### G-2006-79

# An Anytime Deduction Algorithm for the Probabilistic Logic and Entailment Problems

## Brigitte Jaumard and Anderson D. Parreira

We study two basic problems of probabilistic reasoning: the probabilistic logic and the probabilistic entailment problems. The first one can be defined as follows. Given a set of logical sentences and probabilities that these sentences are true, the aim is to determine whether these probabilities are consistent or not. Given a consistent set of logical sentences and probabilities, the probabilistic entailment problem consists in determining the range of the possible values of the probability associated with additional sentences while maintaining a consistent set of sentences and probabilities.

This paper proposes a general approach based on an anytime deduction method that allows the follow-up of the reasoning when checking consistency for the probabilistic logic problem or when determining the probability intervals for the probabilistic entailment problem. Considering a series of subsets of sentences and probabilities, the approach proceeds by computing increasingly narrow probability intervals that either show a contradiction or that contain the tightest entailed probability interval. Computational experience have been conducted to compare the proposed anytime deduction method, called AD-PSAT with an exact one, PSATCOL, using column generation techniques, both with respect to the range of the probability intervals and the computing times.

Published **December 2006**
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27 pages