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A ProofTheoretic Analysis of GoalDirected Provability
 Journal of Logic and Computation
, 1992
"... One of the distinguishing features of logic programming seems to be the notion of goaldirected provability, i.e. that the structure of the goal is used to determine the next step in the proof search process. It is known that by restricting the class of formulae it is possible to guarantee that a ..."
Abstract

Cited by 14 (7 self)
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One of the distinguishing features of logic programming seems to be the notion of goaldirected provability, i.e. that the structure of the goal is used to determine the next step in the proof search process. It is known that by restricting the class of formulae it is possible to guarantee that a certain class of proofs, known as uniform proofs, are complete with respect to provability in intuitionistic logic. In this paper we explore the relationship between uniform proofs and classes of formulae more deeply. Firstly we show that uniform proofs arise naturally as a normal form for proofs in firstorder intuitionistic sequent calculus. Next we show that the class of formulae known as hereditary Harrop formulae are intimately related to uniform proofs, and that we may extract such formulae from uniform proofs in two different ways. We also give results which may be interpreted as showing that hereditary Harrop formulae are the largest class of formulae for which uniform proo...
On GoalDirected Provability in Classical Logic
, 1994
"... this paper we explore the possibilities for a notion of goaldirected 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 mo ..."
Abstract

Cited by 5 (1 self)
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this paper we explore the possibilities for a notion of goaldirected 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 nonatomic 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 goaldirected, and there is no obvious reason to insist on the stronger version. We will refer to the free choice as rightreductive proofs, and to the restricted one as rightdirected proofs. Thus there seems to be more than one notion of goaldirected 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.