Ricochet Robots with Infinite Horizontal Board is Turing-complete

Abstract

This paper investigates the Ricochet Robots game problem from a complexity standpoint. The problem consists of moving robots in a rectangular square-tiled board, from initial tiles to reach specific target tiles. A robot can only move vertically or horizontally and when it starts to move in a given direction, the robot follows this direction, until being blocked by a wall or another robot. This paper proves that the corresponding decision problem to Ricochet Robots is Turing-complete for endless board game and an infinite number of robots. A reduction from a universal Turing machine to Ricochet Robots is exhibited.