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- W2093963957 abstract "In this paper we examine a variant, k-HSAT, of the well-known Satisfiability problem, wherein formula instances are limited to CNF formulae having exactly k literals in each clause at most one of which is un-negated. Such formulae correspond to sets of Horn clauses, each of which involves exactly k propositional variables. In addition, rather than seeking instantiations of the propositional variables that satisfy the formula, satisfying instantiations having at least k−1 variables set to 1 are required (such instantiations being referred to as acceptable). Viewing formulae as sets of Horn clauses, `acceptable' instantiations are exactly those that correspond to `non-trivial' models, i.e., those in which at least one rule has a true antecedent. We show, using analytic methods, that k-HSAT exhibits a phase transition whose behavior is significantly different from that of the classical k-SAT problem concerning which various researchers have presented empirical evidence for the existence of a constant θk such that randomly chosen instances of k-SAT on n variables having m(n) clauses are `almost certainly' satisfiable if m(n)/n<θk and `almost certainly' unsatisfiable if m(n)/n>θk. In this paper we prove a sharp threshold result for k-HSAT for all fixed k⩾2. Specifically, for θk=(k−1)(k+1)/k! it is proved that if φ is chosen uniformly at random from instances of k-HSAT on n variables and m(n) clauses then limn→∞Prob[φhasanacceptableinstantiation]=1ifm(n)nk−1logen<θk,limn→∞Prob[φhasanacceptableinstantiation]=0ifm(n)nk−1logen>θk." @default.
- W2093963957 created "2016-06-24" @default.
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- W2093963957 date "2001-01-01" @default.
- W2093963957 modified "2023-09-23" @default.
- W2093963957 title "A sharp threshold for the phase transition of a restricted Satisfiability problem for Horn clauses" @default.
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- W2093963957 doi "https://doi.org/10.1016/s1567-8326(00)00002-3" @default.
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