Perpetuality and Uniform Normalization in Orthogonal Rewrite Systems

Abstract

We study perpetuality of reduction steps, as well as perpetuality of redexes, in orthogonal rewrite systems. A perpetual step is a reduction step which retains the possibility of infinite reductions. A perpetual redex is a redex which, when put into an arbitrary context, yields a perpetual step. We generalize and refine existing criteria for the perpetuality of reduction steps and redexes in orthogonal term rewriting systems and the λ-calculus due to Bergstra and Klop and others. We first introduce context-sensitive conditional expression reduction systems (CCERSs) and define a concept of orthogonality (which implies confluence) for them. In particular, several important λ-calculi and their extensions and restrictions can naturally be embedded into orthogonal CCERSs. We then define a perpetual reduction strategy which enables one to construct minimal (w.r.t. Lévy's permutation ordering on reductions) infinite reductions in orthogonal fully-extended CCERSs. Using the properties of the minimal perpetual strategy, we prove 1.perpetuality of any reduction step that does not erase potentially infinite arguments, which are arguments that may become, via substitution, infinite after a number of outside steps, and 2.perpetuality (in every context) of any safe redex, which is a redex whose substitution instances may discard infinite arguments only when the corresponding contracta remain infinite. We prove both these perpetuality criteria for orthogonal fully-extended CCERSs and then specialize and apply them to restricted λ-calculi, demonstrating their usefulness. In particular, we prove the equivalence of weak and strong normalization (whose equivalence is here called uniform normalization) for various restricted λ-calculi, most of which cannot be derived from previously known perpetuality criteria.