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New Foundations for Fixpoint Computations: FIXHyperdoctrines and the FIXLogic
"... This paper introduces a new higherorder typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi [Mog 89] and contains a version of Martin Löf’s ‘iteration type’ [MarL 83]. The type system enforces a separation of com ..."
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Cited by 44 (8 self)
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This paper introduces a new higherorder typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi [Mog 89] and contains a version of Martin Löf’s ‘iteration type’ [MarL 83]. The type system enforces a separation of computations from values. The logic contains a novel form of fixpoint induction and can express partial and total correctness statements about evaluation of computations to values. The constructive nature of the logic is witnessed by strong metalogical properties which are proved using a categorytheoretic version of the ‘logical relations’ method [Plo 85].
New Foundations for Fixpoint Computations
, 1990
"... This paper introduces a new higherorder typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi [8] and contains a strong version of MartinLöf’s ‘iteration type’ [ll]. The type system enforces a separation of comput ..."
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Cited by 13 (4 self)
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This paper introduces a new higherorder typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi [8] and contains a strong version of MartinLöf’s ‘iteration type’ [ll]. The type system enforces a separation of computations from values. The logic contains a novel form of fixpoint induction and can express partial and total correctness statements about evaluation of computations to values. The constructive nature of the logic is witnessed by strong metalogical properties which are proved using a categorytheoretic version of the ‘logical relations’ method.
Execution Time of λTerms via Denotational Semantics and Intersection Types. Research Report RR6638
, 2008
"... The multiset based relational model of linear logic induces a semantics of the type free λcalculus, which corresponds to a nonidempotent intersection type system, System R. We prove that, in System R, the size of the type derivations and the size of the types are closely related to the execution t ..."
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Cited by 6 (0 self)
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The multiset based relational model of linear logic induces a semantics of the type free λcalculus, which corresponds to a nonidempotent intersection type system, System R. We prove that, in System R, the size of the type derivations and the size of the types are closely related to the execution time of λterms in a particular environment machine, Krivine’s machine.
Some Observations on the Proof Theory of Second Order Propositional Multiplicative Linear Logic (Extended Abstract)
, 2007
"... We present two new aspects of the proof theory of MLL2. First, we will give a novel proof system in the framework of the calculus of structures. The main feature of the new system is the consequent use of deep inference. Due to the new freedom of permuting inference rules, we are able to observe a d ..."
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We present two new aspects of the proof theory of MLL2. First, we will give a novel proof system in the framework of the calculus of structures. The main feature of the new system is the consequent use of deep inference. Due to the new freedom of permuting inference rules, we are able to observe a decomposition theorem, which is not visible in the sequent calculus. Second, we show a new notion of (boxfree) proof nets which is inspired by the deep inference proof system. Nonetheless, the proof nets are independent from the deductive system. We have “sequentialisation” into the calculus of structures as well as into the sequent calculus. We present a notion of cut elimination which is terminating and confluent, and thus gives us a category of proof nets.
On Fixpoint Objects and Gluing Constructions
, 1997
"... This article 1 has two parts: In the first part, we present some general results about fixpoint objects. The minimal categorical structure required to model soundly the equational type theory which combines higher order recursion and computation types (introduced by [4]) is shown to be precisely a ..."
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Cited by 1 (1 self)
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This article 1 has two parts: In the first part, we present some general results about fixpoint objects. The minimal categorical structure required to model soundly the equational type theory which combines higher order recursion and computation types (introduced by [4]) is shown to be precisely a letcategory possessing a fixpoint object. Functional completeness for such categories is developed. We also prove that categories with fixpoint operators do not necessarily have a fixpoint object. In the second part, we extend Freyd's gluing construction for cartesian closed categories to cartesian closed letcategories, and observe that this extension does not obviously apply to categories possessing fixpoint objects. We solve this problem by giving a new gluing construction for a limited class of categories with fixpoint objects; this is the main result of the paper. We use this categorytheoretic construction to prove a typetheoretic conservative extension result. A version of this pap...
Categorical models of linear logic revisited
"... In this survey, we review the existing categorical axiomatizations of linear logic, with a special emphasis on Seely and Lafont presentations. In a first part, we explain why Benton, Bierman, de Paiva and Hyland had to replace Seely categories by a more complicated axiomatization, and how a while la ..."
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In this survey, we review the existing categorical axiomatizations of linear logic, with a special emphasis on Seely and Lafont presentations. In a first part, we explain why Benton, Bierman, de Paiva and Hyland had to replace Seely categories by a more complicated axiomatization, and how a while later, Benton managed to simplify this axiomatization. In a second part, we show how Lafont axiomatization may be relaxed, in order to admit exponential interpretations different from the free one. Finally, we illustrate with a few examples what categorical models can teach us about linear logic and its models. 1