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

An exercise on transition systems

Labelled transition systems admit different but equivalent characterizations either as relational structures or coalgebras for the powerset functor, each of them with their own merits. Notions of simulation and bisimulation, for example, are expressed in the pointfree relational calculus in a very concise and precise way. On the other hand, the coalgebraic perspective regards processes as inhabitants of a final universe and allows for an intuitive definition of the semantics of process’ combinators. This paper is an exercise on such a dual characterisation. In particular, it discusses how a notion of weak bisimilarity can be lifted from the relational to the coalgebraic level, to become an effective reasoning tool on coinductively defined process algebras.(undefined