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A Characterisation of Functions Computable in Polynomial Time and Space over the Reals with Discrete Ordinary Differential Equations: Simulation of Turing Machines with Analytic Discrete ODEs

Authors Manon Blanc, Olivier Bournez

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Manon Blanc
  • Institut Polytechnique de Paris, Ecole Polytechnique, LIX, Palaiseau, France
Olivier Bournez
  • Institut Polytechnique de Paris, Ecole Polytechnique, LIX, Palaiseau, France

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Manon Blanc and Olivier Bournez. A Characterisation of Functions Computable in Polynomial Time and Space over the Reals with Discrete Ordinary Differential Equations: Simulation of Turing Machines with Analytic Discrete ODEs. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.MFCS.2023.21

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, and we prove 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Computability
  • Theory of computation → Complexity classes
  • Mathematics of computing → Ordinary differential equations
Keywords
  • Discrete ordinary differential equations
  • Finite Differences
  • Implicit complexity
  • Recursion scheme
  • Ordinary differential equations
  • Models of computation
  • Analog Computations

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References

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