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⁎→NA⁎→N satisfying∀α∈Aωliminfn→∞K(α↾n)<∞. Every coloring defines a subset of AωAω by the standard parity condition[K]={α∈Aω:liminfn→∞K(α↾n)≡0mod2}. We show that sets defined by colorings are exactly all Δ30 sets in the standard product topology on AωAω. Furthermore, when considering natural subfamilies of all colorings, we obtain families BC(Σ20), Δ20, and BC(Σ10). 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 limsup condition is used instead of liminf. It turns out that such colorings have smaller expressive power — they cannot define all Δ30 sets.