Diagonalization Games

N. Alon, O. Bousquet, Kasper Green Larsen, S. Moran, S. Moran
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引用次数: 0

Abstract

We study several variants of a combinatorial game which is based on Cantor's diagonal argument. The game is between two players called Kronecker and Cantor. The names of the players are motivated by the known fact that Leopold Kronecker did not appreciate Georg Cantor's arguments about the infinite, and even referred to him as a"scientific charlatan". In the game Kronecker maintains a list of m binary vectors, each of length n, and Cantor's goal is to produce a new binary vector which is different from each of Kronecker's vectors, or prove that no such vector exists. Cantor does not see Kronecker's vectors but he is allowed to ask queries of the form"What is bit number j of vector number i?"What is the minimal number of queries with which Cantor can achieve his goal? How much better can Cantor do if he is allowed to pick his queries \emph{adaptively}, based on Kronecker's previous replies? The case when m=n is solved by diagonalization using n (non-adaptive) queries. We study this game more generally, and prove an optimal bound in the adaptive case and nearly tight upper and lower bounds in the non-adaptive case.
对角化的游戏
本文研究了基于康托尔对角论证的组合对策的几种变体。这场比赛是在克罗内克和康托两位选手之间进行的。这些球员的名字源于一个众所周知的事实,即利奥波德·克罗内克不欣赏乔治·康托尔关于无限的论点,甚至称他为“科学骗子”。在这个博弈中,Kronecker维护了一个包含m个二进制向量的列表,每个向量的长度为n, Cantor的目标是产生一个新的二进制向量,它不同于Kronecker的每个向量,或者证明不存在这样的向量。康托没有看到克罗内克的向量,但他可以问这样的问题:“向量i的位j是多少?”康托能达到目标的最小查询数是多少?如果允许康托根据克罗内克之前的回复自\emph{适应地}选择问题,他能做得多好?当m=n的情况是通过使用n(非自适应)查询的对角化解决的。我们对该对策进行了更一般的研究,并证明了在自适应情况下的最优界和在非自适应情况下的近紧上界和下界。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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