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Efficient resource management for linear logic proof search
 Proceedings of the 5th International Workshop on Extensions of Logic Programming
, 1996
"... The design of linear logic programming languages and theorem provers opens a number of new implementation challenges not present in more traditional logic languages such as Horn clauses (Prolog) and hereditary Harrop formulas (λProlog and Elf). Among these, the problem of efficiently managing the li ..."
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Cited by 54 (11 self)
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The design of linear logic programming languages and theorem provers opens a number of new implementation challenges not present in more traditional logic languages such as Horn clauses (Prolog) and hereditary Harrop formulas (λProlog and Elf). Among these, the problem of efficiently managing the linear context when solving a goal is of crucial importance for the use of these systems in nontrivial applications. This paper studies this problem in the case of Lolli [HM94], though its results have application to other systems. We first give a prooftheoretic presentation of the operational semantics of this language as a resolution calculus. We then present a series of resource management systems designed to eliminate the nondeterminism in the distribution of linear formulas that undermines the efficiency of a direct implementation of this system. 1
Programming in Lygon: An Overview
 ALGEBRAIC METHODOLOGY AND SOFTWARE TECHNOLOGY
, 1996
"... Recently, there has been much interest in the derivation of logic programming languages based on linear logic, a logic of resourceconsumption. Such languages provide a notion of resourceoriented programming, often leading to programs that are more elegant and concise than their equivalents in la ..."
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Cited by 40 (18 self)
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Recently, there has been much interest in the derivation of logic programming languages based on linear logic, a logic of resourceconsumption. Such languages provide a notion of resourceoriented programming, often leading to programs that are more elegant and concise than their equivalents in languages, such as Prolog, based on classical logics. We discuss, with examples, the design, implementation and applications of Lygon, a linear logic programming language. Lygon is based on a prooftheoretic analysis of which occurrences of the linear connectives provide an adequate basis for programming. In common with other linear logic programming languages, Lygon allows clauses to be used exactly once in a computation, thereby avoiding the need for the explicit resourcecounting often necessary in Prologlike languages. Indeed, it appears that resourcesensitivity leads to significant differences between the natural programming methodologies in Lygon and Prolog. Just as linear logic...
Focusing the inverse method for linear logic
 Proceedings of CSL 2005
, 2005
"... 1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10 ..."
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Cited by 38 (11 self)
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1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10
On Bunched Predicate Logic
 Proceedings of the IEEE Symposium on Logic in Computer Science
, 1999
"... We present the logic of bunched implications, BI, in which a multiplicative (or linear) and an additive (or intuitionistic) implication live sidebyside. The propositional version of BI arises from an analysis of the prooftheoretic relationship between conjunction and implication, and may be viewe ..."
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Cited by 29 (17 self)
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We present the logic of bunched implications, BI, in which a multiplicative (or linear) and an additive (or intuitionistic) implication live sidebyside. The propositional version of BI arises from an analysis of the prooftheoretic relationship between conjunction and implication, and may be viewed as a merging of intuitionistic logic and multiplicative, intuitionistic linear logic. The predicate version of BI includes, in addition to usual additive quantifiers, multiplicative (or intensional) quantifiers 8new and 9new , which arise from observing restrictions on structural rules on the level of terms as well as propositions. Moreover, these restrictions naturally allow the distinction between additive predication and multiplicative predication for each propositional connective. We provide a natural deduction system, a sequent calculus, a Kripke semantics and a BHK semantics for BI. We mention computational interpretations, based on locality and sharing, at both the propositiona...
Resourcedistribution via Boolean constraints
 Proceedings of CADE14
, 1997
"... We consider the problem of searching for proofs in sequential presentations of logics with multiplicative (or intensional) connectives. Specifically, we start with the multiplicative fragment of linear logic and extend, on the one hand, to linear logic with its additives and, on the other, to the ..."
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Cited by 28 (8 self)
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We consider the problem of searching for proofs in sequential presentations of logics with multiplicative (or intensional) connectives. Specifically, we start with the multiplicative fragment of linear logic and extend, on the one hand, to linear logic with its additives and, on the other, to the additives of the logic of bunched implications, BI. We give an algebraic method for calculating the distribution of the sideformul in multiplicative rules which allows the occurrence or nonoccurrence of a formula on a branch of a proof to be determined once sufficient information is available. Each formula in the conclusion of such a rule is assigned a Boolean expression. As a search proceeds, a set of Boolean constraint equations is generated. We show that a solution to such a set of equations determines a proof corresponding to the given search. We explain a range of strategies, from the lazy to the eager, for solving sets of constraint equations. We indicate how to apply our methods systematically to large family of relevant systems. 1
A Relevant Analysis of Natural Deduction
 Journal of Logic and Computation
, 1999
"... Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and ..."
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Cited by 23 (7 self)
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Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and Plotkin's LF, with a representation mechanism: the language of RLF is the lLcalculus; the representation mechanism is judgementsastypes, developed for relevant logics. The lLcalculus type theory is a firstorder dependent type theory with two kinds of dependent function spaces: a linear one and an intuitionistic one. We study a natural deduction presentation of the type theory and establish the required prooftheoretic metatheory. The RLF framework is a conservative extension of LF. We show that RLF uniformly encodes (fragments of) intuitionistic linear logic, Curry's l I calculus and ML with references. We describe the CurryHowardde Bruijn correspondence of the lLcalculus with a s...
Implementing the Linear Logic Programming Language Lygon
 INTERNATIONAL LOGIC PROGRAMMING SYMPOSIUM
, 1995
"... There has been considerable work aimed at enhancing the expressiveness of logic programming languages. To this end logics other than classical first order logic have been considered, including intuitionistic, relevant, temporal, modal and linear logic. Girard's linear logic has formed the basis of a ..."
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Cited by 22 (8 self)
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There has been considerable work aimed at enhancing the expressiveness of logic programming languages. To this end logics other than classical first order logic have been considered, including intuitionistic, relevant, temporal, modal and linear logic. Girard's linear logic has formed the basis of a number of logic programming languages. These languages are successful in enhancing the expressiveness of (pure) Prolog and have been shown to provide natural solutions to problems involving concurrency, natural language processing, database processing and various resource oriented problems. One of the richer linear logic programming languages is Lygon. In this paper we investigate the implementation of Lygon. Two significant problems that arise are the division of resources between subbranches of the proof and the selection of the formula to be decomposed. We present solutions to both of these problems.
On the Intuitionistic Force of Classical Search
 THEORETICAL COMPUTER SCIENCE
, 1996
"... The combinatorics of classical propositional logic lies at the heart of both local and global methods of proofsearch enabling the achievement of leastcommitment search. Extension of such methods to the predicate calculus, or to nonclassical systems, presents us with the problem of recovering ..."
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Cited by 19 (5 self)
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The combinatorics of classical propositional logic lies at the heart of both local and global methods of proofsearch enabling the achievement of leastcommitment search. Extension of such methods to the predicate calculus, or to nonclassical systems, presents us with the problem of recovering this leastcommitment principle in the context of noninvertible rules. One successful approach is to view the nonclassical logic as a perturbation on search in classical logic and characterize when a leastcommitment (classical) search yields sufficient evidence for provability in the (nonclassical) logic. This technique has been successfully applied to both local and global methods at the cost of subsidiary searches and is the analogue of the standard treatment of quantifiers via skolemization and unification. In this paper, we take a typetheoretic view of this approach for the case in which the nonclassical logic is intuitionistic. We develop a system of realizers (proofobje...
Deterministic Resource Management for the Linear Logic Programming Language Lygon
, 1994
"... Recently there has been significant interest in the logic programming community in linear logic, a logic designed with bounded resources in mind. As linear logic is a generalisation of classical logic, a logic programming language based on linear logic subsumes and extends (pure) Prolog. One such la ..."
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Cited by 14 (5 self)
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Recently there has been significant interest in the logic programming community in linear logic, a logic designed with bounded resources in mind. As linear logic is a generalisation of classical logic, a logic programming language based on linear logic subsumes and extends (pure) Prolog. One such language is Lygon, a language based on a certain kind of proof in the linear sequent calculus. However these proofs, whilst providing a logical characterization of the language, still retain some of the nondeterminism of the sequent calculus, and hence require further analysis before an implementation can be attempted. In this report we define and discuss a more detailed proof system, which is more deterministic than the original. In particular, this system handles the allocation of resources to different branches of the proof in a lazy manner. The resulting system differs significantly from the original sequent calculus, and so we discuss its properties in some detail. We prove the soundness...
ProofTerms for Classical and Intuitionistic Resolution (Extended Abstract)
, 1996
"... We exploit a system of realizers for classical logic, and a translation from resolution into the sequent calculus, to assess the intuitionistic force of classical resolution for a fragment of intuitionistic logic. This approach is in contrast to formulating locally intuitionistically sound resol ..."
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Cited by 12 (3 self)
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We exploit a system of realizers for classical logic, and a translation from resolution into the sequent calculus, to assess the intuitionistic force of classical resolution for a fragment of intuitionistic logic. This approach is in contrast to formulating locally intuitionistically sound resolution rules. The techniques use the fflcalculus, a development of Parigot's calculus.