Languages of Dot-depth One over Infinite Words

TL;DR

This paper extends the decidability of dot-depth one languages from finite to infinite words, providing an algebraic and topological characterization of the definable languages in this fragment.

Contribution

It offers an effective characterization of dot-depth one languages over infinite words, combining algebraic and topological properties, and generalizes Knast's finite word results.

Findings

Decidability of dot-depth one over infinite words

Characterization using algebraic and topological properties

Unified approach for finite and infinite words

Abstract

Over finite words, languages of dot-depth one are expressively complete for alternation-free first-order logic. This fragment is also known as the Boolean closure of existential first-order logic. Here, the atomic formulas comprise order, successor, minimum, and maximum predicates. Knast (1983) has shown that it is decidable whether a language has dot-depth one. We extend Knast's result to infinite words. In particular, we describe the class of languages definable in alternation-free first-order logic over infinite words, and we give an effective characterization of this fragment. This characterization has two components. The first component is identical to Knast's algebraic property for finite words and the second component is a topological property, namely being a Boolean combination of Cantor sets. As an intermediate step we consider finite and infinite words simultaneously. We then obtain the results for infinite words as well as for finite words as special cases. In particular, we give a new proof of Knast's Theorem on languages of dot-depth one over finite words.