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26
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 ..."
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Cited by 23 (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.
linTAP: A Tableau Prover for Linear Logic
 International Conference TABLEAUX’99
, 1999
"... linTAP is a tableau prover for the multiplicative and exponential fragment M?LL of Girards linear logic. It proves the validity of a given formula by constructing an analytic tableau and ensures the linear validity using prex unication. We present the tableau calculus used by linTAP, an algorithm fo ..."
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Cited by 17 (6 self)
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linTAP is a tableau prover for the multiplicative and exponential fragment M?LL of Girards linear logic. It proves the validity of a given formula by constructing an analytic tableau and ensures the linear validity using prex unication. We present the tableau calculus used by linTAP, an algorithm for prex unication in linear logic, the linTAP implementation, and some experimental results obtained with linTAP. 1
ConnectionBased Proof Construction in Linear Logic
 14 th Conference on Automated Deduction, Lecture Notes in Artificial Intelligence 1249
, 1997
"... Abstract. We present a matrix characterization of logical validity in the multiplicative fragment of linear logic. On this basis we develop a matrixbased proof search procedure for this fragment and a procedure which translates the machinefound proofs back into the usual sequent calculus for linea ..."
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Cited by 14 (8 self)
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Abstract. We present a matrix characterization of logical validity in the multiplicative fragment of linear logic. On this basis we develop a matrixbased proof search procedure for this fragment and a procedure which translates the machinefound proofs back into the usual sequent calculus for linear logic. Both procedures are straightforward extensions of methods which originally were developed for a uniform treatment of classical, intuitionistic and modal logics. They can be extended to further fragments of linear logic once a matrix characterization has been found. 1
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...
Static Analysis of Linear Logic Programming
 NEW GENERATION COMPUTING
, 1997
"... Linear Logic is gaining momentum in computer science because it offers a unified framework and a common vocabulary for studying and analyzing different aspects of programming and computation. We focus here on models where computation is identified with proof search in the sequent system of Linear ..."
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Cited by 13 (3 self)
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Linear Logic is gaining momentum in computer science because it offers a unified framework and a common vocabulary for studying and analyzing different aspects of programming and computation. We focus here on models where computation is identified with proof search in the sequent system of Linear Logic. A proof normalization procedure, called "focusing", has been proposed to make the problem of proof search tractable. Correspondingly,
Connection Methods in Linear Logic and Proof Nets Construction
 Theoretical Computer Science
, 1999
"... Linear logic (LL) is the logical foundation of some typetheoretic languages and also of environments for specification and theorem proving. In this paper, we analyse the relationships between the proof net notion of LL and the connection notion used for efficient proofsearch in different logics. A ..."
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Cited by 12 (2 self)
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Linear logic (LL) is the logical foundation of some typetheoretic languages and also of environments for specification and theorem proving. In this paper, we analyse the relationships between the proof net notion of LL and the connection notion used for efficient proofsearch in different logics. Aiming at using proof nets as a tool for automated deduction in linear logic, we define a connectionbased characterization of provability in Multiplicative Linear Logic (MLL). We show that an algorithm for proof net construction can be seen as a proofsearch connection method. This central result is illustrated with a specific algorithm that is able to construct, for a provable MLL sequent, a set of connections, a proof net and a sequent proof. From these results we expect to extend to other LL fragments, we analyse what happens with the additive connectives of LL by tackling the additive fragment in a similar way.
Proof nets Construction and Automated Deduction in NonCommutative Linear Logic (Extended Abstract)
 Electronic Notes in Theoretical Computer Science
, 1998
"... Proof nets can be seen as a multiple conclusion natural deduction system for Linear Logic (LL) and form a good formalism to analyze some computation mechanisms, for instance in typetheoretic interpretations. This paper presents an algorithm for automated proof nets construction in the noncommutati ..."
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Cited by 6 (5 self)
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Proof nets can be seen as a multiple conclusion natural deduction system for Linear Logic (LL) and form a good formalism to analyze some computation mechanisms, for instance in typetheoretic interpretations. This paper presents an algorithm for automated proof nets construction in the noncommutative multiplicative linear logic that is useful for applications including planning, concurrency or sequentiality. The properties of this algorithm can be proved from a recently defined graphtheoretic characterization of noncommutative proof nets. Involving simple construction principles improved in the commutative case, it leads also to a new proof search method for the noncommutative fragment. Moreover because of the relationships between the noncommutative linear logic and the Lambek calculus we can derive from it an alternate method for automatic construction of proof nets in this calculus. 1 Introduction Linear Logic (LL) can offer a framework to study and analyze various notions or ...
Calculi with dependency relations for Mixed Linear Logic (Extended Abstract)
 In International Workshop on Logic and Complexity in Computer Science, LCCS'2001
"... In this paper we propose new calculi for the multiplicative fragment of Mixed Linear Logic (MMLL) which is a logic that combines both commutative and noncommutative connectives. These basedon sequent and proof net calculi, that can be seen as a new prooftheoretical formulation of MMLL, are based o ..."
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Cited by 6 (3 self)
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In this paper we propose new calculi for the multiplicative fragment of Mixed Linear Logic (MMLL) which is a logic that combines both commutative and noncommutative connectives. These basedon sequent and proof net calculi, that can be seen as a new prooftheoretical formulation of MMLL, are based on the definition of dependency relations. We provide a proofsearch procedure for MMLL that is based on proof nets construction with associated sets of dependencies.
A Matrix Characterization for MELL
 6 th European Workshop on Logics in Artificial Intelligence, LNAI 1489
, 1998
"... We present a matrix characterization of logical validity in the multiplicative fragment of linear logic with exponentials. In the process we elaborate a methodology for proving matrix characterizations correct and complete. Our characterization provides a foundation for matrixbased proof search ..."
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Cited by 5 (3 self)
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We present a matrix characterization of logical validity in the multiplicative fragment of linear logic with exponentials. In the process we elaborate a methodology for proving matrix characterizations correct and complete. Our characterization provides a foundation for matrixbased proof search procedures for as well as for procedures which translate machinefound proofs back into the usual sequent calculus.
Representing Scope in Intuitionistic Deductions
 THEORETICAL COMPUTER SCIENCE
, 1997
"... Intuitionistic proofs can be segmented into scopes which describe when assumptions can be used. In standard descriptions of intuitionistic logic, these scopes occupy contiguous regions of proofs. This leads to an explosion in the search space for automated deduction, because of the difficulty of pla ..."
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Cited by 4 (3 self)
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Intuitionistic proofs can be segmented into scopes which describe when assumptions can be used. In standard descriptions of intuitionistic logic, these scopes occupy contiguous regions of proofs. This leads to an explosion in the search space for automated deduction, because of the difficulty of planning to apply a rule inside a particular scoped region of the proof. This paper investigates an alternative representation which assigns scope explicitly to formulas, and which is inspired in part by semanticsbased translation methods for modal deduction. This calculus is simple and is justified by direct prooftheoretic arguments that transform proofs in the calculus so that scopes match standard descriptions. A Herbrand theorem, established straightforwardly, lifts this calculus to incorporate unification. The resulting system has no impermutabilities whatsoeverrules of inference may be used equivalently anywhere in the proof. Nevertheless, a natural specification describes how terms...