Probabilistic satisfiability

Knowledge Representation in Probabilistic Spatio-Temporal Knowledge Bases

“…The reduction shown in the membership proof, from the consistency checking problem to PSAT, would allow us to define additional tractable cases if the PSAT instances resulting from such a reduction were tractable. However, as will be discussed in Section 6, the tractable cases that have been identified for PSAT (Georgakopoulos et al, 1988;Andersen & Pretolani, 2001) do not carry over to our framework.…”

“…K, where is a "small" non-zero constant whose size is polynomial w.r.t. the size of K. To show the existence of , we observe that (i) checking the consistency of K can be reduced to deciding the PSAT instance φ K consisting of m clauses, where m = |A| + |F| · |ID| · |T | · |Space| + |ID| · |T | + |ID| · |T | · |Space|, as shown in the membership proof of Theorem 1; and (ii) if a satisfying probability distribution for an instance of PSAT with m clauses exists, then there is one with at most m + 1 non-zero probabilities, and with entries consisting of rational numbers with precision O(m 2 ) (Georgakopoulos et al, 1988;Papadimitriou & Steiglitz, 1982); Then, choosing equal to the smallest rational number with precision m 3 suffices to obtain a sufficiently small non-zero constant, whose size is polynomial w.r.t. the size of K, such that A ∪ {loc (id, q, t)…”

“…Given a conjunction φ of clauses, each of them associated with a probability, and an additional clause C with given lower and upper bounds on the admissible values for its probability, decide whether there is a probability distribution satisfying the probability of all the clauses in φ and the lower and upper bounds for C. As in probabilistic logic, where the entailment problem can be reduced to PSAT (Nilsson, 1986;Georgakopoulos et al, 1988), we have shown that the query answering problem in PST KBs can be reduced to the consistency checking problem. However, there is an important difference between the query answering problem and the probabilistic entailment problem: we assume that the input PST KBs is consistent, while the entailment problem is defined for any φ (even if φ is not satisfiable).…”

“…(Membership). We show that checking the consistency of K can be reduced to deciding an instance φ K of (an extension to) the Probabilistic Satisfiability (PSAT) problem (Hailperin, 1984;Nilsson, 1986), which is in NP (Georgakopoulos, Kavvadias, & Papadimitriou, 1988). Given a set of m clauses C 1 , .…”

“…Our query answering problem is related to the decision version of the entailment problem in probabilistic logic (Nilsson, 1986;Georgakopoulos et al, 1988), which is as follows. Given a conjunction φ of clauses, each of them associated with a probability, and an additional clause C with given lower and upper bounds on the admissible values for its probability, decide whether there is a probability distribution satisfying the probability of all the clauses in φ and the lower and upper bounds for C. As in probabilistic logic, where the entailment problem can be reduced to PSAT (Nilsson, 1986;Georgakopoulos et al, 1988), we have shown that the query answering problem in PST KBs can be reduced to the consistency checking problem.…”