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Analog and Symbolic Computation through the Koopman Framework

Francesco Caravelli, Jean-Charles Delvenne

TL;DR

By recasting computation as evolution of observables under the Koopman operator, the paper shows halting and reachability can be read off from the resolvent $(\lambda I-\mathcal{K})^{-1}$ with $|\lambda|>1$; for Cantor-space symbolic dynamics, the test $g_A(\lambda I-\mathcal{K})^{-1}g_B\neq 0$ encodes reachability, while equicontinuous symbolic systems yield decidability. Halting states appear as eigenfunctions with eigenvalue $1$, and their multiplicity equals the number of absorbing basins; cycles impose algebraic relations among eigenvalues. The non-autonomous extension with inputs embeds finite automata and coarse-grained automata into the same operator framework via extended state $Z=X\times U$ and $(\mathcal{K}\phi)(z)=\phi(G(z))$. Overall, the Koopman viewpoint connects computation, spectral theory, and dynamical structure, suggesting signatures of computational hardness in Koopman spectra and extending to analog and polynomial models and potential broader applicability.

Abstract

We develop a Koopman operator framework for studying the {computational properties} of dynamical systems. Specifically, we show that the resolvent of the Koopman operator provides a natural abstraction of halting, yielding a ``Koopman halting problem that is recursively enumerable in general. For symbolic systems, such as those defined on Cantor space, this operator formulation captures the reachability between clopen sets, while for equicontinuous systems we prove that the Koopman halting problem is decidable. Our framework demonstrates that absorbing (halting) states {in finite automata} correspond to Koopman eigenfunctions with eigenvalue one, while cycles in the transition graph impose algebraic constraints on spectral properties. These results provide a unifying perspective on computation in symbolic and analog systems, showing how computational universality is reflected in operator spectra, invariant subspaces, and algebraic structures. Beyond symbolic dynamics, this operator-theoretic lens opens pathways to analyze {computational power of} a broader class of dynamical systems, including polynomial and analog models, and suggests that computational hardness may admit dynamical signatures in terms of Koopman spectral structure.

Analog and Symbolic Computation through the Koopman Framework

TL;DR

By recasting computation as evolution of observables under the Koopman operator, the paper shows halting and reachability can be read off from the resolvent with ; for Cantor-space symbolic dynamics, the test encodes reachability, while equicontinuous symbolic systems yield decidability. Halting states appear as eigenfunctions with eigenvalue , and their multiplicity equals the number of absorbing basins; cycles impose algebraic relations among eigenvalues. The non-autonomous extension with inputs embeds finite automata and coarse-grained automata into the same operator framework via extended state and . Overall, the Koopman viewpoint connects computation, spectral theory, and dynamical structure, suggesting signatures of computational hardness in Koopman spectra and extending to analog and polynomial models and potential broader applicability.

Abstract

We develop a Koopman operator framework for studying the {computational properties} of dynamical systems. Specifically, we show that the resolvent of the Koopman operator provides a natural abstraction of halting, yielding a ``Koopman halting problem that is recursively enumerable in general. For symbolic systems, such as those defined on Cantor space, this operator formulation captures the reachability between clopen sets, while for equicontinuous systems we prove that the Koopman halting problem is decidable. Our framework demonstrates that absorbing (halting) states {in finite automata} correspond to Koopman eigenfunctions with eigenvalue one, while cycles in the transition graph impose algebraic constraints on spectral properties. These results provide a unifying perspective on computation in symbolic and analog systems, showing how computational universality is reflected in operator spectra, invariant subspaces, and algebraic structures. Beyond symbolic dynamics, this operator-theoretic lens opens pathways to analyze {computational power of} a broader class of dynamical systems, including polynomial and analog models, and suggests that computational hardness may admit dynamical signatures in terms of Koopman spectral structure.

Paper Structure

This paper contains 14 sections, 6 theorems, 59 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Computation from the Koopman point of view
  3. Koopman operator for dynamical systems
  4. Computers as dynamical systems
  5. The Cantor space
  6. Koopman operator and the reachability problem
  7. Equicontinuous Symbolic Dynamical Systems
  8. Koopman Operators in the Non-Autonomous Case
  9. External inputs
  10. Koopman formalization of coarse-grained automata and halting
  11. Koopman and Set-Theoretic Automata
  12. Eigenvalues of the Koopman Operator in the Presence of Halting States
  13. Dynamical Distance, Topology, and Graph-Theoretic Representation
  14. Conclusion

Key Result

Theorem 1

Let $\mathbf{F}: X \to X$ be a dynamical system on a compact metric space $X$, with associated Koopman operator $\mathcal{K}$ acting on complex-valued observables. Suppose there exists a countable family $(g_k)_{k \in \mathbb{N}} \subset \mathcal{C}(X)$ such that: in the sense of Pour-El and Richards. Then the Koopman Halting Problem is recursively enumerable.

Figures (2)

  • Figure 1: Set-to-set reachability (left) with intermediate sets $M_1, M_2$ is reformulated in the Koopman framework (right) as the resolvent condition $g_A(\lambda I - \mathcal{K})^{-1} g_B \neq 0$, connecting halting/reachability to operator theory.
  • Figure 2: Mapping between classical automata (left) and Koopman operator framework (right). States correspond to partitions, transitions to Koopman action, and halting states to absorbing eigenfunctions.

Theorems & Definitions (17)

  • Definition 1: Computable operator
  • Definition 2: Koopman Halting Problem
  • Theorem 1
  • proof
  • Definition 3: Equicontinuity
  • Theorem 2
  • proof
  • Definition 4: Halting state
  • Remark 1
  • Theorem 3
  • ...and 7 more