V. Alexiev. Applications of Linear Logic to Computation: An Overview. Logic Journal of the IGPL, 2(1):77-107, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....may be thought of as generalisations of the boolean value true, and for O and are written as and 0, and which may be thought of as generalisations of the boolean value false. There is far more to linear logic than can be discussed in this paper; for a more complete introduction see the papers [15, 16, 42, 2], among others. The sequent calculus for linear logic is given in Appendix A. 2.3. Logic Programming in Linear Logic There has been a signi cant amount of work on logic programming languages based on linear logic, including LO [6] Lolli [20] LinLog [4] Forum [31] and Lygon [17] Such ....

V. Alexiev, Applications of Linear Logic to Computation: An Overview, Bulletin of the IGPL 2:1:77-107, 1994.

....be thought of as generalisations of the boolean value true, and for O and are written as and 0, and which may be thought of as generalisations of the boolean value false. There is far more to linear logic than can be discussed in this paper; for a more complete introduction see the papers [2, 10, 11, 31], among others. A part of the sequent calculus for linear logic is given in Appendix A. 2.3 Logic Programming in Linear Logic The execution models on which these languages are based have generally been based on backward chaining, i.e. given a number of statements (or formulae) which make up the ....

V. Alexiev, Applications of Linear Logic to Computation: An Overview, Bulletin of the IGPL 2:1:77-107, 1994.

....as many times as we like. Thus classical logic may be seen as a particular fragment of linear logic, in that there is a class of linear formulae which precisely matches classical formulae. There is far more to linear logic than can be discussed here; for a more complete introduction see the papers [1, 3, 4], among others. 3. NEGOTIATIONS IN LINEAR LOGIC Turning now to negotiation, the technical crux for us is to show how the notion of requirements and supply can be appropriately modelled in linear logic, and then how we can easily supplement this model, particularly where choices are concerned. ....

V. Alexiev. Applications of linear logic to computation: An overview. Bulletin of the IGPL, 2(1):77--107, 1994.

....LL has enjoyed increasing attention both from proof theorists and computer scientists. Therefore, because of its maturity, LL is useful as formal representation of planning system kernel. Good tutorials to LL are [32] and [21] One of the rst overviews of LL applications is presented in [1]. There exist several ecient formal method tools for proving LL sequents [31] From the complete set of LL connectives and operators we are using multiplicative conjunction ( additive disjunction ( and of course ( Whilst the connectives and are needed to describe pre and ....

V. Alexiev. Applications of Linear Logic to Computation: An Overview. Bulletin of the IGPL, Vol. 2, No. 1, March 1994.

....be thought of as generalisations of the boolean value true, and for O and are written as and 0, and which may be thought of as generalisations of the boolean value false. There is far more to linear logic than can be discussed in this paper; for a more complete introduction see the papers [2, 10, 11, 31], among others. A part of the sequent calculus for linear logic is given in Appendix A. 2.3 Logic Programming in Linear Logic The execution models on which these languages are based have generally been based on backward chaining, i.e. given a number of statements (or formulae) which make up the ....

V. Alexiev, Applications of Linear Logic to Computation: An Overview, Bulletin of the IGPL 2:1:77-107, 1994.

....one. Since its introduction LL has enjoyed increasing attention both from proof theorists and computer scientists. Therefore, because of its maturity, LL is useful as a declarative language. Good tutorials to LL are [92] and [64] One of the rst overviews of LL applications is presented in [1]. There exist several ecient formal method tools for proving LL sequents [90] From the complete set of LL connectives and operators we are using here multiplicative conjunction ( additive disjunction ( and of course ( Whilst the connectives and are needed to describe pre and ....

V. Alexiev. Applications of Linear Logic to Computation: An Overview. Bulletin of the IGPL, Vol. 2, No. 1, March 1994.

....LL has enjoyed increasing attention both from proof theorists and computer scientists. Therefore, because of its maturity, LL is useful as formal representation of planning system kernel. Good tutorials to LL are [27] and [17] One of the rst overviews of LL applications is presented in [1]. There exist several ecient formal method tools for proving LL sequents [26] From the complete set of LL connectives and operators we are using multiplicative conjunction ( additive disjunction ( and of course ( Whilst connectives and are needed to describe pre and postconditions ....

V. Alexiev. Applications of Linear Logic to Computation: An Overview. Bulletin of the IGPL, Vol. 2, No. 1, March 1994.

.... 1 Introduction Many works have been recently devoted to the study of linear logic [10] or some of its fragments) and its ability, as a logic of resources, to be easily and strongly related with di erent topics in computer science, like process calculi or concurrent constraint programming [3]. In many cases, it appears that commutative linear logic is not enough expressive to specify processes having both asynchronous and synchronous behaviors and thus a possible solution can be the use of non commutative logic [2] or non commutative connectives to deal with sequentiality. Recently, a ....

V. Alexiev. Applications of Linear Logic to Computation: An Overview. Logic Journal of the IGPL, 2(1):77-107, 1994.

.... operators and of the structure of proofs which is arguably more in tune with the needs of applications and is, therefore, proving useful in several applications of logic to Computer Science and AI (see, for instance, 22] for an application of relevance logic to modular reasoning systems, and [1] for an overview of computational applications of Linear Logic) Our research showed how a uniform and transparent system of analytic deduction for all these logics can be obtained via a generalization of the classical KE system via the LDS methodology 2 . We believe that our methods will prove ....

V. Alexiev. Applications of linear logic to computation: An overview. Bulletin of the IGPL, 2(1):77--107, 1994.

....three structural rules that determine its expressive power. In particular, two of these rules, weakening and contraction, restrict formulae so they can directly model only information that does not change, and cannot account for finite resources, such as money or computer memory. Alexiev [2] explains that this came about because the methodological setting of classical logic is Platonic so formulae are regarded as eternal truths that try to describe and mirror an ideal static world. 2 In 1987 Girard introduced linear logic, which controls the use of weakening and contraction [3] ....

Alexiev, V., "Applications of linear logic to computation: an overview," available via anonymous ftp, ftp.cs.ualberta.ca:pub/TechReports/TR93-18, December 1993, pp. 1--27

....thought of as generalisations of the boolean value true, and for # and # are written as # and 0, and which may be thought of as generalisations of the boolean value false. There is far more to linear logic than can be discussed in this paper; for a more complete introduction see the papers [11, 12, 36, 2], among others. The sequent calculus for linear logic is given in Appendix A. 2.3 Logic Programming in Linear Logic The execution models on which these languages are based have generally been based on backward chaining, i.e. given a number of statements (or formulae) which make up the program, ....

V. Alexiev, Applications of Linear Logic to Computation: An Overview, Bulletin of the IGPL 2:1:77-107, 1994.

....attention both from proof theorists and computer scientists. Therefore linear logic could be useful as a strictly formal representation of planning system kernel. A good tutorial to linear logic is presented in [36] and [19] One of the rst overviews of linear logic applications is presented in [1]. There exist several ecient formal methods tools to prove the correctness of linear logic formulas [35] Linear logic theorem proving has been used by Jacopin [16] as a classical AI planning kernel. As only the multiplicative conjunction is used in formulas there, the framework has a little ....

....P 1 ; CheckStoreg presents another plan, which uses a previously canned plan P 1 . In this way an hierarchical plan representation is achieved and plan reuse is implemented. 5 Linear logic, planning and Petri nets One of the rst models of concurrency represented in linear logic were Petri nets [1]. It has been shown [12] that Petri nets can be presented in the form of linear logic formulas. Thus at least a part of a set of linear logic formulas can be translated into Petri nets which in turn can be used for planning. One of the rst surveys on Petri nets is [31] where an overview of basic ....

V. Alexiev. Applications of Linear Logic to Computation: An Overview. Bulletin of the IGPL, Vol. 2, No. 1, March 1994.

....no simple and direct semantics for O. Girard himself has referred to this connective as mysterious and its semantics has been mainly as a De Morgan dual to the connective Omega . In section 2 a brief intuitive introduction to certain aspects of linear logic is provided. The reader is referred to [1] along with [4, 5] for the complete technical details. In this paper we focus on our approach to understanding linear logic. Here we present a CCS [7] like process calculus where the principal focus is on a process semantics for O. We then show that other linear logic connectives can be ....

V. Alexiev. Applications of linear logic to computation: An overview. Technical Report TR93-18, University of Alberta, Edmonton, Alberta T6G 2H1, September 1994.

....available to the logic programmer the features of the extended logic. One such extension is linear logic, a resource sensitive refinement of classical logic. Linear logic has many applications in computer science, including the semantics of concurrency, natural language processing and updates [1, 14]. When looking for logic programming languages, the fundamental question of what is meant by a logic programming language naturally arises. Whilst any definition should include the use of Horn clauses in classical logic as a special case, it is not clear exactly what characterizes a logic ....

....than acquire each one. For similar reasons, the courses are joined together with O, rather than Omega . The internal and external choices are also, of course, swapped. There is far more to linear logic than can be discussed in this paper; for a more complete introduction see, for example, [8, 24, 23, 1]. 2.2 Goal directedness in linear logic In order to exploit the features of linear logic in a logic programming context, it is necessary to have some criterion for determining what a logic programming language is in a given logic. One such criterion is the completeness of a class of proofs known ....

V. Alexiev. Applications of Linear Logic to Computation: An Overview. Bulletin of the IGPL 2(1):77-107, 1994.

....Petri nets and prove that the reachability problem between two states in such a net corresponds to the provability in a fragment of F ILL. 2 Linear logic and concurrent computations Linear Logic (LL) is recognized as relevant for computation issues, especially concurrency and state change [2]. We brieAEy outline the main characteristics of LL, the formal system of which being in appendix A. From the sequent calculus point of view, LL can be seen as a substructural logic where the contraction and weakening rules are discarded in order that each assumption must be spent exactly once and ....

V. Alexiev. Applications of Linear Logic to Computation: An Overview. Bulletin of the IGPL, 2(1):77107, 1994.

....we obtain a new semantics based on ordered monoids and also new completeness results for ILL. 1 Introduction Linear logic (denoted LL) 8] is a powerful and expressive logic with connections to a variety of topics in computer science as logic programming, concurrency or functional programming [2]. In this context, Intuitionistic Linear Logic (ILL) 4,14] and some of its sub fragments are often used as the underlying logic of logical frameworks. There exists di erent semantics of ILL based on phases spaces [8] quantales [20] and Petri nets [15] The completeness for ILL with respect to ....

V. Alexiev. Applications of Linear Logic to Computation: An Overview. Logic Journal of the IGPL, 2(1):77-107, 1994.

....of the semantics of programming languages. Mathematical Structures in Computer Science 3 2. Linear Logic and Aspects of Programming Linear logic (in the following abbreviated as LL) is a powerful and expressive logic which presents connections with a variety of topics in computer science (Alexiev 1994). From a proof theoretical point of view, LL derives from Classical Logic (CL) by eliminating the structural rules of weakening and contraction. The result is a logic in which, intuitively, it is possible to treat formulae as resources. Contraction and weakening are re introduced in a restricted ....

V. Alexiev. Applications of Linear Logic to Computation: An Overview. Bulletin of the IGLP, 2(1):77--107, 1994.

....and the logical points of view. 2 Linear logic and proofs The linear logic (LL) introduced by J.Y. Girard [27] is a logic of actions and resources. The main de nitions, concepts and results about LL are given in [29] and a rst overview of its applications in computer science is presented in [2]. We brie y recall the language (and the di erent logical fragments) and the linear one sided sequent calculus is presented in Figure 1. 2.1 Linear Logic The multiplicative fragment of propositional linear logic (MLL) is based on the constants 1 and , the unary connective ( linear ....

....also a two sided sequent calculus having inference rules for negation. Because of the negation properties we can write as ; with negation at the atom level. LL has a rich and well developed proof theory that could be directly applied to have computational interpretations of the logic [2]. Works have been devoted to automated proof construction in various fragments and sequent calculi of LL [3, 25] A way to improve the inherent di culty of dealing with proof search in sequent calculus (and the bureaucracy of taxonomy [27] consists in using the notion of proof net, a graph with ....

V. Alexiev. Applications of Linear Logic to Computation: An Overview. L. J. of the IGPL, 2(1):77107, 1994.

.... resource sensitive logic that can be seen as a refinement of classical logic sequent calculus where contraction and weakening rules are forbidden [22] Both classical linear logic (CLL) and its intuitionistic fragment, called intuitionistic linear logic (ILL) are studied as models of computation [2] and of semantics of parallelism [20] 3.1 CLASSICAL LINEAR LOGIC The usual presentation of CLL consists in starting from the CL sequent calculus with suppression of the contraction and weakening rules. Then the equivalence between the additive and multiplicative presentations of the and ....

V. Alexiev. Applications of Linear Logic to Computation: An Overview. Bulletin of the IGPL, 2(1):77--107, 1994. 28 LABELLED DEDUCTION

....Patient Monitoring, Health Care 2. 2 Linear Logic In 1987 Jean Yves Girard introduced Linear Logic (LL) Gir87] From the start it was recognised as relevant to issues of computation, evidence of which is that the paper appeared not in a journal of logic, but in Theoretical Computer Science [Ale94] Linear Logic is a new constructive logic, following the development of intuitionistic logic. It is a refinement of classical logic and sometimes is described as resource sensitive because it provides an intrinsic and natural accounting 5 of resources. In this section we will discuss the ....

....computation. In parallel we will endeavour to point out why linear logic would be beneficial to AOP. 2.2. 1 Logic of Resources Linear logic is a logic of resources, which makes it directly applicable to many computer science tasks, including concurrency, updates, and natural language processing [Ale94] Linear logic inherits all the properties of classical logic and in addition it has some important characteristics that are not present in classical logic. The main point of difference is that in linear logic (usually) each formula must be used exactly once in a proof; a formula can be neither ....

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Vladimir Alexiev. Applications of linear logic to computation: An overview. Bulletin of the IGPL, 2(1):77--107, March 1994.

.... through proof construction, some applications to computation as well for concurrent logic programming [5, 12] or functional programming [3] as for automated theorem proving (dedicated for instance to prove program properties) Many works on linear logic and their applications to computer science [1] involve methods and techniques to deal with the problem of proof construction in some linear sequent calculi. In this context we have previously proposed algorithms and techniques for proof construction in various fragments of linear logic [9] mainly based on the definition of canonical proofs ....

V. Alexiev. Applications of Linear Logic to Computation: An Overview. Bulletin of the IGPL, 2(1):77--107, 1994.

....which allows one to use them as programming constructs inside an ideal interpreter for the considered logic language. This metaphor, proofs as computations, has been investigated from many different point of views in LL as shown by the works in [3, 2, 24, 27, 29, 18] The reader can refer to [1, 37] to have an extensive bibliography and survey on the topic. We will focus on a particular presentation of LL proof theory based on previous work on Logic Programming and non standard logics [39] 2.2 Forum: Higher Order Linear Logic Forum [36] can be considered as the point of convergence of ....

V. Alexiev. Applications of Linear Logic to Computation: An Overview. Bulletin of the IGLP, 2(1):77--107, 1994.

.... Delta C Gamma; A; B; Delta C exchange Gamma; B; A; Delta C Gamma B weakening Gamma; A B Gamma; A; A B contraction Gamma; A B Figure 1: Structural rules of sequent calculus 3 2 Further work On the whole, the extensive research summarized above and elsewhere [1] points to the possible adequacy of using a substructural logic as a basis for reasoning about dynamic systems. Much of this research has been carried out from a theoretical perspective, and we therefore have a good understanding of the semantic issues of representing specification formalisms into ....

Vladimir Alexiev. Applications of Linear Logic to computation: An overview. Technical Report TR93-18, University of Alberta, Edmonton, 1993.

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V. Alexiev. Applications of Linear Logic to Computation: An Overview. Logic Journal of the IGPL, 2(1):77-107, 1994.

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Alexiev V., Applications of Linear Logic to Computation: An Overview, Bulletin of the Interest Group in Pure and Applied Logics (IGPL), Volume 2 (1994), pp. 77-107. 40

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