The Knuth-Bendix completion algorithm attempts to transform a finite set of identities into a finitely terminating, confluent term rewriting system whose reductions preserve identity. This term rewriting system serves a decision procedure for validating identities. As defined in universal algebra, identities are equalities of two terms: x = y. Presumably, the values of the two terms are equal for all values of variables occurring in them. A reduction order is another input to the completion algorithm provided that every identity is viewed as two candidates for rewrite rules transforming the left-hand side into the right-hand side and vice versa.
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