In tissue P systems several cells (elementary membranes) commu-
nicate through symport/antiport rules, thus carrying out a computation. We
add to such systems the basic feature of (cell) P systems with active membranes
{ the possibility to divide cells. As expected (as it is the case for P systems
with active membranes), in this way we get the possibility to solve computa-
tionally hard problems in polynomial time; we illustrate this possibility with
SAT problem.Ministerio de Ciencia y Tecnología TIC2002-04220-C03-0

Paun, Gheorghe

Pérez Jiménez, Mario de Jesús

Riscos Núñez, Agustín

idUS. Depósito de Investigación Universidad de Sevilla

Tissue P Systems with Cell DivisionGheorghe PA˘UN1,2, Mario PE´REZ-JIME´NEZ2,Agust´ın RISCOS-NU´N˜EZ21Institute of Mathematics of the Romanian AcademyPO Box 1-764, 014700 Bucures¸ti, Romania2 Research Group on Natural ComputingDepartment of Computer Science and Artificial IntelligenceTechnical Higher School of Computer Science EngineeringUniversity of SevillaAvda. Reina Mercedes s/n, 41012 Sevilla, SpainE-mail: {gpaun, marper, ariscosn}@us.esAbstract. In tissue P systems several cells (elementary membranes) commu-nicate through symport/antiport rules, thus carrying out a computation. Weadd to such systems the basic feature of (cell) P systems with active membranes– the possibility to divide cells. As expected (as it is the case for P systemswith active membranes), in this way we get the possibility to solve computa-tionally hard problems in polynomial time; we illustrate this possibility withSAT problem.1 IntroductionIn membrane computing, there are two main classes of P systems: with the membranesarranged hierarchically, inspired from the structure of the cell, and with the membranesplaced in the nodes of a graph, all of them at the same level, inspired from the cellinter-communication in tissues. A particularly interesting sub-class of the first class arethe systems with active membranes, where the membrane division can be used in or-der to solve hard problems, e.g., NP-complete problems, in polynomial or even lineartime, by a space-time trade-off. In the tissue P systems, the communication amongcells is performed by means of symport/antiport rules, well-known in biology. Detailscan be found in [2], [3], as well as in the comprehensive page from the web addresshttp://psystems.disco.unimib.it).In this paper we combine the two definitions, and consider tissue P systems (with thecommunication done through symport/antiport rules) with cell division rules of the sameform as in P systems with active membranes, but without using polarizations. The rulesare used in the non-deterministic maximally parallel way, with the restriction that if adivision rule is used for dividing a cell, then this cell does not participate in any otherrule, for division or communication (when dividing, the interaction of the cell with othercells or with the environment is blocked); the cells obtained by division have the samelabels as the mother cell, hence the rules to be used for evolving them or their objects areinherited (the label precisely identifies the available rules).380This natural extension of tissue P systems provides the possibility of solving SAT prob-lem in polynomial time, in a confluent way: at precise times, one of the objects yes or nois sent to the environment, giving the answer to the question whether the input proposi-tional formula is satisfiable. The construction is uniform: in a polynomial time, a familyof recognizing tissue P systems with cell division is constructed, which, receiving as inputsencodings of instances of SAT, tells us whether or not these instances are satisfiable.2 Tissue P Systems with Cell DivisionWe assume the reader to be familiar with basic elements of membrane computing and wedirectly define the class of P systems which is investigated in this paper.A tissue P system with cell division is a constructΠ = (O,w1, . . . , wm, E,R, io),where:1. m ≥ 1 (the initial degree of the system; the system contains m cells, labeled with1, 2, . . . ,m);2. O is the alphabet of objects;3. w1, . . . , wm are strings over O, describing the multisets of objects placed in the mcells of the system;4. E ⊆ O is the set of objects present in the environment in arbitrarily many copieseach;5. R is a finite set of developmental rules, of the following forms:(a) (i, x/y, j), for i, j ∈ {0, 1, 2, . . . ,m}, i 6= j, and x, y ∈ O∗;communication rules; 1, 2, . . . ,m identify the cells of the system, 0 is the envi-ronment; when applying a rule (i, x/y, j), the objects of the multiset representedby x are sent from region i to region j and simultaneously the objects of themultiset y are sent from region j to region i;(b) [ a ] i → [ b ] i[ c ] i, where i ∈ {1, 2, . . . ,m} and a, b, c ∈ O;division rules; under the influence of object a, the cell with label i is divided intwo cells with the same label; in the first copy the object a is replaced by b, inthe second copy the object a is replaced by c; all other objects are replicatedand copies of them are placed in the two new cells.Therefore, we use antiport rules for communication (for a rule (i, x/y, j) we say thatthe maximum of the lengths of x and y is the weight of the rule), and division rules as inP systems with active membranes.The rules of a system as above are used in the non-deterministic maximally parallelmanner as customary in membrane computing. In each step, all objects and all cells whichcan evolve must evolve (that is, in each step we apply a set of rules which is maximal, nofurther rule can be added), with the following important mentioning: if a cell is divided,then the division rule is the only one which is applied for that cell in that step, its objectsdo not evolve by means of communication rules. This is like saying that a cell which dividesfirst cuts all its communication channels with the other cells and with the environment;the dotter cells will participate to the interaction with other cells or with the environmentonly in the next step – providing that they are not divided once again.381The computation starts from the initial configuration and proceeds as defined above;only halting computations give a result, and the result is the number of objects presentin the halting configuration in cell io; the set of numbers computed in this way by thevarious halting computations in Π is denoted by N(Π).In the present paper we are not interested in the computing power of systems as above– already systems without membrane division are known to be Turing complete (see [2],[1], etc.), but in their computing efficiency. That is why we introduce a variant of tissue Psystems with membrane division, namely recognizing systems with input. Such a systemhas the form Π = (O,w1, . . . , wm, E,R, iin), with the set O containing two distinguishedobjects, yes and no, present in at least one copy in w1w2 . . . wm but not present in E,and with iin ∈ {1, 2, . . . ,m} being the input cell. The computations of the system Πstart from configurations of the form (w1, w2, . . . , winw, . . . , wm;E), where w ∈ O∗ (thatis, after adding the multiset w to the contents of the input cell); all computations halt; inthe last step of a computation either a copy of the object yes or a copy of the object no issent into the environment; we say that the multiset w is recognized/accepted by Π if andonly if the object yes was sent out.If the multiset w codifies an instance Q(n) of a decision problem Q, then we say that Πanswers the question whether or not Q(n) has an affirmative answer (w is accepted if andonly if the codified instance Q(n) has the answer yes). Thus, we say that the problem Qis solved in a time bounded by a mapping f if and only if a class of recognizing P systemswith input, Π(Q,n), can be constructed in polynomial time by a Turing machine startingfrom Q and n, such that Π(Q,n) halts in less than f(n) steps when starting with theinput w(Q,n), which codifies Q(n), and provides the answer yes if and only if Q(n) hasthe affirmative answer. (The code w(Q,n) should be obtained in polynomial time by aTuring machine, starting from the instance Q(n).) The set of all decision problems whichcan be solved as above in a number of steps bounded by a mapping f form the complexityclass MCTD(f). Here we are interested in polynomial time solutions, hence we considerthe class PMCTD, obtained as the union of all classes MCTD(f), for all polynomials f .More precise definitions of complexity classes in terms of membrane computing can befound in [3]. We close this section with an important remark about the previous way ofsolving decision problems. Specifically, we have said nothing about the way the compu-tations proceed; in particular, they can be non-deterministic, as standard in membranecomputing. It is important however that the systems always stop and always they send outan object which is the correct answer to the input problem. In other terms, the systemsare confluent, sound, and complete.3 Solving SAT in Polynomial TimeAs expected, the possibility to divide cells means the possibility to create an exponentialspace in a linear time, and this space can be used in order to obtain fast solutions tocomputationally hard problems.Theorem 3.1 Tissue P systems with active membranes can solve SAT in polynomial time.(Otherwise stated, SAT ∈ PMCTD.)Proof. Let us consider a propositional formula γ = C1 ∧ . . . ∧ Cm, consisting of mclauses Cj = yj,1 ∨ . . . ∨ yj,kj , 1 ≤ j ≤ m, where yj,i ∈ {xl,¬xl | 1 ≤ l ≤ n}, 1 ≤ i ≤ kj(there are used n variables). Without loss of generality, we may assume that no clause382contains two occurrences of some xi or two occurrences of some ¬xi (the formula is notredundant at the level of clauses), or both xi and ¬xi (otherwise such a clause is triviallysatisfiable, hence can be removed).We codify γ, which is an instance of SAT with size parameters n and m, by the multisetw(γ) = {si,j | yj,r = xi, 1 ≤ i ≤ n, 1 ≤ j ≤ m, 1 ≤ r ≤ kj}∪ {s′i,j | yj,r = ¬xi, 1 ≤ i ≤ n, 1 ≤ j ≤ m, 1 ≤ r ≤ kj}.(We replace each variable xi from each clause Cj with si,j and each negated variable ¬xifrom each clause Cj with s′i,j , then we remove all parentheses and connectives. In this waywe pass from γ to w(γ) in a number of steps which is linear with respect to n ·m.)We construct the recognizing tissue P system (of degree 2) with inputΠ = (O,w1, w2, E,R, 2),with the following components:O = {ai, ti, fi | 1 ≤ i ≤ n} ∪ {ri | 1 ≤ i ≤ m}∪ {Ti, Fi | 1 ≤ i ≤ n} ∪ {Ti,j , Fi,j | 1 ≤ i ≤ n, 1 ≤ j ≤ m+ 1}∪ {bi | 1 ≤ i ≤ 3n+m+ 1} ∪ {ci | 1 ≤ i ≤ n+ 1}∪ {di | 1 ≤ i ≤ 3n+ nm+m+ 2} ∪ {ei | 1 ≤ i ≤ 3n+ nm+m+ 4}∪ {f, g, yes, no},w1 = yes no b1c1d1e1,w2 = fga1a2 . . . an,E = O − {yes, no},and the following rules.1. Division rules:[ ai ]2 → [ Ti ]2[ Fi ]2, for all i = 1, 2, . . . , n.(Membrane 2 is repeatedly divided, each time expanding one object ai, correspondingto a variable xi, into Ti and Fi, corresponding to the values true and false which thisvariable may assume. In this way, in n steps, we get 2n cells with label 2, each onecontaining one of the 2n truth-assignments possible for the n variables. The objectsf, g are duplicated, hence a copy of each of them will appear in each cell.)2. Communication rules:(1, bi/b2i+1, 0), for all i = 1, 2, . . . , n+ 1,(1, ci/c2i+1, 0), for all i = 1, 2, . . . , n+ 1,(1, di/d2i+1, 0), for all i = 1, 2, . . . , n+ 1,(1, ei/ei+1, 0), for all i = 1, 2, . . . , 3n+ nm+m+ 3.(In parallel with the operation of dividing cell 2, the counters bi, ci, di, ei from cell1 grow their subscripts. In each step, the number of copies of objects of the firstthree types is doubled, hence after n steps we get 2n copies of bn+1, cn+1, and dn+1.Objects bi will check which clauses are satisfied by a given truth-assignment, objectsci are used in order to multiply the number of copies of ti, fi as we will see imme-diately, di are used to check whether there is at least one truth-assignment which383satisfies all clauses, and ei will be used in order to produce the object no, if this willbe the case, in the end of the computation.)(1, bn+1cn+1/f, 2),(1, dn+1/g, 2).(In step n + 1, the counters bn+1, cn+1, dn+1 are brought in cells with label 2, inexchange of f and g. Because we have 2n copies of each object of these types and 2ncells 2, each one containing exactly one copy of f and one of g, due to the maximalityof the parallelism of using the rules, each cell 2 gets precisely one copy of each ofbn+1, cn+1, dn+1. Note that cells 2 cannot divide any more, because the objects aiwere exhausted.)(2, cn+1Ti/cn+1Ti, 0),(2, cn+1Fi/cn+1Fi,1, 0), for each i = 1, 2, . . . , n,(2, Ti,j/tiTi,j+1, 0),(2, Fi,j/fiFi,j+1, 0), for each i = 1, 2, . . . , n and j = 1, 2, . . . ,m.(In the presence of cn+1, the objects Ti, Fi introduce the objects Ti,1 and Fi,1, re-spectively, which initiates the possibility of introducing m copies of each ti and fiin each cell 2. The idea is that because we have m clauses, in order to check theirvalues for a given truth-assignment of variables, it is possible to need one value foreach valiable for each clause. Note that this phase needs 2n steps for introducingthe double-subscripted objects Ti,1, Fi,1 – for each one we need one step, because wehave only one copy of cn+1 available – then further m steps are necessary for eachTi,1, Fi,1 to grow its second subscript; all these steps are done in parallel, but forthe last introduced Ti,1, Fi,1 we have to continue m steps after the 2n necessary forpriming. In total, we perform 2n+m steps.)(2, bi/bi+1, 0),(2, di/di+1, 0), for all i = n+ 1, . . . , (n+ 1) + (2n+m)− 1.(In parallel with the previous operations, the counters bi and di increase their sub-scripts, until reaching the value 3n +m + 1. This is done in all cells 2 at the sametime. Simultaneously, ei increases its subscript in cell 1.)(2, b3n+m+1tisi,j/b3n+m+1rj , 0),(2, b3n+m+1fis′i,j/b3n+m+1rj , 0), for all 1 ≤ i ≤ n and 1 ≤ j ≤ m,(2, di/di+1, 0), for all i = 3n+m+ 1, . . . , (3n+m+ 1) + nm− 1.(In the presence of b3n+m+1 – and not before – we check the values assumed by clausesfor the truth-assignments from each cell 2. We have only one copy of b3n+m+1 ineach cell, hence we need at most nm steps for this: each clause contains at most nliterals, and we have m clauses. In parallel, d increases the subscript, until reachingthe value 3n+ nm+m+ 1.)(2, d3n+nm+m+iri/d3n+nm+m+i+1, 0), for all i = 1, 2, . . . ,m.(In each cell with label 2 we check whether or not all clauses are satisfied by thecorresponding truth-assignment. For each clause which is satisfied, we increase byone the subscript of d, hence the subscript reaches the value 3n + nm + 2m + 1 ifand only if all clauses are satisfied.)(2, d3n+nm+2m+1/f yes, 1).384(If one of the truth-assignments from a cell 2 has satisfied all clauses, then we reachd3n+nm+2m+1, which is sent to cell 1 in exchange of the objects yes and f .)(2, yes/λ, 0).(In the next step, the object yes leaves the system, signaling the fact that the for-mula is satisfiable. In cell 1, the counter e will increase one more step its subscript,but after that it will remain unchanged – it can leave cell 1 only in the presence off , but this object was already moved to cell 2.)(1, e3n+nm+2m+2f no/λ, 2),(2,no/λ, 0).(If the counter e reaches the subscript 3n+ nm+ 2m+ 2 and the object f is still incell 1, then the object no can be moved to a cell 2, randomly chosen, and from hereit exits the system, signaling that the formula is not satisfiable.)From the previous explanations, one can see that, starting with the multiset w(γ)added to cell 2, which is the input cell, the system correctly answers the question whetheror not γ is satisfiable. The duration of the computation is polynomial in terms of n andm: the answer yes is sent out in step 3n+ nm+ 2m+ 2, while the answer no is sent outin step 3n+ nm+ 2m+ 4. This concludes the proof. 2The antiport rules from the previous construction are of weight at most 3, butthe weight can be reduced to two, at the expense of some slowdown of the system.For instance, instead of the rule (1, e3n+nm+2m+2f no/λ, 2) we can consider the rules(1, e3n+nm+2m+2f/h, 0), (1, h no/λ, 2), where h is a new object. We can proceed in thesame way with the rules (2, b3n+m+1tisi,j/b3n+m+1rj , 0), (2, b3n+m+1fis′i,j/b3n+m+1rj , 0),for 1 ≤ i ≤ n and 1 ≤ j ≤ m, but in this way instead of at most nm steps for finding thesatisfied clauses we will need at most 2nm steps. The details are left to the reader.4 Final RemarksWe have proven that by adding the membrane division feature to tissue P systems (withthe communication done by antiport rules of a small weight) we can solve NP-completeproblems in polynomial time. We exemplify this possibility with SAT problem.It remains as a research topic to consider the same extension for other types of systems,for instance, for cell P systems with symport/antiport rules, or for neural P systems (withstates associated with cells and multiset rewriting rules for processing the objects. Thedifficulty in the case of cell P systems with symport/antiport comes from the fact thatonly the skin membrane can communicate with the environment; on the other hand,the skin membrane cannot be divided, hence we need exponentially many objects forcommunication with inner membranes, and such objects should be brought in from theenvironment. In turn, neural P systems with the maximal use of rules and replicatedcommunication are already known to be able to solveNP-complete problems in polynomialtime; the challenge now is not to use replication). In spite of these difficulties, we expectresults similar to the above one also these cases.Another problem which remains open is to consider tissue P systems with the commu-nication using only symport rules.385Acknowledgements. The support of this research through the project TIC2002-04220-C03-01 of the Ministerio de Ciencia y Tecnolog´ıa of Spain, cofinanced by FEDERfunds, is gratefully acknowledged.References[1] P. Frisco, H.J. Hoogeboom, Simulating counter automata by P systems with sym-port/antiport. In Gh. Pa˘un, G. Rozenberg, A. Salomaa, C. Zandron, eds., MembraneComputing. International Workshop WMC 2002, Curtea de Arges¸, Romania, RevisedPapers. Lecture Notes in Computer Science 2597, Springer-Verlag, Berlin, 2003, 288–301.[2] Gh. Pa˘un, Computing with Membranes: An Introduction. Springer-Verlag, Berlin,2002.[3] M. Pe´rez-Jime´nez, A. Romero-Jime´nez, F. Sancho-Caparrini, Teor´ıa de la Complejidaden Modelos de Computatio´n Celular con Membranas. Editorial Kronos, Sevilla, 2002.386

Tissue P Systems with Cell Division

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Tissue P Systems with Cell Division

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