Complexity of terms, composition, and hypersubstitution

Abstract

We consider four useful measures of the complexity of a term: the maximum depth (usually called the depth), the minimum depth, the variable count, and the operation count. For each of these, we produce a formula for the complexity of the compositionSmn(s,t1,…,tn)in terms of the complexity of the inputss,t1,…,tn. As a corollary, we also obtain formulas for the complexity ofσˆ[t]in terms of the complexity oftwhentis a compound term andσis a hypersubstitution. We then apply these formulas to the theory ofM-solid varieties, examining thek-normalization chains of a variety with respect to the four complexity measures.