On The Problem Of Generating Small Convergent Systems

A sequence (Rn,m)n,mε of normalized string-rewriting systems on a fixed finite alphabet Σ is constructed such that, for all n,m ε , 1. Rn,m contains 44 rules, it is of size O(n + m ), and it is compatible with a length-lexicographical ordering > on Σ*, but 2. given the system Rn,m and the ordering > as input, the Knuth-Bendix completion procedure will generate more than A(n, m) intermediate rules before a finite convergent system Sn,m of size O(n + m) is obtained. Here A denotes Ackermannʹs function