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Probabilistic satisfiability. (English) Zbl 0647.68049
Summary: We study the following computational problem proposed by N. J. Nilsson [Artif. Intell. 28, 71-87 (1986; Zbl 0589.03007)]: Several clauses (disjunctions of literals) are given, and for each clause the probability that the clause is true is specified. We are asked whether these probabilities are consistent. They are if there is a probability distribution on the truth assignments such that the probability of each clause is the measure of its satisfying set of assignments. Since this problem is a generalization of the satisfiability problem for propositional calculus it is immediately NP-hard. We show that it is NP- complete even when there are at most two literals per clause (a case which is polynomial time solvable in the non-probabilitic case). We use arguments from linear programming and graph theory to derive polynomial- time algorithms for some interesting special cases.

MSC:
68Q25 Analysis of algorithms and problem complexity
03B48 Probability and inductive logic
90C05 Linear programming
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