A note on Automatic Baire property

TL;DR

The paper analyzes which $\omega$-languages beyond regular ones admit the Automatic Baire property (ABP). It uses measure-category duality for regular $\omega$-languages and the notion of disjunctive words to characterize when an $\omega$-language has ABP, proving a full characterization for languages of first Baire category and for finite language cases. It shows ABP holds for all regular $\omega$-languages and that the ABP class $\mathcal{A}$ is a Boolean algebra, but that ABP does not follow from modest computational power, as demonstrated by one-counter automata producing open or closed languages without ABP. The paper also provides concrete counter-examples with irrational measures to illustrate limitations, and discusses the instability of ABP under countable unions, clarifying the boundary between automata-theoretic definability and topological regularity.

Abstract

Automatic Baire property is a variant of the usual Baire property which is fulfilled for subsets of the Cantor space accepted by finite automata. We consider the family $\mathcal{A}$ of subsets of the Cantor space having the Automatic Baire property. In particular we show that not all finite subsets have the Automatic Baire property, and that already a slight increase of the computational power of the accepting device may lead beyond the class $\mathcal{A}$.

Theorems & Definitions (17)

• Theorem 1
• Definition 1: Balance condition
• Theorem 3: Theorem 3 of csl/St97
• Definition 2
• Theorem 4
• Definition 3: Automatic Baire property
• Theorem 5: LATA20/finkelijfcs21/finkel
• Corollary 6
• Lemma 7
• Lemma 8