### Abstract

In this paper we give several generalized theorems concerning reducibility notions to certain complexity classes. We study classes that are either (I) closed under NP many-one reductions and polynomial-time conjunctive reductions or (II) closed under coNP many-one reductions and polynomial-time disjunctive reductions. We prove that, for such a class K, (1) reducibility notions of sets to K under polynomial-time constant-round truth-table reducibility, polynomial-time log-Turing reducibility, logspace constant-round truth-table reducibility, logspace log-Turing reducibility, and logspace Turing reducibility are all equivalent and (2) every set that is polynomial-time positive Turing reducible to a set in K is already in K. From these results, we derive some observations on the reducibility notions to C_{=}P and NP.

Original language | English (US) |
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Pages (from-to) | 189-200 |

Number of pages | 12 |

Journal | Mathematical Systems Theory |

Volume | 27 |

Issue number | 3 |

DOIs | |

State | Published - May 1994 |

Externally published | Yes |

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### ASJC Scopus subject areas

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

### Cite this

**Generalized theorems on relationships among reducibility notions to certain complexity classes.** / Ogihara, Mitsunori.

Research output: Contribution to journal › Article

*Mathematical Systems Theory*, vol. 27, no. 3, pp. 189-200. https://doi.org/10.1007/BF01578841

}

TY - JOUR

T1 - Generalized theorems on relationships among reducibility notions to certain complexity classes

AU - Ogihara, Mitsunori

PY - 1994/5

Y1 - 1994/5

N2 - In this paper we give several generalized theorems concerning reducibility notions to certain complexity classes. We study classes that are either (I) closed under NP many-one reductions and polynomial-time conjunctive reductions or (II) closed under coNP many-one reductions and polynomial-time disjunctive reductions. We prove that, for such a class K, (1) reducibility notions of sets to K under polynomial-time constant-round truth-table reducibility, polynomial-time log-Turing reducibility, logspace constant-round truth-table reducibility, logspace log-Turing reducibility, and logspace Turing reducibility are all equivalent and (2) every set that is polynomial-time positive Turing reducible to a set in K is already in K. From these results, we derive some observations on the reducibility notions to C=P and NP.

AB - In this paper we give several generalized theorems concerning reducibility notions to certain complexity classes. We study classes that are either (I) closed under NP many-one reductions and polynomial-time conjunctive reductions or (II) closed under coNP many-one reductions and polynomial-time disjunctive reductions. We prove that, for such a class K, (1) reducibility notions of sets to K under polynomial-time constant-round truth-table reducibility, polynomial-time log-Turing reducibility, logspace constant-round truth-table reducibility, logspace log-Turing reducibility, and logspace Turing reducibility are all equivalent and (2) every set that is polynomial-time positive Turing reducible to a set in K is already in K. From these results, we derive some observations on the reducibility notions to C=P and NP.

UR - http://www.scopus.com/inward/record.url?scp=0039271615&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0039271615&partnerID=8YFLogxK

U2 - 10.1007/BF01578841

DO - 10.1007/BF01578841

M3 - Article

AN - SCOPUS:0039271615

VL - 27

SP - 189

EP - 200

JO - Theory of Computing Systems

JF - Theory of Computing Systems

SN - 1432-4350

IS - 3

ER -