The Phase Transition Analysis for Random Regular Exact (s, c, k)—SAT Problem

Abstract

The length of each clause in a regular (s, c, k) − CN F formula is k. Each argument occurrences s times, among them, positive occurrences (c * s) times and negative occurrences (s − c * s) times, where 0 < c < 1. A regular exact (s, c, k) − SAT question refers to whether there is a set of Boolean variable assignment such that exactly one literal in each clause of the regular (s, c, k) − CN F formula is true. Obviously, the problem is a NP-hard problem. To understand the hardness and the distribution of satisfiable solutions of regular exact (s, c, k) − SAT problem, we introduce a random instance generation model, use the first moment and second moment methods to analyze the satisfiable phase transition of the solutions. Set s * is the phase transition point, we show that the stochastic regular exact (s, c, k) − SAT problem instance is satisfiable with high probability if s < s * and unsatisfiable with high probability if s > s * , among them, s * is a function about a parameter c. Further, we discuss the phase transition point of the random regular exact (s, r, k) − SAT problem-the difference between the positive and negative occurrences of the variable is r-is s *. Finally, through several groups of experimental data to verify, the experimental results are consistent with the theory, and prove the correctness of the theory. INDEX TERMS first moment method, phase transition, satisfiability problem, second moment method.

The Phase Transition Analysis for the Random Regular Exact 2-(d, k)-SAT Problem

“…Definition 2. Suppose a model M, a property Q, and a probability function p : N → [0, 1], if there are two probability functions satisfying the following conditions [18]:…”