Diagonalization Games

TL;DR

This paper analyzes combinatorial diagonalization games inspired by Cantor's diagonal argument, determining optimal query strategies for players with different information and adaptivity constraints.

Contribution

It provides the first comprehensive bounds for both adaptive and non-adaptive variants of the diagonalization game, extending classical diagonalization concepts.

Findings

Optimal bounds for adaptive query strategies.

Nearly tight bounds for non-adaptive strategies.

Extension of diagonalization techniques to combinatorial game settings.

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.