Introducing a space complexity measure for P systems

2012

We show that exponential-space P systems with active membranes characterize the complexity class EXPSPACE. This result is proved by simulating Turing machines working in exponential space via uniform families of P systems with restricted elementary active membranes; the simulation is efficient, in the sense that the time and space required are at most polynomial with respect to the resources employed by the simulated Turing machine.

International Journal of Computer Mathematics, 2006

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 nonnegative integers, only using rules of rewriting and sending objects out. We show that accepting can be done by deterministic systems.

Lecture Notes in Computer Science, 2006

We consider P systems with symport/ antiport rules and small numbers of symbols and membranes and present several results for P systems with symport/ antiport rules simulating register machines with the number of registers depending on the number s of symbols and the number m of membranes. For instance, any recursively enumerable set of natural numbers can be generated (accepted) by systems with s ≥ 2 symbols and m ≥ 1 membranes such that m + s ≥ 6. In particular, the result of the original paper [17] proving universality for three symbols and four membranes is improved (e.g., three symbols and three membranes are sufficient). The general results that P systems with symport/ antiport rules with s symbols and m membranes are able to simulate register machines with max {m (s − 2) , (m − 1) (s − 1)} registers also allows us to give upper bounds for the numbers s and m needed to generate/accept any recursively enumerable set of k-dimensional vectors of non-negative integers or to compute any partial recursive function f : N α → N β . Finally, we also study the computational power of P systems with symport/ antiport rules and only one symbol: with one membrane, we can exactly generate the family of finite sets of non-negative integers; with one symbol and two membranes, we can generate at least all semilinear sets. The most interesting open question is whether P systems with symport/ antiport rules and only one symbol can gain computational completeness (even with an arbitrary number of membranes) as it was shown for tissue P systems in [1].

Fundamenta Informaticae, 2014

We continue the investigation of the computational power of spaceconstrained P systems. We show that only a constant amount of space is needed in order to simulate a polynomial-space bounded Turing machine. Due to this result, we propose an alternative definition of space complexity for P systems, where the amount of information contained in individual objects and membrane labels is also taken into account. Finally, we prove that, when less than a logarithmic number of membrane labels is available, moving the input objects around the membrane structure without rewriting them is not enough to even distinguish inputs of the same length.

Natural Computing, 2010

We consider recognizer P systems having three polarizations associated to the membranes, and we show that they are able to solve the PSPACEcomplete problem Quantified 3SAT when working in polynomial space and exponential time. The solution is uniform (all the instances of a fixed size are solved by the same P system) and uses only communication rules: evolution rules, as well as membrane division and dissolution rules, are not used. Our result shows that, as it happens with Turing machines, this model of P systems can solve in exponential time and polynomial space problems that cannot be solved in polynomial time, unless P = PSPACE.

Enjoying Natural Computing, 2018

We present some high-level open problems in the complexity theory of membrane systems, related to the actual computing power of confluence vs determinism, semi-uniformity vs uniformity, deep vs shallow membrane structures, membrane division vs internal evolution of membranes. For each of these problems we present some reasonable approaches that are, however, unable to be employed "as-is" to provide a complete solution. This will hopefully sparkle new ideas that will allow tackling these open problems.

Journal of Complexity, 2010

Membrane systems, also called P systems, are biologically inspired theoretical models of distributed and parallel computing. This paper presents a new class of tissue-like P systems with cell separation, a feature which allows the generation of new workspace. We study the efficiency of the class of P systems and draw a conclusion that only tractable problems can be efficiently solved by using cell separation and communication rules with the length of at most 1. We further present an efficient (uniform) solution to SAT by using cell separation and communication rules with length at most 6. We conclude that a borderline between efficiency and non-efficiency exists in terms of the length of communication rules (assuming P = NP). We discuss future research topics and open problems.

Two proofs have been shown for P systems with active membranes in previously published papers, demonstrating that these P systems can solve in polynomial time exactly the class of problems PSPACE. Consequently, these P systems are equivalent (up to a polynomial time reduction) to Second Machine Class models as the alternating Turing machine or the PRAM computer. These proofs were based on a modified definition of uniform families of P systems. Here we demonstrate that the results remain valid also in the case of standard definitions.

Bio-Inspired Computing Models and Algorithms, 2019

Lecture Notes in Computer Science, 2009

Membrane computing is a formal framework of distributed parallel multiset processing. Due to massive parallelism and exponential space some intractable computational problems can be solved by P systems with active membranes in a polynomial number of steps. In this paper we generalize this approach from decisional problems to the computational ones, by providing a solution of a #P-complete problem, namely to compute the permanent of a binary matrix. The implication of this result to the PP complexity class is discussed and compared to known results about NP ∪ co − NP.