无限通信复杂性与KW博弈
Infinite Communication Complexity and KW Games
查看机构详情
- Department of Mathematics, UCLA（加州大学洛杉矶分校数学系）
机构由 AI 辅助整理，请以论文原文为准。
浏览论文内容
中文总结 AI 辅助
该研究将KW博弈扩展至无限版本，建立了Cantor空间子集为Borel集与无限通信博弈可解性的等价关系，为描述集合论经典结果提供新组合证明，还证明非良基树集与无限奇偶函数均非Borel集。
中文摘要 AI 辅助
我们通过Karchmer和Wigderson博弈的无限版本（该博弈将有限电路复杂性与通信复杂性关联起来）对Borel集进行了刻画。为此，我们构建了通信复杂性的无限版本，并证明Cantor空间的给定子集是Borel集当且仅当某个无限通信博弈可解。我们利用这种关联为描述集合论中的一些经典结果提供了新的初等纯组合证明，包括解析分离定理以及单调Borel集与正Borel集的等价性。另一项成果是通过某类“切与选”博弈的获胜者对Borel可分性进行了刻画，我们据此得到新的组合证明：既非非良基树集合，也非任何无限奇偶函数是Borel集。
英文摘要
We characterize the Borel sets with an infinite version of Karchmer and Wigderson's game linking finite circuit complexity to communication complexity. To this end, we formulate an infinite version of communication complexity and prove that a given subset of the Cantor space is Borel if and only if a certain infinite communication game is solvable. We utilize this connection to provide new elementary and purely combinatorial proofs of some classical results in descriptive set theory, including the analytic separation theorem and the equivalence of monotone and positive Borel sets. Another consequence is a characterization of Borel separability via the winner of a certain "cut-and-choose" game, which we use to obtain new combinatorial proofs that neither the set of ill-founded trees nor any infinite parity function is Borel.
