In this paper we introduce the complexity class PMC∗
F of all decision
problems solvable in polynomial time by a family of P systems belonging
to a prefixed class of recognizer membrane systems, F.Ministerio de Ciencia y Tecnología TIC2002-04220-C03-0

Pérez Jiménez, Mario de Jesús

Romero Jiménez, Álvaro

Sancho Caparrini, Fernando

idUS. Depósito de Investigación Universidad de Sevilla

Complexity Classes in Cellular Computingwith MembranesMario J. PE´REZ-JIME´NEZ, A´lvaro ROMERO JIME´NEZ,Fernando SANCHO–CAPARRINIDpto. Ciencias de la Computacio´n e Inteligencia ArtificialE.T.S. Ingenier´ıa Informa´tica. Universidad de SevillaAvda. Reina Mercedes s/n. 41012 - Sevilla. Espan˜aE-mail: Mario.Perez,Alvaro.Romero,Fernando.Sancho}@cs.us.esAbstract. In this paper we introduce the complexity class PMC∗F of all deci-sion problems solvable in polynomial time by a family of P systems belongingto a prefixed class of recognizer membrane systems, F .1 IntroductionOne of the main goals of Complexity Theory is to provide tools allowing to classify theproblem regarding to the amount of resources needed for their resolution.In this paper the first complexity classes for P systems are presented, inspired in someideas that Gh. Pa˘un [1] discussed with the authors. Theses classes allows us to detectsome of the inherent difficulties to the computational resolution of certain problems, andthey provide a classification of the abstract problems according the resources they needto be solved in a given model of Cellular Computing with Membranes. Of course, sucha classification demands a precise and formal definition about the concept of abstractproblem and the model to be considered.The following parameters used to specify a complexity class:• The model of computation (in our case, recognizer P systems).• The mode of computation (in our case, deterministic and parallel).• The measure of complexity (time and space).• An upper bound of the resources (a total recursive function from N+ to N+).The paper is organized as follows: in the next section we define what a decision problemis as well as we understand by a recognizer P system without or with input which willthe models considered to study the complexity classes. In section 3 the complexity classesassociated with P system without input is introduced. Section 4 is devoted to study thecomplexity classes associated with P systems with input.2702 PreliminariesDefinition 2.1 A decision problem, X, is a pair (IX , θX) such that IX is a language overa finite alphabet (whose elements are called instances) and θX is a total boolean functionover IX .Before giving the precise definition of the classes, we present some previous notionsabout P systems.Definition 2.2 A recognizer P system is a pair(Π, θΠ)such that Π is a P systems andθΠ is a total boolean function over the halting computations of Π. Let C be a haltingcomputation of Π. If θΠ(C) = 1 we will say that C is an accepting computation. IfθΠ(C) = 0, we will say that C is a rejecting computation.Usually, we will identify the recognizer P system(Π, θΠ)with the P system Π, becausethe function θΠ is usually given in an implicit way by means of the semantic of Π.Definition 2.3 Let Π =(Π(i))i∈Ia family of P systems. Then,1. We will say that Π is D-consistent if for each i ∈ I, Π(i) ∈ D.2. We will say that Π is uniform, by Turing machines, if there exists such a machinethat, for a given i ∈ I, it constructs deterministically the P system Π(i). If thisconstruction is made in polynomial time in the size of i, we will say that a Π ispolynomially uniform, by Turing machines.Moreover, we will demand to the P systems in this paper to be confluent, in thefollowing sense.Definition 2.4 We say that a recognizer P system,(Π, θΠ), is confluent, if it verifies thefollowing conditions:1. Every computation of Π is a halting computation.2. If C1 and C2 are two computations of Π with the same initial configuration, thenθΠ(C1) = θΠ(C2).That is, a recognizer P system is confluent if, either all its computations with identicalinitial configurations are accepting computations, or all of them are rejecting ones. Notethat a deterministic recognizer P system is confluent if and only if every computation is ahalting computation.Definition 2.5 A P system with input is a tuple (Π,Σ, iΠ), where:• Π is a P system, with working alphabet Γ and initial multisets M1, . . . , Mp (asso-ciated with membranes labeled by 1, . . . , p, respectively).• Σ is an (input) alphabet strictly contained in Γ.• M1, . . . ,Mp are multisets over Γ− Σ.• iΠ is the label of a distinguished membrane (of input).271Definition 2.6 Let (Π,Σ, iΠ) be a P system with input and without external output. LetΓ be the working alphabet of Π, µ the membrane structure and M1, . . . ,Mp the initialmultisets (over Γ−Σ) of Π. Let m ∈M(Σ) be a multiset over Σ. The initial configurationof (Π,Σ, iΠ) with input m is (µ0,M0), where µ0 = µ, M0(j) = Mj, for each j 6= iΠ, andM0(iΠ) =MiΠ ∪m.Note 2.1 Let us denote by IΠ the set of all inputs of the P system Π. That is, IΠ = M(Σ),the collection of the multisets over Σ.The computations of a P system with input m ∈ M(Σ) are defined in a natural way.The only novelty is that the initial configuration must be the initial configuration of thesystem associated with the input multiset m ∈M(Σ).In the case of P systems with input and with external output (where we can imaginethat the internal processes are unknown, and we only obtain the information that thesystem sends out to the environment), the concept of computation is introduced in asimilar. In the configurations, we will not work directly with the membrane structure µbut with another structure associated with it including, in some sense, the environment.Definition 2.7 Let µ = (V (µ), E(µ)) be a membrane structure. The membrane structurewith external environment associated with µ is the rooted tree Ext(µ) such that: (a) theroot of the tree is a new node that we will denote env; (b) the set of nodes is V (µ) ∪{env}; and (c) the set of edges is E(µ) ∪{{env, skin}}. The node env is called externalenvironment of the structure µ.Note that we have only included a new node representing the environment which isonly connected with the skin, while the original membrane structure remains unchanged.In this way, every configuration of the system informs about the environment and itscontent.Definition 2.8 A recognizer P system with input is a P system with input, (Π,Σ, iΠ),and with external output such that:1. The working alphabet contains two distinguished elements Y es, No.2. All of its computations halt.3. If C is a computation of Π, then either some object Y es or some object No (butnot both) must have been sent out to the external environment, and only in the laststep of the computation. We say that C is an accepting computation (respectively,rejecting computation) if the object Y es (respectively, No) appears in the externalenvironment associated with the corresponding halting configuration of C.Definition 2.9 We say that a recognizer P system with input is confluent, if for everyC1, C2 computations of Π with the same initial configuration with input m, it is verifiedthat θΠ(C1) = θΠ(C2).2723 Complexity classes associated with P systems without in-putThe first results about resolution ofNP-complete problems in polynomial time (even lineartime) by using cellular computing devices have been obtained by means of variants of Psystems without input membrane. Because of this, in the proofs of these results a systemis constructed for each instance of the problem, solving it. From this idea, we present thefollowing definition.Definition 3.1 Let D be a class of recognizer P systems without input, g : N+ → N+ atotal recursive function, and X = (IX , θX) a decision problem. We say that X ∈MCfD(g)if there exists a family, Π = (Π(w))w∈IX , of P systems such that:• Π is D–consistent.• Π is polynomially uniform by Turing machines.• The family Π is (X, g)-bounded; that is, for each w ∈ IX , all computations of Π(w)halt in, at most, g(|w|) steps.• The family Π is X-sound; that is, for each w ∈ IX , if θX(w) = 1, then everycomputation of Π(w) is an accepting computation.• The family Π is X-complete; that is, for each w ∈ IX , if there exists an acceptingcomputation of Π(w), then θX(w) = 1.If we consider in the above definition the collection of polynomial functions as boundedfunctions, then we obtain the polynomial complexity class for families of P systems withoutinput.Definition 3.2 The class of decision problems solvable in polynomial time by a family ofP systems without input is the classPMCfD=⋃k>0MCfD(nk)It must be noted that this class is closed under polynomial time reduction.Proposition 3.1 Let X = (IX , θX) and Y = (IY , θY ) two decision problems such that Xis polynomial time reducible to Y . If Y ∈ PMCfDthen X ∈ PMCfD.Proof. Since X is polynomial time reducible to Y , there exists a polynomial com-putable function h : IX → IY , such that for every w ∈ IX , θX(w) = 1 if and only ifθY (h(w)) = 1. Hence, there exists a natural number kh > 0 such that |h(w)| ≤ |w|kh , forevery w ∈ IX .Since, by hypothesis, Y ∈ PMCfD, there exist a natural number k > 0 and a family ofP systems, ΠY =(ΠY (w′))w′∈IYsuch that:1. The family ΠY is D-consistent.2. For each w′ ∈ IY , the system Π(w′) has no input membrane.2733. The family ΠY is polynomially uniform.4. The family ΠY is (Y, nk)-bounded.5. The family ΠY is Y -sound.6. The family ΠY is Y -complete.Let us consider the family Π =(Π(w))w∈IX, where Π(w) = ΠY (h(w)), for everyw ∈ IX . Then it is verified that:1. The family Π is polynomially uniform: it is enough to have in mind that the familyΠY is polynomially uniform and the function h is polynomial time computable.2. The family Π is (X,nkh·k)-bounded: let w ∈ IX , since Π(w) = ΠY (h(w)), everycomputation of Π(w) is a halting computation, and it works in less than |h(w)|k ≤(|w|kh)k= |w|kh·k steps.3. The family Π is X-sound: let w ∈ IX such that θX(w) = 1; then θY (h(w)) = 1,hence every computation of ΠY (h(w)) = Π(w) is an accepting one.4. The family Π is X-complete: let w ∈ IX such that there exists an accepting com-putation of Π(w) = ΠY (h(w)). Then θY (h(w)) = 1, and hence θX(w) = 1.Consequently, X ∈ PMCfD. 24 Complexity classes associated with P systems with inputIn the above definition of complexity class we had to work with a family of recognizer Psystems without input, because each P system can only decide about one instance of theproblem.If we consider P systems with input membrane, it is possible to define a complexityclass in some similar way to that defined for classic models (i.e., Turing Machines).Definition 4.1 Let D be a class of recognizer P systems with input, g : N+ → N+ a totalrecursive function, and X = (IX , θX) a decision problem. We say that X ∈ MCiD(g) ifthere exists a P system, Π such that:• Π ∈ D.• Π can be constructed in polynomially time by a deterministic Turing machine, de-pending on the problem.• There exists a polynomially computable function, cod : IX → IΠ, such that:– Π is (X, g)-bounded regarding to cod; that is, for each w ∈ IX , every computa-tion of Π with input cod(w) is a halting computation, and it halts in less thang(|w|) steps.– Π is X-sound regarding to cod; that is, for each w ∈ IX , if θX(w) = 1 thenevery computation of Π with input cod(w) is an accepting computation.274– Π is X-complete regarding to cod; that is, for each w ∈ IX , if there exists anaccepting computation of Π with input cod(w) then θX(w) = 1.If we consider as bound functions the polynomial functions, we obtain the polynomialcomplexity class for P systems with input.Definition 4.2 The class of decision problems solved in polynomial time for a P systemwith input is the classPMCiD =⋃k>0MCiD(nk)This class is, also, closed under polynomial time reduction.Proposition 4.1 Let X = (IX , θX) and Y = (IY , θY ) two decision problems such that Xis reducible to Y in polynomial time. If Y ∈ PMCiD then X ∈ PMCiD.Proof. Since X is reducible to Y in polynomial time, there exists a computablefunction h : IX → IY in polynomial time, such that for every w ∈ IX , θX(w) = 1if and only if θY (h(w)) = 1. Hence, there exists a natural number kh > 0 such that|h(w)| ≤ |w|kh , for every w ∈ IX .Since, by hypothesis, Y ∈ PMCiD, there exist a natural number k > 0 and a P system,Π, such that:1. Π ∈ D.2. Π can be constructed in polynomial time by a Turing Machine.3. There exists a polynomially computable function, codY : IY → IΠ, such that:• Π is (Y, nk)-bounded regarding to codY .• Π is Y -sound regarding to codY .• Π is Y -complete regarding to codY .Let us consider the function cod = codY ◦ h : IX → IΠ. Then cod is a computablefunction in polynomial time. Moreover, the following conditions are verified:• Π is (X,nkh·k)-bounded regarding to cod: let w ∈ IX ; then every computation ofΠ with input cod(w) = codY (h(w)) halts in, at most, |h(w)|k ≤ (|w|kh)k= |w|kh·ksteps.• Π is X-sound regarding to cod: let w ∈ IX such that θX(w) = 1; then θY (h(w)) = 1,hence every computation of Π with input codY (h(w)) = cod(w) is a halting compu-tation.• Π is X-complete regarding to cod: let w ∈ IX such that there exists an accept-ing computation of Π with input cod(w) = codY (h(w)), hence θY (h(w)) = 1, andtherefore θX(w) = 1.Consequently, X ∈ PMCiD. 2Although the above definition of complexity class is rather natural, it is very usual inpractice to find not one P system with input solving a decision problem, but a family of Psystems with input, where one of its P systems solves the instances of similar size. Thiscarried out the third complexity class introduced in this work.275Definition 4.3 Let X = (IX , θX) a decision problem, D a class of recognizer P systemswith input and Π = (Π(n))n∈N+ a family of P systems of D. A polynomial encoding of IXin Π is a pair, (cod, s), of polynomial time computable functions, cod : IX →⋃n∈N+IΠ(n)and s : IX → N+, such that for every w ∈ IX we have cod(w) ∈ IΠ(s(w)).Definition 4.4 Let D be a class of recognizer P systems with input, g : N+ → N+ a totalrecursive function, and X = (IX , θX) a decision problem. We say that X ∈ MCfiD(g) ifthere exists a P system, Π =(Π(n))n∈N+such that:• The family Π is D-consistent.• The family Π is polynomially uniform.• There exist a polynomial encoding, (cod, s), of IX in Π, such that– The family Π is (X, g)-bounded regarding to (cod, s); that is, for each w ∈ IX ,every computation of Π(s(w)) with input cod(w) halts in, at most, s(|w|) steps.– The family Π is X-sound regarding to (cod, s); that is, for every w ∈ IX , ifθX(w) = 1 then every computation of Π(s(w)) with input cod(w) is an acceptingcomputation.– The family Π is X-complete regarding to (cod, s); that is, for every w ∈ IX ,if there exists an accepting computation of Π(s(w)) with input cod(w) thenθX(w) = 1.As usual, the associated polynomial class can be obtained by using the polynomialfunctions as bound functions.Definition 4.5 The class of decision problems solved by a family of P systems with inputis the classPMCfiD=⋃k>0MCfiD(nk)This class is closed under polynomial time reduction.Proposition 4.2 Let X = (IX , θX) and Y = (IY , θY ) two decision problems such that Xis reducible to Y in polynomial time. If Y ∈ PMCfiDthen X ∈ PMCfiD.Proof. Since X is reducible to Y in polynomial time, there exists a total computablefunction h : IX → IY in polynomial time, such that, for every w ∈ IX , θX(w) = 1if and only if θY (h(w)) = 1. Hence, there exists a natural number kh > 0 such that|h(w)| ≤ |w|kh , for every w ∈ IX .Since, by hypothesis, Y ∈ PMCfiD, there exist a natural number k > 0 and a familyof P systems, Π =(Π(n))n∈N+, such that:• The family Π is D-consistent.• The family Π is polynomially uniform.• There exist a polynomial encoding, (codY , sY ), of IY in Π.• Π is (Y, nk)-bounded regarding to (codY , sY ).276• Π is Y -sound regarding to (codY , sY ).• Π is Y -complete regarding to (codY , sY ).Let us consider the functions cod = codY ◦h : IX →⋃n∈N+ IΠ(n) y s = sY ◦h. Then codand s are computable in polynomial time. Moreover, the following conditions are verified:• (cod, s) is a polynomial encoding of IX inΠ: for each w ∈ IX , cod(w) = codY (h(w)) ∈IΠ(sY (h(w))) = IΠ(s(w)).• Π is (X,nkh·k)-bounded regarding to (cod, s): let w ∈ IX , then every computation ofΠ(s(w)) = Π(sY (h(w))) with input cod(w) = codY (h(w)) halts in, at most, |h(w)|k ≤(|w|kh)k= |w|kh·k steps.• Π isX-sound regarding to (cod, s): let w ∈ IX such that θX(w) = 1, then θY (h(w)) =1, hence every computation of Π(sY (h(w))) = Π(s(w)) with input codY (h(w)) =cod(w) is an accepting computation.• Π is X-complete regarding to (cod, s): let w ∈ IX such that there exists an acceptingcomputation of Π(s(w)) = Π(sY (h(w))) with input cod(w) = codY (h(w)), henceθY (h(w)) = 1, and therefore θX(w) = 1.Consequently, X ∈ PMCfiD. 2It is easy to prove that the following relations between above classes are verified.Proposition 4.3 Let D be a class of recognizer P systems with input membrane and letD′ be the same class but considered as without input membrane. It is verified that:1. If X ∈MCfiD(g) then X ∈MCfD′(g).2. If X ∈MCiD(g) then X ∈MCfiD(g).Proof.1. Let us suppose that X ∈ MCfiD(g). Then, there exists a family of P systems,Π =(Π(n))n∈N+, verifying:• The family Π is D-consistent.• Π is polynomially uniform.• There exist a polynomial encoding, (cod, s), of IX in Π.• Π is (X, g)-bounded regarding to (cod, s).• Π is X-sound regarding to (cod, s).• Π is X-complete regarding to (cod, s).Let us consider the family Π′ =(Π′(w))w∈IXin such way that, for each w ∈ IX theP system Π′(w) is obtained considerending the P system Π(s(w)) as a P systemswithout input membrane, and with the multiset cod(w) added to its input membrane.Then, it is easy to check that such a family verifies the conditions for X ∈MCfD′.2772. Let us suppose that X ∈ MCiD(g). Then there exist a P system, Π, verifying thefollowing conditions:• Π ∈ D.• Π can be constructed in polynomial time by a Turing machine.• There exists a computable function, cod : IX → IΠ, in polynomial time, suchthat:– Π is (X, g)-bounded regarding to cod.– Π is X-sound regarding to cod.– Π is X-complete regarding to cod.Let us consider the family Π =(Π(n))n∈N+, where for each n ∈ N+, Π(n) = Π.Then, it is easy to check that this family verifies the conditions for X ∈MCfiD. 25 ConclusionsIn this work several complexity classes associated with membrane systems (with or withoutinput) have been presented. Besides, some inclusion relations between them have beenstudied.We think that this paper establishes a new framework to study computational com-plexity of decision problems through membrane systems, which can provide relevant in-formation about the effective computational power of different variants of P systems.Moreover, we think that these complexity classes in membrane systems offer a newway to attack the P6=NP conjecture, now inside the framework of the cellular computingwith membranes.AcknowledgementThe authors wish to acknowledge the support of the project TIC2002-04220-C03-01 of theMinisterio de Ciencia y Tecnolog´ıa of Spain, cofinanced by FEDER funds.References[1] Pa˘un, Gh.: Membrane Computing. An introduction, Springer-Verlag, Berlin, 2002.[2] Pa˘un Gh. Computing with membranes, Journal of Computer and System Sciences,61, 1 (2000), 108–143, and Turku Center for Computer Science-TUCS Report No 208,1998 (www.tucs.fi).[3] Pa˘un, Gh.; Rozenberg, G. A guide to membrane computing, Theoretical ComputerScience, 287, 2002, 73–100.[4] Pe´rez–Jime´nez, M.J.; Romero-Jime´nez, A.; Sancho-Caparrini, F. Teor´ıa de la Com-plejidad en modelos de computacio´n con membranas, Ed. Kronos, Sevilla, I.S.B.N.84-86273-57-9, 2002.278

Complexity Classes in Cellular Computing with Membranes

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Complexity Classes in Cellular Computing with Membranes

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