On the Borel Inseparability of Game Tree Languages

Borel Games with Lower-Semi-Continuous Payoffs

We prove that every two-player non-zero-sum Borel game with lower-semi-continuous payoffs admits a subgame-perfect $\ep$-equilibrium. This result complements Example 3 in Solan and Vieille (2003), which shows that a subgame-perfect $\ep$-equilibrium need not exists when the payoffs are not lower-semi-continuous.

Highly lopsided information and the Borel hierarchy

In a game where both contestants have perfect information, there is a strict limit on how perfect that information can be. By contrast, when one player is deprived of all information, the limit on the other player's information disappears, admitting a hierarchy of levels of lopsided perfection of information. We turn toward the question of when the player with super-perfect information has a winning strategy, and we exactly answer this question for a specific family of lopsided-information games which we call guessing games.

On Winning Conditions of High Borel Complexity in Pushdown Games

Some decidable winning conditions of arbitrarily high finite Borel complexity for games on finite graphs or on pushdown graphs have been recently presented by O. Serre in [ Games with Winning Conditions of High Borel Complexity, in the Proceedings of the International Conference ICALP 2004, LNCS, Volume 3142, p. 1150-1162 ]. We answer in this paper several questions which were raised by Serre in the above cited paper. We first show that, for every positive integer n, the class C_n(A), which arises in the definition of decidable winning conditions, is included in the class of non-ambiguous cont…

Borel and Hausdorff Hierarchies in Topological Spaces of Choquet Games and Their Effectivization

What parts of classical descriptive set theory done in Polish spaces still hold for more general topological spaces, possibly T0 or T1, but not T2 (i.e. not Hausdorff)? This question has been addressed by Victor Selivanov in a series of papers centered on algebraic domains. And recently it has been considered by Matthew de Brecht for quasi-Polish spaces, a framework that contains both countably based continuous domains and Polish spaces. In this paper we present alternative unifying topological spaces, that we call approximation spaces. They are exactly the spaces for which player Nonempty has…

A determinacy approach to Borel combinatorics

We introduce a new method, involving infinite games and Borel determinacy, which we use to answer several well-known questions in Borel combinatorics.

On Recognizable Tree Languages Beyond the Borel Hierarchy

We investigate the topological complexity of non Borel recognizable tree languages with regard to the difference hierarchy of analytic sets. We show that, for each integer $n \geq 1$, there is a $D_{ω^n}({\bf Σ}^1_1)$-complete tree language L_n accepted by a (non deterministic) Muller tree automaton. On the other hand, we prove that a tree language accepted by an unambiguous Büchi tree automaton must be Borel. Then we consider the game tree languages $W_{(i,k)}$, for Mostowski-Rabin indices $(i, k)$. We prove that the $D_{ω^n}({\bf Σ}^1_1)$-complete tree languages L_n are Wadge reducible to th…

The determinacy of infinite games with eventual perfect monitoring

n infinite two-player zero-sum game with a Borel winning set, in which the opponent's actions are monitored eventually but not necessarily immediately after they are played, is determined. The proof relies on a representation of the game as a stochastic game with perfect information, in which Chance operates as a delegate for the players and performs the randomizations for them, and on Martin's Theorem about the determinacy of such games.

Borel群胚作用的结构性、无点、非Hausdorff拓扑实现

Structural, point-free, non-Hausdorff topological realization of Borel groupoid actions

本文推广Becker--Kechris拓扑实现定理至群胚作用，基于范畴量词的代数性质给出新证明，适用于非Hausdorff及无点上下文，并刻画潜在开关系与一阶结构实现。

An Upper Bound on the Complexity of Recognizable Tree Languages

The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class $\Game (D\_n({\bfΣ}^0\_2))$ for some natural number $n\geq 1$, where $\Game$ is the game quantifier. We first give a detailed exposition of this result. Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space $2^ω$ into the class ${\bfΔ}^1\_2$, and the notions of Wadge degree and Veblen function, we argue that this upper bound on the topological complexity of regular tree languages is much better than the usual ${\bfΔ}^1\_2$.