On the descriptive power of term rewriting systems

Abstract

Given the set ℐ of terms, a congruence ∼ on ℐ and a set N of representatives for ∼, we say a term rewriting system (TRS) R defines (∼, N ) if ∼ is the congruence defined by R and N is the set of R -normal forms. We give necessary and sufficient syntactical conditions on (∼, N ) to be definable by a TRS. It turns out that (∼, N) may be definable by a TRS but not by a uniquely terminating one. In order to find a minimal TRS defining (∼, N ) we construct the reduced TRS for (∼, N ). The reduced TRS R may fail to define (∼, N ), but if every term has an R -normal form, then it is a minimal TRS defining (∼, N ).