First Order Probabilistic Logic

Nilsson revisited in 1986 the early work of Boole (1854) and of Hailperin (1976) on logic and probability, i.e., a generalization of logic in which the truth of sentences are probability values. This led to state precisely several basic problems of artificial intelligence, a paradigm of which is probabilistic satisfiability (PSAT): determine, given a set of clauses (i.e., propositional sentences) and probabilities that these clauses are true, whether these probabilities are consistent. We consider here the extension of PSAT to first order logic, or FOPSAT for short. We propose a delayed column generation algorithm to establish consistency and entail new probability values for a probabilistic satisfiability system to remain consistent when adding a new logical sentence. The progress of the algorithm is illustrated on an example