The Complexity of Testing Ground Reducibility for Linear Word Rewriting Systems with Variables

In 9] we proved that for a word rewriting system with variables R and a word with variables w, it is undecidable if w is ground reducible by R, that is if all the instances of w obtained by substituting its variables by non-empty words are reducible by R. On the other hand, if R is linear, the question is decidable for arbitrary (linear or non-linear) w. In this paper we futher study the complexity of the above problem and prove that it is co-NP-complete if both R and w are restricted to be linear. The proof is based on the construction of a deterministic nite automaton for the language of words reducible by R. The construction generalizes the well-known Aho-Corasick automaton for string matching against a set of keywords.