The algebraic structure of classical realizability models

Etienne Miquey
IRIF, Université de Paris

Date(s) : 06/09/2018   iCal
11 h 00 min - 12 h 30 min

Implicative algebras, developed by Alexandre Miquel, are very simple algebraic structures generalizing at the same time complete Boolean algebras and Krivine realizability algebras, in such a way that they allow to express in a same setting the theory of forcing (in the sense of Cohen) and the theory of classical realizability (in the sense of Krivine). Besides, they have the nice feature of providing a common framework for the interpretation both of types and programs. The main default of these structures is that they are deeply oriented towards the λ-calculus, and that they only allows to faithfully interpret languages in call-by-name. To remediate the situation, we introduce two variants of implicative algebras: disjunctive algebras, centered on the “par” (⅋) connective of linear logic (but in a non-linear framework) and naturally adapted to languages in call-by-name; and conjunctives algebras, centered on the “tensor” (⊗) connective of linear logic and adapted to languages in call-by-value. Amongst other properties, we will see that disjunctive algebras are particular cases of implicative algebras and that conjunctive algebras can be obtained from disjunctive algebras (by reversing the underlying order).


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