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.

翻译：本文提出并研究了一个新颖物理问题——弹球巫师问题的计算复杂性。该问题涉及一个理想化弹球在由单向门（外摆门）、平面墙、抛物面墙、移动平面墙以及能引起加速或减速的缓冲器构成的迷宫中运动。给定弹球的初始位置和速度，任务是判定其是否会击中指定目标点。通过模拟双栈下推自动机，我们证明该问题具有图灵完备性——即使在二维空间中亦然。在我们的构造中，自动机的每一步都对应恒定次数的反射。因此，判定弹球巫师问题至少与停机问题同等困难。此外，我们的构造允许用移动墙替代缓冲器。在这种情况下，即使以恒定速度运动的球——即所谓射线粒子——也可用于模拟，这证明射线粒子追踪问题同样具有图灵完备性。