Strictly orthogonal left linear rewrite systems and primitive recursion

Abstract

Let $\text{F}$ be a signature and $\text{R}$ a strictly orthogonal rewrite system on ground terms of $\text{F}$. We give an effective proof of a bounding condition for $\text{R}$, based on a detailed analysis of how terms are transformed during the rewrite process, which allows us to give recursive bounds on the derivation lengths of terms. We give a syntactic characterisation of the Grzegorczyk hierarchy and a rewriting schema for calculating its functions. As a consequence of this, using results of Elias Tahhan–Bittar, it can be shown that, for $\text{n⩾3}$, the derivation length functions for functions in Grzegorczyk class $\text{E}{}_{\mathrm{n}}$ belong to Grzegorczyk class $\text{E}{}_{\mathrm{n+1}}$. We also give recursive bounds for the derivation lengths of functions defined by parameter recursion.