The complexity of weakly recognizing morphisms

Weakly recognizing morphisms from free semigroups onto finite semigroups are a classical way for defining the class of ω -regular languages, i.e. , a set of infinite words is weakly recognizable by such a morphism if and only if it is accepted by some Buchi automaton. We study the descriptional complexity of various constructions and the computational complexity of various decision problems for weakly recognizing morphisms. The constructions we consider are the conversion from and to Buchi automata, the conversion into strongly recognizing morphisms, as well as complementation. We also show that the fixed membership problem is NC 1 -complete, the general membership problem is in L and that the inclusion, equivalence and universality problems are NL-complete. The emptiness problem is shown to be NL-complete if the input is given as a non-surjective morphism.