Classical billiards can compute (2512.19156v1)

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.

• The paper establishes that two-dimensional billiard dynamics can simulate any Turing machine through precise geometric and topological encoding.
• It details the use of Cantor-set encodings and affine transformations to represent tape states and head movements within billiard boundaries.
• The construction implies that key dynamical questions, such as the existence of periodic orbits, are algorithmically undecidable, impacting kinetic theory and celestial mechanics.

Turing Completeness of Planar Billiards

Introduction

The paper "Classical billiards can compute" (2512.19156) establishes the Turing completeness of classical planar billiard systems. Using the framework of Topological Kleene Field Theory, the authors rigorously demonstrate that the deterministic dynamics of a two-dimensional billiard—an idealized particle reflecting elastically off rigid walls—can simulate any Turing machine. The study situates this universality as intrinsic to classical mechanical systems, including those encountered as hard-wall limits of smooth Hamiltonian systems, and extends the implications of such undecidability phenomena to physically realizable models in kinetic theory and celestial mechanics.

Billiards and the Encoding of Computation

Classical billiards are modeled by a point mass traversing a bounded subset $B \subset \mathbb{R}^2$ under free motion between specular reflections off piecewise smooth boundaries. Despite their simplicity, billiards manifest complex behavior including full integrability, strong chaos, and hyperbolicity.

The authors employ a Cantor-set-based encoding to represent Turing machine tape states as points within a compact interval, utilizing a systematic alternation of digits for left and right halves of the tape. This encoding maps both the tape content and the position of the Turing machine's head into the structure of the interval. Each possible position of the head maps to a distinct subinterval, arranged symmetrically and with exponentially decreasing length for positions far from the origin.

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.

The initial tape state, together with the head's position, is represented by a point within a designated embedding of the interval onto the billiard boundary. The halting state is similarly encoded on a disjoint segment of the boundary.

Construction of the Computational Billiard

The proof of Turing completeness follows from constructing a billiard table whose geometry embodies the transition graph of an arbitrary Turing machine. States are mapped as boundary intervals, while transitions are implemented by "corridors" channeling the billiard's trajectory according to the encoded machine dynamics.

Figure 2: Transforming the graph representation of a Turing machine into a billiard.

The shift operation, which moves the tape head, corresponds to affine transformations between intervals. This is realized with parabolic billiard walls, allowing the mapped point to transfer with the required linear relationship. The explicit map $\tau_k$ is provided so that intervals $I_k$ are embedded with disjointness and ordering preserved. The shift operation is achieved through concatentation of reflections, with left and right motions corresponding to separate affine transformations.

Figure 3: Billiard dynamics implementing a right shift. The green and blue rays correspond to the intervals $I_k$ with $k<0$ and $k\geq0$, respectively.

The read–write operation, crucial for universal computation, is encoded within the billiard wall shape itself. For each possible tape symbol to be read or written at a particular head position, the wall geometry is constructed to direct the billiard trajectory along the correct corridor and perform the required modification to the tape encoding. As the position of the head diverges, the structure of the required wall becomes increasingly intricate, yet always remains physically implementable through differentiable curves joining straight segments.

Figure 4: How the billiard wall implementing the read--write operation changes as $k\to -\infty$.

Rigorous Universality and Undecidability

The main theorem asserts that for every Turing machine, a corresponding two-dimensional billiard table exists whose trajectories precisely simulate that machine's computation. The construction requires only fixed walls and a single moving particle—no memory encoded in velocity or multiple balls.

It follows that for specific billiard tables, qualitative dynamical questions such as reachability or periodicity of trajectories are algorithmically undecidable. This strongly refutes the claim that two-dimensional conservative mechanical systems would inevitably evade algorithmic complexity (bold claim refuted: "planar billiards are too simple to be universal").

Decidability of periodicity is equivalent to the Halting Problem for the embedded machine, and thus there are billiards for which the existence of periodic orbits is formally undecidable.

Physical and Mathematical Implications

The consequences extend far beyond theoretical billiards. Given that billiard flows arise as hard-wall or steep-potential limits of smooth Hamiltonian dynamics, undecidability is not an artifact of idealization but persists in physically natural models. In kinetic theory, billiard-type descriptions capture the motion of hard-sphere gases in configuration space. Similarly, collision-dominated regimes in celestial mechanics, notably in the Newtonian $N$-body problem, are well-approximated by billiard dynamics when modeling close encounters and interaction chains.

The embedding of computation into such models introduces inherent algorithmic limits on long-term prediction, complementing classical chaos and ergodicity. For planetary systems, this implies undecidable outcomes for questions of stability, ejection, or collision sequences, despite fully deterministic underlying equations.

In the context of smooth Hamiltonians with steep walls, billiard-style undecidability emerges in singular perturbation limits, making these barriers robust against regularization and relevant even for real physical systems.

Conclusion

Through explicit constructions, this work establishes that two-dimensional billiard dynamics possesses the full power of universal computation. The foundational undecidability of the Halting Problem is thus inherent in classical planar billiard systems and propagates to a host of physical models in kinetic theory and celestial mechanics. As such, algorithmic complexity and undecidability should be counted among the deep obstructions to predictability in low-dimensional conservative dynamics, alongside integrability and chaos.

Speculative directions include investigating whether traces of classical undecidability persist under quantization in quantum billiard systems, and elucidating the relationship between topological complexity of the billiard domain and computational complexity. The results invite further exploration of the intersection between dynamics, geometry, and theoretical computation, particularly in the context of physically realistic models.