Reasoning about Minimal Belief and Negation as Failure

We investigate the problem of reasoning in the propositional fragment of MBNF, the logic of minimal belief and negation as failure introduced by Lifschitz, which can be considered as a unifying framework for several nonmonotonic formalisms, including default logic, autoepistemic logic, circumscription, epistemic queries, and logic programming. We characterize the complexity and provide algorithms for reasoning in propositional MBNF. In particular, we show that entailment in propositional MBNF lies at the third level of the polynomial hierarchy, hence it is harder than reasoning in all the above mentioned propositional formalisms for nonmonotonic reasoning. We also prove the exact correspondence between negation as failure in MBNF and negative introspection in Moore's autoepistemic logic

Embedding Non-Ground Logic Programs into Autoepistemic Logic for Knowledge Base Combination

In the context of the Semantic Web, several approaches to the combination of ontologies, given in terms of theories of classical first-order logic and rule bases, have been proposed. They either cast rules into classical logic or limit the interaction between rules and ontologies. Autoepistemic logic (AEL) is an attractive formalism which allows to overcome these limitations, by serving as a uniform host language to embed ontologies and nonmonotonic logic programs into it. For the latter, so far only the propositional setting has been considered. In this paper, we present three embeddings of normal and three embeddings of disjunctive non-ground logic programs under the stable model semantics into first-order AEL. While the embeddings all correspond with respect to objective ground atoms, differences arise when considering non-atomic formulas and combinations with first-order theories. We compare the embeddings with respect to stable expansions and autoepistemic consequences, considering the embeddings by themselves, as well as combinations with classical theories. Our results reveal differences and correspondences of the embeddings and provide useful guidance in the choice of a particular embedding for knowledge combination.Comment: 52 pages, submitte

A Description Logic Framework for Commonsense Conceptual Combination Integrating Typicality, Probabilities and Cognitive Heuristics

We propose a nonmonotonic Description Logic of typicality able to account for the phenomenon of concept combination of prototypical concepts. The proposed logic relies on the logic of typicality ALC TR, whose semantics is based on the notion of rational closure, as well as on the distributed semantics of probabilistic Description Logics, and is equipped with a cognitive heuristic used by humans for concept composition. We first extend the logic of typicality ALC TR by typicality inclusions whose intuitive meaning is that "there is probability p about the fact that typical Cs are Ds". As in the distributed semantics, we define different scenarios containing only some typicality inclusions, each one having a suitable probability. We then focus on those scenarios whose probabilities belong to a given and fixed range, and we exploit such scenarios in order to ascribe typical properties to a concept C obtained as the combination of two prototypical concepts. We also show that reasoning in the proposed Description Logic is EXPTIME-complete as for the underlying ALC.Comment: 39 pages, 3 figure

On Properties of Update Sequences Based on Causal Rejection

We consider an approach to update nonmonotonic knowledge bases represented as extended logic programs under answer set semantics. New information is incorporated into the current knowledge base subject to a causal rejection principle enforcing that, in case of conflicts, more recent rules are preferred and older rules are overridden. Such a rejection principle is also exploited in other approaches to update logic programs, e.g., in dynamic logic programming by Alferes et al. We give a thorough analysis of properties of our approach, to get a better understanding of the causal rejection principle. We review postulates for update and revision operators from the area of theory change and nonmonotonic reasoning, and some new properties are considered as well. We then consider refinements of our semantics which incorporate a notion of minimality of change. As well, we investigate the relationship to other approaches, showing that our approach is semantically equivalent to inheritance programs by Buccafurri et al. and that it coincides with certain classes of dynamic logic programs, for which we provide characterizations in terms of graph conditions. Therefore, most of our results about properties of causal rejection principle apply to these approaches as well. Finally, we deal with computational complexity of our approach, and outline how the update semantics and its refinements can be implemented on top of existing logic programming engines.Comment: 59 pages, 2 figures, 3 tables, to be published in "Theory and Practice of Logic Programming