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A functional approach to non‐monotonic logic 1
Author(s) -
Sandewall Erik
Publication year - 1985
Publication title -
computational intelligence
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.353
H-Index - 52
eISSN - 1467-8640
pISSN - 0824-7935
DOI - 10.1111/j.1467-8640.1985.tb00061.x
Subject(s) - monotonic function , axiom , mathematics , mathematical proof , inference , rule of inference , discrete mathematics , set (abstract data type) , binary relation , computer science , artificial intelligence , mathematical analysis , geometry , programming language
Axiom sets and their extensions are viewed as functions from the set of formulas in the language to a set of four truth values, t, f, u for undefined, and k for contradiction. Such functions form a lattice with “contains less information” as the partial order ?, and “combination of several sources of knowledge” as the least‐upper‐bound operation ⊔. Inference rules are expressed as binary relations between such functions. We show that the usual criterium on fixpoints, namely, to be minimal, does not apply correctly in the case of non‐monotonic inference rules. A stronger concept, approachable fixpoints, is introduced and proven to be sufficient for the existence of a derivation of the fixpoint. In addition, the usefulness of our approach is demonstrated by concise proofs for some previously known results about normal default rules.