Knuth-Bendix Algorithm

The Knuth-Bendix algorithm[Knuth, D. E and Bendix, P. E., ``Simple word problems in universal algebras'', in J. Leech (ed.), Computational Problems in Abstract Algebra, Pergammon Press, 1970, pp. 263-297.] describes how to derive a complete set of rewrite rules R from an equational theory E , such that:

If E implies that two terms s and t are equal, then the reductions in R will rewrite both s and t to the same irreducible form in a finite number of steps.

Two properties are needed:

• Confluence: no matter what sequence of transforms is chosen, the final result is the same.

• Termination: the process of applying transforms will terminate.

Unfortunately, rather simple systems do not have a Knuth-Bendix solution.