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A Uniform ProofTheoretic Investigation Of Linear Logic Programming
, 1994
"... In this paper we consider the problem of identifying logic programming languages for linear logic. Our analysis builds on a notion of goaldirected provability, characterized by the socalled uniform proofs, previously introduced for minimal and intuitionistic logic. A class of uniform proofs in lin ..."
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Cited by 69 (21 self)
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In this paper we consider the problem of identifying logic programming languages for linear logic. Our analysis builds on a notion of goaldirected provability, characterized by the socalled uniform proofs, previously introduced for minimal and intuitionistic logic. A class of uniform proofs in linear logic is identified by an analysis of the permutability of inferences in the linear sequent calculus. We show that this class of proofs is complete (for logical consequence) for a certain (quite large) fragment of linear logic, which thus forms a logic programming language. We obtain a notion of resolution proof, in which only one left rule, of clausedirected resolution, is required. We also consider a translation, resembling those of Girard, of the hereditary Harrop fragment of intuitionistic logic into our framework. We show that goaldirected provability is preserved under this translation.
The Uniform Prooftheoretic Foundation of Linear Logic Programming (Extended Abstract)
 Proceedings of the International Logic Programming Symposium
, 1991
"... ) James Harland Department of Computer Science University of Melbourne Parkville, 3052 Australia jah@cs.mu.oz.au David Pym Department of Computer Science University of Edinburgh Edinburgh EH9 3JZ Scotland, U.K. dpym@lfcs.ed.ac.uk Abstract We present a prooftheoretic analysis of a natu ..."
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Cited by 46 (7 self)
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) James Harland Department of Computer Science University of Melbourne Parkville, 3052 Australia jah@cs.mu.oz.au David Pym Department of Computer Science University of Edinburgh Edinburgh EH9 3JZ Scotland, U.K. dpym@lfcs.ed.ac.uk Abstract We present a prooftheoretic analysis of a natural notion of logic programming for Girard's linear logic. This analysis enables us to identify a suitable notion of uniform proof. This in turn enables us to identify choices of classes of definite and goal formulae for which uniform proofs are complete and so to obtain the appropriate formulation of resolution proof for such choices. Resolution proofs in linear logic are somewhat difficult to define. This difficulty arises from the need to decompose definite formulae into a form suitable for the use of the linear resolution rule, a rule which requires the selected clause to be deleted after use, and from the presence of the modality ! (of course). We consider a translation  resembling ...
On Resolution in Fragments of Classical Linear Logic (Extended Abstract)
"... ) J.A. Harland y D.J. Pym z University of Melbourne University of Edinburgh Australia Scotland, U.K. Abstract We present a prooftheoretic foundation for logic programming in Girard's linear logic. We exploit the permutability properties of twosided linear sequent calculus to identify approp ..."
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Cited by 16 (3 self)
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) J.A. Harland y D.J. Pym z University of Melbourne University of Edinburgh Australia Scotland, U.K. Abstract We present a prooftheoretic foundation for logic programming in Girard's linear logic. We exploit the permutability properties of twosided linear sequent calculus to identify appropriate notions of uniform proof, definite formula, goal formula, clause and resolution proof for fragments of linear logic. The analysis of this paper extends earlier work by the present authors to include negative occurrences of z (par) and positive occurrences of ! (of course !) and ? (why not ?). These connectives introduce considerable difficulty. We consider briefly some of the issues related to the mechanical implementation of our resolution proofs. 1 Introduction An interesting recent development in logic of some significance for theoretical computer science is linear logic [3], [4], a relevance logic lacking the both structural rules of weakening and contraction, except via the expo...
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 ..."
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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 ..."
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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.