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Universal Analog Computation: Fraïssé limits of dynamical systems

Levin Hornischer

TL;DR

The paper investigates universality in analog computation by modeling analog devices as dynamical systems and applying category-theoretic Fraïssé limits within algebroidal categories.It proves the existence and uniqueness (up to isomorphism) of a universal, homogeneous nondeterministic system, and explains why no universal deterministic system exists in general.To recover universality for deterministic models, it builds ω-proshifts by closing countable shift families under ω-chains, yielding universal ω-proshifts for shifts of finite type and for sofic shifts, which also enjoy the shadowing property.The work connects to coalgebra and domain theory, and highlights deep links between universality, embedding/factor relations, and dynamical-system classifications, with implications for universal analog computing and automata-theoretic structures.

Abstract

Analog computation is an alternative to digital computation, that has recently re-gained prominence, since it includes neural networks. Further important examples are cellular automata and differential analyzers. While analog computers offer many advantages, they lack a notion of universality akin to universal digital computers. Since analog computers are best formalized as dynamical systems, we review scattered results on universal dynamical systems, identifying four senses of universality and connecting to coalgebra and domain theory. For nondeterministic systems, we construct a universal system as a Fraïssé limit. It not only is universal in many of the identified senses, it also is unique in additionally being homogeneous. For deterministic systems, a universal system cannot exist, but we provide a simple method for constructing subclasses of deterministic systems with a universal and homogeneous system. This way, we introduce sofic proshifts: those systems that are limits of sofic shifts. In fact, their universal and homogeneous system even is a limit of shifts of finite type and has the shadowing property.

Universal Analog Computation: Fraïssé limits of dynamical systems

TL;DR

The paper investigates universality in analog computation by modeling analog devices as dynamical systems and applying category-theoretic Fraïssé limits within algebroidal categories.It proves the existence and uniqueness (up to isomorphism) of a universal, homogeneous nondeterministic system, and explains why no universal deterministic system exists in general.To recover universality for deterministic models, it builds ω-proshifts by closing countable shift families under ω-chains, yielding universal ω-proshifts for shifts of finite type and for sofic shifts, which also enjoy the shadowing property.The work connects to coalgebra and domain theory, and highlights deep links between universality, embedding/factor relations, and dynamical-system classifications, with implications for universal analog computing and automata-theoretic structures.

Abstract

Analog computation is an alternative to digital computation, that has recently re-gained prominence, since it includes neural networks. Further important examples are cellular automata and differential analyzers. While analog computers offer many advantages, they lack a notion of universality akin to universal digital computers. Since analog computers are best formalized as dynamical systems, we review scattered results on universal dynamical systems, identifying four senses of universality and connecting to coalgebra and domain theory. For nondeterministic systems, we construct a universal system as a Fraïssé limit. It not only is universal in many of the identified senses, it also is unique in additionally being homogeneous. For deterministic systems, a universal system cannot exist, but we provide a simple method for constructing subclasses of deterministic systems with a universal and homogeneous system. This way, we introduce sofic proshifts: those systems that are limits of sofic shifts. In fact, their universal and homogeneous system even is a limit of shifts of finite type and has the shadowing property.

Paper Structure

This paper contains 26 sections, 45 theorems, 29 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Four senses of universal analog computation
  3. Dynamical systems
  4. Categories of dynamical systems
  5. Connections to coalgebra and domain theory
  6. Algebroidal categories of systems
  7. Proof of the algebroidality theorem
  8. The universal and homogeneous nondeterministic system
  9. The universality result
  10. The nature of the universal system
  11. Universality for deterministic systems?
  12. Deterministic vs nondeterministic systems
  13. No universal deterministic system
  14. The category of deterministic systems still is semi-algebroidal
  15. Algebroidal categories of deterministic systems
  16. ...and 11 more sections

Key Result

Theorem 1

There is a nondeterministic system $(U,T)$ that is Moreover, $(U, T)$ is unique up to isomorphism with these two properties.

Figures (4)

  • Figure 1: Nondeterministic equivariance: ways of requiring that the $T$-dynamics is matched, via $\varphi$, to the $S$-dynamics, and vice versa. Solid arrows are assumed relations, dashed arrows are relations required to exist.
  • Figure 2: Definitions related to the category-theoretic Fraïssé theorem. Dashed arrows indicate the required existence of a morphism (however, unique existence is not required).
  • Figure 3: Diagram to prove that $(X,T)$ and $(Y,S)$ are isomorphic.
  • Figure 4: Finding a diagonal $\omega$-chain.

Theorems & Definitions (92)

  • Theorem : Corollary \ref{['cor: universal nondeterministic system']} below
  • Theorem : Corollary \ref{['cor: universal and homogeneous proshifts']} below
  • Definition 3.1
  • Definition 4.1
  • Proposition 4.2
  • Definition 4.3
  • Proposition 4.4
  • Definition 4.5
  • Remark 4.6: Limit--colimit coincidence
  • Theorem 5.1
  • ...and 82 more