How Pinball Wizards Simulate a Turing Machine

TL;DR

The paper studies a 2D Pinball Wizard model in which a point ball with variable speed moves through a maze of walls, one-way gates, parabola walls, bumpers, and moving walls to decide whether the ball hits a target. It demonstrates that this 2D system can simulate a $2$-stack pushdown automaton, with each automaton step corresponding to a constant number of reflections, making the problem at least as hard as the Halting problem and thereby Turing-complete when speed is not fixed. The authors further show that replacing bumpers with moving walls preserves universality, implying the Ray Particle Tracing variant (constant speed) is also Turing-complete. This work links physical-geometry computation to classical models of computation, highlighting how two independent stacks can be realized in a planar setting and raising open questions about minimal component sets and the impact of non-ideal physics on such universal constructions.

Abstract

We introduce and investigate the computational complexity of a novel physical problem known as the Pinball Wizard problem. It involves an idealized pinball moving through a maze composed of one-way gates (outswing doors), plane walls, parabolic walls, moving plane walls, and bumpers that cause acceleration or deceleration. Given the initial position and velocity of the pinball, the task is to decide whether it will hit a specified target point. By simulating a two-stack pushdown automaton, we show that the problem is Turing-complete -- even in two-dimensional space. In our construction, each step of the automaton corresponds to a constant number of reflections. Thus, deciding the Pinball Wizard problem is at least as hard as the Halting problem. Furthermore, our construction allows bumpers to be replaced with moving walls. In this case, even a ball moving at constant speed -- a so-called ray particle -- can be used, demonstrating that the Ray Particle Tracing problem is also Turing-complete.

Figures (26)

• Figure 1: a) A perfectly elastic collision of a ball of speed $\vec{v}$ with a wall, after the collision the direction has changed but the speed is preserved. b) A ball has a collision with a one-way gate on the blocking side. This leads to a perfectly elastic collision. c) A ball has a collision with a one-way gate on the pass through side.
• Figure 2: a) A perfectly elastic collision of a ball of speed $\vec{v}_b$ with a wall of speed $\vec{v}_w$ b) relative to the wall the ball has a speed of $\vec{v}_b'=\vec{v}_b+\vec{v}_w'$ c) the collision and the speed of the ball $\vec{v}_b"$ relative to the wall after the collision and d) the balls direction and speed $\vec{v}_b"'=\vec{v}_b"+\vec{v}_w$ if it is not analyzed relative to the moving wall.
• Figure 3: a) A bumper with positive (speed increasing) effect and b) a bumper with negative (speed decreasing) effect.
• Figure 5: Illustration of the stack operations implemented by the time offset.
• Figure 6: Multiplying the time offset by a factor using bumpers: The first red way (way 1) from the input (at offset interval start) to the right hand side bumper wall illustrates the the length $\delta_1$. The green way (way 2) between the two bumper walls illustrates the the length $\delta_2$. This is the only way where the ball has a changed speed. The second red way (way 3) from the left hand side bumper wall to the output (at offset interval end) illustrates the the length $\delta_3$. The way between the two bumper walls is used to implement the multiplication of the time offset by a constant. The vertical shift of the 3 ways is used to increase readability.

Theorems & Definitions (7)

• Definition 1: Pinball Wizard problem
• Definition 2: Ray Particle Tracing problem
• Lemma 1
• Lemma 2
• Theorem 1
• Corollary 1
• Lemma 3