Playing Games in the Baire Space

TL;DR

This paper extends classical game and automata theory to the Baire space, introducing N-memory automata and transducers, and proves the decidability of game solvability with constructive winning strategies.

Contribution

It introduces N-memory automata and transducers for Baire space games and proves the decidability of game solvability with constructive strategies.

Findings

Decidability of game solvability in Baire space with N-memory automata.

Construction of N-memory transducers implementing winning strategies.

Extension of automata theory from Cantor to Baire space.

Abstract

We solve a generalized version of Church's Synthesis Problem where a play is given by a sequence of natural numbers rather than a sequence of bits; so a play is an element of the Baire space rather than of the Cantor space. Two players Input and Output choose natural numbers in alternation to generate a play. We present a natural model of automata ("N-memory automata") equipped with the parity acceptance condition, and we introduce also the corresponding model of "N-memory transducers". We show that solvability of games specified by N-memory automata (i.e., existence of a winning strategy for player Output) is decidable, and that in this case an N-memory transducer can be constructed that implements a winning strategy for player Output.