A polynomial-time algorithm for the automatic Baire property
This work provides an algorithmic foundation for the automatic Baire property, benefiting researchers in automata theory and descriptive set theory by enabling efficient construction of topological covers.
The paper presents a polynomial-time algorithm to construct open and meagre sets (definable by finite automata) that cover a given set defined by a Muller automaton, establishing the automatic Baire property. It also provides a conversion from a restricted class of Muller automata to equivalent Büchi automata with at most quadratic size.
A subset of a topological space possesses the Baire property if it can be covered by an open set up to a meagre set. For the Cantor space of infinite words Finkel introduced the automatic Baire category where both sets, the open and the meagre, can be chosen to be definable by finite automata. Here we show that, given a Muller automaton defining the original set, resulting open and meagre sets can be constructed in polynomial time. Since the constructed sets are of simple topological structure, it is possible to construct not only Muller automata defining them but also the simpler Büchi automata. To this end we give, for a restricted class of Muller automata, a conversion to equivalent Büchi automata of at most quadratic size.