Limits of Rush Hour Logic Complexity

TL;DR

This paper investigates the computational complexity of Rush Hour Logic, demonstrating that even the simplified Size 2 Rush Hour model can simulate polynomial space computations, and explores the complexity of Unit Rush Hour.

Contribution

The paper proves that Size 2 Rush Hour can support polynomial space computation, settling a conjecture, and analyzes the complexity of Unit Rush Hour, linking it to maze puzzles and providing empirical evidence of its hardness.

Findings

Size 2 Rush Hour supports polynomial space computation.

Size 2 Rush Hour can construct necessary computational building blocks.

Unit Rush Hour is computationally hard and related to maze puzzles.

Abstract

Rush Hour Logic was introduced in [Flake&Baum99] as a model of computation inspired by the ``Rush Hour'' toy puzzle, in which cars can move horizontally or vertically within a parking lot. The authors show how the model supports polynomial space computation, using certain car configurations as building blocks to construct boolean circuits for a cpu and memory. They consider the use of cars of length 3 crucial to their construction, and conjecture that cars of size 2 only, which we'll call `Size 2 Rush Hour', do not support polynomial space computation. We settle this conjecture by showing that the required building blocks are constructible in Size 2 Rush Hour. Furthermore, we consider Unit Rush Hour, which was hitherto believed to be trivial, show its relation to maze puzzles, and provide empirical support for its hardness.