A Characterization of Functions over the Integers Computable in Polynomial Time Using Discrete Ordinary Differential Equations

This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations. It presents a new framework using these equations as a central tool for computation and algorithm design. We present the general theory of discrete OD...

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Vydáno v:Computational complexity Ročník 32; číslo 2; s. 7
Hlavní autoři: Bournez, Olivier, Durand, Arnaud
Médium: Journal Article
Jazyk:angličtina
Vydáno: Cham Springer International Publishing 01.12.2023
Springer Nature B.V
Springer Verlag
Témata:
Algorithm Analysis and Problem Complexity
Algorithms
Complexity
Computation
Computational Mathematics and Numerical Analysis
Computer Science
Difference equations
Differential equations
Ordinary differential equations
Polynomials
68Q15
03D20
68Q01
65Q10
discrete ordinary differential equations
recursion scheme
03D15
Implicit complexity
difference equations
65L99
ISSN:1016-3328, 1420-8954
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Shrnutí:This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations. It presents a new framework using these equations 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) ODEs and classical ODE tools such as changes of variables to capture computability and complexity measures, or as a tool for programming many algorithms.
Bibliografie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:1016-3328
1420-8954
DOI:10.1007/s00037-023-00240-1