Universality in computable dynamical systems: Old and new (2507.10725v1)

Abstract: The relationship between computational models and dynamics has captivated mathematicians and computer scientists since the earliest conceptualizations of computation. Recently, this connection has gained renewed attention, fueled by T. Tao's programme aiming to discover blowing-up solutions of the Navier-Stokes equations using an embedded computational model. In this survey paper, we review some of the recent works that introduce novel and exciting perspectives on the representation of computability through dynamical systems. Starting from dynamical universality in a classical sense, we shall explore the modern notions of Turing universality in fluid dynamics and Topological Kleene Field Theories as a systematic way of representing computable functions by means of dynamical bordisms. Finally, we will discuss some important open problems in the area.

• The paper presents a comprehensive survey of universality in computable dynamical systems, embedding arbitrary dynamics into classical and modern frameworks.
• It demonstrates explicit constructions, such as embedding Turing machines into Euler flows and leveraging contact geometry for reproducing computation.
• It introduces Topological Kleene Field Theories as a categorical equivalence between dynamical bordisms and partial recursive functions, bridging topology and computation.

Universality in Computable Dynamical Systems: A Survey of Classical and Modern Perspectives

This paper provides a comprehensive survey of the interplay between computability and dynamical systems, with a particular focus on universality phenomena. The authors systematically review classical results on the embedding of arbitrary dynamics into restricted classes of systems, recent advances in the representation of computation within fluid dynamics, and the emergence of Topological Kleene Field Theories (TKFT) as a categorical framework for computability via dynamical bordisms.

Classical Universality in Dynamical Systems

The survey begins by formalizing the notion of universality: a class $\mathcal{U}$ of dynamical systems is universal for a class $\mathcal{C}$ if every system in $\mathcal{C}$ can be embedded into some system in $\mathcal{U}$ via a smooth embedding that preserves the dynamics. This concept is illustrated by the universality of Hamiltonian systems: any smooth vector field on a manifold $M$ can be embedded as the Hamiltonian flow on the cotangent bundle $T^*M$ with a suitably chosen Hamiltonian. The construction is explicit and leverages the canonical symplectic structure of $T^*M$.

The authors further discuss universality within the subclass of potential-type Hamiltonian systems, where the Hamiltonian is of the form $H_V(q,p) = \frac{1}{2}|p|^2 + V(q)$. Under mild conditions (existence of a nowhere-vanishing vector field and an exactness property), any smooth dynamical system can be embedded into such a system. The extension to infinite-dimensional settings, such as $\infty$-potential dynamics on function spaces, is also addressed, highlighting the flexibility of the universality framework.

Universality in Fluid Dynamics and Euler Flows

A central theme of the paper is the embedding of computational processes into the dynamics of fluids, specifically through the Euler equations on Riemannian manifolds. Building on Tao's constructive embedding theorem, the authors review how any quadratic ODE system with a suitable energy conservation property can be realized as an invariant subsystem of the Euler equations on a high-dimensional manifold. This result demonstrates that Euler flows are at least as complex as any such quadratic system, and, by extension, can exhibit chaotic behavior and encode arbitrary computational processes.

The survey details the geometric and topological structures underlying these embeddings, particularly the role of contact geometry and Beltrami fields. The contact mirror theorem establishes a correspondence between Reeb vector fields of contact forms and stationary solutions to the Euler equations, providing a geometric toolkit for constructing physically meaningful, divergence-free vector fields with prescribed dynamical properties.

Computability and Turing Universality in Dynamics

The authors transition to the representation of discrete computation within continuous dynamical systems. They formalize the embedding of discrete systems (such as Turing machines) into continuous flows via the Poincaré first return map on suitable sections. The construction proceeds by encoding Turing machine configurations as points in a Cantor set, which is then embedded into a disc. Generalized shift maps, which capture the action of Turing machines, are realized as block diffeomorphisms of the disc.

A key result is that Euler flows on orientable 3-manifolds are Turing-universal: for any block diffeomorphism (and thus any Turing machine), there exists a stationary Euler flow whose Poincaré return map on a section reproduces the computation. The construction is explicit, involving the mapping torus of the diffeomorphism, extension to a contact structure, and application of the contact mirror to obtain a stationary Euler flow.

The survey also reviews alternative approaches to embedding computation in continuous systems, including ODE-based characterizations of computable functions, physical devices (e.g., marble runs), and chemical or biological systems capable of universal computation.

Topological Kleene Field Theories and Dynamical Bordisms

A significant portion of the paper is devoted to the development of Topological Kleene Field Theories (TKFT), a categorical framework for representing computable (partial recursive) functions as reaching maps of vector fields on manifolds with boundary (dynamical bordisms). The construction is as follows:

• Given a partial recursive function $f: \mathbb{N} \dashrightarrow \mathbb{N}$, one constructs a dynamical bordism $(M, X)$ between discs such that the flow of $X$ from the incoming boundary to the outgoing boundary realizes $f$ as its reaching function.
• The construction thickens the state graph of a Turing machine computing $f$ into a 3-manifold, with tubes corresponding to transitions and discs to states. The vector field is assembled from local models corresponding to the computational steps.
• The resulting functor from the category of clean dynamical bordisms to the category of partial recursive functions is full, establishing a categorical equivalence between topological and computational complexity.

The authors introduce topological and metric notions of complexity for computable functions, such as the minimal first Betti number of a representing bordism (topological complexity) and the length of flow lines (length complexity). They conjecture that these geometric invariants are asymptotically equivalent to classical computational complexity measures, suggesting a new geometric approach to complexity theory.

Open Problems and Future Directions

The paper concludes with a set of open problems that delineate the frontier of research in this area:

• Topological invariants and computational complexity: Can topological features of bordisms (e.g., Betti numbers) be used to classify the complexity of computable functions?
• Characterization of Turing universality: Is there a fundamental dynamical operation, analogous to the horseshoe in chaos theory, that characterizes Turing universality in continuous systems?
• Physical realizability: To what extent can Turing-universal dynamical systems be implemented in physical systems, such as fluids or electromagnetic fields?
• Comparison with quantum and super-Turing computation: Can continuous dynamical systems, possibly via TKFT constructions, surpass the computational power of quantum computers or realize super-Turing models?
• Universality in celestial mechanics: Is the $n$-body or $n$-centre problem Turing-universal for some $n$ and parameter choices?

Implications and Outlook

The survey demonstrates that the boundary between computation and dynamics is highly permeable: not only can arbitrary computation be embedded in physical systems, but the complexity of computation can be recast in geometric and topological terms. The explicit constructions reviewed in the paper provide a blueprint for implementing computation in a variety of dynamical settings, from fluid flows to contact and symplectic manifolds.

From a practical perspective, these results suggest new avenues for analog computation, the design of physical devices that exploit dynamical universality, and the development of geometric complexity theory. The categorical perspective of TKFTs may also inform the design of new programming paradigms and the analysis of computational processes in natural systems.

Theoretically, the work raises fundamental questions about the limits of computation in continuous systems, the role of topology in complexity, and the potential for new models of computation that transcend the classical Turing framework. The connections to quantum computation and the possibility of physically realizable, efficient universal systems remain open and compelling directions for future research.