Domain-theoretic Semantics for Disjunctive Logic Programs

We propose three equivalent domain-theoretic semantics for dis-junctive logic programs in the style of Gelfond and Lifschitz. These are (i) a resolution-style inference semantics; (ii) a model-theoretic semantics; and (iii) a (nondeterministic) state-transition semantics. We show how these three semantics generalize \for free" to disjunctive logic programming over any Scott domain. We also relate these semantics to default logic, showing how Zhang-Rounds power-default reasoning is isomorphic to clausal default logic, and nally by brieey mentioning stable-model semantics in Scott domains. The main technical tool used is the Smyth powerdomain, a way to specify deno-tational semantics of nondeterminism. To make this tool work, we give a representation theorem for the Smyth powerdomain as a domain of clausal theories over a Scott domain. 1. Introduction Gelfond and Lifschitz, in GL91], introduced a class of disjunctive logic programs incorporating what they called \classical negation", and also incorporating negation as failure. They provided \answer set semantics" for nondisjunctive programs without negation as failure, and also provided a stable model semantics for the \extended disjunctive database" case, where heads could be disjunctive, and the two kinds of negation were present. In this paper we focus on Gelfond-Lifschitz programs with disjunctive heads and \classical" negation, but without negation as failure, which we call simply GL programs. We propose { using Scott's domain theory { a resolution-style semantics for GL programs which bears the same kind of relation to extended disjunctive logic programs as SLD resolution bears to SLDNF resolution in the non-disjunctive case. (For disjunctive logic programs with positive atoms only, such a project is undertaken in SMR97], though the focus there is not on domain theory.) Our reason for doing this is to provide a mathematicalfoundation, based on the extensive research on domain-theoretic semantics of programming languages, for GL-style disjunctive logic programs, and thereby to establish a basis on which to ground the many proposals for extended versions of disjunctive logic programming semantics which have been introduced in the past few years. We actually provide three equivalent semantics for GL programs: (i) a semantics using resolution-style inference; (ii) a model-theoretic semantics, and | the main new result in this paper | (iii) an operational semantics based on nondeterministic state transitions. This shows that GL programs already have a rich mathematical theory, and that further generalizations and augmentations of these programs will likely not have an ad hoc character. We substantiate this claim by showing how …