First-Order Typed Model Counting for Probabilistic Conditional Reasoning at Maximum Entropy

Marco Wilhelm, Marc Finthammer, Gabriele Kern-Isberner, Christoph Beierle

Abstract: First-order typed model counting extends first-order model counting by the ability to distinguish between different types of models. In this paper, we exploit this benefit in order to calculate weighted conditional impacts (WCIs) which play a central role in nonmonotonic reasoning based on conditionals. More precisely, WCIs store information about the verification and the falsification of conditionals with respect to a possible worlds semantics, and therefore serve as sufficient statistics for maximum entropy (ME) distributions as models of probabilistic conditional knowledge bases. Formally, we annotate formulas with algebraic types that encode concisely all structural information needed to compute WCIs, while allowing for a systematic and efficient counting of models. In this way, our approach to typed model counting for ME-reasoning integrates both structural and counting aspects in the same framework.