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

The length of each clause in a regular (s, c, k)-CNF 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)-CNF 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^{*}} .