A Simple Proof that Ricochet Robots is PSPACE-Complete

TL;DR

The paper proves PSPACE-completeness of relocation in Ricochet Robots with fixed geometry by a simpler reduction from Finite Function Generation ($FFGEN$). It develops a reconfiguration framework for individually controlled particle systems and a suite of gadgets—Function Selector, Function Enforcer, Lock Selector, Lock Gadget, Relocation Goal Gadget, and Reconfiguration Goal Gadget—to simulate $FFGEN$ within the tilt model. This gadget-based construction situates the problem among other Tilt-model results and clarifies the complexity landscape for fixed-geometry, addressable robots in reconfiguration tasks. The work strengthens the understanding of computational difficulty in tilt-based puzzles and highlights open questions about removing fixed geometry strategies.

Abstract

In this paper, we seek to provide a simpler proof that the relocation problem in Ricochet Robots (Lunar Lockout with fixed geometry) is PSPACE-complete via a reduction from Finite Function Generation (FFG). Although this result was originally proven in 2003, we give a simpler reduction by utilizing the FFG problem, and put the result in context with recent publications showing that relocation is also PSPACE-complete in related models.

Figures (9)

• Figure 1: The board games Lunar Lockout and Ricochet Robots.
• Figure 2: Ricochet Robots relocation example. Given the starting configuration (Start), a specific tile (green tile g), and a specific location (outlined), this provides a sequence of particle tilts that places g in the location. The tiles are labeled g, r, p, y based on their respective colors of green, red, purple, and yellow. The grey tiles are all blocked locations that can not be moved.
• Figure 3: (a) Flowchart of the relocation process. Purple paths lead to a state that can only be reached with the last domain element. Blue paths lead to states that can only be reached with the first domain element. Red paths can be traversed only once per element. (b) Descriptions of each state in the flowchart.
• Figure 4: Example of a two element system.
• Figure 5: An example of function selector and function enforcer gadgets with domain $D = \{1,2,3\}$ and functions $f_1,f_2, f_3\ s.t.\ f_1(1) = 2, f_1(2) = 3, f_1(3) = 1, f_2(1) = 2, f_2(2) = 1, f_2(3) = 2, f_3(1) = 3, f_3(2) = 2, f_3(3) = 1$. The arrows represent the possible input and output of the gadgets.

Theorems & Definitions (6)

• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Theorem 3.5
• Corollary 3.6