A stochastic limit approach to the SAT problem Luigi Accardi† and Masanori Ohya‡ † Centro V.Volterra, Universit`adi Roma Torvergata, Via Orazio Raimondo, 00173 Roma, Italia e-mail:
[email protected] ‡Department of Information Sciences, Tokyo University of Science, Noda City, Chiba 278-8510, Japan e-mail:
[email protected] I. INTRODUCTION Let L = 0, 1 be a Boolean lattice with usual join and meet {, and} t (x) be the truth value of a literal ∨x ∧ Although the ability of computer is highly progressed, in X. Then the truth value of a clause C is written as there are several problems which may not be solved ef- t (C) x C t (x). Moreover≡ ∨ ∈ the set of all clauses C (j =1, 2, ,m) fectively, namely, in polynomial time. Among such prob- C j ··· lems, NP problem and NP complete problem are fun- is called satisfiable iff the meet of all truth values of Cj is 1; t ( ) m t (C ) = 1. Thus the SAT problem is damental. It is known that all NP complete problems C ≡ ∧j=1 j are equivalent and an essential question is whether there written as follows: exists an algorithm to solve an NP complete problem in Definition 3 SAT Problem: Given a Boolean set X polynomial time. They have been studied for decades ≡ and for which all known algorithms have an exponential x1, , xn and a set = 1, , m of clauses, deter- mine{ ··· whether} is satisfiableC {C or··· not.C } running time in the length of the input so far.