Topological extension of parity automata

The paper presents a concept of a coloring — an extension of deterministic parity automata. A coloring K is a function A∗ → N satisfying ∀α∈Aω lim inf n→∞ K(α n) <∞. Every coloring defines a subset of A by the standard parity condition [K] = { α ∈ A : lim inf n→∞ K(α n) ≡ 0 mod 2 } . We show that sets defined by colorings are exactly all ∆3 sets in the standard product topology on A. Furthermore, when considering natural subfamilies of all colorings, we obtain families BC(Σ2), ∆2, and BC(Σ1). Therefore, colorings can be seen as a characterisation of all these classes with a uniform acceptance condition. Additionally, we analyse a similar definition of colorings where the lim sup condition is used instead of lim inf. It turns out that such colorings have smaller expressive power — they cannot define all ∆3 sets.