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It is well-known that some equational theories such as groups or booleanalgebras can be defined by fewer equational axioms than the original axioms.However, it is not easy to determine if a given set of axioms is the smallestor not. Malbos and Mimram investigated a general method to find a lower boundof the cardinality of the set of equational axioms (or rewrite rules) that isequivalent to a given equational theory (or term rewriting systems), usinghomological algebra. Their method is an analog of Squier's homology theory onstring rewriting systems. In this paper, we develop the homology theory forterm rewriting systems more and provide a better lower bound under a strongernotion of equivalence than their equivalence. The author also implemented aprogram to compute the lower bounds, and experimented with 64 complete TRSs.

A Lower Bound of the Number of Rewrite Rules Obtained by Homological Methods