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.
Most logic-based machine learning algorithms rely on an Occamist bias where textual complexity of hypotheses is minimised. Within Inductive Logic Programming (ILP), this approach fails to distinguish between the efficiencies of hypothesised programs, such as quick sort (O(n log n)) and bubble sort (O(n2)).
In the last few years, there has been a large effort for analyzing the computational properties of reasoning in fuzzy description logics. This has led to a number of papers studying the complexity of these logics, depending on the chosen semantics. Surprisingly, despite being arguably the simplest form of fuzzy semantics, not much is known about the complexity of reasoning in fuzzy description logics w.r.t.
We investigate the complexity of satisfiability for one-agent refinement modal logic (RML), an extension of basic modal logic (ML) obtained by adding refinement quantifiers on structures. RML is known to have the same expressiveness as ML, but the translation of RML into ML is of non-elementary complexity, and RML is at least doubly exponentially more succinct than ML. In this paper we show that RML-satisfiability is only' singly exponentially harder than ML-satisfiability, the latter being a well-known PSPACE-complete problem.
In default reasoning, usually not all possible ways of resolving conflicts between default rules are acceptable. Criteria expressing acceptable ways of resolving the conflicts may be hardwired in the inference mechanism, for example specificity in inheritance reasoning can be handled this way, or they may be given abstractly as an ordering on the default rules. In this article we investigate formalizations of the latter approach in Reiter's default logic. Our goal is to analyze and compare the computational properties of three such formalizations in terms of their computational complexity: the prioritized default logics of Baader and Hollunder, and Brewka, and a prioritized default logic that is based on lexicographic comparison. The analysis locates the propositional variants of these logics on the second and third levels of the polynomial hierarchy, and identifies the boundary between tractable and intractable inference for restricted classes of prioritized default theories.