### Abstract

The computational complexity of the solution h to the ordinary differential equation h(0) = 0, h'(t) = g(t,h(t)) under various assumptions on the function g has been investigated. Kawamura showed in 2010 that the solution h can be PSPACE-hard even if g is assumed to be Lipschitz continuous and polynomial-time computable. We place further requirements on the smoothness of g and obtain the following results: the solution h can still be PSPACE-hard if g is assumed to be of class C^{1}; for each k > 2, the solution h can be hard for the counting hierarchy even if g is of class C^{k}.

Original language | English |
---|---|

Article number | 6 |

Journal | Logical Methods in Computer Science |

Volume | 10 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 11 2014 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Logical Methods in Computer Science*,

*10*(1), [6]. https://doi.org/10.2168/LMCS-10(1:6)2014

**Computational complexity of smooth differential equations.** / Kawamura, Akitoshi; Ota, Hiroyuki; Rösnick, Carsten; Ziegler, Martin.

Research output: Contribution to journal › Article

*Logical Methods in Computer Science*, vol. 10, no. 1, 6. https://doi.org/10.2168/LMCS-10(1:6)2014

}

TY - JOUR

T1 - Computational complexity of smooth differential equations

AU - Kawamura, Akitoshi

AU - Ota, Hiroyuki

AU - Rösnick, Carsten

AU - Ziegler, Martin

PY - 2014/2/11

Y1 - 2014/2/11

N2 - The computational complexity of the solution h to the ordinary differential equation h(0) = 0, h'(t) = g(t,h(t)) under various assumptions on the function g has been investigated. Kawamura showed in 2010 that the solution h can be PSPACE-hard even if g is assumed to be Lipschitz continuous and polynomial-time computable. We place further requirements on the smoothness of g and obtain the following results: the solution h can still be PSPACE-hard if g is assumed to be of class C1; for each k > 2, the solution h can be hard for the counting hierarchy even if g is of class Ck.

AB - The computational complexity of the solution h to the ordinary differential equation h(0) = 0, h'(t) = g(t,h(t)) under various assumptions on the function g has been investigated. Kawamura showed in 2010 that the solution h can be PSPACE-hard even if g is assumed to be Lipschitz continuous and polynomial-time computable. We place further requirements on the smoothness of g and obtain the following results: the solution h can still be PSPACE-hard if g is assumed to be of class C1; for each k > 2, the solution h can be hard for the counting hierarchy even if g is of class Ck.

UR - http://www.scopus.com/inward/record.url?scp=84893876435&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84893876435&partnerID=8YFLogxK

U2 - 10.2168/LMCS-10(1:6)2014

DO - 10.2168/LMCS-10(1:6)2014

M3 - Article

AN - SCOPUS:84893876435

VL - 10

JO - Logical Methods in Computer Science

JF - Logical Methods in Computer Science

SN - 1860-5974

IS - 1

M1 - 6

ER -