@article{SKRZYPCZAK201316,
    Abstract = {The paper presents a concept of a coloring --- an extension of deterministic parity automata. A coloring K is a function A⁎→N satisfying∀{$\alpha$}∈A{$\omega$}liminfn→∞K({$\alpha$}↾n)<∞. Every coloring defines a subset of A{$\omega$} by the standard parity condition[K]={{$\alpha$}∈A{$\omega$}:liminfn→∞K({$\alpha$}↾n)≡0mod2}. We show that sets defined by colorings are exactly all Δ30 sets in the standard product topology on A{$\omega$}. 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.},
    Author = {Skrzypczak, Micha{\l}},
    File = {Topological extension of parity automata - 1-s2.0-S0890540113000667-main.pdf},
    ISSN = {0890-5401},
    Journal = {Information and Computation},
    Keywords = {Borel hierarchy, Parity condition, Deterministic automata},
    Pages = {16-27},
    Title = {Topological extension of parity automata},
    URL = {https://www.sciencedirect.com/science/article/pii/S0890540113000667},
    Volume = {228-229},
    Year = {2013},
    bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0890540113000667},
    bdsk-url-2 = {https://doi.org/10.1016/j.ic.2013.06.004},
    date-added = {2021-06-25 16:28:33 +0200},
    date-modified = {2021-06-25 16:28:33 +0200},
    doi = {10.1016/j.ic.2013.06.004}
}

@article{SKRZYPCZAK201316, Abstract = {The paper presents a concept of a coloring --- an extension of deterministic parity automata. A coloring K is a function A⁎→N satisfying∀{$\alpha$}∈A{$\omega$}liminfn→∞K({$\alpha$}↾n)<∞. Every coloring defines a subset of A{$\omega$} by the standard parity condition[K]={{$\alpha$}∈A{$\omega$}:liminfn→∞K({$\alpha$}↾n)≡0mod2}. We show that sets defined by colorings are exactly all Δ30 sets in the standard product topology on A{$\omega$}. 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.}, Author = {Skrzypczak, Micha{\l}}, File = {Topological extension of parity automata - 1-s2.0-S0890540113000667-main.pdf}, ISSN = {0890-5401}, Journal = {Information and Computation}, Keywords = {Borel hierarchy, Parity condition, Deterministic automata}, Pages = {16-27}, Title = {Topological extension of parity automata}, URL = {https://www.sciencedirect.com/science/article/pii/S0890540113000667}, Volume = {228-229}, Year = {2013}, bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0890540113000667}, bdsk-url-2 = {https://doi.org/10.1016/j.ic.2013.06.004}, date-added = {2021-06-25 16:28:33 +0200}, date-modified = {2021-06-25 16:28:33 +0200}, doi = {10.1016/j.ic.2013.06.004} }