ( easi- ble o in easible) owa d he success ul P sys ems, while REASON uses induc i e me hod ( om simple o com- plex P sys ems) o ob ain he success ul P sys ems. A me aheu is ic app oach may be a gene ic algo i hm, a quan um-inspi ed e olu iona y algo i hm o o he s. •Usage: META is qui e easy o unde s and and mas e o a beginne , while REASON sounds a qui e complex echnique o a beginne . •Gene a ion: META is a mo e gene al echnique han REASON and he e o e i is possible o use META o design di e en P sys ems. While in REASON, di e en P sys ems a e designed by using di e en speci ic ea- soning echniques. •Resou ce: In REASON, i is possible o calcula e he esou ce equi ed by a P sys em wi h espec o com- pu ing ime and wo kspace. While in META, i is qui e ha d o summa ize he esou ce. •So wa e: The e alua ion o a success ul P sys em in META is pe o med by using he well-known P sys em simula o , P-Lingua [28]. REASON does no need any so wa e. •Ex endibili y: REASON can be easily ex ended om a speci ic o a gene al P sys em, e.g., om a low-deg ee o high-deg ee polynomial P sys em, o a kind o memb ane sys em. This ex endibili y is no sui able o META. VI. CONCLUSION This pape ex ends he wo k in [27] om he au oma ic design o de e minis ic ansi ion P sys ems o compu ing polynomials wi h na u al numbe coe icien s o he au oma ic design o such kind o memb ane sys ems o compu ing poly- nomials wi h in ege coe icien s, by analyzing he syn ac ical and seman ics ing edien s o cell-like memb ane sys ems. This is a signi ican s ep o he p og ammabili y o memb ane sys ems, namely how o au oma ically design a P sys em by using p og ams so as o de elop a use ul oolbox o he communi y o memb ane compu ing. As u u e wo k we plan o ex end his me hod in o de o design new a ian s o memb ane sys ems wi h he capabili y o pe o ming mo e complex asks like inding he mini- mal memb ane sys em, wi h espec o he numbe o used objec s, o a gi en assignmen o like p ac ical applica ions such as memb ane con olle s o mobile obo s. On he o he hand, we p opose: (a) o de elop so wa e pla o ms o simula e ansi ion P sys ems using a weak in e p e a ion o he p io i ies as well as FPGA (Field P og ammable Ga e A ay) based ha dwa e o implemen hem; and (b) he use memb ane-inspi ed e olu iona y algo i hms [15], [29], [30] o op imiza ion spiking neu al P sys ems [31] o implemen he au oma ic design o a memb ane sys ems (including spiking neu al P sys ems) o sol ing compu a ionally ha d p oblems. REFERENCES [1] Gh. 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