Technical Perspective: Combining logic and probability

models that satisfy a formula. The former can be used to perform Maximum a Posteriori (MAP) inference , and the latter, called " weighted model counting, " to perform marginal inference or to compute the parti tion function. PTP provides the final step in unifying probabilistic, propositional, and first-order inference. PTP lifts a weighted version of DPLL by allowing it to branch on a logical expression containing un-instantiated variables. In the best case, PTP can perform weighted model counting while only grounding a small part of a large relational probabilistic theory. SRL is a highly active research area, where many of the ideas in PTP appear in various forms. There are lifted versions of other exact inference algorithms such as variable elimination, as well as lifted versions of approximate algorithms such as belief propagation and variational inference. Approximate inference is often the best one can hope for in large, complex domains. Gogate and Domingos suggest how PTP could be turned in a fast approximate algorithm by sampling from the set of children of a branch point. PTP sparks many interesting directions for future research. Algorithms must be developed to quickly identify good literals for lifted branching and decomposition. Approximate versions of PTP need to be fleshed out and evaluated against traditional methods for probabilistic inference. Finally, the development of a lifted version of DPLL suggests that researchers working on logical theorem proving revisit the traditional divide between syntactic methods for quantified logics and model-finding methods for propositional logic. A GOAL OF research in artificial intelligence and machine learning since the early days of expert systems has been to develop automated reasoning methods that combine logic and probability. Probabilistic theorem proving (PTP) unifies three areas of research in computer science: reasoning under uncertainty , theorem-proving in first-order logic, and satisfiability testing for prop-ositional logic. Why is there a need to combine logic and probability? Probability theory allows one to quantify uncertainty over a set of propositions—ground facts about the world—and a probabilistic reasoning system allows one to infer the probability of unknown (hidden) propositions conditioned on the knowledge of other propositions. However, probability theory alone has nothing to say about how propositions are constructed from relationships over entities or tuples of entities, and how general knowledge at the level of relationships is to be represented and applied. Handling relations takes us into the domain of first-order logic. An important case is …