We study the computational complexity of finding maximum a posteriori configurations in Bayesian networks whose probabilities are specified by logical formulas. This approach leads to a fine grained study in which local information such as context-sensitive independence and determinism can be considered. It also allows us to characterize more precisely the jump from tractability to NP-hardness and beyond, and to consider the complexity introduced by evidence alone.
Knowledge compilation is a powerful technique for compactly representing and efficiently reasoning about logical knowledge bases. It has been successfully applied to numerous problems in artificial intelligence, such as probabilistic inference and conformant planning. Conditioning, which updates a knowledge base with observed truth values for some propositions, is one of the fundamental operations employed for reasoning. In the propositional setting, conditioning can be efficiently applied in all cases. Recently, people have explored compilation for first-order knowledge bases. The majority of this work has centered around using first-order d-DNNF circuits as the target compilation language. However, conditioning has not been studied in this setting. This paper explores how to condition a first-order d-DNNF circuit. We show that it is possible to efficiently condition these circuits on unary relations. However, we prove that conditioning on higher arity relations is #P-hard. We study the implications of these findings on the application of performing lifted inference for first-order probabilistic models.This leads to a better understanding of which types of queries lifted inference can address.
First-order model counting emerged recently as a novel reasoning task, at the core of efficient algorithms for probabilistic logics. We present a Skolemization algorithm for model counting problems that eliminates existential quantifiers from a first-order logic theory without changing its weighted model count. For certain subsets of first-order logic, lifted model counters were shown to run in time polynomial in the number of objects in the domain of discourse, where propositional model counters require exponential time. However, these guarantees apply only to Skolem normal form theories (i.e., no existential quantifiers) as the presence of existential quantifiers reduces lifted model counters to propositional ones. Since textbook Skolemization is not sound for model counting, these restrictions precluded efficient model counting for directed models, such as probabilistic logic programs, which rely on existential quantification. Our Skolemization procedure extends the applicability of first-order model counters to these representations. Moreover, it simplifies the design of lifted model counting algorithms.
We examine the meaning and the complexity of probabilistic logic programs that consist of a set of rules and a set of independent probabilistic facts (that is, programs based on Sato's distribution semantics). We focus on two semantics, respectively based on stable and on well-founded models. We show that the semantics based on stable models (referred to as the "credal semantics") produces sets of probability measures that dominate infinitely monotone Choquet capacities; we describe several useful consequences of this result. We then examine the complexity of inference with probabilistic logic programs. We distinguish between the complexity of inference when a probabilistic program and a query are given (the inferential complexity), and the complexity of inference when the probabilistic program is fixed and the query is given (the query complexity, akin to data complexity as used in database theory). We obtain results on the inferential and query complexity for acyclic, stratified, and normal propositional and relational programs; complexity reaches various levels of the counting hierarchy and even exponential levels.
This paper presents complexity analysis and variational methods for inference in probabilistic description logics featuring Boolean operators, quantification, qualified number restrictions, nominals, inverse roles and role hierarchies. Inference is shown to be PEXP-complete, and variational methods are designed so as to exploit logical inference whenever possible.