Peg-Solitaire, String Rewriting Systems and Finite Automata

We consider a class of length-preserving string rewriting systems and show that the set of encodings of pairs of strings such that f can be derived from s using the rewriting rules can be accepted by finite automata. As a consequence, we show the existence of a linear time algorithm for determining the solvability of a given k x n peg-solitaire board, for any fixed k. This result is in contrast to the recent results of [UEHA] and [AVIS] that the same problem is NP-hard for n × n boards. We look at some related string rewriting systems and find conditions under which the encodings of the pairs where f can be derived from s is regular.