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Logic Programming in a Fragment of Intuitionistic Linear Logic
"... When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses. Attempting to prove a goal of the form D ⊃ G from the context (set of formulas) Γ leads to an attempt to prove the goal G in the extended context Γ ..."
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When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses. Attempting to prove a goal of the form D ⊃ G from the context (set of formulas) Γ leads to an attempt to prove the goal G in the extended context Γ ∪ {D}. Thus during the bottomup search for a cutfree proof contexts, represented as the lefthand side of intuitionistic sequents, grow as stacks. While such an intuitionistic notion of context provides for elegant specifications of many computations, contexts can be made more expressive and flexible if they are based on linear logic. After presenting two equivalent formulations of a fragment of linear logic, we show that the fragment has a goaldirected interpretation, thereby partially justifying calling it a logic programming language. Logic programs based on the intuitionistic theory of hereditary Harrop formulas can be modularly embedded into this linear logic setting. Programming examples taken from theorem proving, natural language parsing, and data base programming are presented: each example requires a linear, rather than intuitionistic, notion of context to be modeled adequately. An interpreter for this logic programming language must address the problem of splitting contexts; that is, when attempting to prove a multiplicative conjunction (tensor), say G1 ⊗ G2, from the context ∆, the latter must be split into disjoint contexts ∆1 and ∆2 for which G1 follows from ∆1 and G2 follows from ∆2. Since there is an exponential number of such splits, it is important to delay the choice of a split as much as possible. A mechanism for the lazy splitting of contexts is presented based on viewing proof search as a process that takes a context, consumes part of it, and returns the rest (to be consumed elsewhere). In addition, we use collections of Kripke interpretations indexed by a commutative monoid to provide models for this logic programming language and show that logic programs admit a canonical model.
Logic Programming and Negation: A Survey
 JOURNAL OF LOGIC PROGRAMMING
, 1994
"... We survey here various approaches which were proposed to incorporate negation in logic programs. We concentrate on the prooftheoretic and modeltheoretic issues and the relationships between them. ..."
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Cited by 250 (8 self)
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We survey here various approaches which were proposed to incorporate negation in logic programs. We concentrate on the prooftheoretic and modeltheoretic issues and the relationships between them.
Applications of Linear Logic to Computation: An Overview
, 1993
"... This paper is an overview of existing applications of Linear Logic (LL) to issues of computation. After a substantial introduction to LL, it discusses the implications of LL to functional programming, logic programming, concurrent and objectoriented programming and some other applications of LL, li ..."
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Cited by 42 (3 self)
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This paper is an overview of existing applications of Linear Logic (LL) to issues of computation. After a substantial introduction to LL, it discusses the implications of LL to functional programming, logic programming, concurrent and objectoriented programming and some other applications of LL, like semantics of negation in LP, nonmonotonic issues in AI planning, etc. Although the overview covers pretty much the stateoftheart in this area, by necessity many of the works are only mentioned and referenced, but not discussed in any considerable detail. The paper does not presuppose any previous exposition to LL, and is addressed more to computer scientists (probably with a theoretical inclination) than to logicians. The paper contains over 140 references, of which some 80 are about applications of LL. 1 Linear Logic Linear Logic (LL) was introduced in 1987 by Girard [62]. From the very beginning it was recognized as relevant to issues of computation (especially concurrency and stat...
Linear Logic and Computation: A Survey
 Proof and Computation, Proceedings Marktoberdorf Summer School
, 1993
"... . This is a survey of computational aspects of linear logic related to proof search. Keywords. Linear logic, cut free proof search, logic programming, complexity. 1 Introduction Linear logic, introduced by Girard [14, 36, 32], is a refinement of classical logic. While the central notions of truth ..."
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Cited by 16 (6 self)
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. This is a survey of computational aspects of linear logic related to proof search. Keywords. Linear logic, cut free proof search, logic programming, complexity. 1 Introduction Linear logic, introduced by Girard [14, 36, 32], is a refinement of classical logic. While the central notions of truth (emphasized in classical logic) and proof construction (emphasized in intuitionistic logic) remain important in linear logic, it might be said that the emphasis in linear logic is on state. Linear logic is sometimes described as being resource sensitive because it provides an intrinsic and natural accounting of process states, events, and resources. Linear logic also sheds new light on classical logic and its relationship to intuitionistic logic, see Girard [15, 16] and Danos et al. [11]. An evocative semantic paradigm for linear logic by means of games is proposed by Blass [7] and by Abramsky and Jagadeesan [2]. As an intuitive motivation, let us consider reading logical deductions so tha...
The Declarative Semantics of the Prolog Selection Rule
 SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 1994
"... We axiomatize the Prolog selection rule which always selects the leftmost literal in a goal. We introduce a new completion of a logic program which we call the lcompletion of the program. The lcompletion is formulated as a firstorder theory in a language extended by new predicate symbols which ex ..."
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We axiomatize the Prolog selection rule which always selects the leftmost literal in a goal. We introduce a new completion of a logic program which we call the lcompletion of the program. The lcompletion is formulated as a firstorder theory in a language extended by new predicate symbols which express success, failure and lefttermination of queries. The main results of the paper are the following. If a query succeeds, fails or is leftterminating under the Prolog selection rule, then the corresponding formula in the extended language is provable from the lcompletion. Conversely, if a logic program and a query are correct with respect to some mode assignment and if one can prove in the lcompletion that the query succeeds and is leftterminating, then the goal is successful and Prolog, using its depth first search, will compute an answer substitution for the goal. This result can even be extended to so called nonfloundering queries.
A QuantifierFree Completion of Logic Programs
 Computer Science Logic, selected papers from CSL ’89
, 1990
"... We present a proof theoretic approach to the problem of negation in logic programming. We introduce a quantifierfree sequent calculus which is sound for Negation as Failure. Some extensions of the calculus have 3valued or intuitionistic interpretations. 1 ..."
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We present a proof theoretic approach to the problem of negation in logic programming. We introduce a quantifierfree sequent calculus which is sound for Negation as Failure. Some extensions of the calculus have 3valued or intuitionistic interpretations. 1
A transformation of propositional Prolog programs into classical logic
 Archive for Mathematical Logic
, 1995
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SLDNF, Constructive Negation and Grounding
"... In some approaches to the declarative semantics of logic programming, a program \Pi is treated as shorthand for the propositional program Ground (\Pi) whose rules are the ground instances of the rules of \Pi. We relate the success and failure of a program \Pi with a goal G under SLDNF with construct ..."
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In some approaches to the declarative semantics of logic programming, a program \Pi is treated as shorthand for the propositional program Ground (\Pi) whose rules are the ground instances of the rules of \Pi. We relate the success and failure of a program \Pi with a goal G under SLDNF with constructive negation to the success and failure of Ground (\Pi) with ground instances of G under propositional SLDNF. As a corollary, we get a soundness theorem for SLDNF with constructive negation relative to the answer set semantics. Since this semantics is closely related to default logic and autoepistemic logic, proofs of such theorems help us understand why the computational mechanisms developed in logic programming provide partial implementations of these nonmonotonic formalisms. 1 Introduction In the approaches to the declarative semantics of logic programming developed over the last years, variables in a logic program are often treated as metavariables for ground terms. From this perspecti...
Negation as a Resource: a novel view on Answer Set Semantics ⋆
"... Abstract. In recent work, we provided a formulation of ASP programs in terms of linear logic theories. Answer sets were characterized in terms of maximal tensor conjunctions provable from such theories. In this paper, we propose a full comparison between Answer Set Semantics and its variation obtain ..."
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Abstract. In recent work, we provided a formulation of ASP programs in terms of linear logic theories. Answer sets were characterized in terms of maximal tensor conjunctions provable from such theories. In this paper, we propose a full comparison between Answer Set Semantics and its variation obtained by interpreting literals (including negative literals) as resources, which leads to a different interpretation of negation. We argue that this novel view can be of both theoretical and practical interest, and we propose a modified Answer Set Semantics that we call Resourcebased Answer Set Semantics. One advantage is that of avoiding inconsistencies, so every program has a (possibly empty) resourcebased answer set. This implies however the introduction of a different way of representing constraints.