### Abstract

We characterize the complexity of some natural and important problems in linear algebra. In particular, we identify natural complexity classes for which the problems of (a) determining if a system of linear equations is feasible and (b) computing the rank of an integer matrix, (as well as other problems), are complete under logspace reductions. As an important part of presenting this classification, we show that the "exact counting logspace hierarchy" collapses to near the bottom level. (We review the definition of this hierarchy below.) We further show that this class is closed under NC^{1}-reducibihty, and that it consists of exactly those languages that have logspace uniform span programs (introduced by Karchmer and Wigderson) over the rationals. In addition, we contrast the complexity of these problems with the complexity of determining if a system of linear equations has an integer solution.

Original language | English (US) |
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Title of host publication | Proceedings of the 28th Annual ACM Symposium on Theory of Computing, STOC 1996 |

Publisher | Association for Computing Machinery |

Pages | 161-167 |

Number of pages | 7 |

ISBN (Electronic) | 0897917855 |

DOIs | |

State | Published - Jul 1 1996 |

Event | 28th Annual ACM Symposium on Theory of Computing, STOC 1996 - Philadelphia, United States Duration: May 22 1996 → May 24 1996 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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Volume | Part F129452 |

ISSN (Print) | 0737-8017 |

### Conference

Conference | 28th Annual ACM Symposium on Theory of Computing, STOC 1996 |
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Country | United States |

City | Philadelphia |

Period | 5/22/96 → 5/24/96 |

### ASJC Scopus subject areas

- Software

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## Cite this

*Proceedings of the 28th Annual ACM Symposium on Theory of Computing, STOC 1996*(pp. 161-167). (Proceedings of the Annual ACM Symposium on Theory of Computing; Vol. Part F129452). Association for Computing Machinery. https://doi.org/10.1145/237814.237856