The complexity of testing ground reducibility for linear word rewriting systems with variables

Abstract

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