Generating small convergent systems can be extremely hard

Abstract

A sequence $(R_{n,m} )_{n,m \in \mathbb{N}}$ of normalized string-rewriting systems on some finite alphabet ⌆ is constructed such that, for all n, m ∃ ℕ,

• R_n,m contains 44 rules, it is of size O(n+m), and it is compatible with a length-lexicographical ordering > on ⌆ ^*, but

• given the system R _n,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 S _n,m of size O(n+m) is obtained, where A denotes Ackermann's function.

The results presented here where obtained while the second author was visiting at the Fachbereich Informatik, UniversitÄt Kaiserslautern during his sabbatical 1991/92.