On the Length of Word Chains

is a word chain if for each wi there are indices j, k c i with wi = wjwL. (By convention, wj is a letter of the underlying alphabet if j < 0.) Clearly, addition chains are exactly word chains over a one-letter alphabet. The word chain is said to compute a word w if w belongs to the chain. The chain length of w is the smallest length of a word chain computing w. Besides the idea of generalizing addition chains, the interest of this notion comes from the intuitively clear fact that it lakes into account regularities in words, much better than for instance finite automata can do. This claim will be supported by some of the subsequent examples. It is well known that the length of a shortest