An apparently innocent problem in Membrane Computing

The search for effcient solutions of computationally hard problems by means of families of membrane systems has lead to a wide and prosperous eld of research. The study of computational complexity theory in Membrane Computing is mainly based on the look for frontiers of effciency between different classes of membrane systems. Every frontier provides a powerful tool for tackling the P versus NP problem in the following way. Given two classes of recognizer membrane systems R1 and R2, being systems from R1 non-effcient (that is, capable of solving only problems from the class P) and systems from R2 presumably e cient (that is, capable of solving NP-complete problems), and R2 the same class that R1 with some ingredients added, passing from R1 to R2 is comparable to passing from the non effciency to the presumed effciency. In order to prove that P = NP, it would be enough to, given a solution of an NP-complete problem by means of a family of recognizer membrane systems from R2, try to remove the added ingredients to R2 from R1. In this paper, we study if it is possible to solve SAT by means of a family of recognizer P systems from AM0(�����d;+n), whose non-effciency was demonstrated already

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

A new perspective on computational complexity theory in Membrane Computing

A single Turing machine can solve decision problems with an in nite number of instances. On the other hand, in the framework of membrane computing, a \solution" to an abstract decision problem consists of a family of membrane systems (where each system of the family is associated with a nite set of instances of the problem to be solved). An interesting question is to analyze the possibility to nd a single membrane system able to deal with the in nitely many instances of a decision problem. In this context, it is fundamental to de ne precisely how the instances of the problem are introduced into the system. In this paper, two different methods are considered: pre-computed (in polynomial time) resources and non-treated resources. An extended version of this work will be presented in the 20th International Conference on Membrane Computing.Ministerio de Economía, Industria y Competitividad TIN2017-89842-

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

Characterizing PSPACE with Shallow Non-Confluent P Systems

In P systems with active membranes, the question of understanding the power of non-confluence within a polynomial time bound is still an open problem. It is known that, for shallow P systems, that is, with only one level of nesting, non-con uence allows them to solve conjecturally harder problems than con uent P systems, thus reaching PSPACE. Here we show that PSPACE is not only a bound, but actually an exact characterization. Therefore, the power endowed by non-con uence to shallow P systems is equal to the power gained by con uent P systems when non-elementary membrane division and polynomial depth are allowed, thus suggesting a connection between the roles of non-confluence and nesting depth

A Framework for Complexity Classes in Membrane Computing

The purpose of the present work is to give a general idea about the existing results and open problems concerning the study of complexity classes within the membrane computing framework. To this aim, membrane systems (seen as computing devices) are briefly introduced, providing the basic definition and summarizing the key ideas, trying to cover the various approaches that are under investigation in this area – of course, special attention is paid to the study of complexity classes. The paper concludes with some final remarks that hint the reasons why this field (as well as other unconventional models of computation) is attracting the attention of a growing community.Ministerio de Educación y Ciencia TIN2005-09345-C04-01Junta de Andalucía TIC-58

New applications for an old tool

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-