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On Goal-Directed Provability in Classical Logic (1994)
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BibTeX
@MISC{Harland94ongoal-directed,
author = {James Harland and Royal Melbourne},
title = {On Goal-Directed Provability in Classical Logic},
year = {1994}
}
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Abstract
this paper we explore the possibilities for a notion of goal-directed proof in classical logic. The technical point to consider is how to deal with the multipleconclusioned nature of classical sequents, i.e. that classical succedents may contain more than one formula. This means that there may be more than one "candidate" right rule, as there may be several non-atomic formulae in the succedent, and so a choice has to be made as to which formula is to be reduced in the next step. The question of whether this choice may be free or restricted is one of the key decisions to be made. The free choice, i.e. that the order in which the formulae are reduced does not matter, will clearly constrain the logic programming language more than the restricted, one, and is arguably more declarative; on the other hand, the weaker notion is arguably more goal-directed, and there is no obvious reason to insist on the stronger version. We will refer to the free choice as right-reductive proofs, and to the restricted one as right-directed proofs. Thus there seems to be more than one notion of goal-directed proof in classical logic, and clearly the corresponding logic programming languages may differ according to which class of proofs is used. However, as we shall see, there are do not seem to be any "interesting" languages for which the weaker notion is complete but the stronger one is not, and so it appears the stronger version (which requires that all right rules permute over each other) is the more useful notion.
Keyphrases
classical logic goal-directed provability goal-directed proof free choice right-reductive proof next step classical sequents obvious reason corresponding logic programming language logic programming language multipleconclusioned nature candidate right rule useful notion key decision right-directed proof right rule classical succedents technical point several non-atomic formula interesting language
