Analog and Symbolic Computation through the Koopman Framework

2025-10-07

AI Summary

This paper introduces the “Koopman Halting Problem,” recasting the classical halting problem as a spectral decision problem for the Koopman operator. Methodologically, it integrates Koopman operator theory, spectral analysis, and tools from topological dynamical systems to systematically investigate decidability of this problem on Cantor space and equicontinuous systems. The contributions are threefold: (1) It establishes, for the first time, rigorous correspondences between absorbing states of finite automata and Koopman eigenfunctions, and between cycles in state-transition graphs and algebraic constraints on the operator’s spectrum; (2) It proves decidability of the Koopman Halting Problem for equicontinuous systems, thereby revealing how computational hardness is dynamically encoded in spectral structure; (3) It constructs a unifying framework wherein universality—across symbolic computation (e.g., automata) and analog computation (e.g., dynamical systems)—is characterized jointly by spectral properties, invariant subspaces, and algebraic invariants of the Koopman operator, offering a novel operator-theoretic foundation for computation theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Kernel Methods

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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.

Problem

• Abstracting halting through Koopman operator resolvent
• Capturing reachability in symbolic and equicontinuous systems
• Unifying computation via operator spectra and algebraic structures

Innovation

• Koopman operator framework for dynamical systems computation
• Resolvent abstracts halting as recursively enumerable problem
• Operator spectra unify symbolic and analog computational universality