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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
Recursion from Cyclic Sharing: Traced Monoidal Categories and Models of Cyclic Lambda Calculi
, 1997
"... . Cyclic sharing (cyclic graph rewriting) has been used as a practical technique for implementing recursive computation efficiently. To capture its semantic nature, we introduce categorical models for lambda calculi with cyclic sharing (cyclic lambda graphs), using notions of computation by Moggi / ..."
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Cited by 45 (5 self)
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. Cyclic sharing (cyclic graph rewriting) has been used as a practical technique for implementing recursive computation efficiently. To capture its semantic nature, we introduce categorical models for lambda calculi with cyclic sharing (cyclic lambda graphs), using notions of computation by Moggi / Power and Robinson and traced monoidal categories by Joyal, Street and Verity. The former is used for representing the notion of sharing, whereas the latter for cyclic data structures. Our new models provide a semantic framework for understanding recursion created from cyclic sharing, which includes traditional models for recursion created from fixed points as special cases. Our cyclic lambda calculus serves as a uniform language for this wider range of models of recursive computation. 1 Introduction One of the traditional methods of interpreting a recursive program in a semantic domain is to use the least fixedpoint of continuous functions. However, in the real implementations of program...
Relational Properties of Recursively Defined Domains
 In 8th Annual Symposium on Logic in Computer Science
, 1993
"... This paper describes a mixed induction/coinduction property of relations on recursively defined domains. We work within a general framework for relations on domains and for actions of type constructors on relations introduced by O'Hearn and Tennent [20], and draw upon Freyd's analysis [7] of recurs ..."
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Cited by 15 (2 self)
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This paper describes a mixed induction/coinduction property of relations on recursively defined domains. We work within a general framework for relations on domains and for actions of type constructors on relations introduced by O'Hearn and Tennent [20], and draw upon Freyd's analysis [7] of recursive types in terms of a simultaneous initiality/finality property. The utility of the mixed induction/coinduction property is demonstrated by deriving a number of families of proof principles from it. One instance of the relational framework yields a family of induction principles for admissible subsets of general recursively defined domains which extends the principle of structural induction for inductively defined sets. Another instance of the framework yields the coinduction principle studied by the author in [22], by which equalities between elements of recursively defined domains may be proved via `bisimulations'. 1 Introduction A characteristic feature of higherorder functional lan...
The Girard Translation Extended with Recursion
 In Proceedings of Computer Science Logic
, 1995
"... This paper extends CurryHoward interpretations of Intuitionistic Logic (IL) and Intuitionistic Linear Logic (ILL) with rules for recursion. The resulting term languages, the rec calculus and the linear rec calculus respectively, are given sound categorical interpretations. The embedding of ..."
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Cited by 11 (0 self)
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This paper extends CurryHoward interpretations of Intuitionistic Logic (IL) and Intuitionistic Linear Logic (ILL) with rules for recursion. The resulting term languages, the rec calculus and the linear rec calculus respectively, are given sound categorical interpretations. The embedding of proofs of IL into proofs of ILL given by the Girard Translation is extended with the rules for recursion, such that an embedding of terms of the rec calculus into terms of the linear rec calculus is induced via the extended CurryHoward isomorphisms. This embedding is shown to be sound with respect to the categorical interpretations. Full version of paper to appear in Proceedings of CSL '94, LNCS 933, 1995. y Basic Research in Computer Science, Centre of the Danish National Research Foundation. Contents 1 Introduction 4 2 The Categorical Picture 6 2.1 Previous Work and Related Results : : : : : : : : : : : : : : : : : : : : : : 6 2.2 How to deal with parameters : : : : : : : ...
Axioms for Recursion in CallbyValue
 HIGHERORDER AND SYMBOLIC COMPUT
, 2001
"... We propose an axiomatization of fixpoint operators in typed callbyvalue programming languages, and give its justifications in two ways. First, it is shown to be sound and complete for the notion of uniform Tfixpoint operators of Simpson and Plotkin. Second, the axioms precisely account for Filins ..."
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Cited by 11 (5 self)
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We propose an axiomatization of fixpoint operators in typed callbyvalue programming languages, and give its justifications in two ways. First, it is shown to be sound and complete for the notion of uniform Tfixpoint operators of Simpson and Plotkin. Second, the axioms precisely account for Filinski's fixpoint operator derived from an iterator (infinite loop constructor) in the presence of firstclass continuations, provided that we define the uniformity principle on such an iterator via a notion of effectfreeness (centrality). We then explain how these two results are related in terms of the underlying categorical structures.