Operations on Weakly Recognizing Morphisms

TL;DR

This paper investigates the complexity of various transformations involving weakly recognizing morphisms, B"uchi automata, and strongly recognizing morphisms, providing bounds and insights into their interrelations for omega-regular languages.

Contribution

It offers new bounds and detailed analysis on the descriptional complexity of conversions and operations on weakly recognizing morphisms, especially for binary alphabets and simple semigroups.

Findings

Bounds for conversion between morphisms and automata

Complexity results for complementation of weakly recognizing morphisms

More precise bounds for binary alphabets and simple semigroups

Abstract

Weakly recognizing morphisms from free semigroups onto finite semigroups are a classical way for defining the class of omega-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 B\"uchi automaton. We consider the descriptional complexity of various constructions for weakly recognizing morphisms. This includes the conversion from and to B\"uchi automata, the conversion into strongly recognizing morphisms, and complementation. For some problems, we are able to give more precise bounds in the case of binary alphabets or simple semigroups.