### Abstract

The authors propose and develop a complexity theory of feasible closure properties. For each of the classes #P, SpanP, OptP, and MidP, they establish complete characterizations--in terms of complexity class collapses--of the conditions under which the class has all feasible closure properties. In particular, #P is P-closed if and only if PP = UP; SpanP is P-closed if and only if R-MidP is P-closed if and only if P^{PP} = NP; and OptP is P-closed if and only if NP = co-NP. Furthermore, for each of these classes, the authors show natural operations--such as subtraction and division--to be hard closure properties, in the sense that if a class is closed under one of these, then it has all feasible closure properties. They also study potentially intermediate closure properties for #P. These properties--maximum, minimum, median, and decrement--seem neither to be possessed by #P nor to be #P-hard.

Original language | English (US) |
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Title of host publication | Proc 6 Annu Struct Complexity Theor |

Publisher | Publ by IEEE |

Pages | 16-29 |

Number of pages | 14 |

ISBN (Print) | 0818622555 |

State | Published - 1991 |

Externally published | Yes |

Event | Proceedings of the 6th Annual Structure in Complexity Theory Conference - Chicago, IL, USA Duration: Jun 30 1991 → Jul 3 1991 |

### Other

Other | Proceedings of the 6th Annual Structure in Complexity Theory Conference |
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City | Chicago, IL, USA |

Period | 6/30/91 → 7/3/91 |

### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*Proc 6 Annu Struct Complexity Theor*(pp. 16-29). Publ by IEEE.

**A complexity theory for feasible closure properties.** / Ogihara, Mitsunori; Hemachandra, Lane A.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proc 6 Annu Struct Complexity Theor.*Publ by IEEE, pp. 16-29, Proceedings of the 6th Annual Structure in Complexity Theory Conference, Chicago, IL, USA, 6/30/91.

}

TY - GEN

T1 - A complexity theory for feasible closure properties

AU - Ogihara, Mitsunori

AU - Hemachandra, Lane A.

PY - 1991

Y1 - 1991

N2 - The authors propose and develop a complexity theory of feasible closure properties. For each of the classes #P, SpanP, OptP, and MidP, they establish complete characterizations--in terms of complexity class collapses--of the conditions under which the class has all feasible closure properties. In particular, #P is P-closed if and only if PP = UP; SpanP is P-closed if and only if R-MidP is P-closed if and only if PPP = NP; and OptP is P-closed if and only if NP = co-NP. Furthermore, for each of these classes, the authors show natural operations--such as subtraction and division--to be hard closure properties, in the sense that if a class is closed under one of these, then it has all feasible closure properties. They also study potentially intermediate closure properties for #P. These properties--maximum, minimum, median, and decrement--seem neither to be possessed by #P nor to be #P-hard.

AB - The authors propose and develop a complexity theory of feasible closure properties. For each of the classes #P, SpanP, OptP, and MidP, they establish complete characterizations--in terms of complexity class collapses--of the conditions under which the class has all feasible closure properties. In particular, #P is P-closed if and only if PP = UP; SpanP is P-closed if and only if R-MidP is P-closed if and only if PPP = NP; and OptP is P-closed if and only if NP = co-NP. Furthermore, for each of these classes, the authors show natural operations--such as subtraction and division--to be hard closure properties, in the sense that if a class is closed under one of these, then it has all feasible closure properties. They also study potentially intermediate closure properties for #P. These properties--maximum, minimum, median, and decrement--seem neither to be possessed by #P nor to be #P-hard.

UR - http://www.scopus.com/inward/record.url?scp=0026373298&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0026373298&partnerID=8YFLogxK

M3 - Conference contribution

SN - 0818622555

SP - 16

EP - 29

BT - Proc 6 Annu Struct Complexity Theor

PB - Publ by IEEE

ER -