Banach-Mazur Games on Graphs

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We survey determinacy, deﬁnability, and complexity issues of Banach-Mazur games on ﬁnite and inﬁnite graphs. Inﬁnite games where two players take turns to move a token through a directed graph, thus tracing out an inﬁnite path, have numerous applications in different branches of mathematics and computer science. In the usual format, the possible moves of the players are given by the edges of the graph; in each move a player takes the token from its current position along an edge to a next position. In Banach-Mazur games the players instead select in each move a path of arbitrary ﬁnite length rather than just an edge. In both cases the outcome of a play is an inﬁnite path. A winning condition is thus given by a set of inﬁnite paths which is often speciﬁed by a logical formula, for instance from S1S, LTL, or ﬁrst-order logic. Banach-Mazur games have a long tradition in descriptive set theory and topology, and they have recently been shown to have interesting applicati...

Erich Grädel

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