First, the dependency graph technique, not so far from its current application,
was developed trying to  nd the shortest computations for membrane systems
solving instances of SAT. Certain families of membrane systems have been demonstrated
to be non-effcient by means of the reduction of  nding an accepting computation (respectively,
rejecting computation) to the problem of reaching from a node of the dependency
graph to another one.
In this paper, a novel application to this technique is explained. Supposing that a
problem can be solved by means of a kind of membrane systems leads to a contradiction
by means of using the dependency graph as a reasoning method. In this case, it is demonstrated
that a single system without dissolution, polarizations and cooperation cannot
distinguish a single object from more than one object.
An extended version of this work will be presented in the 20th International Conference
on Membrane Computing.Ministerio de Industria, Economía y Competitividad TIN2017-89842-

Orellana Martín, David

Pérez Hurtado de Mendoza, Ignacio

Pérez Jiménez, Mario de Jesús

Research Group on Natural Computing (Coordinador)

Valencia Cabrera, Luis

English

idUS. Depósito de Investigación Universidad de Sevilla

New applications for an old toolLuis Valencia-Cabrera, David Orellana-Mart´ın,Ignacio Pe´rez-Hurtado, Mario J. Pe´rez-Jime´nezResearch Group on Natural ComputingDepartment of Computer Science and Artificial IntelligenceUniversidad de SevillaAvda. Reina Mercedes s/n, 41012 Sevilla, SpainE-mail: {lvalencia, dorellana, perezh, marper}@us.esSummary. First, the dependency graph technique, not so far from its current appli-cation, was developed trying to find the shortest computations for membrane systemssolving instances of SAT. Certain families of membrane systems have been demonstratedto be non-efficient by means of the reduction of finding an accepting computation (respec-tively, rejecting computation) to the problem of reaching from a node of the dependencygraph to another one.In this paper, a novel application to this technique is explained. Supposing that aproblem can be solved by means of a kind of membrane systems leads to a contradictionby means of using the dependency graph as a reasoning method. In this case, it is demon-strated that a single system without dissolution, polarizations and cooperation cannotdistinguish a single object from more than one object.An extended version of this work will be presented in the 20th International Confer-ence on Membrane Computing.1 IntroductionThe computational efficiency of a model in a computing paradigm refers to its abil-ity to provide polynomial time solutions for computationally hard problems, gener-ally achieved by making use of an exponential workspace constructed in a naturalway. Aspects related to the computational efficiency within membrane computingwere first analyzed in 1999, with the introduction of a new computing model calledP system with active membranes [5]. These systems are non-cooperative (the lefthand side of any rule consists of only one object) and their membranes play arelevant role in computations to the extent that new membranes can be createdby division rules. The membranes of these systems are supposed to have one ofthree possible electrical polarizations: positive, negative or neutral. In this con-text, it was given an ad-hoc solution to the Boolean satisfiability problem (SAT)166 L. Valencia-Cabrera et al.by means of such kind of P systems. More specifically, a P system with activemembranes which makes use of simple object evolution rules (only one object isproduced for this kind of rules), dissolution rules and division rules for elementaryand non-elementary membranes, is associated with every instance ϕ of SAT. Thus,the syntactic structure of the formula is “captured” by the description of the sys-tem and, furthermore, in this context a P system can only process one instance ofthe problem. The solution provided runs in linear time with respect to the size ofthe input formula ϕ, that is, the maximum between the number of variables andthe number of clauses in ϕ.Usually, computational complexity theory deals with decision problems, thatis, problems requiring a yes/no answer. Each decision problem has a languageassociated with it, in a natural way, so that solving such problems is definedthrough the recognition of the corresponding language. Thus, in order to describein a formal way what solving a decision problem means, basic recognizer transitionP systems (initially called decision P systems) were defined [7].Let us recall that an abstract problem can be solved by using a single Turingmachine, that is, for every instance of the problem, the Turing machine receivingthe input corresponding to that instance returns the correct answer. This is dueto the fact that these machines have an unlimited and unrestricted memory, giventhe infinite tape it includes (consisting of an infinite number of cells). Bearing inmind that the ingredients necessary to define a membrane system are finite, anabstract problem should be solved, in general, by an infinite numerable family ofmembrane systems, in such a way that each system in the family is in charge ofprocessing all the instances having the same size.It seems interesting to analyze what kind of membrane systems are capableof solving decision problems through only one unique system. In this context, itis essential to clarify how the instances of the problem are introduced into thesystem. Next, we consider the case in which the instances are directly introducedinside the system (free of resources) by means of a representation of the problemto be solved. It is important to remark that this means that the input alphabetof the P system is the same one that the alphabet of the problem, so there is nopossibility of encoding, for instance, an instance of a problem from P to an objectyes or an object no.Definition 1. Let X = (IX , θX) be a decision problem where IX is a languageover a finite alphabet ΣX . Let R be a class of recognizer membrane systems withinput membrane. We say that problem X is solvable in polynomial time by a singlemembrane system Π from R, free of resources, denoted by X ∈ PMC1fR , if thefollowing holds:• The input alphabet of Π is ΣX .• The system Π is polynomially bounded with regard to X; that is, there existsa polynomial p(r) such that for each instance u ∈ IX , every computation of thesystem Π with input multiset u performs at most p(|u|) steps.New applications for an old tool 167• The system Π is sound with regard to X; that is, for each instance u ∈ IX ,if there exists an accepting computation of the system Π with input multiset uthen θX(u) = 1.• The system Π is complete with regard to X; that is, for each instance u ∈ IXsuch that θX(u) = 1, every computation of the system Π with input multiset uis an accepting computation.From the previous definition it is easy to prove that PMC1fR ⊆ PMCR, for everyclass R of recognizer membrane systems with input membrane.2 Previous uses of the dependency graphThe dynamics of a membrane system provides, in a natural way, a tree of com-putation. More precisely, the computation tree of a membrane system Π, denotedComp(Π), is a rooted labelled maximal tree, whose maximal branches will becalled computations of Π. A computation of Π is a halting computation if andonly if it is a finite branch. The labels of the leaves of Comp(Π) are called haltingconfigurations.Given a semi-uniform or uniform solution (in polynomial time) for a decisionproblem by means of a family of recognizer membrane systems, every instance ofthe problem is processed by a system of the family. This system must be confluent,so in order to know its answer for any instance it is enough to consider only onecomputation of such system. In this context, an exciting challenge would be lookingfor a computation with minimum length. For that, some weak metrics on thedegree of closeness configurations of a membrane system with a fixed structureof membranes have been studied in [2]. In this context, in order to search forthe shortest paths in a graph providing a sound computation of the system, thedependency graph associated with the set of rules of a recognizer membrane systemwas introduced. This concept is based on the dependence among elements of thealphabet with respect to the set of rules of the P system. Several weak metricsover the set of configurations of the system based on the concept of dependencygraph were considered, starting from the notion of distance between two nodes ofthe graph (the length of the shortest path connecting v1 and v2, or infinite if thereis no path from v1 to v2).Also, in some kind of recognizer membrane systems, it is possible to considera directed graph (also called dependency graph) verifying the following properties:(a) it can be constructed from the set or rules of the system in polynomial time,that is, in a time bounded by a polynomial function depending on the total numberof rules and the maximum length of them; and (b) the accepting computationsof such systems can be characterized by means of a “reachability” property inthe dependency graph associated with it (the existence of a path in the graphbetween two specific nodes). Therefore, dependency graphs provide a techniqueto tackle the limits on efficient computations in membrane systems; that is, thenon-efficiency of such systems.168 L. Valencia-Cabrera et al.3 Dependency graph as a technique to prove negativeresults in membrane systemsLet R be a class of recognizer membrane systems such that every system from Ris associated with a dependency graph verifying the following property: a compu-tation of a system from R is an accepting computation if and only if there existsa path between two distinguished nodes in the dependency graph associated withthe system. In this situation, it is possible to show that some decision problemX = (IX , θX) cannot be solved in polynomial time in a uniform way by means of asingle membrane system, free of resources, from R. This remark is illustrated byan example.The ONLY-ONE-OBJECT problem is the decision problem X = (IX , θX) definedas follows: IX = {an | n ∈ N, n ≥ 1} and θX(an) = 1 if and only if n = 1. Itis easy to design a deterministic Turing machine which takes two computationsteps, solving the ONLY-ONE-OBJECT problem. Let us see that ONLY-ONE-OBJECT/∈ PMC1fAM0(−d,+ne).Theorem 1. There is no recognizer membrane system from the class AM0(−d,+ne)solving the ONLY-ONE-OBJECT problem in polynomial time by a single membranesystem and free of resources.Proof. Let us assume that there exists a recognizer membrane system Π fromAM0(−d,+ne) verifying the following: (a) the input alphabet of Π is the sin-gleton {a}; (b) every computation of Π with input multiset {a} is an acceptingcomputation; and (c) every computation of Π with input multiset {an}, for eachn > 1, is a rejecting computation.Let us denote by GΠ+{a} (respectively, GΠ+{an}, for each n > 1) the depen-dency graph associated with the system Π + {a} (resp. Π + {an}). Then, foreach n > 1, we have GΠ+{a} = GΠ+{an}, since there would always be an edge(sΠ , (a, iin)) in the dependency graph, and the rest of the graph would remain thesame. Besides, every computation of Π + {a} is an accepting computation if andonly if every computation of Π + {an}, for each n > 1, is an accepting compu-tation, which is a contradiction of the initial hypothesis, thus there cannot existsuch membrane system.4 ConclusionsAlong this work, some of the main results concerning the use of dependency graphswithin membrane computing to analyze the computational efficiency of computingmodels have been reviewed. It is worth pointing out that, albeit the P versus NPproblem is the most important one in Computer Science, there are other interestingproblems in the field of Computational Complexity Theory, also below P. Whenusing polynomial precomputed resources, problems from P can be easily solved.But considering membrane systems free of precomputed resources, things changeNew applications for an old tool 169in such a way that there cannot be trivial solutions that could be obtained at first.Then, it would be useful to study these kinds of systems to solve problems belowthis complexity class. In this case, with this technique it has been demonstratedthat there is no solution to the ONLY-ONE-OBJECT problem by means of a singlemembrane system from AM0(−d,+ne).Adapting currently used methodologies to new applications is an interestingfuture research line to improve existing results and obtain new ones. Besides, thesearch for new techniques to demonstrate the non-efficiency or the inability for cer-tain membrane systems to solve some decision problems is critical when addressingthe P versus NP and other interesting problems in the field of ComputationalComplexity Theory.AcknowledgementsThis work was supported in part by the research project TIN2017-89842-P, co-financed by Ministerio de Economı´a, Industria y Competitividad (MINECO) ofSpain, through the Agencia Estatal de Investigacio´n (AEI), and by Fondo Eu-ropeo de Desarrollo Regional (FEDER) of the European Union.References1. A. Alhazov, M.J. Pe´rez-Jime´nez. Uniform solution to QSAT using polarizationless ac-tive membranes. In J. Durand-Lose, M. Margenstern (eds.) Machines, Computations,and Universality 5th International Conference, MCU 2007, Orle´ans, France, Septem-ber 10-13, 2007. Proceedings. Lecture Notes in Computer Science, 4664 (2007), 122-133.2. A. Cordo´n-Franco, M. A. Gutie´rrez-Naranjo, M.J. Pe´rez-Jime´nez, A. Riscos-Nu´n˜ez.Weak metrics on configurations of a P system. In Gh. Paun, A. Riscos, A´. Romero,F. Sancho (eds.) Proceedings of the Second Brainstorming Week on Membrane Com-puting, Report RGNC 01/2004, 2004, pp. 139-151.3. M. A. Gutie´rrez-Naranjo, M.J. Pe´rez-Jime´nez, A. Riscos-Nu´n˜ez, F.J. Romero-Campero. On the power of dissolution in P systems with active membranes. In R.Freund, Gh. Pa˘un, G. Rozenberg, A. Salomaa (eds.) Membrane Computing, 6th Inter-national Workshop, WMC 2005, Vienna, Austria, July 18-21, 2005, Revised Selectedand Invited Papers. Lecture Notes in Computer Science, 3850 (2006), 224-240.4. C.H. Papadimitriou. Computational Complexity, Addison–Wesley, Massachusetts,1995.5. Gh. Pa˘un. P systems with active membranes: attacking NP–complete problems,Journal of Automata, Languages and Combinatorics, 6, 1 (2001), 75-90. A prelimi-nary version in Centre for Discrete Mathematics and Theoretical Computer Science,CDMTCS Research Report Series-102, May 1999, 16 pages.6. Gh. Pa˘un. Further twenty six open problems in membrane computing. In M.A. Gutie´rrez-Naranjo, A. Riscos-Nu´n˜ez, F.J. Romero-Campero, D. Sburlan (eds.)Proceedings of the Third Brainstorming Week on Membrane Computing, Fe´nix Edi-tora, Sevilla, 2005, 249-262.170 L. Valencia-Cabrera et al.7. M.J. Pe´rez-Jime´nez, A´. Romero-Jime´nez, F. Sancho-Caparrini. Decision P systemsand the P 6= NP conjecture. In Gh. Pa˘un, Gr. Rozenberg, A. Salomaa, C. Zandron(eds.) Membrane Computing 2002. Lecture Notes in Computer Science, 2597 (2003),388-399. A preliminary version in Gh. Pa˘un, C. Zandron (eds.) Pre-proceedings ofWorkshop on Membrane Computing 2002, MolCoNet project-IST-2001-32008, Pub-lication No. 1, Curtea de Arges, Romanian, August 19-23, 2002, pp. 345-354.

New applications for an old tool

https://idus.us.es/bitstream/11441/90442/1/165_Applications.pdf

New applications for an old tool

Abstract

Similar works

Full text

Available Versions

idUS. Depósito de Investigación Universidad de Sevilla