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340
An extension of system F with subtyping
 Information and Computation
, 1991
"... System F is a wellknown typed lcalculus with polymorphic types, which provides a basis for polymorphic programming languages. We study an extension of F, called F <: (pronounced efsub) that combines parametric polymorphism with subtyping. The main focus of the paper is the equational theory of F ..."
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Cited by 110 (11 self)
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System F is a wellknown typed lcalculus with polymorphic types, which provides a basis for polymorphic programming languages. We study an extension of F, called F <: (pronounced efsub) that combines parametric polymorphism with subtyping. The main focus of the paper is the equational theory of F <: , which is related to PER models and the notion of parametricity. We study some categorical properties of the theory when restricted to closed terms, including interesting categorical isomorphisms. We also investigate prooftheoretical properties, such as the conservativity of typing judgments with respect to F. We demonstrate by a set of examples how a range of constructs may be encoded in F <: . These include record operations and subtyping hierarchies that are related to features of objectoriented languages. Appears in: International Conference on Theoretical Aspects of Computer Software, T.Ito, A.R.Meyer Eds., Lecture Notes in Computer Science n. 526, pp 750770, Springer Verlag, 19...
A Logic of Argumentation for Reasoning under Uncertainty.
 Computational Intelligence
, 1995
"... We present the syntax and proof theory of a logic of argumentation, LA. We also outline the development of a category theoretic semantics for LA. LA is the core of a proof theoretic model for reasoning under uncertainty. In this logic, propositions are labelled with a representation of the arguments ..."
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Cited by 107 (3 self)
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We present the syntax and proof theory of a logic of argumentation, LA. We also outline the development of a category theoretic semantics for LA. LA is the core of a proof theoretic model for reasoning under uncertainty. In this logic, propositions are labelled with a representation of the arguments which support their validity. Arguments may then be aggregated to collect more information about the potential validity of the propositions of interest. We make the notion of aggregation primitive to the logic, and then define strength mappings from sets of arguments to one of a number of possible dictionaries. This provides a uniform framework which incorporates a number of numerical and symbolic techniques for assigning subjective confidences to propositions on the basis of their supporting arguments. These aggregation techniques are also described, with examples. Key words: Uncertain reasoning, epistemic probability, argumentation, nonclassical logics, nonmonotonic reasoning 1. Introd...
Linear Logic, Autonomous Categories and Cofree Coalgebras
 In Categories in Computer Science and Logic
, 1989
"... . A brief outline of the categorical characterisation of Girard's linear logic is given, analagous to the relationship between cartesian closed categories and typed calculus. The linear structure amounts to a autonomous category: a closed symmetric monoidal category G with finite products and a c ..."
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Cited by 103 (7 self)
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. A brief outline of the categorical characterisation of Girard's linear logic is given, analagous to the relationship between cartesian closed categories and typed calculus. The linear structure amounts to a autonomous category: a closed symmetric monoidal category G with finite products and a closed involution. Girard's exponential operator, ! , is a cotriple on G which carries the canonical comonoid structure on A with respect to cartesian product to a comonoid structure on !A with respect to tensor product. This makes the Kleisli category for ! cartesian closed. 0. INTRODUCTION. In "Linear logic" [1987], JeanYves Girard introduced a logical system he described as "a logic behind logic". Linear logic was a consequence of his analysis of the structure of qualitative domains (Girard [1986]): he noticed that the interpretation of the usual conditional ")" could be decomposed into two more primitive notions, a linear conditional "\Gammaffi" and a unary operator "!" (called "of cours...
Compiling with Types
, 1995
"... Compilers for monomorphic languages, such as C and Pascal, take advantage of types to determine data representations, alignment, calling conventions, and register selection. However, these languages lack important features including polymorphism, abstract datatypes, and garbage collection. In contr ..."
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Cited by 102 (14 self)
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Compilers for monomorphic languages, such as C and Pascal, take advantage of types to determine data representations, alignment, calling conventions, and register selection. However, these languages lack important features including polymorphism, abstract datatypes, and garbage collection. In contrast, modern programming languages such as Standard ML (SML), provide all of these features, but existing implementations fail to take full advantage of types. The result is that performance of SML code is quite bad when compared to C. In this thesis, I provide a general framework, called typedirected compilation, that allows compiler writers to take advantage of types at all stages in compilation. In the framework, types are used not only to determine efficient representations and calling conventions, but also to prove the correctness of the compiler. A key property of typedirected compilation is that all but the lowest levels of the compiler use typed intermediate languages. An advantage of this approach is that it provides a means for automatically checking the integrity of the resulting code. An important
The ProofTheory and Semantics of Intuitionistic Modal Logic
, 1994
"... Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpret ..."
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Cited by 102 (0 self)
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Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic metatheory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are selfjustifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic firstorder logic. It is also established that, in many cases, the natural deduction systems induce wellknown intuitionistic modal logics, previously given by Hilbertstyle axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their
Functional interpretations of feasibly constructive arithmetic
 Annals of Pure and Applied Logic
, 1993
"... i ..."
Categorical Logic
 A CHAPTER IN THE FORTHCOMING VOLUME VI OF HANDBOOK OF LOGIC IN COMPUTER SCIENCE
, 1995
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SPECWARE: Formal Support for Composing Software
 In Mathematics of Program Construction
, 1995
"... Specware supports the systematic construction of formal specifications and their stepwise refinement into programs. The fundamental operations in Specware are that of composing specifications (via colimits), the corresponding refinement by composing refinements (via sheaves), and the generation of p ..."
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Cited by 75 (0 self)
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Specware supports the systematic construction of formal specifications and their stepwise refinement into programs. The fundamental operations in Specware are that of composing specifications (via colimits), the corresponding refinement by composing refinements (via sheaves), and the generation of programs by composing code modules (via colimits). The concept of diagram refinement is introduced as a practical realization of composing refinements via sheaves. Sequential and parallel composition of refinements satisfy a distributive law which is a generalization of similar compatibility laws in the literature. Specware is based on a rich categorical framework with a small set of orthogonal concepts. We believe that this formal basis will enable the scaling to systemlevel software construction.
Models of Sharing Graphs: A Categorical Semantics of let and letrec
, 1997
"... To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sha ..."
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Cited by 62 (10 self)
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To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sharing graphs. The simplest is firstorder acyclic sharing graphs represented by letsyntax, and others are extensions with higherorder constructs (lambda calculi) and/or cyclic sharing (recursive letrec binding). For each of four settings, we provide the equational theory for representing the sharing graphs, and identify the class of categorical models which are shown to be sound and complete for the theory. The emphasis is put on the algebraic nature of sharing graphs, which leads us to the semantic account of them. We describe the models in terms of the notions of symmetric monoidal categories and functors, additionally with symmetric monoidal adjunctions and traced
Classical Logic and Computation
, 2000
"... This thesis contains a study of the proof theory of classical logic and addresses the problem of giving a computational interpretation to classical proofs. This interpretation aims to capture features of computation that go beyond what can be expressed in intuitionisticlogic. We introduce several ..."
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Cited by 58 (7 self)
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This thesis contains a study of the proof theory of classical logic and addresses the problem of giving a computational interpretation to classical proofs. This interpretation aims to capture features of computation that go beyond what can be expressed in intuitionisticlogic. We introduce several strongly normalising cutelimination procedures for classicallogic. Our procedures are less restrictive than previous strongly normalising procedures, while at the same time retaining the strong normalisation property, which various standardcutelimination procedures lack. In order to apply proof techniques from term rewriting, including symmetric reducibility candidates and recursive path ordering, we develop termannotations for sequent proofs of classical logic. We then present a sequenceconclusion natural deduction calculus for classical logicand study the correspondence between cutelimination and normalisation. In contrast to earlier work, which analysed this correspondence in various fragments of intuitionisticlogic, we establish the correspondence in classical logic. Finally, we study applications of cutelimination. In particular, we analyse severalclassical proofs with respect to their behaviour under cutelimination. Because our cutelimination procedures impose fewer constraints than previous procedures, we are ableto show how a fragment of classical logic can be seen as a typing system for the simplytyped lambda calculus extended with an erratic choice operator. As a pleasing consequence, we can give a simple computational interpretation to Lafont's example.