Planar Billiards and Turing Completeness

• The paper demonstrates that planar billiards can simulate Turing machines using elastic reflections and gadgets like parabolic reflectors and one-way gates.
• Researchers employ Cantor and stack encoding methods to embed tape and control state into the billiard trajectory, achieving universal computation with fixed reflection steps.
• This work links classical dynamics with computational theory, implying that fundamental trajectory problems such as reachability and periodicity are algorithmically undecidable.

Planar billiards, systems describing the motion of a point particle reflecting elastically inside a two-dimensional region with piecewise-smooth boundary (or equipped with additional geometric/gate-like features), have recently been shown to achieve Turing completeness. This result establishes that the trajectories of such classical systems can simulate any computation that a Turing machine can perform, rendering certain associated decision problems algorithmically undecidable. The underlying constructions exploit the geometric flexibility and dynamical structure of planar billiards to realize universal computation by encoding logical and memory operations in the reflection dynamics.

1. Formal Setting: Planar Billiard Dynamics and Models

In the planar billiard model, a point particle moves freely within a compact domain $B \subset \mathbb{R}^2$, with the boundary $\partial B$ taken to be piecewise %%%%2%%%% and possibly including a finite collection of interior "scatterers" with smooth boundaries. The particle evolves according to Newtonian free motion:

$\ddot{q}(t) = 0, \qquad q(t) \in B,$

and undergoes specular reflection at collisions with $\partial B$, where the outgoing velocity $v^+$ is related to the incoming velocity $v^-$ and the inward-pointing normal $n$ by

$v^{+} = v^{-} - 2 \langle v^{-}, n \rangle n.$

Mechanical analogs such as the "Pinball Wizard" model introduce further geometric elements: planar walls, parabolic reflectors, moving walls (with rational-function trajectories), bumpers (that instantaneously rescale velocity), and one-way gates, each with precise reflection and passage laws. Variations exist where the particle moves at a constant speed (the "ray particle" or "ray-tracing" setting) or where bumpers allow variable speed under explicit modulation (Adejoh et al., 2 Oct 2025). All geometric and control information is specified via rational data.

2. Computational Gadgets: Encodings and Stack Simulation

Simulation of arbitrary Turing machines by planar billiards is constructed by encoding the configuration of computational memory (tape contents, head position, and control state) into the continuous parameters governing the billiard's trajectory. Two principal approaches have emerged:

• Cantor Encoding (Topological Kleene Field Theory/TKFT): Tape and head position encoded into real-valued intervals via middle-third Cantor set representations; individual tape cells map to ternary subintervals, while the head position determines embedding in disjoint intervals of phase space (Miranda et al., 22 Dec 2025).
• Stack/Offset Encoding (Pinball Wizard): Two disjoint binary stacks are realized through real-valued time-delay offsets ($\Delta t$) and spatial offsets ($\Delta x$), manipulated via sequences of reflections through gadgets that effect dyadic transformations corresponding to push and pop operations—e.g., $f(\Delta) = \Delta/2$ (push 0), $f(\Delta) = (1+\Delta)/2$ (push 1), $f(\Delta) = 2\Delta$ (pop 0), $f(\Delta) = 2\Delta - 1$ (pop 1) (Adejoh et al., 2 Oct 2025).

Key gadgets implemented in these constructions include:

Gadget Type	Physical Realization	Computation Role
Parabolic reflectors	Confocal parabolas, rational data	Shift/scale coordinate (e.g., $x \to x/2$)
Moving walls	Prescribed velocity segments	Time-delay manipulation
Bumpers	Speed rescaling (variable $\lambda(t)$)	Time-delay (stack) adjustment
One-way gates	Directional passages	Routing, logical control
Zig-zag walls ($C^1$)	Boundary with modulated slope	Read/write tape operation

The stack-based approach interleaves "tracks" in space and time, routing the ball through a deterministic pipeline of stack-gadget operations, where each push and pop can be simulated via a constant number of billiard reflections independent of stack depth.

3. Simulation of Turing Machines

Turing completeness is achieved by assembling the requisite computational gadgets to effect stepwise simulation of a discrete-time deterministic Turing Machine (TM). The construction enables the following capabilities:

1. Tape and head encoding: The initial configuration of the TM is mapped to a specific starting point or offset in the billiard system.
2. Control flow: Successive billiard reflections encode state transitions, using one-way gates and offset gadgets to decode current symbols and state.
3. Memory manipulation: Push and pop on two grids for stack simulation or ternary digit manipulation for Cantor encoding.
4. Halting and output: A unique geometric "halting" corridor or output boundary is constructed such that the particle reaches this region if and only if the simulated TM halts.

For example, in the Cantor-encoded setup, each reflection at a shift gadget advances the head position via $x_{t,k} \mapsto x_{t,k+1}$, while the read-write gadget selectively manipulates the ternary digit corresponding to the current tape symbol.

The simulation is shown to be efficient, requiring a polynomial-size billiard construction for a given TM, and every TM step is realized by a fixed, bounded number of billiard reflections (Miranda et al., 22 Dec 2025, Adejoh et al., 2 Oct 2025).

4. Undecidability Results

The existence of universal computation within planar billiards implies that fundamental dynamical and geometric questions about their trajectories are algorithmically undecidable. Specific decision problems shown to inherit the computational hardness of the Halting Problem include:

• Point reachability: Deciding whether a billiard trajectory, launched from a specified initial state, will ever pass through a designated target point or exit interval. This is undecidable, as the constructed reachability problem is equivalent to the halting of the encoded Turing machine [(Adejoh et al., 2 Oct 2025), Theorem 3.1; (Miranda et al., 22 Dec 2025), Theorem].
• Periodicity: Given an initial condition, deciding whether the billiard trajectory eventually becomes periodic—i.e., returns to its starting interval and direction after a finite number of reflections—is likewise undecidable, as this is equivalent to halting in the simulated computation [(Miranda et al., 22 Dec 2025), Corollary].
• Output decoding: The output word or configuration resulting from the simulation (if the particle reaches the halting region) directly reconstructs the final TM tape state via unambiguous projection from geometric coordinates.

A plausible implication is that many natural dynamical invariants or questions about trajectory classification in even low-dimensional billiard domains cannot be decided algorithmically.

5. Physical Realization and Model Variants

A variety of models within planar billiards suffice for Turing completeness:

• Piecewise-smooth billiards: Systems with only fixed, smooth ($C^2$) boundaries and scatterers, with no need for non-physical elements like variable-speed or teleportation, can achieve universality via suitable spatial encoding and geometric construction (Miranda et al., 22 Dec 2025).
• Pinball Wizard/Ray Tracing: The "Pinball Wizard" framework allows more engineering freedom, such as moving walls and bumpers (variable-speed via $\lambda(t)$), but universality persists even when restricted to constant-speed rays with only moving mirror segments and one-way gates [(Adejoh et al., 2 Oct 2025), Corollary 4.1].
• Gadget minimality: Worked examples demonstrate universality with compact configurations (e.g., domains $B^*$ with a minimal number of internal scatterers and boundary gadgets), using explicit parabolic reflectors and analytic walls.

These results suggest that undecidable behavior arises generically in broad classes of classical billiard-type models, including those appearing as limiting cases of hard-sphere gases and collision-chain reductions of celestial mechanics (Miranda et al., 22 Dec 2025). The physical ingredients—elastic specular reflection, piecewise-smooth boundaries, and determinism—are standard in mathematical billiards and classical mechanics.

6. Theoretical and Foundational Implications

The demonstration that planar billiards can compute establishes two key facts:

1. Equivalence to classical computation: Planar billiard systems can simulate arbitrary Turing machines, placing the reachability problem in the classical field of recursively inseparable (undecidable) problems.
2. Geometric universality: Universal computation is possible using purely geometric operations—reflections, deterministic scattering, and smooth boundaries—without recourse to quantum effects, randomness, or nonlocality.

Topological Kleene Field Theory provides a formal language connecting low-dimensional dynamical systems and computation: reachability reduces to bordism between input/output manifolds, generalizing Turing machines to continuous flows (Miranda et al., 22 Dec 2025). The simulation of stacks or tapes via real-valued offsets or Cantor-set encodings forms a central methodological insight.

This suggests that undecidability is endemic even in deterministic, frictionless, and physically plausible billiard-type models, with ramifications for both dynamical systems theory and the foundations of computation in physics. Fundamental questions about the periodicity, reachability, or recurrence of orbits in these systems are Turing-hard or harder.

7. Explicit Examples and Classification

Concrete, minimal universal billiard constructions can be described in closed form:

• Bennett’s Universal TM Realization: A two-state, three-symbol reversible Turing machine $M^*$ (Bennett 1973) is embedded as a billiard in a rectangular region with two smooth internal scatterers [(Miranda et al., 22 Dec 2025), §6]. The transition rules are physically encoded in the arrangement and parameterization of parabolic reflectors and $C^1$-graph walls.
• Pinball Wizard Gadget Pipeline: A global pipeline of stack gadgets, each implemented by precise arrangements of parabolic and planar reflectors, moving walls, and one-way gates, realizes full step-by-step simulation of arbitrary single-tape deterministic Turing machines, with complete geometric data provided via rational functions [(Adejoh et al., 2 Oct 2025), §2–4].

These explicit constructions not only demonstrate the abstract possibility of billiard universality, but also provide templates for future engineering or mathematical exploration of physically universal billiard systems.