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

by 1,2 and 1,*
1
College of Computer Science and Technology, Guizhou University, Guiyang 550025, China
2
Department of Electronics and Information Engineering, Anshun University, Anshun 561000, China
*
Author to whom correspondence should be addressed.
Entropy 2020, 22(5), 569; https://doi.org/10.3390/e22050569
Received: 22 March 2020 / Revised: 13 May 2020 / Accepted: 17 May 2020 / Published: 19 May 2020
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 $u ( k , d ) ≤ f ( k , d ) + 1$ . View Full-Text
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Fu, Z.; Xu, D. Uniquely Satisfiable d-Regular (k,s)-SAT Instances. Entropy 2020, 22, 569.