Higher-order rewrite systems

Abstract

Higher-order rewrite systems are an extension of term-rewriting to (simply typed) A-terms. This yields a formalism with a built-in notion of variable binding (λ-abstraction) and substitution (β-reduction) which is capable of describing the manipulation of terms with bound variables. Typical examples are various λ-calculi themselves, logical formulae, proof terms and programs.

In this talk we survey the growing literature in this area with an emphasis on confluence results. In particular we examine some recent results by van Oostrom on very general confluence criteria which go beyond orthogonality and even generalize a result by Huet on parallel reduction between critical pairs. We also present a new critical pair condition for higher-order rewrite systems with arbitrary left-hand sides.