Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Classical billiards can compute

Eva Miranda, Isaac Ramos

TL;DR

This work proves that two-dimensional billiards can simulate any Turing machine, using Topological Kleene Field Theory to encode computation into billiard flows. The authors construct a two-dimensional, one-ball billiard that realizes read/write and head-shift operations via carefully designed walls and corridors, establishing Turing completeness. Consequently, basic decision problems for billiards, such as reachability or periodicity, become undecidable. The results extend to physically natural systems—hard-sphere gases, collision chains in celestial mechanics, and smooth Hamiltonians in steep-wall limits—highlighting intrinsic algorithmic limits to long-term prediction in classical dynamics and related physical regimes.

Abstract

We show that two-dimensional billiard systems are Turing complete by encoding their dynamics within the framework of Topological Kleene Field Theory. Billiards serve as idealized models of particle motion with elastic reflections and arise naturally as limits of smooth Hamiltonian systems under steep confining potentials. Our results establish the existence of undecidable trajectories in physically natural billiard-type models, including billiard-type models arising in hard-sphere gases and in collision-chain limits of celestial mechanics.

Classical billiards can compute

TL;DR

This work proves that two-dimensional billiards can simulate any Turing machine, using Topological Kleene Field Theory to encode computation into billiard flows. The authors construct a two-dimensional, one-ball billiard that realizes read/write and head-shift operations via carefully designed walls and corridors, establishing Turing completeness. Consequently, basic decision problems for billiards, such as reachability or periodicity, become undecidable. The results extend to physically natural systems—hard-sphere gases, collision chains in celestial mechanics, and smooth Hamiltonians in steep-wall limits—highlighting intrinsic algorithmic limits to long-term prediction in classical dynamics and related physical regimes.

Abstract

We show that two-dimensional billiard systems are Turing complete by encoding their dynamics within the framework of Topological Kleene Field Theory. Billiards serve as idealized models of particle motion with elastic reflections and arise naturally as limits of smooth Hamiltonian systems under steep confining potentials. Our results establish the existence of undecidable trajectories in physically natural billiard-type models, including billiard-type models arising in hard-sphere gases and in collision-chain limits of celestial mechanics.
Paper Structure (14 sections, 6 theorems, 20 equations, 6 figures)

This paper contains 14 sections, 6 theorems, 20 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Billiards
  4. Turing machines
  5. Turing completeness of two-dimensional billiards
  6. Cantor-based encodings
  7. Embedding Turing machines into billiards
  8. Proof of the main theorem
  9. Physical Applications
  10. Hard-sphere gases and kinetic theory
  11. Collision chains in celestial mechanics
  12. Singular perturbation and small-mass limits
  13. Smooth Hamiltonian realizations
  14. Conclusions

Key Result

Theorem 1

Bennet73 For every Turing machine, there exists an equivalent reversible Turing machine.

Figures (6)

  • Figure 1: Encoding into the one-dimensional interval. Each interval represents a position of the head and the tape state is represented by a point in the interval.
  • Figure 2: Different trajectories of a computational billiard.
  • Figure 3: Transforming the graph representation of a Turing machine into a billiard.
  • Figure 4: Billiard dynamics implementing a right shift. The green and blue rays correspond to the intervals $I_k$ with $k<0$ and $k\geq0$, respectively.
  • Figure 5: Billiard dynamics implementing the read--write operation. The blue and red rays correspond to reading a $0$ or a $1$, respectively.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Definition 1
  • Theorem 1
  • Definition 2: Computational billiard
  • Remark 1
  • Theorem 2
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 5 more