# Combining Probabilistic Logic Programming with the Power of Maximum Entropy

*Gabriele Kern−Isberner and Thomas Lukasiewicz*

### Abstract

This paper is on the combination of two powerful approaches to uncertain reasoning: logic programming in a probabilistic setting, on the one hand, and the information-theoretical principle of maximum entropy, on the other hand. More precisely, we present two approaches to probabilistic logic programming under maximum entropy. The first one is based on the usual notion of entailment under maximum entropy, and is defined for the very general case of probabilistic logic programs over Boolean events. The second one is based on a new notion of entailment under maximum entropy, where the principle of maximum entropy is coupled with the closed world assumption (CWA) from classical logic programming. It is only defined for the more restricted case of probabilistic logic programs over conjunctive events. We then analyze the nonmonotonic behavior of both approaches along benchmark examples and along general properties for default reasoning from conditional knowledge bases. It turns out that both approaches have very nice nonmonotonic features. In particular, they realize some inheritance of probabilistic knowledge along subclass relationships, without suffering from the problem of inheritance blocking and from the drowning problem. They both also satisfy the property of rational monotonicity and several irrelevance properties. We finally present algorithms for both approaches, which are based on generalizations of recent techniques for probabilistic logic programming under logical entailment. The algorithm for the first approach still produces quite large weighted entropy maximization problems, while the one for the second approach generates optimization problems of the same size as the ones produced in probabilistic logic programming under logical entailment.