Bounded Quantification Is Undecidable

AbstractF≤ is a typed λ-calculus with subtyping and bounded second-order polymorphism. First introduced by Cardelli and Wegner, it has been widely studied as a core calculus for type systems with subtyping. We use a reduction from the halting problem for two-counter Turing machines to show that the subtyping and typing relations of F≤ are undecidable

Semantic Predicate Types and Approximation for Class-based Object Oriented Programming

We apply the principles of the intersection type discipline to the study of class-based object oriented programs and; our work follows from a similar approach (in the context of Abadi and Cardelli's Varsigma-object calculus) taken by van Bakel and de'Liguoro. We define an extension of Featherweight Java, FJc and present a predicate system which we show to be sound and expressive. We also show that our system provides a semantic underpinning for the object oriented paradigm by generalising the concept of approximant from the Lambda Calculus and demonstrating an approximation result: all expressions to which we can assign a predicate have an approximant that satisfies the same predicate. Crucial to this result is the notion of predicate language, which associates a family of predicates with a class.Comment: Proceedings of 11th Workshop on Formal Techniques for Java-like Programs (FTfJP'09), Genova, Italy, July 6 200

A Coq Library of Undecidable Problems

International audienceWe propose a talk on our library of mechanised reductions to establish undecidability results in Coq. The library is a collaborative effort, growing constantly and we are seeking more outside contributors willing to work on undecidability results in Coq

Dinaturality Meets Genericity: A Game Semantics of Bounded Polymorphism

We study subtyping and parametric polymorphism, with the aim of providing direct and tractable semantic representations of type systems with these expressive features. The liveness order uses the Player-Opponent duality of game semantics to give a simple representation of subtyping: we generalize it to include graphs extracted directly from second-order intuitionistic types, and use the resulting complete lattice to interpret bounded polymorphic types in the style of System F_<:, but with a more tractable subtyping relation. To extend this to a semantics of terms, we use the type-derived graphs as arenas, on which strategies correspond to dinatural transformations with respect to the canonical coercions ("on the nose" copycats) induced by the liveness ordering. This relationship between the interpretation of generic and subtype polymorphism thus provides the basis of the semantics of our type system

Subtype Universes

We introduce a new concept called a subtype universe, which is a collection of subtypes of a particular type. Amongst other things, subtype universes can model bounded quantification without undecidability. Subtype universes have applications in programming, formalisation and natural language semantics. Our construction builds on coercive subtyping, a system of subtyping that preserves canonicity. We prove Strong Normalisation, Subject Reduction and Logical Consistency for our system via transfer from its parent system UTT[?]. We discuss the interaction between subtype universes and other sorts of universe and compare our construction to previous work on Power types