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How Pinball Wizards Simulate a Turing Machine

Authors Rosemary U. Adejoh , Andreas Jakoby , Sneha Mohanty , Christian Schindelhauer

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Pinball Wizard problem
  • Halting problem
  • Turing-complete

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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.

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Rosemary U. Adejoh, Andreas Jakoby, Sneha Mohanty, and Christian Schindelhauer. How Pinball Wizards Simulate a Turing Machine. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 4:1-4:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.FSTTCS.2025.4

Author Details

Rosemary U. Adejoh
  • Faculty of Media, Bauhaus-Universität Weimar, Germany
Andreas Jakoby
  • Faculty of Media, Bauhaus-Universität Weimar, Germany
Sneha Mohanty
  • Computer Networks and Telematics, University of Freiburg, Germany
Christian Schindelhauer
  • Computer Networks and Telematics, University of Freiburg, Germany

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