Technical Report LCL9406

The complexity of a particular term-rewrite system is considered: the rule of associativity (x \ast y) \ast z \rhd x \ast (y \ast z). Algorithms and exact calculations are given for the longest and shortest sequences of applications of \rhd that result in normal form (NF). The shortest NF sequence for a term x is always n - d_{\mbox{rm}}(x), where n is the number of occurrences of \ast in x and d_{\mbox{rm}}(x) is the depth of the rightmost leaf of x. The longest NF sequence for any term is of length n(n-1)/2.