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Maps II: Chasing Diagrams in Categorical Proof Theory
, 1996
"... In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositionsastypes and proofsasconstructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, ..."
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Cited by 7 (4 self)
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In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositionsastypes and proofsasconstructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, a third paradigm arises, not available elsewhere: logicaloperationsasadjunctions. It offers an answer to the notorious question of the equality of proofs. So we chase diagrams in algebra of proofs. On the basis of these ideas, the present paper investigates proof theory of regular logic: the f; 9gfragment of the first order logic with equality. The corresponding categorical structure is regular fibration. The examples include stable factorisations, sites, triposes. Regular logic is exactly what is needed to talk about maps, as total and singlevalued relations. However, when enriched with proofsasarrows, this familiar concept must be supplied with an additional conversion rule, conn...
Categorical Logic Of Concurrency And Interaction. I: Synchronous Processes
, 1995
"... This is a report on a mathematician's effort to understand some concurrency theory. The starting point is a logical interpretation of Nielsen and Winskel's [30] account of the basic models of concurrency. Upon the obtained logical structures, we build a calculus of relations which yields, ..."
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Cited by 4 (3 self)
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This is a report on a mathematician's effort to understand some concurrency theory. The starting point is a logical interpretation of Nielsen and Winskel's [30] account of the basic models of concurrency. Upon the obtained logical structures, we build a calculus of relations which yields, when cut down by bisimulations, Abramsky's interaction category of synchronous processes [2]. It seems that all interaction categories arise in this way. The obtained presentation uncovers some of their logical contents and perhaps sheds some more light on the original idea of processes as relations extended in time. The sequel of this paper will address the issues of asynchrony, preemption, noninterleaving and linear logic in the same setting. 1 Introduction Concurrency in computation is modelled in many different ways. Several attempts at unification have been made. Most recently, Abramsky [1, 2] has proposed the paradigm of relations extended in time as a foundation for theory of processes. His in...
Containers Constructing Strictly Positive Types
"... with disjoint coproducts and initial algebras of container functors (the categorical analogue of Wtypes) — and then establish that nested strictly positive inductive and coinductive types, which we call strictly positive types, exist in any MartinLöf category. Central to our development are the n ..."
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with disjoint coproducts and initial algebras of container functors (the categorical analogue of Wtypes) — and then establish that nested strictly positive inductive and coinductive types, which we call strictly positive types, exist in any MartinLöf category. Central to our development are the notions of containers and container functors, introduced in Abbott, Altenkirch, and Ghani (2003a). These provide a new conceptual analysis of data structures and polymorphic functions by exploiting dependent type theory as a convenient way to define constructions in MartinLöf categories. We also show that morphisms between containers can be full and faithfully interpreted as polymorphic functions (i.e. natural transformations) and that, in the presence of Wtypes, all strictly positive types (including nested inductive and coinductive types) give rise to containers.