Abstract:
Uncertainty is a fundamental and irreducible aspect of our knowledge about the world. Probability is the most well-understood and widely applied logic for computational scientific reasoning under uncertainty. As theory and practice advance, general-purpose languages are beginning to emerge for which the fundamental logical basis is probability. However, such languages have lacked a logical foundation that fully integrates classical first-order logic with probability theory. This paper presents such an integrated logical foundation. A formal specification is presented for multi-entity Bayesian networks (MEBN), a knowledge representation language based on directed graphical probability models. A proof is given that a probability distribution over interpretations of any consistent, finitely axiomatizable first-order theory can be defined using MEBN. A semantics based on random variables provides a logically coherent foundation for open world reasoning and a means of analyzing tradeoffs between accuracy and computation cost. Furthermore, the underlying Bayesian logic is inherently open, having the ability to absorb new facts about the world, incorporate them into existing theories, and/or modify theories in the light of evidence. Bayesian inference provides both a proof theory for combing prior knowledge with observations, and a learning theory for refining a representation as evidence accrues. The results of this paper provide a logical foundation for the rapidly evolving literature on first-order Bayesian knowledge representation, and point the way toward Bayesian languages suitable for general-purpose knowledge representation and computing. Because first-order Bayesian logic contains classical first-order logic as a deterministic subset, it is a natural candidate as a universal representation for integrating domain ontologies expressed in languages based on classical first-order logic or subsets thereof.
