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Recursion Schemes, Discrete Differential Equations and Characterization of Polynomial Time Computations

Authors Olivier Bournez , Arnaud Durand

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  • 14 pages

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Author Details

Olivier Bournez
  • Laboratoire d'Informatique de l'X (LIX), CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France
Arnaud Durand
  • Université Paris Diderot, IMJ-PRG, CNRS UMR 7586, Case 7012, 75205 Paris cedex 13, France


We would like to thank Sabrina Ouazzani for many scientific discussions about the results in this article.

Cite AsGet BibTex

Olivier Bournez and Arnaud Durand. Recursion Schemes, Discrete Differential Equations and Characterization of Polynomial Time Computations. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs). It presents a new framework using discrete ODEs as a central tool for computation and algorithm design. We present the general theory of discrete ODEs for computation theory, we illustrate this with various examples of algorithms, and we provide several implicit characterizations of complexity and computability classes. The proposed framework presents an original point of view on complexity and computation classes. It unifies several constructions that have been proposed for characterizing these classes including classical approaches in implicit complexity using restricted recursion schemes, as well as recent characterizations of computability and complexity by classes of continuous ordinary differential equations. It also helps understanding the relationships between analog computations and classical discrete models of computation theory. At a more technical point of view, this paper points out the fundamental role of linear (discrete) ordinary differential equations and classical ODE tools such as changes of variables to capture computability and complexity measures, or as a tool for programming many algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Theory of computation → Models of computation
  • Theory of computation → Computability
  • Theory of computation → Recursive functions
  • Computer systems organization → Analog computers
  • Theory of computation → Complexity classes
  • Theory of computation → Complexity theory and logic
  • Mathematics of computing → Differential equations
  • Mathematics of computing → Ordinary differential equations
  • Mathematics of computing → Differential calculus
  • Implicit complexity
  • discrete ordinary differential equations
  • recursion scheme


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