Relating only knowing to minimal belief and negation as failure

Document Type

Article

Publication Date

1-1-1994

Abstract

In this paper, we relate Lifschitz’s logic of minimal belief and negation as failure (MBNF) to Levesque’s logic of only knowing (OL). MBNF can be viewed as an integration of Lin and Shoham’s logic of grounded knowledge and the theory of epistemic queries by Levesque and Reiter. Lifschitz showed that MBNF can be used as a general framework to compare different non-monotonic formalisms such as default logic and circumscription, as well as several forms of logic programs, including disjunctive logic programs with classical negation. Levesque’s logic of only knowing is an epistemic formalism which can be used to model agent’s knowledge and belief and to answer epistemic queries. The OL logic allows one to express the statement is all that is known’. Levesque showed that autoepistemic logic of Moore can be embedded in OL and he also gave a proof theory for OL. We show that a substantial subset of propositional MBNF can be embedded in OL (and hence in autoepistemic logic). In particular, the class of theories in MBNF which corresponds to the translation of various logic programs, a large subclass of default theories, as well as circumscription, can be embedded in OL. This result has two implications: (1) it shows that the logic of only-knowing (OL) can also be used as a general framework that encompasses various kinds of non-monotonic reasoning formalisms and several forms of logic programs. In particular, it shows that extended logic programs with classical negation can be embeded in autoepistemic logic, due to the close connection between autoepistemic logic and OL; (2) for a large class of theories in MBNF, the query-answering task (and thus the inferencing task in the related formalisms) can be reduced to inferencing in OL. Since the proof theory for the propositional OL is sound and complete, this gives a sound and complete proof theory for a large sub-class of propositional MBNF. We also expect that the proof theory for the general OL logic (with predicates and equality) will give a sound inference procedure for answering queries in first order MBNF theories. © 1994 Taylor & Francis.

Publication Source (Journal or Book title)

Journal of Experimental and Theoretical Artificial Intelligence

First Page

409

Last Page

429

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