
A Note on Generalized Algebraic Theories and Categories with Families
We give a new syntax independent definition of the notion of a generaliz...
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A sequent calculus with dependent types for classical arithmetic
In a recent paper, Herbelin developed dPA^ω, a calculus in which constru...
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A Diagrammatic Calculus for Algebraic Effects
We introduce a new, diagrammatic notation for representing the result of...
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Types are Internal ∞Groupoids
By extending type theory with a universe of definitionally associative a...
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Substitution Principle and semidirect products
In the classical theory of regular languages the concept of recognition ...
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Duality for Normal Lattice Expansions and Sorted, Residuated Frames with Relations
We revisit the problem of Stone duality for lattices with various quasio...
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Condition/Decision Duality and the Internal Logic of Extensive Restriction Categories
In flowchart languages, predicates play an interesting double role. In t...
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Revisiting the duality of computation: an algebraic analysis of classical realizability models
In an impressive series of papers, Krivine showed at the edge of the last decade how classical realizability provides a surprising technique to build models for classical theories. In particular, he proved that classical realizability subsumes Cohen's forcing, and even more, gives rise to unexpected models of set theories. Pursuing the algebraic analysis of these models that was first undertaken by Streicher, Miquel recently proposed to lay the algebraic foundation of classical realizability and forcing within new structures which he called implicative algebras. These structures are a generalization of Boolean algebras based on an internal law representing the implication. Notably, implicative algebras allow for the adequate interpretation of both programs (i.e. proofs) and their types (i.e. formulas) in the same structure. The very definition of implicative algebras takes position on a presentation of logic through universal quantification and the implication and, computationally, relies on the callbyname λcalculus. In this paper, we investigate the relevance of this choice, by introducing two similar structures. On the one hand, we define disjunctive algebras, which rely on internal laws for the negation and the disjunction and which we show to be particular cases of implicative algebras. On the other hand, we introduce conjunctive algebras, which rather put the focus on conjunctions and on the callbyvalue evaluation strategy. We finally show how disjunctive and conjunctive algebras algebraically reflect the wellknown duality of computation between callbyname and callbyvalue.
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