Curry-style type Isomorphisms and Game Semantics

Curry-style system F, ie. system F with no explicit types in terms, can be seen as a core presentation of polymorphism from the point of view of programming languages. This paper gives a characterisation of type isomorphisms for this language, by using a game model whose intuitions come both from the syntax and from the game semantics universe. The model is composed of: an untyped part to interpret terms, a notion of game to interpret types, and a typed part to express the fact that an untyped strategy plays on a game. By analysing isomorphisms in the model, we prove that the equational system corresponding to type isomorphisms for Curry-style system F is the extension of the equational system for Church-style isomorphisms with a new, non-trivial equation: forall X.A = A[forall Y.Y/X] if X appears only positively in A.Comment: Accept\'e \`a Mathematical Structures for Computer Science, Special Issue on Type Isomorphism

Filter Models: Non-idempotent Intersection Types, Orthogonality and Polymorphism

This paper revisits models of typed lambda calculus based on filters of intersection types: By using non-idempotent intersections, we simplify a methodology that produces modular proofs of strong normalisation based on filter models. Building such a model for some type theory shows that typed terms can be typed with intersections only, and are therefore strongly normalising. Non-idempotent intersections provide a decreasing measure proving a key termination property, simpler than the reducibility techniques used with idempotent intersections. Such filter models are shown to be captured by orthogonality techniques: we formalise an abstract notion of orthogonality model inspired by classical realisability, and express a filter model as one of its instances, along with two term-models (one of which captures a now common technique for strong normalisation). Applying the above range of model constructions to Curry-style System F describes at different levels of detail how the infinite polymorphism of System F can systematically be reduced to the finite polymorphism of intersection types

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$\mu$-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$\mu$-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

Towards an Implicit Calculus of Inductive Constructions. Extending the Implicit Calculus of Constructions with Union and Subset Types.

International audienceWe present extensions of Miquel's Implicit Calculus of Constructions (ICC) and Barras and Bernardo's decidable Implicit Calculus of Constructions (ICC*) with union and subset types. The purpose of these systems is to solve the problem of interaction betweeen logical and computational data. This is a work in progress and our long term goal is to add the whole inductive types to ICC and ICC* in order to define a complete framework for theorem proving

On completeness of reducibility candidates as a semantics of strong normalization

This paper defines a sound and complete semantic criterion, based on reducibility candidates, for strong normalization of theories expressed in minimal deduction modulo \`a la Curry. The use of Curry-style proof-terms allows to build this criterion on the classic notion of pre-Heyting algebras and makes that criterion concern all theories expressed in minimal deduction modulo. Compared to using Church-style proof-terms, this method provides both a simpler definition of the criterion and a simpler proof of its completeness.Comment: 24 page