Prefix-Rewriting on Context-Free Groups

Abstract.

 A class of groups that has received much attention is the class of context-free groups. This class of groups can be characterized algebraically as well as through some language-theoretical properties as well as through certain combinatorial properties of presentations. Here we use the fact that a finitely generated group is context-free if and only if it admits a finite complete presentation of a certain form that we call a virtually free presentation.

It is known that the generalized word problem for context-free groups is decidable. Here we show how prefix-rewriting systems can be used to solve this problem. In fact, we show that the Knuth-Bendix completion procedure always terminates when applied to prefix-rewriting systems on virtually free presentations of context-free groups. In addition, we present a specialized completion algorithm for prefix-rewriting systems on virtually free presentations which has polynomial-time complexity. Thus, the uniform generalized word problem for virtually free presentations of context-free groups is decidable in polynomial-time.

Since finitely generated subgroups of context-free groups are again context-free, they admit presentations of the same form. We present a polynomial-time algorithm that, given a virtually free presentation of a context-free group G and a finite subset U of G as input, computes a virtually free presentation for the subgroup of G that is generated by U.