Hardness results for constant-free pattern languages and word equations

Abstract

<p>We study constant-free versions of the inclusion problem of pattern languages and the satisfiability problem of word equations. The inclusion problem of pattern languages is known to be undecidable for both erasing and nonerasing pattern languages, but decidable for constant-free erasing pattern languages. We prove that it is undecidable for constant-free nonerasing pattern languages. The satisfiability problem of word equations is known to be in PSPACE and NP-hard. We prove that the nonperiodic satisfiability problem of constant-free word equations is NP-hard. Additionally, we prove a polynomial-time reduction from the satisfiability problem of word equations to the problem of deciding whether a given constant-free equation has a solution morphism α such that α(xy) ≠ α(yx) for given variables x and y. </p>