Complexity of terms, superpositions, and generalized hypersubstitutions

In this paper, we consider the four useful measurements of the complexity of a term, called the maximum depth, the minimum depth, the variable count, and the operation count. We construct a formula for the complexity of the superposition Sm.s; t1; : : : ; tm/ in terms of complexity of the inputs s; t1; : : : ; tm for each of these measurements. We also obtain formulas for the complexity of O TtU in terms of the complexity where t is a compound term and is a generalized hypersubstitution. We apply these formulas to the theory of M-strongly solid varieties, examining the k-normalization chains of a variety with respect to these complexity measurements.