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32
Simple Consequence Relations
 Information and Computation
, 1991
"... We provide a general investigation of Logic in which the notion of a simple consequence relation is taken to be fundamental. Our notion is more general than the usual one since we give up monotonicity and use multisets rather than sets. We use our notion for characterizing several known logics (incl ..."
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Cited by 100 (18 self)
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We provide a general investigation of Logic in which the notion of a simple consequence relation is taken to be fundamental. Our notion is more general than the usual one since we give up monotonicity and use multisets rather than sets. We use our notion for characterizing several known logics (including Linear Logic and nonmonotonic logics) and for a general, semanticsindependent classification of standard connectives via equations on consequence relations (these include Girard's "multiplicatives" and "additives"). We next investigate the standard methods for uniformly representing consequence relations: Hilbert type, Natural Deduction and Gentzen type. The advantages and disadvantages of using each system and what should be taken as good representations in each case (especially from the implementation point of view) are explained. We end by briefly outlining (with examples) some methods for developing nonuniform, but still efficient, representations of consequence relations.
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...
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.
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.
Mutable Object State for ObjectOriented Logic Programming: A Survey
, 1993
"... One of the most difficult problems on the way to an integration of ObjectOriented and Logic Programming is the modeling of changeable object state (i.e. object dynamics) in a particular logic in order not to forfeit the declarative nature of LP. Classical logic is largely unsuitable for such a task ..."
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Cited by 11 (2 self)
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One of the most difficult problems on the way to an integration of ObjectOriented and Logic Programming is the modeling of changeable object state (i.e. object dynamics) in a particular logic in order not to forfeit the declarative nature of LP. Classical logic is largely unsuitable for such a task, because it adopts a general (both temporally and spatially), Platonic notion of validity, whereas object state changes over time and is local to an object. This paper presents the problem and surveys the stateoftheart approaches to its solution, as well as some emerging, promising new approaches. The paper tries to relate the different approaches, to evaluate their merits and deficiencies and to identify promising directions for development. Keywords: ObjectOriented Logic Programming, mutable object state, survey. 1 The Problem: Dynamics of Objects From the research literature on integration of ObjectOriented Programming (OOP) and Logic Programming (LP) one gets the impression that ...
Quantales as completions of ordered monoids: Revised semantics for Intuitionistic Linear Logic
 Electronic Notes in Theoretical Computer Science
, 2000
"... The aim of this paper is to propose a unified analysis of the relationships between the notions of order and closure and to relate it to different semantics of Intuitionistic Linear Logic (ILL). We study the embedding of ordered monoids into quantales and then we propose general constructions and re ..."
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Cited by 7 (4 self)
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The aim of this paper is to propose a unified analysis of the relationships between the notions of order and closure and to relate it to different semantics of Intuitionistic Linear Logic (ILL). We study the embedding of ordered monoids into quantales and then we propose general constructions and results about such an embedding. Therefore we obtain a new semantics based on ordered monoids and also new completeness results for ILL.
Grafting Modalities onto substructural implication systems
 Studia Logica
, 1996
"... We investigate the semantics of the logical systems obtained by introducing the modalities 2 and 3 into the family of substructural implication logics (including relevant, linear and intuitionistic implication) . Then, in the spirit of the LDS (Labelled Deductive Systems) methodology, we "import" th ..."
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Cited by 7 (3 self)
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We investigate the semantics of the logical systems obtained by introducing the modalities 2 and 3 into the family of substructural implication logics (including relevant, linear and intuitionistic implication) . Then, in the spirit of the LDS (Labelled Deductive Systems) methodology, we "import" this semantics into the classical proof system KE. This leads to the formulation of a uniform labelled refutation system for the new logics which is a natural extension of a system for substructural implication developed by the first two authors in a previous paper. Keywords: Kripke semantics, Labelled Deductive Systems, KE system. 1 Introduction The notion of modality is central in pure and applied logic. Many systems presented to formalise some application area require the addition of modality to the language for a variety of reasons: to cater for changes of the system in time, or perhaps for the dependency of the system on the context, or even to bring metalevel notions into the object l...
Agent Negotiation as Proof Search in Linear Logic
, 2002
"... this paper we show how a basis in linear logic enables a richer (and arguably more natural) basis for reasoning about negotiations. In particular (i) the representation of conditionals is direct and natural in linear logic; (ii) linear logic allows for consumables to be modelled; and (iii) linear lo ..."
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Cited by 7 (0 self)
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this paper we show how a basis in linear logic enables a richer (and arguably more natural) basis for reasoning about negotiations. In particular (i) the representation of conditionals is direct and natural in linear logic; (ii) linear logic allows for consumables to be modelled; and (iii) linear logic allows varying types of choices to be modelled in a clearer way than using classical logic. This last is of particular importance  there is an important distinction between an agent that is willing to provide clothing or food where another agent makes the choice, and an agent that is willing to provide either clothing or food but where that agent chooses between them. Hence the contribution of this paper is to provide a more natural basis for the reasoning required. In this sense our focus is on the outcome of the negotiation, rather than the (possibly argumentative) process. In particular, we assume that each party specifies what it desires, and the negotiation process presented finds solutions that satisfy all parties
Labelled Deduction
, 2000
"... In this paper, we propose new labelled proof systems to analyse the intuitionistic provability in classical and linear logics. An important point is to understand how search in a nonclassical logic can be viewed as a perturbation of search in classical logic. Therefore, suitable characterizations o ..."
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Cited by 6 (3 self)
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In this paper, we propose new labelled proof systems to analyse the intuitionistic provability in classical and linear logics. An important point is to understand how search in a nonclassical logic can be viewed as a perturbation of search in classical logic. Therefore, suitable characterizations of intuitionistic provability and related labelled sequent calculi are defined for linear logic. An alternative approach, based on the notion of proofnet and on the definition of suitable labelled classical proofnets, allows to directly study the intuitionistic provability by constructing intuitionistic proofnets for sequents of classical linear logic. Keywords: Proof theory, intuitionistic logic, linear logic, labelled sequent calculus, proofnets, automated deduction. 1. INTRODUCTION Many proofsearch methods (sequent calculus, tableaux, resolution, connections) have been naturally developed in classical logic (CL) with a view 1 2 LABELLED DEDUCTION to avoiding the possible redundanc...
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.