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

First Steps Towards Linking Membrane Depth and the Polynomial Hierarchy

In this paper we take the first steps in studying possible connections between non-elementary division with limited membrane depth and the levels of the Polynomial Hierarchy. We present a uniform family with a membrane structure of depth d + 1 that solves a problem complete for level d of the Polynomial Hierarchy

Simulating counting oracles with cooperation

We prove that monodirectional shallow chargeless P systems with active membranes and minimal cooperation working in polynomial time precisely characterise P#P k , the complexity class of problems solved in polynomial time by deterministic Turing machines with a polynomial number of parallel queries to an oracle for a counting problem

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

Introducing a Space Complexity Measure for P Systems

We define space complexity classes in the framework of membrane computing, giving some initial results about their mutual relations and their connection with time complexity classes, and identifying some potentially interesting problems which require further research

Complete Problems for a Variant of P Systems with Active Membranes

We identify a family of decision problems that are hard for some complexity classes defined in terms of P systems with active membranes working in polynomial time. Furthermore, we prove the completeness of these problems in the case where the systems are equipped with a form of priority that linearly orders their rules. Finally, we highlight some possible connections with open problems related to the computational complexity of P systems with active membranes

An Optimal Frontier of the Efficiency of Tissue P Systems with Cell Division

In the framework of tissue P systems with cell division, the length of communication rules provides a frontier for the tractability of decision problems. On the one hand, the limitation on the efficiency of tissue P systems with cell division and communication rules of length 1 has been established. On the other hand, polynomial time solutions to NP–complete problems by using families of tissue P systems with cell division and communication rules of length at most 3 has been provided. In this paper, we improve the previous result by showing that the HAM-CYCLE problem can be solved in polynomial time by a family of tissue P systems with cell division by using communication rules with length at most 2. Hence, a new tractability boundary is given: passing from 1 to 2 amounts to passing from non–efficiency to efficiency, assuming that P ̸= NP.Ministerio de Ciencia e Innovación TIN2009-13192Junta de Andalucía P08 – TIC 0420

Subroutines in P Systems and Closure Properties of Their Complexity Classes

The literature on membrane computing describes several variants of P systems whose complexity classes C are "closed under exponentiation", that is, they satisfy the inclusion PC C, where PC is the class of problems solved by polynomial-time Turing machines with oracles for problems in C. This closure automatically implies closure under many other operations, such as regular operations (union, concatenation, Kleene star), intersection, complement, and polynomial-time mappings, which are inherited from P. Such results are typically proved by showing how elements of a family of P systems can be embedded into P systems simulating Turing machines, which exploit the elements of as subroutines. Here we focus on the latter construction, abstracting from the technical details which depend on the speci c variant of P system, in order to describe a general strategy for proving closure under exponentiation

A Toolbox for Simpler Active Membrane Algorithms

We show that recogniser P systems with active membranes can be augmented with a priority over their set of rules and any number of membrane charges without loss of generality, as they can be simulated by standard P systems with active membranes, in particular using only two charges. Furthermore, we show that more general accepting conditions, such as sending out several, possibly contradictory results and keeping only the first one, or rejecting by halting without output, are also equivalent to the standard accepting conditions. The simulations we propose are always without significant loss of efficiency, and thus the results of this paper can hopefully simplify the design of algorithms for P systems with active membranes

Improving Universality Results on Parallel Enzymatic Numerical P Systems

We improve previously known universality results on enzymatic numerical P systems (EN P systems, for short) working in all-parallel and one-parallel modes. By using a attening technique, we rst show that any EN P system working in one of these modes can be simulated by an equivalent one-membrane EN P system working in the same mode. Then we show that linear production functions, each depending upon at most one variable, su ce to reach universality for both computing modes. As a byproduct, we propose some small deterministic universal enzymatic numerical P systems