Efficient Algorithms for Morphisms over Omega-Regular Languages

TL;DR

This paper presents efficient algorithms for computing minimal recognizing morphisms and solving decision problems for omega-regular languages, improving the understanding and processing of these languages in automata theory.

Contribution

It introduces a quadratic-time algorithm for minimizing strongly recognizing morphisms and provides methods for decision problems related to weakly recognizing morphisms.

Findings

Quadratic-time algorithm for computing the syntactic morphism

Efficient decision algorithms for weakly recognizing morphisms

Experimental results on converting MSO formulas into recognizing morphisms

Abstract

Morphisms to finite semigroups can be used for recognizing omega-regular languages. The so-called strongly recognizing morphisms can be seen as a deterministic computation model which provides minimal objects (known as the syntactic morphism) and a trivial complementation procedure. We give a quadratic-time algorithm for computing the syntactic morphism from any given strongly recognizing morphism, thereby showing that minimization is easy as well. In addition, we give algorithms for efficiently solving various decision problems for weakly recognizing morphisms. Weakly recognizing morphism are often smaller than their strongly recognizing counterparts. Finally, we describe the language operations needed for converting formulas in monadic second-order logic (MSO) into strongly recognizing morphisms, and we give some experimental results.