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Solvable Initial Value Problems Ruled by Discontinuous Ordinary Differential Equations

Olivier Bournez, Riccardo Gozzi

TL;DR

This work investigates IVPs governed by discontinuous ODEs that nonetheless admit a unique solution, introducing the notion of solvable IVPs. It proves that for solvable right-hand sides, the solution can be obtained analytically through transfinite recursion up to a countable ordinal, providing a constructive framework and a transfinite search method (α-Monkeys) to realize the solution. The authors demonstrate the capacity of solvable IVPs to encode noncomputable information, exemplified by a construction in which the solution at a finite time encodes the Turing halting set, and discuss the implications for transfinite computation and hypercomputation. This approach leverages Denjoy-style totalization ideas extended to ODEs and situates these systems in a computability-theoretic context, suggesting a structured hierarchy of solvable dynamics based on ordinal ranks. Overall, the paper broadens the landscape of analog computation by showing how discontinuous ODEs with unique solutions can simulate transfinite processes and potentially more powerful computational models."

Abstract

We study initial value problems having dynamics ruled by discontinuous ordinary differential equations with the property of possessing a unique solution. We identify a precise class of such systems that we call solvable intitial value problems and we prove that for this class of problems the unique solution can always be obtained analytically via transfinite recursion. We present 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 is related to ordinal Turing computations.

Solvable Initial Value Problems Ruled by Discontinuous Ordinary Differential Equations

TL;DR

This work investigates IVPs governed by discontinuous ODEs that nonetheless admit a unique solution, introducing the notion of solvable IVPs. It proves that for solvable right-hand sides, the solution can be obtained analytically through transfinite recursion up to a countable ordinal, providing a constructive framework and a transfinite search method (α-Monkeys) to realize the solution. The authors demonstrate the capacity of solvable IVPs to encode noncomputable information, exemplified by a construction in which the solution at a finite time encodes the Turing halting set, and discuss the implications for transfinite computation and hypercomputation. This approach leverages Denjoy-style totalization ideas extended to ODEs and situates these systems in a computability-theoretic context, suggesting a structured hierarchy of solvable dynamics based on ordinal ranks. Overall, the paper broadens the landscape of analog computation by showing how discontinuous ODEs with unique solutions can simulate transfinite processes and potentially more powerful computational models."

Abstract

We study initial value problems having dynamics ruled by discontinuous ordinary differential equations with the property of possessing a unique solution. We identify a precise class of such systems that we call solvable intitial value problems and we prove that for this class of problems the unique solution can always be obtained analytically via transfinite recursion. We present 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 is related to ordinal Turing computations.
Paper Structure (20 sections, 19 theorems, 28 equations)

This paper contains 20 sections, 19 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Remark
  3. Structure of the paper
  4. Historical accounts and related work
  5. Denjoy's totalization method for integration.
  6. ODEs as analog model of computation.
  7. What is known about polynomial ODEs: lower bounds.
  8. What is known about ODEs: upper bounds.
  9. Preliminaries
  10. Notation
  11. Basic concepts in analysis and useful theorems
  12. Ordinary Differential Equations and Initial Value Problems
  13. Functions of class Baire one
  14. Solvable functions
  15. Search method
  16. ...and 5 more sections

Key Result

Theorem 1

Let $[a,b]$ an interval in $\mathbb{R}$, $r \in \mathbb{N}$ and let $f: [a,b] \to \mathbb{R}^r$ be an absolutely continuous function. Then $f$ is differentiable almost everywhere and $f(x)= \int_{a}^{x} f'(t) dt$ for all $x \in [a,b]$.

Theorems & Definitions (44)

  • Definition 1: Globally Lipschitz function
  • Definition 2: Absolutely continuous function
  • Theorem 1
  • Definition 3: Uniformly bounded
  • Definition 4: Equicontinuous
  • Theorem 2: Ascoli
  • Theorem 3: Uniform limit theorem
  • Theorem 4: Differentiable limit theorem
  • Theorem 5
  • Theorem 6: Cantor-Baire stationary principle
  • ...and 34 more