James Harland and David Pym. A uniform proof-theoretic investigation of linear logic programming. Journal of Logic and Computation, 4(2):175--207, April 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... A further case of structural control of inference that has attracted particular interest is linear logic, where linear disjunction must be understood to specify synchronization between concurrent processes rather than proof by case analysis; see, e.g. Andreoli, 1992, Hodas and Miller, 1994, Pym and Harland, 1994, Miller, 1996, Kobayashi et al. 1999] The investigation of fragments of linear logic remains essential, as linear logic has no analogue of an explicitly scoped proof system, and so unlike intuitionistic logic and modal logic must be understood as a refinement of classical logic rather than ....

Pym, D. and Harland, J. (1994). A uniform proof-theoretic investigation of linear logic programming. Journal of Logic and Computation, 4:175--207.

....language based on linear logic: LO [2] Lolli [8] 3] Lygon [6] 7] and ACL [9] But, none of them have not been implemented on a compiler system. Lolli and Lygon are implemented as interpreter system on Prolog or SML, and their resource management methods are described in [4] and [10, 7] for Lolli and Lygon respectively. When we consider compiler system based on these methods, we need to reconstruct a new resource table after the resource consumption, and maintain multiple resource tables. This will slow down the execution speed. In this paper, we describe an extension of WAM ....

....not be used later. 4.5 Execution Method of The execution of arises non determinism, and it is difficult to treat it completely. Therefore, we define the operational semantics of as follows, although it is not complete. Consumes all consumable resources. The treatment in [4] or [10, 7] should be considered for further enhancement. 4.6 Consumption of Exponentials Exponential resources 1 and A can be consumed arbitrary times. Therefore, we don t need to change their consumption levels except when they are removed as related resource. Now we describe the first version of the ....

D. Pym and J. Harland. A uniform proof-theoretic investigation of linear logic programming. Journal of Logic and Computation, 4(2):175--207, April 1994.

....because it consumes hypotheses and consumed hypotheses can never be used again. Therefore, resource can be represented as a formula in linear logic rather than using a term. There have been several proposals for logic programming language based on linear logic: LO [2] Lolli [8] 3] Lygon [6][7] and ACL [9] But, none of them have not been implemented on a compiler system. Lolli and Lygon are implemented as interpreter system on Prolog or SML, and their resource management methods are described in [4] and [10, 7] for Lolli and Lygon respectively. When we consider compiler ....

James Harland and David Pym. A uniform proof-theoretic investigation of linear logic programming. Journal of Logic and Computation, 4(2):175--207, April 1994.

....programming, Linear logic, Warren Abstract Machine 1 INTRODUCTION Linear logic developed by J. Y. Girard [3] can be described as a logic of resources. There have been several proposals for logic programming language based on linear logic: LO [2] LinLog [1] ACL [5] Lolli [4] Lygon [7], and Forum [6] Lolli and Lygon are implemented as interpreter systems (on SML and #Prolog for Lolli, on Prolog for Lygon) But, none of them have been implemented as a compiler system. We developed a compiler system of a linear logic programming language called LLP [8] LLP programs are ....

D. Pym and J. Harland. A uniform proof-theoretic investigation of linear logic programming. Journal of Logic and Computation, 4(2):175--207, April 1994.

....which states that detours through lemmata can be avoided, and it can be used to show many important logical properties like consistency, interpolation, and Beth definability. Cut free sequent calculi are also useful for automated deduction [14] nonclassical extensions of logic programming [22], and studying deep connections between cut elimination, lambda calculi and functional programming. Sequent calculi, and their extensions, therefore play an important role in theoretical computer science. Display Logic [5] is a generalised sequent framework for non classical logics. Since it is ....

Pym, D and J Harland. A Uniform Proof-theoretic Investigation of Linear Logic Programming. J. of Logic and Computation 4 (1994), 175--207.

....a forward chaining approach seems particularly suitable for reactive systems, as this provides a simple and natural way to express conditions which are dependent on the dynamics of the environment. The techniques for backward chaining (both classically and for linear logic) are well known [15, 20, 25, 27]; the integration of forward chaining techniques into such a system was investigated in [13] In particular, this allows a combination of don t know nondeterminism (common in logic programming) via backwardchaining with don t care nondeterminism via forward chaining. Such an integrated system is ....

D. Pym and J. Harland, A Uniform Proof-Theoretic Investigation of Linear Logic Programming, Journal of Logic and Computation 4:2, April, 1994.

....a forward chaining approach seems particularly suitable for reactive systems, as this provides a simple and natural way to express conditions which are dependent on the dynamics of the environment. The techniques for backward chaining (both classically and for linear logic) are well known [15, 20, 25, 27]; the integration of forward chaining techniques into such a system was investigated in [13] In particular, this allows a combination of don t know nondeterminism (common in logic programming) via backward chaining with don t care nondeterminism via forward chaining. Such an integrated system is ....

D. Pym and J. Harland, A Uniform Proof-Theoretic Investigation of Linear Logic Programming, Journal of Logic and Computation 4:2, April, 1994.

....Logic Programming paradigm, that identi es programs as formulae and execution asproof search. This paradigm has been previously applied to linear logic with the notion of uniform proofs [16, 12] and focusing proofs [1] as well as to the design of concurrent languages based on proof search in LL [2, 20, 14, 18] or in NL [19, 11] However our approach is analytical in that we study an existing programming language CC, and model CC computations in a fragment of LL or NL. Moreover we model properties of in nite CC computations through the observation of accessible stores which have no counterpart in the ....

D. Pym and J. Harland. A uniform proof-theoretic investigation of linear loigc programming. Jounral of Logic and Computation, 4:2:175{ 207, 1994.

....Originally the specifications were taken to be first order Horn theories, with a restricted proof search, resolution, as the computational engine. However, the demand for more expressive power and efficiency has led language designers to consider logical extensions to the original Horn Clause core [26, 48, 51, 50, 46, 28]) and to add control features and constructs drawn from other language paradigms (e.g. types [55, 32, 49] partial evaluation [31] and constraints [10, 29, 30] to name just a few) The effect has been to expand the boundaries of the subject and of the very notion of declarative content of a ....

J. Harland and D. Pym. A uniform proof-theoretic investigation of linear logic programming. Journal of Logic and Computation, 4(2), 1994.

....Logic Programming paradigm, that identi es programs as formulae and execution as proofsearch. This paradigm has been previously applied to linear logic with the notion of uniform proofs [16, 12] and focusing proofs [1] as well as to the design of concurrent languages based on proof search in LL [2, 20, 14, 18] or in NL [19, 11] However our approach is analytical in that we study an existing programming language CC, and model CC computations in a fragment of LL or NL. Moreover we model properties of in nite CC computations through the observation of accessible stores which have no counterpart in the ....

D. Pym and J. Harland. A uniform proof-theoretic investigation of linear loigc programming. Jounral of Logic and Computation, 4:2:175-207, 1994.

....because it consumes hypotheses and consumed hypotheses can never be used again. Therefore, resource can be represented as a formula in linear logic rather than using a term. There have been several proposals for logic programming language based on linear logic: LO [2] Lolli [6] 3] Lygon [8][5] and ACL [7] But, none of them have not been implemented on a compiler system. Lolli 1 and Lygon 2 are implemented as interpreter system on Prolog or SML, and their resource management methods are described in [3] and [8, 5] for Lolli and Lygon respectively. When we consider compiler ....

....language based on linear logic: LO [2] Lolli [6] 3] Lygon [8] 5] and ACL [7] But, none of them have not been implemented on a compiler system. Lolli 1 and Lygon 2 are implemented as interpreter system on Prolog or SML, and their resource management methods are described in [3] and [8, 5] for Lolli and Lygon respectively. When we consider compiler system based on these methods, we need to reconstruct a new resource table after the resource consumption, and maintain multiple resource tables. This will slow down the execution speed. In this paper, we describe an extension of WAM ....

[Article contains additional citation context not shown here]

D. Pym and J. Harland. A uniform proof-theoretic investigation of linear logic programming. Journal of Logic and Computation, 4(2):175--207, April 1994.

....Linear logic was introduced by J. Y.Girard in 1987 [4] as a resource conscious refinement of classical logic. Since then a number of logic programming languages 2 M. Banbara, K. Kang, T. Hirai, and N. Tamura based on linear logic have been proposed: LO[1] ACL[12] Lolli[3] 8] 9] Lygon[5], Forum[13] and LLP[2] 15] These languages suggest a direction to extend logic programming to be more expressive and more e#cient. The treatment of formulas as resources gives us not only powerful expressiveness, but also e#cient access to a large set of data. However, in linear logic, whole ....

James Harland and David Pym. A uniform proof-theoretic investigation of linear logic programming. Journal of Logic and Computation, 4(2):175--207, April 1994.

....logic between the left and the righthand sides of the consequence relation and the symmetry that is observed in CLL. Note that embeddings using a positive and a negative translation have been used by several people when embedding calculi in linear logic. See for example, Tro92] HM94] [HP94]. As noted several times above, we have only given the embedding for the oe; 8 fragment of intuitionistic logic. This is because our interest is in the induced calculi and these are unattractive outside of this fragment. We leave it as an open problem how to embed disjunction on the left, ....

....example with respect to goal directed proof search) and so we are not surprised that this is the largest fragment that can easily be embedded to give MJ. Harland and Pym have also embedded hereditary Harrop formulae into linear logic using a two function, positive and negative, embedding (see [HP94]) Their embedding into ILL doesn t induce a uniform proof calculus for hereditary Harrop formulae. If, however, the embedding is into a uniform proof calculus for linear logic, then the calculus induced will be a uniform proof calculus. Embedding intuitionistic logic into linear logic has also ....

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J. A. Harland and D. J. Pym. A Uniform Proof-Theoretic Investigation of Linear Logic Programming. Journal of Logic and Computation, 4(2):175--207, 1994. BIBLIOGRAPHY 214

....Linlog extends LO by adding several additional operators. The execution of Linlog is a search goaldirected proof called focusing proof. The focusing proofs are based on the concept of the goal directed proof which is related to the uniform proof. Lygon was proposed by James Harland and David Pym [14, 15]. Lygon is based on a fragment of multiple conclusion (classical) linear logic, and a variant of uniform proof search is complete for that fragment, like Lolli and Forum. However, the definition of uniform proof di#ers from that used in the design of Forum. In general, Lygon execution is more ....

J. Harland and D. Pym. A uniform proof-theoretic investigation of linear logic programming. Journal of Logic and Computation, 4(2):175--207, Apr. 1994.

....and all the branches terminate at the top with axiom schemes. For full sequent calculi for classical logic, see Gentzen [5] or Kleene [6] By modifying the sequent calculus for classical logic, the syntax for linear logic can be presented. For sequent calculi for linear logic, see Pym and Harland [7] or Lincoln et al. 8] 3. Structural rules of classical logic Classical logic contains three structural rules, weakening, contraction, and commutativity. Commutativity, also known as interchange or exchange, says that the order in which formulae appear in a deduction does not matter. For ....

Pym, D. J., and Harland, J. A., "A uniform proof-theoretic investigation of linear logic programming," J. Logic Computat., Vol. 4. No.2, 1994, pp. 175--207

....a forward chaining approach seems particularly suitable for reactive systems, as this provides a simple and natural way to express conditions which are dependent on the dynamics of the environment. The techniques for backward chaining (both classically and for linear logic) are well known [26, 32, 38, 20]; the integration of forwardchaining techniques into such a system was investigated in [18] In particular, this allows a combination of don t know nondeterminism (common in logic programming) via backward chaining with don t care nondeterminism via forward chaining. Such an integrated system is ....

D. Pym and J. Harland, A Uniform Proof-Theoretic Investigation of Linear Logic Programming, Journal of Logic and Computation 4:2, April, 1994.

....and their computation may be identified as searching for proofs: given a program P and a goal G we attempt to satisfy G by searching for a proof of P G using the inference rules of a given logic. There have been various proof theoretic approaches to the design of logic programming languages [1, 2, 5, 7, 12, 16, 21] and a corresponding variety of languages implemented (Prolog [24] Lolli[12] LinLog[1] Prolog[20] Forum[18] Lygon[26] among others) However, despite many similarities in such analyses, the issue of a criterion for the identification of logic programming languages remains problematic. The ....

.... The properties of such proofs have been thoroughly investigated for various fragments of first and higher order intuitionistic logic (for Horn clauses and for a generalisation of them known as hereditary Harrop formulas [17] as well as for fragments of classical logic [19, 8] and linear logic [12, 1, 2, 21]. These analyses exhibit many points of similarity, as well as some specific points of difference, which can generally be traced to particular inference rules and their properties. For example, linear logic inherits a number of the properties of the classical inference rules (and in some cases, ....

[Article contains additional citation context not shown here]

Pym D., Harland J. A Uniform Proof-theoretic Investigation of Linear Logic Programming, Journal of Logic and Computation, 4:2:175-207, 1994.

....and then the solution which has been found is broadcast to all other branches. Another branch is then selected and the process continues. Such strategies have properties similar to depth first search and are typically used in the implementation of linear logic programming languages such as Lygon [9, 23, 19] and Lolli [11] Eager strategies are those in which all branches are searched in parallel and one single set of equations is collected. This single set of equations is then solved once all equations have been found. Intermediate strategies solve a fixed number of branches worth of equations at ....

.... our approach to be an algebraic view of (the resource distribution problem in) proof search; in particular, this approach may be considered as a form of labelled deduction [2] Our approach is similar in spirit to that of proof nets [7] and has its origins in the notion of path introduced in [19]; however, it should be noted that in our case we recover multiplicative consequences from additive rules by the use of Boolean constraints, rather than recover additive consequences from an underlying multiplicative system. We have suggested that it is helpful to consider the formul to be ....

D. Pym and J. Harland, A Uniform Proof-theoretic Investigation of Linear Logic Programming, Journal of Logic and Computation 4:2:175-207, April, 1994.

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James Harland and David Pym. A uniform proof-theoretic investigation of linear logic programming. Journal of Logic and Computation, 4(2):175--207, April 1994.

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J. Harland and D. Pym. A uniform proof-theoretic investigation of linear logic programming. Journal of Logic and Computation, 4(2):175--207, Apr. 1994.

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James Harland and David Pym. A uniform proof-theoretic investigation of linear logic programming. Journal of Logic and Computation, 4(2):175--207, April 1994.

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D. Pym and J. Harland. A uniform proof-theoretic investigation of linear logic programming. Journal of Logic and Computation, 4(2):175--207, April 1994.

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J. A. Harland and D. J. Pym. A uniform proof-theoretic investigation of linear logic programming. Journal of Logic and Computation, 4(2):175--207, 1994.

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J. Harland and D. Pym. A uniform proof-theoretic investigation of linear logic programming. Journal of Logic and Computation, 4(2):175--207, Apr. 1994.

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James Harland and David Pym. A uniform prooftheoretic investigation of linear logic programming. Journal of Logic Programming, 4(2):175-207, April 1994 20

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