Decision questions concerning semilinearity, morphisms, and commutation of languages

Abstract

Let ℓ be a class of automata (in a precise sense to be defined) and Cℓ the class obtained by augmenting each automaton in ℓ with finitely many reversal-bounded counters. We first show that if the languages defined by ℓ are effectively semilinear, then so are the languages defined by Cc, and, hence, their emptiness problem is decidable. This result is then used to show the decidability of various problems concerning morphisms and commutation of languages. We also prove a surprising undecidability result: given a fixed two element code K, it is undecidable whether a given context-free language L commutes with K, i.e., LK = KL. © 2011 Springer-Verlag Berlin Heidelberg.