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The Complexity of Computing in Continuous Time: Space Complexity Is Precision

Authors Manon Blanc , Olivier Bournez

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Manon Blanc
  • Institut Polytechnique de Paris, Ecole Polytechnique, LIX, 91128 Palaiseau Cedex, France
  • Université Paris-Saclay, LISN, 91190 Gif-sur-Yvette, France
Olivier Bournez
  • Institut Polytechnique de Paris, Ecole Polytechnique, LIX, 91128 Palaiseau Cedex, France

Funding

This work has been partially supported by ANR Project ∂IFFERENCE and Labex Digicosme.

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Manon Blanc and Olivier Bournez. The Complexity of Computing in Continuous Time: Space Complexity Is Precision. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 129:1-129:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.129

Abstract

Models of computations over the integers are equivalent from a computability and complexity theory point of view by the (effective) Church-Turing thesis. It is not possible to unify discrete-time models over the reals. The situation is unclear but simpler for continuous-time models, as there is a unifying mathematical model, provided by ordinary differential equations (ODEs). Each model corresponds to a particular class of ODEs. For example, the General Purpose Analog Computer model of Claude Shannon, introduced as a mathematical model of analogue machines (Differential Analyzers), is known to correspond to polynomial ODEs. However, the question of a robust complexity theory for such models and its relations to classical (discrete) computation theory is an old problem. There was some recent significant progress: it has been proved that (classical) time complexity corresponds to the length of the involved curves, i.e. to the length of the solutions of the corresponding polynomial ODEs. The question of whether there is a simple and robust way to measure space complexity remains. We argue that space complexity corresponds to precision and conversely. Concretely, we propose and prove an algebraic characterisation of FPSPACE, using continuous ODEs. Recent papers proposed algebraic characterisations of polynomial-time and polynomial-space complexity classes over the reals, but with a discrete-time: those algebras rely on discrete ODE schemes. Here, we use classical (continuous) ODEs, with the classic definition of derivation and hence with the more natural context of continuous-time associated with ODEs. We characterise both the case of polynomial space functions over the integers and the reals. This is done by proving two inclusions. The first is obtained using some original polynomial space method for solving ODEs. For the other, we prove that Turing machines, with a proper representation of real numbers, can be simulated by continuous ODEs and not just discrete ODEs. A major consequence is that the associated space complexity is provably related to the numerical stability of involved schemas and the associated required precision. We obtain that a problem can be solved in polynomial space if and only if it can be simulated by some numerically stable ODE, using a polynomial precision.

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
  • Models of computation
  • Ordinary differential equations
  • Real computations
  • Analog computations
  • Complexity theory
  • Implicit complexity
  • Recursion scheme

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