Computing with cells: membrane systems - some complexity issues.

Membrane computing is a branch of natural computing which abstracts computing models from the structure and the functioning of the living cell. The main ingredients of membrane systems, called P systems, are (i) the membrane structure, which consists of a hierarchical arrangements of membranes which delimit compartments where (ii) multisets of symbols, called objects, evolve according to (iii) sets of rules which are localised and associated with compartments. By using the rules in a nondeterministic/deterministic maximally parallel manner, transitions between the system configurations can be obtained. A sequence of transitions is a computation of how the system is evolving. Various ways of controlling the transfer of objects from one membrane to another and applying the rules, as well as possibilities to dissolve, divide or create membranes have been studied. Membrane systems have a great potential for implementing massively concurrent systems in an efficient way that would allow us to solve currently intractable problems once future biotechnology gives way to a practical bio-realization. In this paper we survey some interesting and fundamental complexity issues such as universality vs. nonuniversality, determinism vs. nondeterminism, membrane and alphabet size hierarchies, characterizations of context-sensitive languages and other language classes and various notions of parallelism

P Systems with Symport/Antiport of Rules

Moving \instructions" instead of \data", using transport mecha- nisms inspired by biology { this could represent, shortly, the basic idea of the computing device presented in this paper. Speci¯cally, we propose a new class of P systems that use, at the same time, evolution rules and symport/antiport rules. The idea of this kind of systems is simple: during a computation symbol- objects (the \data") evolve using evolution rules but they cannot be moved; on the other hand, the evolution rules (the \instructions") can be moved across the membranes using classical symport/antiport rules. We present di®erent results using di®erent combinations between the power of the evolution rules (catalytic, non-cooperative rules) and the weight of the symport/antiport rules. In particular, we show that, using non-cooperative rules and antiports of un- bounded weight is possible to obtain at least the Parikh set of ET0L languages. On the other hand, using catalytic rules (one catalyst) and antiports of weight 2, the system becomes universal. Several open problems are also presented

Membrane Fission: A Computational Complexity Perspective

Membrane fission is a process by which a biological membrane is split into two new ones in the manner that the content of the initial membrane is separated and distributed between the new membranes. Inspired by this biological phenomenon, membrane separation rules were considered in membrane computing. In this work, we investigate cell-like P systems with symport/antiport rules and membrane separation rules from a computational complexity perspective. Specifically, we establish a limit on the efficiency of such P systems which use communication rules of length at most two, and we prove the computational efficiency of this kind of models when using communication rules of length at most three. Hence, a sharp borderline between tractability and NP–hardness is provided in terms of the length of communication rules.Ministerio de Economía y Competitividad TIN2012-3743

Catalytic and communicating Petri nets are Turing complete

In most studies about the expressiveness of Petri nets, the focus has been put either on adding suitable arcs or on assuring that a complete snapshot of the system can be obtained. While the former still complies with the intuition on Petri nets, the second is somehow an orthogonal approach, as Petri nets are distributed in nature. Here, inspired by membrane computing, we study some classes of Petri nets where the distribution is partially kept and which are still Turing complete

The Computational Complexity of Tissue P Systems with Evolutional Symport/Antiport Rules

Tissue P systems with evolutional communication (symport/antiport) rules are computational models inspired by biochemical systems consisting of multiple individuals living and cooperating in a certain environment, where objects can be modified when moving from one region to another region. In this work, cell separation, inspired from membrane fission process, is introduced in the framework of tissue P systems with evolutional communication rules.The computational complexity of this kind of P systems is investigated. It is proved that only problems in class P can be efficiently solved by tissue P systems with cell separation with evolutional communication rules of length at most (��, 1), for each natural number �� ≥ 1. In the case where that length is upper bounded by (3, 2), a polynomial time solution to the SAT problem is provided, hence, assuming that P ̸= NP a new boundary between tractability and NP-hardness on the basis of the length of evolutional communication rules is provided. Finally, a new simulator for tissue P systems with evolutional communication rules is designed and is used to check the correctness of the solution to the SAT problem

Narrowing Frontiers of Efficiency with Evolutional Communication Rules and Cell Separation

In the framework of Membrane Computing, several efficient solutions to computationally hard problems have been given. To find new borderlines between families of P systems that can solve them and the ones that cannot is an important way to tackle the P versus NP problem. Adding syntactic and/or semantic ingredients can mean passing from non-efficiency to presumably efficiency. Here, we try to get narrow frontiers, setting the stage to adapt efficient solutions from a family of P systems to another one. In order to do that, a solution to the SAT problem is given by means of a family of tissue P systems with evolutional symport/antiport rules and cell separation with the restriction that both the left-hand side and the right-hand side of the rules have at most two objects.Ministerio de Economía y Competitividad TIN2017-89842-PNational Natural Science Foundation of China No 6132010600

Minimization Strategies for Maximally Parallel Multiset Rewriting Systems

Maximally parallel multiset rewriting systems (MPMRS) give a convenient way to express relations between unstructured objects. The functioning of various computational devices may be expressed in terms of MPMRS (e.g., register machines and many variants of P systems). In particular, this means that MPMRS are computationally complete; however, a direct translation leads to quite a big number of rules. Like for other classes of computationally complete devices, there is a challenge to find a universal system having the smallest number of rules. In this article we present different rule minimization strategies for MPMRS based on encodings and structural transformations. We apply these strategies to the translation of a small universal register machine (Korec, 1996) and we show that there exists a universal MPMRS with 23 rules. Since MPMRS are identical to a restricted variant of P systems with antiport rules, the results we obtained improve previously known results on the number of rules for those systems.Comment: This article is an improved version of [1

Limits on Efficient Computation in P Systems with Symport/Antiport Rules

Classical membrane systems with symport/antiport rules observe the con- servation law, in the sense that they compute by changing the places of objects with respect to the membranes, and not by changing the objects themselves. In these systems the environment plays an active role because the systems not only send objects to the environment, but also bring objects from the environment. In the initial configuration of a system, there is a special alphabet whose elements appear in an arbitrary large number of copies. The ability of these computing devices with infinite copies of some objects has been widely exploited in the design of efficient solutions to computationally hard problems. This paper deals with computational aspects of P systems with symport/antiport rules and membrane division rules or membrane separation rules. Specifically, we study the limitations of such P systems when the only communication rules allowed have length 1.Ministerio de Ciencia e Innovación TIN2012-3743

Further Remarks on Trace Languages in P Systems with Symport/Antiport

P systems are parallel molecular computing models which process multisets of objects in cell-like membrane structures. In this paper we consider the trace languages of a special symbol, the traveler, in symport/antiport P systems where, instead of multisets of objects, sets of objects were considered. Two different ways to define the trace language are proposed. One of the families of languages obtained in this way is proved to be equal to the family of regular languages and the other one to be strictly smaller. Some ideas for further research are also considered

Minimal Cooperation in P Systems with Symport/Antiport: A Complexity Approach

Membrane systems with symport/antiport rules compute by just moving objects among membranes, and not by changing the objects themselves. In these systems the environment plays an active role because, not only it receives objects from the system, but it also sends objects into the system. Actually, in this framework it is commonly assumed that an arbitrarily large number of copies of some objects are initially available in the environment. This special feature has been widely exploited for the design of e cient solutions to computationally hard problems in the framework of tissue like P systems able to create an exponential workspace in polynomial time (e.g. via cell division or cell separation rules). This paper deals with cell-like P systems which use symport/antiport rules as communication rules, and the role played by the minimal cooperation is studied from a computational complexity point of view. Speci cally, the limitations on the e ciency of P systems with membrane separation whose symport/antiport rules involve at most two objects are established. In addition, a polynomial time solution to HAM-CYCLE problem, a well known NP-complete problem, by using a family of such kind of P systems with membrane division, is provided. Therefore, in the framework of cell-like P systems with minimal cooperation in communication rules, passing from membrane separation to membrane division amounts to passing from tractability to NP{hardness.Ministerio de Economía y Competitividad TIN2012-3743