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 |
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| Abstract | 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. |
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| AbstractList | 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. 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. |
| ArticleNumber | 7 |
| Author | Bournez, Olivier Durand, Arnaud |
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| References | Garrett Birkhoff & Gian-Carlo Rota (1989). Ordinary Differential Equations. John Wiley & Sons, 4th edition. David Gleich (2005). Finite calculus: A tutorial for solving nasty sums. Stanford University . Manuel L. Campagnolo (2001). Computational Complexity of Real Valued Recursive Functions and Analog Circuits. Ph.D. thesis, Universidade T´ecnica de Lisboa. Olivier Bournez, Daniel S. Graςa & Amaury Pouly (2017). Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length. Journal of the ACM 64(6), 38:1–38:76. Earl A. Coddington & Norman Levinson (1955).Theory of Ordinary Differential Equations . Mc-Graw-Hill. Manuel L. Campagnolo, Cristopher Moore & José Félix Costa (2002b).An analog characterization of the Grzegorczyk hierarchy . Journal of Complexity 18(4), 977–1000. Ker-I Ko (1983). On the Computational Complexity of Ordinary Differential Equations. Information and Control 58(1–3), 157–194. Daniel Leivant (1994). Predicative recurrence and computational complexity I: Word recurrence and poly-time. In Feasible Mathematics II, Peter Clote & Jeffery Remmel, editors, 320–343. Birkhäuser. Ronald L. Graham, Donald E. Knuth, Oren Patashnik & Stanley Liu (1989). Concrete mathematics: a foundation for computer science. Computers in Physics 3(5), 106–107. Stephen Bellantoni & Stephen Cook (1992). A new recursion_ theoretic characterization of the poly-time functions. Computational Complexity 2, 97–110. A.O. Gelfand (1963). Calcul des différences finies. Dunod. László. Kalmár (1943). Egyzzerü példa eldönthetetlen aritmetikai problémára. Mate és Fizikai Lapok 50, 1–23. H.E. Rose (1984).Subrecursion . Oxford university press. David B Thompson (1972). Subrecursiveness: Machine-independent notions of computability in restricted time and storage. Mathematical Systems Theory 6(1-2), 3–15. Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing Company. Gustavo Lau (2018). URL http://www.acm.ciens.ucv.ve/main/entrenamiento/material/DiscreteCalculus.pdf.Discrete calculus. Thomas H Cormen, Charles E Leiserson, Ronald L Rivest & Clifford Stein (2009). Introduction to algorithms (third edition). MIT press. Jerzy Mycka & José Félix Costa (2005). What lies beyond the mountains? Computational systems beyond the Turing limit. European Association for Theoretical Computer Science Bulletin 85, 181–189. F.A. Izadi, N. Aliev & G Bagirov (2009). Discrete Calculus by Analogy. Bentham Science Publishers. V.I. Arnold (1978). Ordinary Differential Equations. MIT Press. Olivier Bournez & Arnaud Durand (2019). Recursion Schemes, Discrete Differential Equations and Characterization of Polynomial Time Computations. In 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, August 26–30, 2019, Aachen, Germany, Peter Rossmanith, Pinar Heggernes & Joost- Pieter Katoen, editors, volume 138 of LIPIcs, 23:1–23:14. Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik. URL https://doi.org/10.4230/LIPIcs.MFCS.2019.23. Pieter Collins & Daniel S Graςa (2008). Effective computability of solutions of ordinary differential equations the thousand monkeys approach. Electronic Notes in Theoretical Computer Science 221, 103– 114. Piergiorgio Odifreddi (1992). Classical Recursion Theory, volume 125 of Studies in Logic and the foundations of mathematics .North- Holland. Olivier Bournez & Amaury Pouly (2018). A Survey on Analog Models of Computation. In Handbook of Computability and Complexity in Analysis, Vasco Brattka & Peter Hertling, editors. Springer. To appear. Bruno Loff, José Félix Costa & Jerzy Mycka (2007). The New Promise of Analog Computation. In Computability in Europe 2007: Computation and Logic in the Real World. Jerzy Mycka & José Félix Costa (2006). The P≠NP conjecture in the context of real and complex analysis. Journal of Complexity 22(2), 287–303. Manuel L. Campagnolo, Cristopher Moore & José Félix Costa (2002a). An analog characterization of the Grzegorczyk hierarchy 18(4), 977–1000. Peter Clote & Evangelos Kranakis (2013).Boolean functions and computation models . Springer Science & Business Media. P. Clote (1998). Computational Models and Function Algebras. In Handbook of Computability Theory, Edward R. Griffor, editor, 589– 681. North-Holland, Amsterdam. ISBN 0-444-89882-4. Amaury Pouly (2015). Continuous models of computation: from computability to complexity. Ph.D. thesis, Ecole Polytechnique and Unidersidade Do Algarve. https://pastel.archives-ouvertes.fr/tel-01223284, Ackermann Award 2017. Cristopher Moore (1996). Recursion theory on the reals and continuous-time computation. Theoretical Computer Science 162(1), 23–44. Daniel Leivant & Jean-Yves Marion (1993). Lambda Calculus Characterizations of Poly-Time. Fundamenta Informatica 19(1,2), 167,184. Alan Cobham (1962). The intrinsic computational difficulty of functions. In Proceedings of the International Conference on Logic, Methodology, and Philosophy of Science, Y. Bar-Hillel, editor, 24–30. North- Holland, Amsterdam. Charles Jordan & Károly Jordán (1965).Calculus of finite differences , volume 33. American Mathematical Soc. Akitoshi Kawamura (2009). Lipschitz continuous ordinary differential equations are polynomial-space complete. In 2009 24th Annual IEEE Conference on Computational Complexity, 149–160. IEEE. Daniel Leivant & Jean-Yves Marion (1995). Ramified recurrence and computational complexity II: substitution and poly-space. In Computer Science Logic, 8th Workshop, CSL’94, L. Pacholski & J. Tiuryn, editors, volume 933 of Lecture Notes in Computer Science, 369–380. Springer, Kazimierz, Poland. 240_CR12 240_CR34 240_CR11 240_CR33 240_CR14 240_CR36 240_CR13 240_CR35 240_CR16 240_CR15 240_CR18 240_CR17 240_CR19 240_CR4 240_CR5 240_CR6 240_CR7 240_CR1 240_CR2 240_CR3 240_CR8 240_CR9 240_CR21 240_CR20 240_CR23 240_CR22 240_CR25 240_CR24 240_CR27 240_CR26 240_CR29 240_CR28 240_CR30 240_CR10 240_CR32 240_CR31 |
| References_xml | – reference: Manuel L. Campagnolo (2001). Computational Complexity of Real Valued Recursive Functions and Analog Circuits. Ph.D. thesis, Universidade T´ecnica de Lisboa. – reference: Earl A. Coddington & Norman Levinson (1955).Theory of Ordinary Differential Equations . Mc-Graw-Hill. – reference: Olivier Bournez, Daniel S. Graςa & Amaury Pouly (2017). Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length. Journal of the ACM 64(6), 38:1–38:76. – reference: V.I. Arnold (1978). Ordinary Differential Equations. MIT Press. – reference: Charles Jordan & Károly Jordán (1965).Calculus of finite differences , volume 33. American Mathematical Soc. – reference: Garrett Birkhoff & Gian-Carlo Rota (1989). Ordinary Differential Equations. John Wiley & Sons, 4th edition. – reference: Akitoshi Kawamura (2009). Lipschitz continuous ordinary differential equations are polynomial-space complete. In 2009 24th Annual IEEE Conference on Computational Complexity, 149–160. IEEE. – reference: Stephen Bellantoni & Stephen Cook (1992). A new recursion_ theoretic characterization of the poly-time functions. Computational Complexity 2, 97–110. – reference: Bruno Loff, José Félix Costa & Jerzy Mycka (2007). The New Promise of Analog Computation. In Computability in Europe 2007: Computation and Logic in the Real World. – reference: P. Clote (1998). Computational Models and Function Algebras. In Handbook of Computability Theory, Edward R. Griffor, editor, 589– 681. North-Holland, Amsterdam. ISBN 0-444-89882-4. – reference: Peter Clote & Evangelos Kranakis (2013).Boolean functions and computation models . Springer Science & Business Media. – reference: Jerzy Mycka & José Félix Costa (2005). What lies beyond the mountains? Computational systems beyond the Turing limit. European Association for Theoretical Computer Science Bulletin 85, 181–189. – reference: Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing Company. – reference: David Gleich (2005). Finite calculus: A tutorial for solving nasty sums. Stanford University . – reference: Piergiorgio Odifreddi (1992). Classical Recursion Theory, volume 125 of Studies in Logic and the foundations of mathematics .North- Holland. – reference: Daniel Leivant & Jean-Yves Marion (1993). Lambda Calculus Characterizations of Poly-Time. Fundamenta Informatica 19(1,2), 167,184. – reference: A.O. Gelfand (1963). Calcul des différences finies. Dunod. – reference: Olivier Bournez & Amaury Pouly (2018). A Survey on Analog Models of Computation. In Handbook of Computability and Complexity in Analysis, Vasco Brattka & Peter Hertling, editors. Springer. To appear. – reference: David B Thompson (1972). Subrecursiveness: Machine-independent notions of computability in restricted time and storage. Mathematical Systems Theory 6(1-2), 3–15. – reference: Pieter Collins & Daniel S Graςa (2008). Effective computability of solutions of ordinary differential equations the thousand monkeys approach. Electronic Notes in Theoretical Computer Science 221, 103– 114. – reference: Ronald L. Graham, Donald E. Knuth, Oren Patashnik & Stanley Liu (1989). Concrete mathematics: a foundation for computer science. Computers in Physics 3(5), 106–107. – reference: F.A. Izadi, N. Aliev & G Bagirov (2009). Discrete Calculus by Analogy. Bentham Science Publishers. – reference: Thomas H Cormen, Charles E Leiserson, Ronald L Rivest & Clifford Stein (2009). Introduction to algorithms (third edition). MIT press. – reference: Manuel L. Campagnolo, Cristopher Moore & José Félix Costa (2002b).An analog characterization of the Grzegorczyk hierarchy . Journal of Complexity 18(4), 977–1000. – reference: Daniel Leivant & Jean-Yves Marion (1995). Ramified recurrence and computational complexity II: substitution and poly-space. In Computer Science Logic, 8th Workshop, CSL’94, L. Pacholski & J. Tiuryn, editors, volume 933 of Lecture Notes in Computer Science, 369–380. Springer, Kazimierz, Poland. – reference: Olivier Bournez & Arnaud Durand (2019). Recursion Schemes, Discrete Differential Equations and Characterization of Polynomial Time Computations. In 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, August 26–30, 2019, Aachen, Germany, Peter Rossmanith, Pinar Heggernes & Joost- Pieter Katoen, editors, volume 138 of LIPIcs, 23:1–23:14. Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik. URL https://doi.org/10.4230/LIPIcs.MFCS.2019.23. – reference: Jerzy Mycka & José Félix Costa (2006). The P≠NP conjecture in the context of real and complex analysis. Journal of Complexity 22(2), 287–303. – reference: László. Kalmár (1943). Egyzzerü példa eldönthetetlen aritmetikai problémára. Mate és Fizikai Lapok 50, 1–23. – reference: H.E. Rose (1984).Subrecursion . Oxford university press. – reference: Daniel Leivant (1994). Predicative recurrence and computational complexity I: Word recurrence and poly-time. In Feasible Mathematics II, Peter Clote & Jeffery Remmel, editors, 320–343. Birkhäuser. – reference: Manuel L. Campagnolo, Cristopher Moore & José Félix Costa (2002a). An analog characterization of the Grzegorczyk hierarchy 18(4), 977–1000. – reference: Amaury Pouly (2015). Continuous models of computation: from computability to complexity. Ph.D. thesis, Ecole Polytechnique and Unidersidade Do Algarve. https://pastel.archives-ouvertes.fr/tel-01223284, Ackermann Award 2017. – reference: Gustavo Lau (2018). URL http://www.acm.ciens.ucv.ve/main/entrenamiento/material/DiscreteCalculus.pdf.Discrete calculus. – reference: Cristopher Moore (1996). Recursion theory on the reals and continuous-time computation. Theoretical Computer Science 162(1), 23–44. – reference: Ker-I Ko (1983). On the Computational Complexity of Ordinary Differential Equations. Information and Control 58(1–3), 157–194. – reference: Alan Cobham (1962). The intrinsic computational difficulty of functions. In Proceedings of the International Conference on Logic, Methodology, and Philosophy of Science, Y. Bar-Hillel, editor, 24–30. 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| Snippet | This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations. It... |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithms Complexity Computation Computational Mathematics and Numerical Analysis Computer Science Difference equations Differential equations Ordinary differential equations Polynomials |
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| Title | A Characterization of Functions over the Integers Computable in Polynomial Time Using Discrete Ordinary Differential Equations |
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