Recursion Schemes, Discrete Differential Equations and Characterization of Polynomial Time Computations
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.
Implicit complexity
discrete ordinary differential equations
recursion scheme
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
23:1-23:14
Regular Paper
We would like to thank Sabrina Ouazzani for many scientific discussions about the results in this article.
Olivier
Bournez
Olivier Bournez
Laboratoire d'Informatique de l'X (LIX), CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France
https://orcid.org/0000-0002-9218-1130
Supported by RACAF Project from Agence National de la Recherche and Labex Digicosme Project ACDC.
Arnaud
Durand
Arnaud Durand
Université Paris Diderot, IMJ-PRG, CNRS UMR 7586, Case 7012, 75205 Paris cedex 13, France
10.4230/LIPIcs.MFCS.2019.23
V. I. Arnold. Ordinary Differential Equations. MIT Press, 1978.
S. Bellantoni and S. Cook. A new recursion-theoretic characterization of the poly-time functions. Computational Complexity, 2:97-110, 1992.
G. Birkhoff and G.-C. Rota. Ordinary Differential Equations. John Wiley & Sons, 4th edition, 1989.
O. Bournez, D. S. Graça, and A. Pouly. Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length. Journal of the ACM, 64(6):38:1-38:76, 2017. URL: https://doi.org/10.1145/3127496.
https://doi.org/10.1145/3127496
Olivier Bournez and Amaury Pouly. A Survey on Analog Models of Computation. In Vasco Brattka and Peter Hertling, editors, Handbook of Computability and Complexity in Analysis. Springer. To appear, 2018.
Manuel L. Campagnolo. Computational Complexity of Real Valued Recursive Functions and Analog Circuits. PhD thesis, Universidade Técnica de Lisboa, 2001.
Manuel L. Campagnolo, Cristopher Moore, and José Félix Costa. An analog characterization of the Grzegorczyk hierarchy. Journal of Complexity, 18(4):977-1000, 2002.
A. Cobham. The intrinsic computational difficulty of functions. In Y. Bar-Hillel, editor, Proceedings of the International Conference on Logic, Methodology, and Philosophy of Science, pages 24-30. North-Holland, Amsterdam, 1962.
E. A. Coddington and N. Levinson. Theory of Ordinary Differential Equations. Mc-Graw-Hill, 1955.
Pieter Collins and Daniel S Graça. Effective computability of solutions of ordinary differential equations the thousand monkeys approach. Electronic Notes in Theoretical Computer Science, 221:103-114, 2008.
Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, and Clifford Stein. Introduction to algorithms (third edition). MIT press, 2009.
AO Gelfand. Calcul des différences finies. Dunod, 1963.
David Gleich. Finite calculus: A tutorial for solving nasty sums. Stanford University, 2005.
Ronald L Graham, Donald E Knuth, Oren Patashnik, and Stanley Liu. Concrete mathematics: a foundation for computer science. Computers in Physics, 3(5):106-107, 1989.
FA Izadi, N Aliev, and G Bagirov. Discrete Calculus by Analogy. Bentham Science Publishers, 2009.
Charles Jordan and Károly Jordán. Calculus of finite differences, volume 33. American Mathematical Soc., 1965.
L. Kalmár. Egyzzerü példa eldönthetetlen aritmetikai problémára. Mate és Fizikai Lapok, 50:1-23, 1943.
A. Kawamura. Lipschitz continuous ordinary differential equations are polynomial-space complete. In 2009 24th Annual IEEE Conference on Computational Complexity, pages 149-160. IEEE, 2009.
Ker-I Ko. On the Computational Complexity of Ordinary Differential Equations. Information and Control, 58(1-3):157-194, 1983.
Gustavo Lau. Discrete calculus. URL: http://www.acm.ciens.ucv.ve/main/entrenamiento/material/DiscreteCalculus.pdf.
http://www.acm.ciens.ucv.ve/main/entrenamiento/material/DiscreteCalculus.pdf
D. Leivant. Predicative recurrence and computational complexity I: Word recurrence and poly-time. In Peter Clote and Jeffery Remmel, editors, Feasible Mathematics II, pages 320-343. Birkhäuser, 1994.
D. Leivant and J-Y Marion. Lambda Calculus Characterizations of Poly-Time. Fundamenta Informatica, 19(1,2):167,184, September 1993.
Daniel Leivant and Jean-Yves Marion. Ramified recurrence and computational complexity II: substitution and poly-space. In L. Pacholski and J. Tiuryn, editors, Computer Science Logic, 8th Workshop, CSL'94, volume 933 of Lecture Notes in Computer Science, pages 369-380, Kazimierz, Poland, 1995. Springer.
Bruno Loff, José Félix Costa, and Jerzy Mycka. The New Promise of Analog Computation. In Computability in Europe 2007: Computation and Logic in the Real World., 2007.
Cristopher Moore. Recursion theory on the reals and continuous-time computation. Theoretical Computer Science, 162(1):23-44, August 1996.
Jerzy Mycka and José Félix Costa. What lies beyond the mountains? Computational systems beyond the Turing limit. European Association for Theoretical Computer Science Bulletin, 85:181-189, February 2005.
Jerzy Mycka and José Félix Costa. The P ̸ = NP conjecture in the context of real and complex analysis. Journal of Complexity, 22(2):287-303, 2006.
P. Odifreddi. Classical Recursion Theory, volume 125 of Studies in Logic and the foundations of mathematics. North-Holland, April 1992.
Amaury Pouly. Continuous models of computation: from computability to complexity. PhD thesis, Ecole Polytechnique and Unidersidade Do Algarve, 2015. https://pastel.archives-ouvertes.fr/tel-01223284, Ackermann Award 2017.
H. E. Rose. Subrecursion, Functions and Hierarchies. Clarendon Press, Oxford, 1984.
Olivier Bournez and Arnaud Durand
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode