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Proof search for programming in Intuitionistic Linear Logic (Extended Abstract)
 CADE12 Workshop on Proof Search in TypeTheoretic Languages
, 1994
"... Introduction Linear logic (denoted LL) [6] is a powerful and expressive logic with connections to a variety of topics in computer science as logic programming, concurrency or functional programming. From the logical side, LL combines the constructive content of Intuitionistic Logic with the symmetr ..."
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Introduction Linear logic (denoted LL) [6] is a powerful and expressive logic with connections to a variety of topics in computer science as logic programming, concurrency or functional programming. From the logical side, LL combines the constructive content of Intuitionistic Logic with the symmetries of Classical Logic and from the computation side, it offers a control on resource management and evaluation order. Concerning functional programming, applications of LL to computation can be seen through the CurryHoward isomorphism in which propositions are interpreted as types, proofs as programs and proof normalization process as computation. Works have been recently devoted to term assignment for intuitionistic linear logic (ILL) [3, 12] and full LL [1] with proposals of linear lambda calculi having important properties as subjectreduction or substitution property. Having natural deduction and sequent calculus proof systems of ILL (that are proved equivalent), we can investi
Agents via Mixedmode Computation in Linear Logic
 Proposal, Proceedings of the ICLP'01 Workshop on Computational Logic in MultiAgent Systems (CLIMA01), Paphos
, 2001
"... Agent systems based on the Belief, Desire and Intention model of Rao and Georgeff have been used for a number of successful applications. However, it is often difficult to learn how to apply such systems, due to the complexity of both the semantics of the system and the computational model. In add ..."
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Agent systems based on the Belief, Desire and Intention model of Rao and Georgeff have been used for a number of successful applications. However, it is often difficult to learn how to apply such systems, due to the complexity of both the semantics of the system and the computational model. In addition, there is a gap between the semantics and the concepts that are presented to the programmer. In this paper we address these issues by recasting the foundations of such systems into a logic programming framework. In particular we show how the integration of backward and forwardchaining techniques for linear logic provides a natural starting point for this investigation. We discuss how the integrated system provides for the interaction between the proactive and reactive parts of the system, and we discuss several aspects of this interaction. In particular, one perhaps surprising outcome is that goals and plans may be thought of as declarative and procedural aspects of the same concept. We also discuss the language design issues for such a system, and particularly the way in which the potential choices for rule evaluation in a forwardchaining manner is crucial to the behaviour of the system.
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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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.
Forward and Backward Chaining in Linear Logic
"... . Logic programming languages based on linear logic are of both theoretical and practical interest, particulaly because such languages can be seen as providing a logical basis for programs which execute within a dynamic environment. Most linear logic programming languages are implemented using stand ..."
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. Logic programming languages based on linear logic are of both theoretical and practical interest, particulaly because such languages can be seen as providing a logical basis for programs which execute within a dynamic environment. Most linear logic programming languages are implemented using standard resolution or backward chaining techniques. However, there are many applications in which the combination of such techniques with forward chaining ones are desirable. We develop a prooftheoretic foundation for a system which combines both forms of reasoning in linear logic. 1 Introduction Backward chaining is a standard technique in automated deduction, particularly in logic programming systems, often taking the form of a version of Robinson's resolution rule [18]. The fundamental question is to determine whether or not a given formula follows from a given set of formul, and there are various techniques which can be used to guide the search for a proof. An instance of this approach is...
Towards the Automation of the Design of Logic Programming Languages
 Department of Computer Science, RMIT
, 1997
"... Logic programs consist of formulas of mathematical logic and various prooftheoretic techniques can be used to design and analyse execution models for such programs. We briefly review the main problems, which are questions that are still elusive in the design of logic programming languages, from a p ..."
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Logic programs consist of formulas of mathematical logic and various prooftheoretic techniques can be used to design and analyse execution models for such programs. We briefly review the main problems, which are questions that are still elusive in the design of logic programming languages, from a prooftheoretic point of view. Existing approaches and analyses which lead to the various languages are all rather sophisticated and involve complex manipulations of proofs. All are designed for analysis on paper by a human and many of them are ripe for automation. We aim to perform the automation of some aspects of prooftheoretic analyses, in order to assist in the design of logic programming languages. In this paper we describe the first steps towards the design of such an automatic analysis tool. We investigate the usage of particular proof manipulations for the analysis of logic programming strategies. We propose a more precise specification of sequent calculi inference rules that we use ...
Canonical Proofs for Linear Logic Programming Frameworks
 ProofTheoretical Extensions of Logic Programming 210, PostConference Workshop for ICLP'94, Santa Margherita Ligure
, 1994
"... We discuss here the prooftheoretic foundations for theorem proving and logic programming in linear logic, mainly studying how to define canonical proofs (that are complete) for efficient proof search in fragments of linear logic. We analyze the conception of such proof forms, for frameworks based o ..."
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We discuss here the prooftheoretic foundations for theorem proving and logic programming in linear logic, mainly studying how to define canonical proofs (that are complete) for efficient proof search in fragments of linear logic. We analyze the conception of such proof forms, for frameworks based on proofconstruction as computation, emphasizing the relationship between the logical fragment and its proof search strategies. This point is essential for the definition and implementation of logic programming languages within linear logic.
Correspondences between Classical, Intuitionistic and Uniform Provability
 Theoretical Computer Science
, 2000
"... Based on an analysis of the inference rules used, we provide a characterization of the situations in which classical provability entails intuitionistic provability. We then examine the relationship of these derivability notions to uniform provability, a restriction of intuitionistic provability that ..."
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Based on an analysis of the inference rules used, we provide a characterization of the situations in which classical provability entails intuitionistic provability. We then examine the relationship of these derivability notions to uniform provability, a restriction of intuitionistic provability that embodies a special form of goaldirectedness. We determine, first, the circumstances in which the former relations imply the latter. Using this result, we identify the richest versions of the socalled abstract logic programming languages in classical and intuitionistic logic. We then study the reduction of classical and, derivatively, intuitionistic provability to uniform provability via the addition to the assumption set of the negation of the formula to be proved. Our focus here is on understanding the situations in which this reduction is achieved. However, our discussions indicate the structure of a proof procedure based on the reduction, a matter also considered explicitly elsewhere.
Proofsearch in typetheoretic languages: an introduction
 Theoretical Computer Science
, 2000
"... We introduce the main concepts and problems in the theory of proofsearch in typetheoretic languages and survey some specific, connected topics. We do not claim to cover all of the theoretical and implementation issues in the study of proofsearch in typetheoretic languages; rather, we present som ..."
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We introduce the main concepts and problems in the theory of proofsearch in typetheoretic languages and survey some specific, connected topics. We do not claim to cover all of the theoretical and implementation issues in the study of proofsearch in typetheoretic languages; rather, we present some key ideas and problems, starting from wellmotivated points of departure such as a definition of a typetheoretic language or the relationship between languages and proofobjects. The strong connections between different proofsearch methods in logics, type theories and logical frameworks, together with their impact on programming and implementation issues, are central in this context.
A Constructive Type System to Integrate Logic and Functional Programming
 CADE Workshop on Proofsearch in Typetheoretic Languages
, 1994
"... In this work we present a type system called HH def that extends the theory of simply typed hereditary Harrop formulae [Mil90] with definitions and strong \Sigmatypes. The use of definitions permits the construction of clearer programs and of shorter proofs by using a rule (the def rule) similar ..."
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In this work we present a type system called HH def that extends the theory of simply typed hereditary Harrop formulae [Mil90] with definitions and strong \Sigmatypes. The use of definitions permits the construction of clearer programs and of shorter proofs by using a rule (the def rule) similar to Gentzen's cut rule. Proofsearch for HH def is performed in a goaldirected manner with occurrences of defined constants in a goal triggering instances of the def rule. Such a search procedure is shown to be complete for HH def . 1 Introduction The motivation for development of the calculus HH def is to provide a logical foundation on which to develop a programming language that integrates logic and functional programming. This work develops the ideas outlined in [Pin94], which are intended to be a first step towards a prooftheoretic characterisation of such programming. The central idea is that the execution mechanisms both for logic and for functional programming can be seen as ...
GoalDirected Proof Search in MultipleConclusioned Intuitionistic Logic
 In Proceedings of the First International Conference on Computational Logic, volume LNAI 1861
, 2000
"... . A key property in the definition of logic programming languages is the completeness of goaldirected proofs. This concept originated in the study of logic programming languages for intuitionistic logic in the (singleconclusioned) sequent calculus LJ, but has subsequently been adapted to multip ..."
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. A key property in the definition of logic programming languages is the completeness of goaldirected proofs. This concept originated in the study of logic programming languages for intuitionistic logic in the (singleconclusioned) sequent calculus LJ, but has subsequently been adapted to multipleconclusioned systems such as those for linear logic. Given these developments, it seems interesting to investigate the notion of goaldirected proofs for a multipleconclusioned sequent calculus for intuitionistic logic, in that this is a logic for which there are both singleconclusioned and multipleconclusioned systems (although the latter are less well known). In this paper we show that the language obtained for the multipleconclusioned system differs from that for the singleconclusioned case, show how hereditary Harrop formulae can be recovered, and investigate contractionfree fragments of the logic. 1 Introduction Logic programming is based upon the observation that if ...