Abstract
Presumably efficient computing models are characterized by their capability to provide polynomial-time solutions for NP-complete problems. Given a class ℛ of recognizer membrane systems, ℛ denotes the set of decision problems solvable by families from ℛ in polynomial time and in a uniform way. PMCℛ is closed under complement and under polynomial-time reduction. Therefore, if ℛ is a presumably efficient computing model of recognizer membrane systems, then NP ∪ co-NP ⊆ PMCℛ. In this paper, the lower bound NP ∪ co-NP for the time complexity class PMCℛ is improved for any presumably efficient computing model ℛ of recognizer membrane systems verifying some simple requirements. Specifically, it is shown that DP ∪ co-DP is a lower bound for such PMCℛ, where DP is the class of differences of any two languages in NP. Since NP ∪ co-NP ⊆ DP ∩ co-DP, this lower bound for PMCℛ delimits a thinner frontier than that with NP ∪ co-NP.
Highlights
Membrane Computing is a computing paradigm inspired by some basic biological features
A recognizer membrane system has the following additional syntactic and semantic peculiarities: (a) the working alphabet has two distinguished objects; (b) there exist an input compartment, and there is an input alphabet strictly contained in the working alphabet; (c) the initial content of each compartment is a multiset of objects from the working alphabet not belonging to the input alphabet; (d) all computations halt; and (e) for each computation and only at its last step, either object yes or object no must have been released to the environment
The Methodology e main result of this paper refers to the robustness of the complexity class PMCR associated with some computing models of recognizer membrane systems, with respect to the operation product of two decision problems
Summary
Membrane Computing is a computing paradigm inspired by some basic biological features. We say that a family Π {Π(n) | n ∈ N} of recognizer membrane systems solves a decision problem X in polynomial time in a uniform way if such family Π can be generated by a deterministic Turing machine working in polynomial time, and there exists a pair (cod, s) of polynomial-time computable functions over the set of instances of X verifying the following: (a) for each instance u ∈ IX, s(u) is a natural number and cod(u) is an input multiset of the system Π(s(u)); (b) for each n ∈ N, the set s− 1({n}) is a finite set; and (c) the family Π is polynomially bounded, sound, and complete with regard to (X, cod, s). E main contribution of this paper is to provide a lower bound for PMCR delimiting a frontier with such class that is thinner to the one found with respect to NP ∪ co-NP, with R being a presumably efficient computing model of recognizer membrane systems verifying some simple requirements (e.g., in the case of cell-like membrane systems, allowing object evolution rules, dissolution rules, and communication rules).
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