Modal Logic and the Approximation Induction Principle

We prove a compactness theorem in the context of Hennessy-Milner logic. It is used to derive a sufficient condition on modal characterizations for the Approximation Induction Principle to be sound modulo the corresponding process equivalence. We show that this condition is necessary when the equivalence in question is compositional with respect to the projection operators

Relational Graph Models at Work

We study the relational graph models that constitute a natural subclass of relational models of lambda-calculus. We prove that among the lambda-theories induced by such models there exists a minimal one, and that the corresponding relational graph model is very natural and easy to construct. We then study relational graph models that are fully abstract, in the sense that they capture some observational equivalence between lambda-terms. We focus on the two main observational equivalences in the lambda-calculus, the theory H+ generated by taking as observables the beta-normal forms, and H* generated by considering as observables the head normal forms. On the one hand we introduce a notion of lambda-K\"onig model and prove that a relational graph model is fully abstract for H+ if and only if it is extensional and lambda-K\"onig. On the other hand we show that the dual notion of hyperimmune model, together with extensionality, captures the full abstraction for H*

CoqMTU: a higher-order type theory with a predicative hierarchy of universes parametrized by a decidable first-order theory

International audienceWe study a complex type theory, a Calculus of Inductive Constructions with a predicative hierarchy of universes and a first-order theory T built in its conversion relation. The theory T is specified abstractly, by a set of constructors, a set of defined symbols, axioms expressing that constructors are free and defined symbols completely defined, and a generic elimination principle relying on crucial properties of first-order structures satisfying the axioms. We first show that CoqMTU enjoys all basic meta-theoretical properties of such calculi, confluence, subject reduction and strong normalization when restricted to weak-elimination, implying the decidability of type-checking in this case as well as consistency. The case of strong elimination is left open