### 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 - Dec 1 1991 |

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

### Publication series

Name | Proc 6 Annu Struct Complexity Theor |
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### 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). (Proc 6 Annu Struct Complexity Theor). Publ by IEEE.