Logarithmic SAT Solution with Membrane Computing

Radu Nicolescu,Yezhou Liu,James Cooper,Alec Henderson,Michael Dinneen

doi:10.3390/axioms11020066

Abstract

P systems have been known to provide efficient polynomial (often linear) deterministic solutions to hard problems. In particular, cP systems have been shown to provide very crisp and efficient solutions to such problems, which are typically linear with small coefficients. Building on a recent result by Henderson et al., which solves SAT in square-root-sublinear time, this paper proposes an orders-of-magnitude-faster solution, running in logarithmic time, and using a small fixed-sized alphabet and ruleset (25 rules). To the best of our knowledge, this is the fastest deterministic solution across all extant P system variants. Like all other cP solutions, it is a complete solution that is not a member of a uniform family (and thus does not require any preprocessing). Consequently, according to another reduction result by Henderson et al., cP systems can also solve k-colouring and several other NP-complete problems in logarithmic time.

Highlights

The P-versus-NP problem remains one of the most important unsolved problems in computational complexity theory
To the best of our knowledge, this represents a significant breakthrough in membrane computing, being orders-of-magnitude faster than all previous deterministic solutions
We show that cP systems can theoretically solve satisfiability problem (SAT) in sublinear logarithmic time

Summary

Introduction

The P-versus-NP problem remains one of the most important unsolved problems in computational complexity theory. The big theoretical question is whether every problem of which the solution can be verified in polynomial time (NP) can be solved in polynomial time (P). Many studies have investigated different approaches to solve such hard problems in a reasonable amount of time (e.g., polynomial or even linear time). Such methods include approximation [2], fixing parameters [3], or the use of alternative theoretical models, such as P systems [4,5,6,7,8,9]. We show that cP systems can theoretically solve SAT in sublinear logarithmic time. A literal is a variable or its negation (here indicated by overbars)

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