Axiomatic Domain Theory in Categories of Partial Maps

Fiore, Marcelo

doi:10.1017/cbo9780511526565

Cited by 62 publications

(111 citation statements)

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“…Nevertheless, our applications to axiomatically given classes of models demonstrate that our results should be viewed equally much as a contribution to the field of axiomatic domain theory [9,10,39,4,3,5,6]. It is the author's view that embedding categories of predomains within models of intuitionistic set theory is the correct approach to obtaining an axiomatic account of domain-theoretic constructions that applies uniformly across the different types of model.…”

Section: Introductionmentioning

confidence: 68%

“…[3,Lemma 8.4.4]. However, because of the direct interpretation of recursive types using (·) † , we can strengthen the isomorphism of loc.…”

Section: The Interpretation Of Fpcmentioning

confidence: 91%

“…Using Freyd's reduction of recursive types to algebraic compactness [9,3], Theorem 1 does guarantee that, for any predomain Y , a solution X to X ∼ = X Y exists in pP. Indeed, one can solve arbitrary recursive domain equations in pP.…”

Section: Algebraic Compactnessmentioning

confidence: 99%

“…Although it seems possible to apply Pitts' method of defining relations [31], doing so would require the development of further machinery. Because we already have the technology of suitable categories at our disposal, it seems easier to adapt the techniques of [33,3].…”

Section: Internal Computational Adequacymentioning

confidence: 99%

“…In contrast to previous approaches to computational adequacy for recursive types [33,3,25], the models we consider are not required to be order-enriched. Full details of the results outlined in this section will appear in [44].…”

Section: Applicationsmentioning

confidence: 99%

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Computational adequacy for recursive types in models of intuitionistic set theory

Simpson

Proceedings 17th Annual IEEE Symposium on Logic in Computer Science

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This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domain-theoretic 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 Zermelo-Fraenkel set theory. We then apply this result to obtain a denotational interpretation of FPC, a recursively typed lambda-calculus with callby-value 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 1-consistency. 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 domain-theoretic models.

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Section: Introductionmentioning

confidence: 68%

“…[3,Lemma 8.4.4]. However, because of the direct interpretation of recursive types using (·) † , we can strengthen the isomorphism of loc.…”

Section: The Interpretation Of Fpcmentioning

confidence: 91%

Section: Algebraic Compactnessmentioning

confidence: 99%

Section: Internal Computational Adequacymentioning

confidence: 99%

Section: Applicationsmentioning

confidence: 99%

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Computational adequacy for recursive types in models of intuitionistic set theory

Simpson

Proceedings 17th Annual IEEE Symposium on Logic in Computer Science

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Games and Full Abstraction for FPC

McCusker

2000

Information and Computation

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We present a new category of games, G, and build from it a cartesian closed category I and its extensional quotient E. E represents an improvement over existing categories of games in that it has sums as well as products, function spaces and recursive types. A model of the language fpc, a sequential functional language with just this type structure, in E is described and shown to be fully abstract.

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