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Simulation of Turing machines with analytic discrete ODEs: FPTIME and FPSPACE over the reals characterised with discrete ordinary differential equations

Manon Blanc, Olivier Bournez

TL;DR

This work demonstrates that functions over the reals computable in polynomial time or polynomial space can be captured by discrete ODEs, via the $\mathbb{LDL}^\circ$ formalism and its real extension. The authors simulate Turing machines with analytic discrete ODEs, leveraging Cantor-like tape encodings and robust numerical schemes, and introduce an effective limit operator ELim to convert approximate ODE solutions into exact computable functions. They show that both $\mathbf{FPTIME}$ and $\mathbf{FPSPACE}$ characterizations extend to real-valued inputs, removing the need for non-analytic primitives and enabling a programmable ODE-based paradigm with explicit complexity bounds. The results bridge analog and discrete computation on the real line and open up avenues for ODE-based programming with rigorous complexity guarantees.

Abstract

We prove that functions over the reals computable in polynomial time can be characterised using discrete ordinary differential equations (ODE), also known as finite differences. We also provide a characterisation of functions computable in polynomial space over the reals. In particular, this covers space complexity, while existing characterisations were only able to cover time complexity, and were restricted to functions over the integers. We prove furthermore that no artificial sign or test function is needed even for time complexity. At a technical level, this is obtained by proving that Turing machines can be simulated with analytic discrete ordinary differential equations. We believe this result opens the way to many applications, as it opens the possibility of programming with ODEs, with an underlying well-understood time and space complexity.

Simulation of Turing machines with analytic discrete ODEs: FPTIME and FPSPACE over the reals characterised with discrete ordinary differential equations

TL;DR

This work demonstrates that functions over the reals computable in polynomial time or polynomial space can be captured by discrete ODEs, via the formalism and its real extension. The authors simulate Turing machines with analytic discrete ODEs, leveraging Cantor-like tape encodings and robust numerical schemes, and introduce an effective limit operator ELim to convert approximate ODE solutions into exact computable functions. They show that both and characterizations extend to real-valued inputs, removing the need for non-analytic primitives and enabling a programmable ODE-based paradigm with explicit complexity bounds. The results bridge analog and discrete computation on the real line and open up avenues for ODE-based programming with rigorous complexity guarantees.

Abstract

We prove that functions over the reals computable in polynomial time can be characterised using discrete ordinary differential equations (ODE), also known as finite differences. We also provide a characterisation of functions computable in polynomial space over the reals. In particular, this covers space complexity, while existing characterisations were only able to cover time complexity, and were restricted to functions over the integers. We prove furthermore that no artificial sign or test function is needed even for time complexity. At a technical level, this is obtained by proving that Turing machines can be simulated with analytic discrete ordinary differential equations. We believe this result opens the way to many applications, as it opens the possibility of programming with ODEs, with an underlying well-understood time and space complexity.
Paper Structure (18 sections, 42 theorems, 77 equations, 8 figures)

This paper contains 18 sections, 42 theorems, 77 equations, 8 figures.

Table of Contents

  1. Introduction
  2. More on related works
  3. Organisation of the article
  4. About appendices
  5. Note about current version vs final version
  6. Some concepts related to discrete ODEs
  7. Some results about various functions
  8. Some properties of sigmoids
  9. Simulating Turing machines with functions of $\mathbb{LDL}^\circ$
  10. Converting integers and dyadics to words, and conversely
  11. Proofs and applications
  12. Concepts from computable analysis
  13. On computable analysis: Complexity
  14. Some results from $MFCS2019MFCSJournalBlancBournezMCU22vantardise$
  15. Some general statements from MFCS2019MFCSJournal
  16. ...and 3 more sections

Key Result

Theorem 1.1

For functions over the reals, we have $\mathbb{LDL}= \mathbf{FPTIME}$ where

Figures (8)

  • Figure 1: Graphical representation of $xY(x,2^{2+2})$ obtained with maple.
  • Figure 2: Graphical representation of $\operatorname{\mathfrak{\mathcal{C}-s}}(2,\frac{1}{2},\frac{3}{4},x)$ and $\operatorname{\mathfrak{\mathcal{C}-s}}(2^{5},\frac{1}{2},\frac{3}{4},x)$ obtained with maple.
  • Figure 3: Graphical representations of $\xi(2,4,x)$ obtained with maple: some details on the right.
  • Figure 4: Graphical representation of $\xi_{1}(2,4,x)$ and $\xi_{2}(2,4,x)$ obtained with maple.
  • Figure 5: Graphical representation of $\sigma_{1}(2,4,x)$ and $\sigma_{2}(2,4,x)$ obtained with maple.
  • ...and 3 more figures

Theorems & Definitions (103)

  • Remark 1
  • Remark 2
  • Theorem 1.1: MFCS2019
  • Corollary 1.2: Functions over the integers
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 1.3
  • Theorem 1.4: Generic functions over the reals
  • Theorem 1.5
  • ...and 93 more