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108
The Grammar and Processing of Order and Dependency: a Categorial Approach
, 1990
"... This thesis presents accounts of a range of linguistic phenomena in an extended categorial framework, and develops proposals for processing grammars set within this framework. Linguistic phenomena whose treatment we address include word order, grammatical relations and obliqueness, extraction and is ..."
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Cited by 68 (6 self)
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This thesis presents accounts of a range of linguistic phenomena in an extended categorial framework, and develops proposals for processing grammars set within this framework. Linguistic phenomena whose treatment we address include word order, grammatical relations and obliqueness, extraction and island constraints, and binding. The work is set within a flexible categorial framework which is a version of the Lambek calculus (Lambek, 1958) extended by the inclusion of additional typeforming operators whose logical behaviour allows for the characterization of some aspect of linguistic phenomena. We begin with the treatment of extraction phenomena and island constraints. An account is developed in which there are many interrelated notions of boundary, and where the sensitivity of any syntactic process to a particular class of boundaries can be addressed within the grammar. We next present a new categorial treatment of word order which factors apart the specification of the order of a h...
Possible Worlds and Resources: The Semantics of BI
 THEORETICAL COMPUTER SCIENCE
, 2003
"... The logic of bunched implications, BI, is a substructural system which freely combines an additive (intuitionistic) and a multiplicative (linear) implication via bunches (contexts with two combining operations, one which admits Weakening and Contraction and one which does not). BI may be seen to a ..."
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Cited by 47 (17 self)
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The logic of bunched implications, BI, is a substructural system which freely combines an additive (intuitionistic) and a multiplicative (linear) implication via bunches (contexts with two combining operations, one which admits Weakening and Contraction and one which does not). BI may be seen to arise from two main perspectives. On the one hand, from prooftheoretic or categorical concerns and, on the other, from a possibleworlds semantics based on preordered (commutative) monoids. This semantics may be motivated from a basic model of the notion of resource. We explain BI's prooftheoretic, categorical and semantic origins. We discuss in detail the question of completeness, explaining the essential distinction between BI with and without ? (the unit of _). We give an extensive discussion of BI as a semantically based logic of resources, giving concrete models based on Petri nets, ambients, computer memory, logic programming, and money.
Applications of Linear Logic to Computation: An Overview
, 1993
"... This paper is an overview of existing applications of Linear Logic (LL) to issues of computation. After a substantial introduction to LL, it discusses the implications of LL to functional programming, logic programming, concurrent and objectoriented programming and some other applications of LL, li ..."
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Cited by 42 (3 self)
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This paper is an overview of existing applications of Linear Logic (LL) to issues of computation. After a substantial introduction to LL, it discusses the implications of LL to functional programming, logic programming, concurrent and objectoriented programming and some other applications of LL, like semantics of negation in LP, nonmonotonic issues in AI planning, etc. Although the overview covers pretty much the stateoftheart in this area, by necessity many of the works are only mentioned and referenced, but not discussed in any considerable detail. The paper does not presuppose any previous exposition to LL, and is addressed more to computer scientists (probably with a theoretical inclination) than to logicians. The paper contains over 140 references, of which some 80 are about applications of LL. 1 Linear Logic Linear Logic (LL) was introduced in 1987 by Girard [62]. From the very beginning it was recognized as relevant to issues of computation (especially concurrency and stat...
Geometry of Interaction and Linear Combinatory Algebras
, 2000
"... this paper was quite di#erent, stemming from the axiomatics of categories of tangles (although the authors were aware of possible connections to iteration theories. In fact, similar axiomatics in the symmetric case, motivated by flowcharts and "flownomials" had been developed some years ea ..."
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Cited by 42 (10 self)
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this paper was quite di#erent, stemming from the axiomatics of categories of tangles (although the authors were aware of possible connections to iteration theories. In fact, similar axiomatics in the symmetric case, motivated by flowcharts and "flownomials" had been developed some years earlier by Stefanescu (Stefanescu 2000).) However, the first author realized, following a stimulating discussion with Gordon Plotkin, that traced monoidal categories provided a common denominator for the axiomatics of both the Girardstyle and AbramskyJagadeesanstyle versions of the Geometry of Interaction, at the basic level of the multiplicatives. This insight was presented in (Abramsky 1996), in which Girardstyle GoI was dubbed "particlestyle", since it concerns information particles or tokens flowing around a network, while the AbramskyJagadeesan style GoI was dubbed "wavestyle", since it concerns the evolution of a global information state or "wave". Formally, this distinction is based on whether the tensor product (i.e. the symmetric monoidal structure) in the underlying category is interpreted as a coproduct (particle style) or as a product (wave style). This computational distinction between coproduct and product interpretations of the same underlying network geometry turned out to have been partially anticipated, in a rather di#erent context, in a pioneering paper by E. S. Bainbridge (Bainbridge 1976), as observed by Dusko Pavlovic. These two forms of interpretation, and ways of combining them, have also been studied recently in (Stefanescu 2000). He uses the terminology "additive" for coproductbased (i.e. our "particlestyle") and "multiplicative" for productbased (i.e. our "wavestyle"); this is not suitable for our purposes, because of the clash with Linear Logic term...
Noncommutative logic I : the multiplicative fragment
, 1998
"... INTRODUCTION Unrestricted exchange rules of Girard's linear logic [8] force the commutativity of the multiplicative connectives\Omega (times, conjunction) and & (par, disjunction) , and henceforth the commutativity of all logic. This a priori commutativity is not always desirable  it is ..."
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Cited by 35 (7 self)
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INTRODUCTION Unrestricted exchange rules of Girard's linear logic [8] force the commutativity of the multiplicative connectives\Omega (times, conjunction) and & (par, disjunction) , and henceforth the commutativity of all logic. This a priori commutativity is not always desirable  it is quite problematic in applications like linguistics or computer science , and actually the desire of a noncommutative logic goes back to the very beginning of LL [9]. Previous works on noncommutativity deal essentially with noncommutative fragments of LL, obtained by removing the exchange rule at all. At that point, a simple remark on the status of exchange in the sequent calculus is necessary to be clear: there are two presentations of exchange in commutative LL, either sequents are finite sets of occurrences of formulas and exchange is obviously implicit, or sequents are fini
A Semantic analysis of control
, 1998
"... This thesis examines the use of denotational semantics to reason about control flow in sequential, basically functional languages. It extends recent work in game semantics, in which programs are interpreted as strategies for computation by interaction with an environment. Abramsky has suggested that ..."
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Cited by 32 (5 self)
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This thesis examines the use of denotational semantics to reason about control flow in sequential, basically functional languages. It extends recent work in game semantics, in which programs are interpreted as strategies for computation by interaction with an environment. Abramsky has suggested that an intensional hierarchy of computational features such as state, and their fully abstract models, can be captured as violations of the constraints on strategies in the basic functional model. Nonlocal control flow is shown to fit into this framework as the violation of strong and weak ‘bracketing ’ conditions, related to linear behaviour. The language µPCF (Parigot’s λµ with constants and recursion) is adopted as a simple basis for highertype, sequential computation with access to the flow of control. A simple operational semantics for both callbyname and callbyvalue evaluation is described. It is shown that dropping the bracketing condition on games models of PCF yields fully abstract models of µPCF.
Games and full abstraction for nondeterministic languages
, 1999
"... Abstract Nondeterminism is a pervasive phenomenon in computation. Often it arises as an emergent property of a complex system, typically as the result of contention for access to shared resources. In such circumstances, we cannot always know, in advance, exactly what will happen. In other circumstan ..."
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Cited by 32 (3 self)
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Abstract Nondeterminism is a pervasive phenomenon in computation. Often it arises as an emergent property of a complex system, typically as the result of contention for access to shared resources. In such circumstances, we cannot always know, in advance, exactly what will happen. In other circumstances, nondeterminism is explicitly introduced as a means of abstracting away from implementation details such as precise command scheduling and control flow. However, the kind of behaviours exhibited by nondeterministic computations can be extremely subtle in comparison to those of their deterministic counterparts and reasoning about such programs is notoriously tricky as a result. It is therefore important to develop semantic tools to improve our understanding of, and aid our reasoning about, such nondeterministic programs. In this thesis, we extend the framework of game semantics to encompass nondeterministic computation. Game semantics is a relatively recent development in denotational semantics; its main novelty is that it views a computation not as a static entity, but rather as a dynamic process of interaction. This perspective makes the theory wellsuited to modelling many aspects of computational processes: the original use of game semantics in modelling the simple functional language PCF has subsequently been extended to handle more complex control structures such as references and continuations.
Geometry of Interaction III: Accommodating the Additives
 In: Advances in Linear Logic, LNS 222,CUP, 329–389
, 1995
"... The paper expounds geometry of interaction, for the first time in the full case, i.e. for all connectives of linear logic, including additives and constants. The interpretation is done within a C ∗algebra which is induced by the rule of resolution of logic programming, and therefore the execution f ..."
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Cited by 30 (5 self)
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The paper expounds geometry of interaction, for the first time in the full case, i.e. for all connectives of linear logic, including additives and constants. The interpretation is done within a C ∗algebra which is induced by the rule of resolution of logic programming, and therefore the execution formula can be presented as a simple logic programming loop. Part of the data is public (shared channels) but part of it can be viewed as private dialect (defined up to isomorphism) that cannot be shared during interaction, thus illustrating the theme of communication without understanding. One can prove a nilpotency (i.e. termination) theorem for this semantics, and also its soundness w.r.t. a slight modification of familiar sequent calculus in the case of exponentialfree conclusions. 1
Abstract scalars, loops, and free traced and strongly compact closed categories
 PROCEEDINGS OF CALCO 2005, VOLUME 3629 OF SPRINGER LECTURE NOTES IN COMPUTER SCIENCE
, 2005
"... We study structures which have arisen in recent work by the present author and Bob Coecke on a categorical axiomatics for Quantum Mechanics; in particular, the notion of strongly compact closed category. We explain how these structures support a notion of scalar which allows quantitative aspects of ..."
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Cited by 26 (6 self)
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We study structures which have arisen in recent work by the present author and Bob Coecke on a categorical axiomatics for Quantum Mechanics; in particular, the notion of strongly compact closed category. We explain how these structures support a notion of scalar which allows quantitative aspects of physical theory to be expressed, and how the notion of strong compact closure emerges as a significant refinement of the more classical notion of compact closed category. We then proceed to an extended discussion of free constructions for a sequence of progressively more complex kinds of structured category, culminating in the strongly compact closed case. The simple geometric and combinatorial ideas underlying these constructions are emphasized. We also discuss variations where a prescribed monoid of scalars can be ‘glued in ’ to the free construction.