Untyped Recursion Schemes and Infinite Intersection Types

A new framework for higher-order program verification has been recently proposed, in which higher-order functional programs are modelled as higher-order recursion schemes and then model-checked. As recursion schemes are essentially terms of the simply-typed lambda-calculus with recursion and tree constructors, however, it was not clear how the new framework applies to programs written in languages with more advanced type systems. To circumvent the limitation, this paper introduces an untyped version of recursion schemes and develops an infinite intersection type system that is equivalent to the model checking of untyped recursion schemes, so that the model checking can be reduced to type checking as in recent work by Kobayashi and Ong for typed recursion schemes. The type system is undecidable but we can obtain decidable subsets of the type system by restricting the shapes of intersection types, yielding a sound (but incomplete in general) model checking algorithm.