A major portion of the work and time involved in completing an incomplete set of reductions using an E-completion procedure such as the one described by Knuth and Bendix  or its extension to associative-commutative equational theories as described by Peterson and Stickel [PS81] is spent calculating critical pairs and subsequently testing them for coherence. A pruning technique which removes from consideration those critical pairs that represent redundant or superfluous information, either before, during, or after their calculation, can therefore make a marked difference in the run time and efficiency of an E-completion procedure to which it is applied.
The exploitation of term symmetry is one such pruning technique. The calculation of redundant critical pairs can be avoided by detecting the term symmetries that can occur between the subterms of the left-hand side of the major reduction being used, and later between the unifiers of these subterms with the left-hand side of the minor reduction. After calculation, and even after reduction to normal form, the observation of term symmetries can lead to significant savings.
The results in this paper were achieved through the development and use of a flexible E-unification algorithm which is currently written to process pairs of terms which may contain any combination of Null-E, C (Commutative), AC (Associative-Commutative) and ACI (Associative-Commutative with Identity) operators. One characteristic of this E-unification algorithm that we have not observed in any other to date is the ability to process a pair of terms which have different ACI top-level operators. In addition, the algorithm is a modular design which is a variation of the Yelick model [Ye85], and is easily extended to process terms containing operators of additional equational theories by simply "plugging in" a unification module for the new theory.
Mayfield, Blayne E. and Wilkerson, Ralph W., "The Role of Term Symmetry in E-Unification and E-Completion" (1988). Computer Science Technical Reports. 87.
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