A typed store-passing translation for general references

Pottier, François

doi:10.1145/1925844.1926403

Cited by 4 publications

(1 citation statement)

References 31 publications

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“…We show in Section 5 that the category BiCBUlt of bisected, complete bounded ultrametric spaces is a co-reflective subcategory of S. Thus, our present work can be seen as an extension of the work of Nakano to include the full internal language of a topos, in particular dependent types, and an associated higher-order logic. Pottier [32] presents an extension of System F with recursive kinds based on Nakano's calculus; hence S also models the kind language of his system. Di Gianantonio and Miculan [10] studied guarded recursive definitions of functions in certain sheaf toposes over well-founded complete Heyting algebras, thus including S. Our work extends the work of Di Gianantonio and Miculan by also including guarded recursive definitions of types, by emphasizing the use of the internal logic (this was suggested as future work in [10]), and by including an extensive example application.…”

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confidence: 99%

First steps in synthetic guarded domain theory: step-indexing in the topos of trees

Birkedal¹,

Møgelberg²,

Schwinghammer³

et al. 2012

Logical Methods in Computer Science

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Abstract. We present the topos S of trees as a model of guarded recursion. We study the internal dependently-typed higher-order logic of S and show that S models two modal operators, on predicates and types, which serve as guards in recursive definitions of terms, predicates, and types. In particular, we show how to solve recursive type equations involving dependent types. We propose that the internal logic of S provides the right setting for the synthetic construction of abstract versions of step-indexed models of programming languages and program logics. As an example, we show how to construct a model of a programming language with higher-order store and recursive types entirely inside the internal logic of S. Moreover, we give an axiomatic categorical treatment of models of synthetic guarded domain theory and prove that, for any complete Heyting algebra A with a wellfounded basis, the topos of sheaves over A forms a model of synthetic guarded domain theory, generalizing the results for S.

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confidence: 99%

First steps in synthetic guarded domain theory: step-indexing in the topos of trees

Birkedal¹,

Møgelberg²,

Schwinghammer³

et al. 2012

Logical Methods in Computer Science

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Sequent Calculus in the Topos of Trees

Clouston

Goré

2015

Lecture Notes in Computer Science

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Abstract. Nakano's "later" modality, inspired by Gödel-Löb provability logic, has been applied in type systems and program logics to capture guarded recursion. Birkedal et al modelled this modality via the internal logic of the topos of trees. We show that the semantics of the propositional fragment of this logic can be given by linear converse-well-founded intuitionistic Kripke frames, so this logic is a marriage of the intuitionistic modal logic KM and the intermediate logic LC. We therefore call this logic KM lin . We give a sound and cut-free complete sequent calculus for KM lin via a strategy that decomposes implication into its static and irreflexive components. Our calculus provides deterministic and terminating backward proof-search, yields decidability of the logic and the coNP-completeness of its validity problem. Our calculus and decision procedure can be restricted to drop linearity and hence capture KM.

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