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Set Descriptive Complexity of Solvable Functions

Riccardo Gozzi, Olivier Bournez

TL;DR

This paper investigates the descriptive complexity of solvable functions arising from dynamical systems governed by discontinuous ODEs. It introduces a transfinite solvable ranking based on the sequence of $f$-removed sets of the derivative, and proves that every countable ordinal occurs as a solvable rank via a Borel encoding from well-founded trees, establishing unboundedness below $ω_1$. The authors situate the solvable ranking relative to classical differentiability-based rankings (Kechris-Woodin, Denjoy, Zalcwasser), showing dominance in relevant regimes and linking to coanalytic structure; the set of solvable functions is shown to be a $\boldsymbol{Π}_1^1$ subset of $C([0,1])$ with the ranking forming a coanalytic norm. The work also contextualizes solvable ODEs as a framework for simulating ordinal computations, underscoring that solving discontinuous IVPs requires the totality of countable ordinals and suggesting directions for extending these ideas to higher dimensions and lightface complexity analyses.

Abstract

In a recent article, we introduced and studied a precise class of dynamical systems called solvable systems. These systems present a dynamic ruled by discontinuous ordinary differential equations with solvable right-hand terms and unique evolution. They correspond to a class of systems for which a transfinite method exist to compute the solution. We also presented several examples including a nontrivial one whose solution yields, at an integer time, a real encoding of the halting set for Turing machines; therefore showcasing that the behavior of solvable systems might describe ordinal Turing computations. In the current article, we study in more depth solvable systems, using tools from descriptive set theory. By establishing a correspondence with the class of well-founded trees, we construct a coanalytic ranking over the set of solvable functions and discuss its relation with other existing rankings for differentiable functions, in particular with the Kechris-Woodin, Denjoy and Zalcwasser ranking. We prove that our ranking is unbounded below the first uncountable ordinal.

Set Descriptive Complexity of Solvable Functions

TL;DR

This paper investigates the descriptive complexity of solvable functions arising from dynamical systems governed by discontinuous ODEs. It introduces a transfinite solvable ranking based on the sequence of -removed sets of the derivative, and proves that every countable ordinal occurs as a solvable rank via a Borel encoding from well-founded trees, establishing unboundedness below . The authors situate the solvable ranking relative to classical differentiability-based rankings (Kechris-Woodin, Denjoy, Zalcwasser), showing dominance in relevant regimes and linking to coanalytic structure; the set of solvable functions is shown to be a subset of with the ranking forming a coanalytic norm. The work also contextualizes solvable ODEs as a framework for simulating ordinal computations, underscoring that solving discontinuous IVPs requires the totality of countable ordinals and suggesting directions for extending these ideas to higher dimensions and lightface complexity analyses.

Abstract

In a recent article, we introduced and studied a precise class of dynamical systems called solvable systems. These systems present a dynamic ruled by discontinuous ordinary differential equations with solvable right-hand terms and unique evolution. They correspond to a class of systems for which a transfinite method exist to compute the solution. We also presented several examples including a nontrivial one whose solution yields, at an integer time, a real encoding of the halting set for Turing machines; therefore showcasing that the behavior of solvable systems might describe ordinal Turing computations. In the current article, we study in more depth solvable systems, using tools from descriptive set theory. By establishing a correspondence with the class of well-founded trees, we construct a coanalytic ranking over the set of solvable functions and discuss its relation with other existing rankings for differentiable functions, in particular with the Kechris-Woodin, Denjoy and Zalcwasser ranking. We prove that our ranking is unbounded below the first uncountable ordinal.
Paper Structure (18 sections, 15 theorems, 14 equations)

This paper contains 18 sections, 15 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Polynomial ODEs and models of computations
  3. Computing with more general ODEs
  4. About solvable ODEs
  5. Relations to other rankings and hiearchies
  6. Contributions
  7. Structure of the paper
  8. Preliminaries
  9. Notation
  10. Hyperarithmetical hierarchy, trees and coanalytic norms
  11. IVPs with ordinary differential equations
  12. Solvable functions and examples
  13. Examples
  14. Set descriptive complexity of solvable functions
  15. Comparison with differentiability rankings
  16. ...and 3 more sections

Key Result

Proposition 1

Every $\boldsymbol{\Pi}_1^1$ set $A$ in a Polish space admits a $\boldsymbol{\Pi}_1^1$ norm $\varphi: A \rightarrow \omega_1$. If $\varphi: A \rightarrow \alpha$ is a $\boldsymbol{\Pi}_1^1$ norm, then for each $\xi<\alpha$ let Then $A_{\xi}$ is Borel, $A_{\xi} \subseteq A_\eta$ if $\xi \leq \eta$, and $A=\bigcup_{\xi<\alpha} A_{\xi}$. In particular, every $\boldsymbol{\Pi}_1^1$ set is the union o

Theorems & Definitions (37)

  • Definition 1: Kleene's $\mathcal{O}$
  • Definition 2: Constructive ordinals
  • Definition 3: Hyperarithmetical hierarchy
  • Definition 4: $\boldsymbol{\Pi}_1^1$ norm
  • Proposition 1
  • Definition 5: Limsup rank
  • Definition 6: Sequence of $f$-removed sets on $E$
  • Definition 7: Solvable function
  • Theorem 1
  • Definition 8: $(\alpha)$Monkeys approach
  • ...and 27 more