### Abstract

For various polynomial-time reducibilities r, the authors ask whether being r-reducible to a sparse set is a broader notion than being r-equivalent to a sparse set. Although distinguishing equivalence and reducibility to sparse sets, for many-one or 1-truth-table reductions, would imply that P ≠ NP, the authors show that for k-truth-table reductions, k ≥ 2, equivalence and reducibility to sparse sets provably differ. Though R. Gavaldaand D. Watanabe have shown that, for any polynomial-time computable unbounded function f(·), some sets f(n)-truth-table reducible to sparse sets are not even Turing equivalent to sparse sets, the authors show that extending their result to the 2-truth-table case would provide a proof that P ≠ NP. Additionally, the authors study the relative power of different notions of reducibility and show that disjunctive and conjunctive truth-table reductions to sparse sets are surprisingly powerful, refuting a conjecture of K. Ko (1989).

Original language | English (US) |
---|---|

Title of host publication | Proceedings of the Sixth Annual Structure in Complexity Theory Conference |

Publisher | Publ by IEEE |

Pages | 220-229 |

Number of pages | 10 |

ISBN (Print) | 0818622555, 9780818622557 |

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 |
---|---|

City | Chicago, IL, USA |

Period | 6/30/91 → 7/3/91 |

### Fingerprint

### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*Proceedings of the Sixth Annual Structure in Complexity Theory Conference*(pp. 220-229). Publ by IEEE.

**Relating equivalence and reducibility to sparse sets.** / Allender, Eric; Hemachandra, Lane A.; Ogihara, Mitsunori; Watanabe, Osamu.

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

*Proceedings of the Sixth Annual Structure in Complexity Theory Conference.*Publ by IEEE, pp. 220-229, Proceedings of the 6th Annual Structure in Complexity Theory Conference, Chicago, IL, USA, 6/30/91.

}

TY - GEN

T1 - Relating equivalence and reducibility to sparse sets

AU - Allender, Eric

AU - Hemachandra, Lane A.

AU - Ogihara, Mitsunori

AU - Watanabe, Osamu

PY - 1991

Y1 - 1991

N2 - For various polynomial-time reducibilities r, the authors ask whether being r-reducible to a sparse set is a broader notion than being r-equivalent to a sparse set. Although distinguishing equivalence and reducibility to sparse sets, for many-one or 1-truth-table reductions, would imply that P ≠ NP, the authors show that for k-truth-table reductions, k ≥ 2, equivalence and reducibility to sparse sets provably differ. Though R. Gavaldaand D. Watanabe have shown that, for any polynomial-time computable unbounded function f(·), some sets f(n)-truth-table reducible to sparse sets are not even Turing equivalent to sparse sets, the authors show that extending their result to the 2-truth-table case would provide a proof that P ≠ NP. Additionally, the authors study the relative power of different notions of reducibility and show that disjunctive and conjunctive truth-table reductions to sparse sets are surprisingly powerful, refuting a conjecture of K. Ko (1989).

AB - For various polynomial-time reducibilities r, the authors ask whether being r-reducible to a sparse set is a broader notion than being r-equivalent to a sparse set. Although distinguishing equivalence and reducibility to sparse sets, for many-one or 1-truth-table reductions, would imply that P ≠ NP, the authors show that for k-truth-table reductions, k ≥ 2, equivalence and reducibility to sparse sets provably differ. Though R. Gavaldaand D. Watanabe have shown that, for any polynomial-time computable unbounded function f(·), some sets f(n)-truth-table reducible to sparse sets are not even Turing equivalent to sparse sets, the authors show that extending their result to the 2-truth-table case would provide a proof that P ≠ NP. Additionally, the authors study the relative power of different notions of reducibility and show that disjunctive and conjunctive truth-table reductions to sparse sets are surprisingly powerful, refuting a conjecture of K. Ko (1989).

UR - http://www.scopus.com/inward/record.url?scp=0026398366&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0026398366&partnerID=8YFLogxK

M3 - Conference contribution

SN - 0818622555

SN - 9780818622557

SP - 220

EP - 229

BT - Proceedings of the Sixth Annual Structure in Complexity Theory Conference

PB - Publ by IEEE

ER -