Undecidability of Two-dimensional Robot Games

TL;DR

This paper proves that determining the winning strategy in a two-dimensional robot game, a vector addition game on the integer lattice, is undecidable, resolving a question posed in 2011 and bridging the gap between known decidable and undecidable cases.

Contribution

It establishes the undecidability of the two-dimensional robot game, a fundamental problem in vector addition games, answering an open question from 2011.

Findings

Decidability of robot games in higher dimensions remains open.

The paper proves undecidability specifically for the 2D case.

This result closes the gap between known decidable and undecidable cases.

Abstract

Robot game is a two-player vector addition game played on the integer lattice $\mathbb{Z}^n$. Both players have sets of vectors and in each turn the vector chosen by a player is added to the current configuration vector of the game. One of the players, called Eve, tries to play the game from the initial configuration to the origin while the other player, Adam, tries to avoid the origin. The problem is to decide whether or not Eve has a winning strategy. In this paper we prove undecidability of the robot game in dimension two answering the question formulated by Doyen and Rabinovich in 2011 and closing the gap between undecidable and decidable cases.