Based on the paper ``Existence and construction of rewrite systems'' by \textit{N. Dershowitz} and \textit{L. Marcus} [ATR-82(8478)-3, The Aerospace Corporation (1982)], this paper gives partial answers to the following questions: when does a given decidable equational theory have a decision procedure in the form of a canonical rewrite system, and when does the Knuth-Bendix procedure generate such a rewrite system for a given equational theory? More precisely, in the third section it is shown that there are decidable equational theories which require an expanded language in the order to obtain a canonical rewrite system to decide them, and others which do not have canonical rewrite systems in any language. The section four establishes conditions which imply the uniqueness of a term-rewriting system modulo a congruence and the section five considers the relationship between the Knuth-Bendix completion procedure and the existence of a finite canonical rewrite system for an equational theory.