Rational consequence relation

Abstract: We introduce a generalization of epsilon-probability functions and a new correspondence with rational consequence relations. The rational consequence relations corresponding to the generalized epsilon-probability functions of maximum entropy are investigated and it is shown that in this case the analogous notion of `maximum entropy closure' equals the rational closure, thus reconfirming the rational closure of a conditional knowledge base as the simplest, least prejudiced, rational consequence relation satisfying that knowledge base.

Abstract: The paper describes a preferential approach for dealing with exceptions in KLM preferential logics, based on the rational closure. It is well known that the rational closure does not allow an independent handling of the inheritance of different defeasible properties of concepts. Several solutions have been proposed to face this problem and the lexicographic closure is the most notable one. In this work, we consider an alternative closure construction, called the Multi Preference closure (MP-closure), that has been first considered for reasoning with exceptions in DLs. Here, we reconstruct the notion of MP-closure in the propositional case and we show that it is a natural variant of Lehmann's lexicographic closure. Abandoning Maximal Entropy (an alternative route already considered but not explored by Lehmann) leads to a construction which exploits a different lexicographic ordering w.r.t. the lexicographic closure, and determines a preferential consequence relation rather than a rational consequence relation. We show that, building on the MP-closure semantics, rationality can be recovered, at least from the semantic point of view, resulting in a rational consequence relation which is stronger than the rational closure, but incomparable with the lexicographic closure. We also show that the MP-closure is stronger than the Relevant Closure.

Abstract: The question of conditional inference, i.e., of which conditional sentences of the form “if A then, normally, B” should follow from a set KB of such sentences, has been one of the classic questions of AI, with several well-known solutions proposed. Perhaps the most notable is the rational closure construction of Lehmann and Magidor, under which the set of inferred conditionals forms a rational consequence relation, i.e., satisfies all the rules of preferential reasoning, plus Rational Monotonicity. However, this last named rule is not universally accepted, and other researchers have advocated working within the larger class of disjunctive consequence relations, which satisfy the weaker requirement of Disjunctive Rationality. While there are convincing arguments that the rational closure forms the “simplest” rational consequence relation extending a given set of conditionals, the question of what is the simplest disjunctive consequence relation has not been explored. In this paper, we propose a solution to this question and explore some of its properties.

Abstract: Datalog is a powerful language that can be used to represent explicit knowledge and compute inferences in knowledge bases. Datalog cannot, however, represent or reason about contradictory rules. This is a limitation as contradictions are often present in domains that contain exceptions. In this paper, we extend Datalog to represent contradictory and defeasible information. We define an approach to efficiently reason about contradictory information in Datalog and show that it satisfies the KLM requirements for a rational consequence relation. We introduce DDLV, a defeasible Datalog reasoning system that implements this approach. Finally, we evaluate the performance of DDLV.