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18
Structural Induction and Coinduction in a Fibrational Setting
 Information and Computation
, 1997
"... . We present a categorical logic formulation of induction and coinduction principles for reasoning about inductively and coinductively defined types. Our main results provide sufficient criteria for the validity of such principles: in the presence of comprehension, the induction principle for in ..."
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Cited by 86 (16 self)
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. We present a categorical logic formulation of induction and coinduction principles for reasoning about inductively and coinductively defined types. Our main results provide sufficient criteria for the validity of such principles: in the presence of comprehension, the induction principle for initial algebras is admissible, and dually, in the presence of quotient types, the coinduction principle for terminal coalgebras is admissible. After giving an alternative formulation of induction in terms of binary relations, we combine both principles and obtain a mixed induction/coinduction principle which allows us to reason about minimal solutions X = oe(X) where X may occur both positively and negatively in the type constructor oe. We further strengthen these logical principles to deal with contexts and prove that such strengthening is valid when the (abstract) logic we consider is contextually/functionally complete. All the main results follow from a basic result about adjunc...
CallbyValue Games
, 1997
"... . A general construction of models of callbyvalue from models of callbyname computation is described. The construction makes essential use of the properties of sum types in common denotational models of callbyname. When applied to categories of games, it yields fully abstract models of the cal ..."
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Cited by 77 (7 self)
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. A general construction of models of callbyvalue from models of callbyname computation is described. The construction makes essential use of the properties of sum types in common denotational models of callbyname. When applied to categories of games, it yields fully abstract models of the callbyvalue functional language PCFv , which can be extended to incorporate recursive types, and of a language with local references as in Standard ML. 1 Introduction In recent years game semantics has emerged as a novel and intuitively appealing approach to modelling programming languages. Its first success was in providing a syntaxfree description of a fully abstract model of PCF [10, 1, 15]; full abstraction results have also been obtained for untyped and recursively typed functional languages, as well as languages with imperative features [12, 3]. However, none of this work addressed the problem of modelling callbyvalue languagesa major shortcoming, given that many reallife langua...
Presheaf Models for Concurrency
, 1999
"... In this dissertation we investigate presheaf models for concurrent computation. Our aim is to provide a systematic treatment of bisimulation for a wide range of concurrent process calculi. Bisimilarity is defined abstractly in terms of open maps as in the work of Joyal, Nielsen and Winskel. Their wo ..."
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Cited by 49 (19 self)
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In this dissertation we investigate presheaf models for concurrent computation. Our aim is to provide a systematic treatment of bisimulation for a wide range of concurrent process calculi. Bisimilarity is defined abstractly in terms of open maps as in the work of Joyal, Nielsen and Winskel. Their work inspired this thesis by suggesting that presheaf categories could provide abstract models for concurrency with a builtin notion of bisimulation. We show how
Complete Axioms for Categorical Fixedpoint Operators
 In Proceedings of 15th Annual Symposium on Logic in Computer Science
, 2000
"... We give an axiomatic treatment of fixedpoint operators in categories. A notion of iteration operator is defined, embodying the equational properties of iteration theories. We prove a general completeness theorem for iteration operators, relying on a new, purely syntactic characterisation of the fre ..."
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Cited by 42 (6 self)
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We give an axiomatic treatment of fixedpoint operators in categories. A notion of iteration operator is defined, embodying the equational properties of iteration theories. We prove a general completeness theorem for iteration operators, relying on a new, purely syntactic characterisation of the free iteration theory. We then show how iteration operators arise in axiomatic domain theory. One result derives them from the existence of sufficiently many bifree algebras (exploiting the universal property Freyd introduced in his notion of algebraic compactness) . Another result shows that, in the presence of a parameterized natural numbers object and an equational lifting monad, any uniform fixedpoint operator is necessarily an iteration operator. 1. Introduction Fixed points play a central role in domain theory. Traditionally, one works with a category such as Cppo, the category of !continuous functions between !complete pointed partial orders. This possesses a leastfixedpoint oper...
A Theory of Recursive Domains with Applications to Concurrency
 In Proc. of LICS ’98
, 1997
"... Marcelo Fiore , Glynn Winskel (1) BRICS , University of Aarhus, Denmark (2) LFCS, University of Edinburgh, Scotland December 1997 Abstract We develop a 2categorical theory for recursively defined domains. ..."
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Cited by 24 (14 self)
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Marcelo Fiore , Glynn Winskel (1) BRICS , University of Aarhus, Denmark (2) LFCS, University of Edinburgh, Scotland December 1997 Abstract We develop a 2categorical theory for recursively defined domains.
A Convenient Category of Domains
 GDP FESTSCHRIFT ENTCS, TO APPEAR
"... We motivate and define a category of topological domains, whose objects are certain topological spaces, generalising the usual ωcontinuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also su ..."
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Cited by 18 (3 self)
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We motivate and define a category of topological domains, whose objects are certain topological spaces, generalising the usual ωcontinuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, can be used as the basis for a theory of computability, and provides a model of parametric polymorphism.
Presheaf Models for CCSlike Languages
 THEORETICAL COMPUTER SCIENCE
, 1999
"... The aim of this paper is to harness the mathematical machinery around presheaves for the purposes of process calculi. Joyal, Nielsen and Winskel proposed a general definition of bisimulation from open maps. Here we show that openmap bisimulations within a range of presheaf models are congruences ..."
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Cited by 8 (2 self)
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The aim of this paper is to harness the mathematical machinery around presheaves for the purposes of process calculi. Joyal, Nielsen and Winskel proposed a general definition of bisimulation from open maps. Here we show that openmap bisimulations within a range of presheaf models are congruences for a general process language, in which CCS and related languages are easily encoded. The results are then transferred to traditional models for processes. By first establishing the congruence results for presheaf models, abstract, general proofs of congruence properties can be provided and the awkwardness caused through traditional models not always possessing the cartesian liftings, used in the breakdown of process operations, are sidestepped. The abstract results are applied to show that hereditary historypreserving bisimulation is a congruence for CCSlike languages to which is added a refinement operator on event structures as proposed by van Glabbeek and Goltz.
Computational Adequacy for Recursive Types in Models of Intuitionistic Set Theory
 In Proc. 17th IEEE Symposium on Logic in Computer Science
, 2003
"... This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domaintheoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown ..."
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Cited by 8 (2 self)
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This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domaintheoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown that the existence of solutions to recursive domain equations depends upon the strength of the set theory. We observe that the internal set theory of an elementary topos is not strong enough to guarantee their existence. In contrast, as our first main result, we establish that solutions to recursive domain equations do exist when the category of sets is a model of full intuitionistic ZermeloFraenkel set theory. We then apply this result to obtain a denotational interpretation of FPC, a recursively typed lambdacalculus with callbyvalue operational semantics. By exploiting the intuitionistic logic of the ambient model of intuitionistic set theory, we analyse the relationship between operational and denotational semantics. We first prove an “internal ” computational adequacy theorem: the model always believes that the operational and denotational notions of termination agree. This allows us to identify, as our second main result, a necessary and sufficient condition for genuine “external ” computational adequacy to hold, i.e. for the operational and denotational notions of termination to coincide in the real world. The condition is formulated as a simple property of the internal logic, related to the logical notion of 1consistency. We provide useful sufficient conditions for establishing that the logical property holds in practice. Finally, we outline how the methods of the paper may be applied to concrete models of FPC. In doing so, we obtain computational adequacy results for an extensive range of realizability and domaintheoretic models.
An Equational Notion of Lifting Monad
 TITLE WILL BE SET BY THE PUBLISHER
, 2003
"... We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category ..."
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Cited by 5 (1 self)
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We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of the former fully embeds in the partial category of the latter. Thus equational lifting monads precisely capture the equational properties of partial maps as induced by partial map classifiers. The representation theorem also provides a tool for transferring nonequational properties of partial map classifiers to equational lifting monads. It is proved using a direct axiomatization of Kleisli categories of equational lifting monads. This axiomatization is of interest in its own right. 1
Coalgebraic Representation Theory of Fractals (Extended Abstract)
"... We develop a representation theory in which a point of a fractal specified by metric means (by a variant of an iterated function system, IFS) is represented by a suitable equivalence class of infinite streams of symbols. The framework is categorical: symbolic representatives carry a final coalgebra; ..."
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We develop a representation theory in which a point of a fractal specified by metric means (by a variant of an iterated function system, IFS) is represented by a suitable equivalence class of infinite streams of symbols. The framework is categorical: symbolic representatives carry a final coalgebra; an IFSlike metric specification of a fractal is an algebra for the same functor. Relating the two there canonically arises a representation map, much like in America and Rutten’s use of metric enrichment in denotational semantics. A distinctive feature of our framework is that the canonical representation map is bijective. In the technical development, gluing of shapes in a fractal specification is a major challenge. On the metric side we introduce the notion of injective IFS to be used in place of conventional IFSs. On the symbolic side we employ Leinster’s presheaf framework that uniformly addresses necessary identification of streams—such as.0111... =.1000... in the binary expansion of real numbers. Our leading example is the unit interval I = [0, 1].