Uniquely Satisfiable d-Regular (k,s)-SAT Instances

Unique k-SAT is the promised version of k-SAT where the given formula has 0 or 1 solution and is proved to be as difficult as the general k-SAT. For any k≥3, s≥f(k,d) and (s+d)/2>k−1, a parsimonious reduction from k-CNF to d-regular (k,s)-CNF is given. Here regular (k,s)-CNF is a subclass of CNF, where each clause of the formula has exactly k distinct variables, and each variable occurs in exactly s clauses. A d-regular (k,s)-CNF formula is a regular (k,s)-CNF formula, in which the absolute value of the difference between positive and negative occurrences of every variable is at most a nonnegative integer d. We prove that for all k≥3, f(k,d)≤u(k,d)+1 and f(k,d+1)≤u(k,d). The critical function f(k,d) is the maximal value of s, such that every d-regular (k,s)-CNF formula is satisfiable. In this study, u(k,d) denotes the minimal value of s such that there exists a uniquely satisfiable d-regular (k,s)-CNF formula. We further show that for s≥f(k,d)+1 and (s+d)/2>k−1, there exists a uniquely satisfiable d-regular (k,s+1)-CNF formula. Moreover, for k≥7, we have that u(k,d)≤f(k,d)+1.