6 research outputs found
Retractions in Intersection Types
This paper deals with retraction - intended as isomorphic embedding - in
intersection types building left and right inverses as terms of a lambda
calculus with a bottom constant. The main result is a necessary and sufficient
condition two strict intersection types must satisfy in order to assure the
existence of two terms showing the first type to be a retract of the second
one. Moreover, the characterisation of retraction in the standard intersection
types is discussed.Comment: In Proceedings ITRS 2016, arXiv:1702.0187
Proof Systems for Retracts in Simply Typed Lambda Calculus
Abstract. This paper concerns retracts in simply typed lambda calculus assuming βη-equality. We provide a simple tableau proof system which characterises when a type is a retract of another type and which leads to an exponential decision procedure.
Second-Order Type Isomorphisms Through Game Semantics
The characterization of second-order type isomorphisms is a purely
syntactical problem that we propose to study under the enlightenment of game
semantics. We study this question in the case of second-order
λ-calculus, which can be seen as an extension of system F to
classical logic, and for which we define a categorical framework: control
hyperdoctrines. Our game model of λ-calculus is based on polymorphic
arenas (closely related to Hughes' hyperforests) which evolve during the play
(following the ideas of Murawski-Ong). We show that type isomorphisms coincide
with the "equality" on arenas associated with types. Finally we deduce the
equational characterization of type isomorphisms from this equality. We also
recover from the same model Roberto Di Cosmo's characterization of type
isomorphisms for system F. This approach leads to a geometrical comprehension
on the question of second order type isomorphisms, which can be easily extended
to some other polymorphic calculi including additional programming features.Comment: accepted by Annals of Pure and Applied Logic, Special Issue on Game
Semantic
Retracts in Simple Types
International audienceWe prove the decidability of the existence of a definable retraction between two given simple types when both types are built over a unique ground type. Instead of defining some extension of a former type system from which these retractions could be inferred, we obtain this result as a corollary of the decidability of the minimal model of simply typed λ-calculus