scholarly journals Enhanced Membrane Computing Algorithm for SAT Problems Based on the Splitting Rule

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1412
Author(s):  
Hao ◽  
Liu
Keyword(s):  
Membrane Computing ◽  
Space Complexity ◽  
Natural Computing ◽  
Sat Problem ◽  
Computing Algorithm ◽  
Complete Problems ◽  
Np Complete ◽  
Splitting Rule ◽  
Np Problems ◽  
Time And Space Complexity

Boolean propositional satisfiability (SAT) problem is one of the most widely studied NP-complete problems and plays an outstanding role in many domains. Membrane computing is a branch of natural computing which has been proven to solve NP problems in polynomial time with a parallel compute mode. This paper proposes a new algorithm for SAT problem which combines the traditional membrane computing algorithm of SAT problem with a classic simplification rule, the splitting rule, which can divide a clause set into two axisymmetric subsets, deal with them respectively and simultaneously, and obtain the solution of the original clause set with the symmetry of their solutions. The new algorithm is shown to be able to reduce the space complexity by distributing clauses with the splitting rule repeatedly, and also reduce both time and space complexity by executing one-literal rule and pure-literal rule as many times as possible.

Membrane Computing

2011 ◽  
pp. 1-31 ◽  
Author(s):  
Gheorghe Paun
Keyword(s):  
Membrane Computing ◽  
Living Cells ◽  
Point Of View ◽  
Natural Computing ◽  
Historical Evolution ◽  
Biological Phenomena ◽  
Complete Problems ◽  
Hard Problems ◽  
Np Complete ◽  
And Mathematics

Membrane computing is a branch of natural computing whose initial goal was to abstract computing models from the structure and the functioning of living cells. The research was initiated about five years ago (at the end of 1998), and since that time the area has been developed significantly from a mathematical point of view. The basic types of results of this research concern the computability power (in comparison with the standard Turing machines and their restrictions) and the efficiency (the possibility to solve computationally hard problems, typically NP-complete problems, in a feasible time and typically polynomial). However, membrane computing has recently become attractive also as a framework for devising models of biological phenomena, with the tendency to provide tools for modelling the cell itself, not only the local processes. This chapter surveys the basic elements of membrane computing, somewhat in its “historical” evolution: from biology to computer science and mathematics and back to biology. The presentation is informal, without any technical detail, and an invitation to membrane computing intended to acquaint the nonmathematician reader with the main directions of research of the domain, the type of central results, and the possible lines of future development, including the possible interest of the biologist looking for discrete algorithmic tools for modelling cell phenomena.

Method for Organizing Grover's Quantum Oracle

2014 ◽  
Vol 21 (04) ◽  
pp. 1450011
Author(s):  
Hideaki Ito ◽  
Saburou Iida
Keyword(s):  
Quantum Computation ◽  
Knapsack Problem ◽  
Time Complexity ◽  
Quantum Circuit ◽  
Space Complexity ◽  
Complete Problems ◽  
Black Boxes ◽  
Np Complete ◽  
Quantum Oracle ◽  
Toffoli Gates

In a quantum computation, some algorithms use oracles (black boxes) for abstract computational objects. This paper presents an example for organizing Grover's quantum oracle by synthesizing several unitary gates such as CNOT gates, Toffoli gates, and Hadamard gates. As an example, we show a concrete quantum circuit for the knapsack problem, which belongs to the class of NP-complete problems. The time complexity of an oracle for the knapsack problem is estimated to be O(n2), where n is the number of variables. And the same order is obtained for space complexity.

scholarly journals Solving Vertex Cover Problem by Means of Tissue P Systems with Cell Separation

2010 ◽  
Vol 5 (4) ◽  
pp. 540 ◽  
Author(s):  
Chun Lu ◽  
Xingyi Zhang
Keyword(s):  
Cell Separation ◽  
Membrane Computing ◽  
Linear Time ◽  
Vertex Cover ◽  
P Systems ◽  
Computing Model ◽  
Vertex Cover Problem ◽  
Complete Problems ◽  
Np Complete ◽  
Cover Problem

Tissue P systems is a computing model in the framework of membrane computing inspired from intercellular communication and cooperation between neurons. Many different variants of this model have been proposed. One of the most important models is known as tissue P systems with cell separation. This model has the ability of generating an exponential amount of workspace in linear time, thus it allows us to design cellular solutions to NP-complete problems in polynomial time. In this paper, we present a solution to the Vertex Cover problem via a family of such devices. This is the first solution to this problem in the framework of tissue P systems with cell separation.

The Future

2017 ◽  
Author(s):  
Lance Fortnow
Keyword(s):  
Big Data ◽  
Ground State ◽  
New Technologies ◽  
Economic Environment ◽  
Inherent Difficulty ◽  
Complete Problems ◽  
Np Complete ◽  
Np Problems ◽  
Tools And Techniques ◽  
Np Problem

This chapter explores some of today's great challenges of computing. These challenges include parallel computation, dealing with big data, and the networking of everything. The chapter then argues that P versus NP goes well beyond a simple mathematical puzzle. The P versus NP problem is a way of thinking, a way to classify computational problems by their inherent difficulty. P versus NP also brings communities together. There are NP-complete problems in physics, biology, economics, and many other fields. Physicists and economists work on very different problems, but they share a commonality that can give great benefits from sharing tools and techniques. Tools developed to find the ground state of a physical system can help find equilibrium behavior in a complex economic environment. Ultimately, the inherent difficulty of NP problems leads to new technologies.

P SYSTEMS WITH INPUT IN BINARY FORM

2006 ◽  
Vol 17 (01) ◽  
pp. 127-146 ◽  
Author(s):  
ALBERTO LEPORATI ◽  
CLAUDIO ZANDRON ◽  
MIGUEL A. GUTIÉRREZ-NARANJO
Keyword(s):  
Complexity Theory ◽  
Binary Form ◽  
P Systems ◽  
Turing Machines ◽  
Simple Method ◽  
Sat Problem ◽  
Numerical Problem ◽  
Complete Problems ◽  
Np Complete

Current P systems which solve NP–complete numerical problems represent the instances of the problems in unary notation. However, in classical complexity theory, based upon Turing machines, switching from binary to unary encoded instances generally corresponds to simplify the problem. In this paper we show that, when working with P systems, we can assume without loss of generality that instances are expressed in binary notation. More precisely, we propose a simple method to encode binary numbers using multisets, and a family of P systems which transforms such multisets into the usual unary notation. Such a family could thus be composed with the unary P systems currently proposed in the literature to obtain (uniform) families of P systems which solve NP–complete numerical problems with instances encoded in binary notation. We introduce also a framework which can be used to design uniform families of P systems which solve NP–complete problems (both numerical and non-numerical) working directly on binary encoded instances, i.e., without first transforming them to unary notation. We illustrate our framework by designing a family of P systems which solves the 3-SAT problem. Next, we discuss the modifications needed to obtain a family of P systems which solves the PARTITION numerical problem.

scholarly journals Set-based SAT-solving

2007 ◽  
Vol 20 (3) ◽  
pp. 395-414 ◽  
Author(s):  
Bernd Steinbach ◽  
Christian Posthoff
Keyword(s):  
Data Structure ◽  
Binary Solution ◽  
Solution Process ◽  
Efficient Algorithms ◽  
Sat Solving ◽  
Sat Problem ◽  
Complete Problems ◽  
Binary Coding ◽  
Np Complete ◽  
Problem Lists

The 3-SAT problem is one of the most important and interesting NP-complete problems with many applications in different areas. In several previous papers we showed the use of ternary vectors and set-theoretic considerations as well as binary codings and bit-parallel vector operations in order to solve this problem. Lists of orthogonal ternary vectors have been the main data structure, the intersection of ternary vectors (representing sets of binary solution candidates) was the most important set theoretic operation. The parallelism of the solution process has been established on the register level, i.e. related to the existing hardware, by using a binary coding of the ternary vectors, and it was also possible to transfer the solution process to a hierarchy of processors working in parallel. This paper will show further improvements of the existing algorithms which are easy to understand and result in very efficient algorithms and implementations. Some examples will be presented that will show the recent results.

scholarly journals P Systems with Evolutional Communication and Division Rules

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 327
Author(s):  
David Orellana-Martín ◽  
Luis Valencia-Cabrera ◽  
Mario J. Pérez-Jiménez
Keyword(s):  
Computational Complexity ◽  
Complexity Theory ◽  
Membrane Computing ◽  
P Systems ◽  
Membrane Systems ◽  
Sat Problem ◽  
Complete Problem ◽  
Intractable Problems ◽  
Np Complete ◽  
The One

A widely studied field in the framework of membrane computing is computational complexity theory. While some types of P systems are only capable of efficiently solving problems from the class P, adding one or more syntactic or semantic ingredients to these membrane systems can give them the ability to efficiently solve presumably intractable problems. These ingredients are called to form a frontier of efficiency, in the sense that passing from the first type of P systems to the second type leads to passing from non-efficiency to the presumed efficiency. In this work, a solution to the SAT problem, a well-known NP-complete problem, is obtained by means of a family of recognizer P systems with evolutional symport/antiport rules of length at most (2,1) and division rules where the environment plays a passive role; that is, P systems from CDEC^(2,1). This result is comparable to the one obtained in the tissue-like counterpart, and gives a glance of a parallelism and the non-evolutionary membrane systems with symport/antiport rules.

Complexity, computational

2018 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Propositional Logic ◽  
Common Type ◽  
Combinatorial Games ◽  
Np Completeness ◽  
Complete Problems ◽  
Np Complete ◽  
Feasible Solutions ◽  
Np Problems

The theory of computational complexity is concerned with estimating the resources a computer needs to solve a given problem. The basic resources are time (number of steps executed) and space (amount of memory used). There are problems in logic, algebra and combinatorial games that are solvable in principle by a computer, but computationally intractable because the resources required by relatively small instances are practically infeasible. The theory of NP-completeness concerns a common type of problem in which a solution is easy to check but may be hard to find. Such problems belong to the class NP; the hardest ones of this type are the NP-complete problems. The problem of determining whether a formula of propositional logic is satisfiable or not is NP-complete. The class of problems with feasible solutions is commonly identified with the class P of problems solvable in polynomial time. Assuming this identification, the conjecture that some NP problems require infeasibly long times for their solution is equivalent to the conjecture that P≠NP. Although the conjecture remains open, it is widely believed that NP-complete problems are computationally intractable.

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