### Abstract

In answer to Ko's question raised in 1983, we show that an initial value problem given by a polynomial-time computable, Lipschitz continuous function can have a polynomial-space complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differential equation is weak. We define a class of polynomial-space computation tableaux with equally weak feedback, and show that they are still polynomial-space complete. The same technique also settles Ko's two later questions on Volterra integral equations.

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

Pages (from-to) | 305-332 |

Number of pages | 28 |

Journal | Computational Complexity |

Volume | 19 |

Issue number | 2 |

DOIs | |

Publication status | Published - May 1 2010 |

Externally published | Yes |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Mathematics(all)
- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

**Lipschitz continuous ordinary differential equations are polynomial-space complete.** / Kawamura, Akitoshi.

Research output: Contribution to journal › Article

*Computational Complexity*, vol. 19, no. 2, pp. 305-332. https://doi.org/10.1007/s00037-010-0286-0

}

TY - JOUR

T1 - Lipschitz continuous ordinary differential equations are polynomial-space complete

AU - Kawamura, Akitoshi

PY - 2010/5/1

Y1 - 2010/5/1

N2 - In answer to Ko's question raised in 1983, we show that an initial value problem given by a polynomial-time computable, Lipschitz continuous function can have a polynomial-space complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differential equation is weak. We define a class of polynomial-space computation tableaux with equally weak feedback, and show that they are still polynomial-space complete. The same technique also settles Ko's two later questions on Volterra integral equations.

AB - In answer to Ko's question raised in 1983, we show that an initial value problem given by a polynomial-time computable, Lipschitz continuous function can have a polynomial-space complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differential equation is weak. We define a class of polynomial-space computation tableaux with equally weak feedback, and show that they are still polynomial-space complete. The same technique also settles Ko's two later questions on Volterra integral equations.

UR - http://www.scopus.com/inward/record.url?scp=77953542311&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953542311&partnerID=8YFLogxK

U2 - 10.1007/s00037-010-0286-0

DO - 10.1007/s00037-010-0286-0

M3 - Article

VL - 19

SP - 305

EP - 332

JO - Computational Complexity

JF - Computational Complexity

SN - 1016-3328

IS - 2

ER -