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

Looking for Simple Common Schemes to Design Recognizer P Systems with Active Membranes That Solve Numerical Decision Problems

Earlier solutions to decision problems by means of P systems used many counter objects to control the synchronization of different stages in a computation (usually as many counters as the stage must last in the worst case). In this paper we propose a way to replace those counters with some spacial objects for each stage. Furthermore, following the ideas presented in [1], in order to have a common scheme to attack numerical problems, all instances of a problem with the same size are solved by the same P system (which depends on the size) given an input which describes the corresponding instance of the problem. We illustrate these ideas with a cellular solution to the Subset-Sum problem

From SAT to SAT-UNSAT using P systems with dissolution rules

DP is the class of problems that are the differences between two languages from NP. Most difficult problems from DP are called DP-complete problems, that can be seen as the conjunction of an NP-complete problem and a co-NP-complete problem. It is easy to see that the problem P vs NP is equivalent to the problem P vs DP, and therefore DP-complete problems would be better candidates to attack the conjecture, since they seem to be harder than NP-complete problems. In this paper, a methodology to transform an efficient solution of an NP-complete problem into an efficient solution of a DP-complete problem is applied. More precisely, a solution to SAT is given by means of a uniform family of recognizer polarizationless P systems with active membranes with dissolution rules and division rules for both elementary and non-elementary membranes, and later it is transformed into a solution to the problem SAT-UNSAT.Ministerio de Ciencia e Innovación TIN2017-89842-

Membrane division, restricted membrane creation and object complexity in P systems

We improve, by using register machines, some existing universality results for specific models of P systems. P systems with membrane creation are known to generate all recursively enumerable sets of vectors of non-negative integers, even when no region (except the environment) contains more than one object of the same kind.We showhere that they generate all recursively enumerable languages, and that two membrane labels are sufficient (the same result holds for accepting all recursively enumerable vectors of non-negative integers). Moreover, at most two objects are present inside the system at any time in the generative case.We then prove that 10 + msymbols are sufficient to generate any recursively enumerable language over m symbols. P systems with active membranes without polarizations are known to generate all recursively enumerable sets of vectors of non-negative integers. We show that they generate all recursively enumerable languages; four starting membranes with three labels or seven starting membranes with two labels are sufficient. P systems with active membranes and two polarizations are known to generate/accept all recursively enumerable sets of vectors of non-negative integers, using only rules of rewriting and sending objects out.We show that accepting can be done by deterministic systems. Finally, we show that P systems with restricted membrane creation (the newly created membrane can only be of the same kind as the parent one) generate at least matrix languages, even when having at most one object in the configuration (except the environment). We conclude by presenting a summary of the main results obtained in this paper and a list of open questions.Ministerio de Ciencia y Tecnología TIC2002-04220-C03-0

One and Two Polarizations, Membrane Creation and Objects Complexity in P Systems

We improve, by using register machines, some existing universality results for specific models of P systems. P systems with membrane creation are known to generate all recursively enumerable sets of vectors of non-negative integers, even when no region (except the environment) contains more than one object of the same kind. We here show that they generate all recursively enumerable languages, and two membrane labels are sufficient (the same result holds for accepting all recursively enumerable vectors of non-negative integers). Moreover, at most two objects are present inside the system at any time in the generative case. Then we prove that 10 + m symbols are enough to generate any recursively enumerable language over m symbols. P systems with active membranes without polarizations are known to generate all recursively enumerable sets of vectors of non-negative integers. We show that they generate all recursively enumerable languages; four starting membranes with three labels or seven starting membranes with two labels are sufficient. P systems with active membranes and two polarizations are known to generate/accept all recursively enumerable sets of vectors of non-negative integers, only using rules of rewriting and sending objects out. We show that accepting can be done by deterministic systems. Finally, remarks and open questions are presented.Ministerio de Ciencia y Tecnología TIC2002-04220-C03-0

A New Strategy to Improve the Performance of PDP-Systems Simulators

One of the major challenges that current P systems simulators have to deal with is to be as efficient as possible. A P system is syntactically described as a membrane structure delimiting regions where multisets of objects evolve by means of evolution rules. According to that, on each computation step, the applicability of the rules for the current P system configuration must be calculated. In this paper we extend previous works that use Rete-based simulation algorithm in order to improve the time consumed during the checking phase in the selection of rules. A new approach is presented, oriented to the acceleration of Population Dynamics P Systems simulations.Ministerio de Economía y Competitividad TIN2012- 3743

A Linear-Time Solution to the Knapsack Problem Using P Systems with Active Membranes

Up to now, P systems dealing with numerical problems have been rarely considered in the literature. In this paper we present an effective solution to the Knapsack problem using a family of deterministic P systems with active membranes using 2-division. We show that the number of steps of any computation is of linear order, but polynomial time is required for pre-computing resources.Ministerio de Ciencia y Tecnología TIC2002-04220-C03-0

Tissue P Systems with Cell Division

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

Computing Partial Recursive Functions by Virus Machines

Virus Machines are a computational paradigm inspired by the manner in which viruses replicate and transmit from one host cell to another. This paradigm provides non-deterministic sequential devices. Non-restricted Virus Machines are unbounded Virus Machines, in the sense that no restriction on the number of hosts, the number of instructions and the number of viruses contained in any host along any computation is placed on them. The computational completeness of these machines has been obtained by simulating register machines. In this paper, Virus Machines as function computing devices are considered. Then, the universality of non-restricted virus machines is proved by showing that they can compute all partial recursive functions.Ministerio de Economía y Competitividad TIN2012- 3743

Rete Algorithm for P System Simulators

The Rete algorithm is a well-known algorithm in rule-based production systems which builds directed acyclic graphs that represent higher-level rule sets. This allows the rule-based systems to avoid complete re-evaluation of all conditions of the rules each step in order to check the applicability of the rules and, therefore, the computational e ciency of the production systems is improved. In this paper we study how these ideas can be applied in the improvement of the design of computational simulators in the framework of Membrane Computing.Junta de Andalucía P08-TIC-04200Ministerio de Economía y Competitividad TIN2012-3743