Strongly sequential and inductively sequential term rewriting systems

Abstract

The concept of definitional tree by Antoy serves to introduce control information into the bare set of rules of a constructor-based term rewriting system (TRS). TRSs whose rules can be arranged into a definitional tree are called inductively sequential. By relying on the existence of such a definitional tree, an optimal rewriting strategy, the outermost-needed strategy, is defined. Optimality was proved with respect to the Huet-Lévy theory of neededness. In this paper, we prove that strongly sequential and inductively sequential constructor-based TRSs coincide. We also show that outermost-needed rewriting only reduces strongly needed redexes.