### Abstract

We distinguish self-reducibility of a language L with the question of whether search reduces to decision for L. Results include: (i) If NE ≠ E, then there exists a set t in NP - P such that search reduces to decision for L, search does not nonadaptively reduce to decision for L and L is not self-reducible, (ii) If UE ≠ E, then there exists a language L ∈ UP - P such that search nonadaptively reduces to decision for L, but L is not self-reducible, (iii) If UE ∩ co-UE ≠ E, then there is a disjunctive self-reducible language L ∈ UP - P for which search does not nonadaptively reduce to decision. We prove that if NE ⊈ BPE, then there is a language L ∈ NP - BPP such that L is randomly self-reducible, not nonadaptively randomly self-reducible, and not self-reducible. We obtain results concerning trade-offs in multiprover interactive proof systems and results that distinguish checkable languages from those that are nonadaptively checkable. Many of our results are proven by constructing p-selective sets. We obtain a p-selective set that is not ≤ ^{P}
_{tt}-equivalent to any tally language, and we show that if P = PP, then every p-selective set is ≤ ^{P}
_{T}-equivalent to a tally language. Similarly, if P = NP, then every cheatable set is ≤ ^{P}
_{m}-equivalent to a tally language. We construct a recursive p-selective tally set that is not cheatable.

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

Pages (from-to) | 194-209 |

Number of pages | 16 |

Journal | Journal of Computer and System Sciences |

Volume | 53 |

Issue number | 2 |

DOIs | |

State | Published - Oct 1996 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Computational Theory and Mathematics

### Cite this

*Journal of Computer and System Sciences*,

*53*(2), 194-209. https://doi.org/10.1006/jcss.1996.0061

**P-selective sets and reducing search to decision vs self-reducibility.** / Hemaspaandra, Edith; Naik, Ashish V.; Ogihara, Mitsunori; Selman, Alan L.

Research output: Contribution to journal › Article

*Journal of Computer and System Sciences*, vol. 53, no. 2, pp. 194-209. https://doi.org/10.1006/jcss.1996.0061

}

TY - JOUR

T1 - P-selective sets and reducing search to decision vs self-reducibility

AU - Hemaspaandra, Edith

AU - Naik, Ashish V.

AU - Ogihara, Mitsunori

AU - Selman, Alan L.

PY - 1996/10

Y1 - 1996/10

N2 - We distinguish self-reducibility of a language L with the question of whether search reduces to decision for L. Results include: (i) If NE ≠ E, then there exists a set t in NP - P such that search reduces to decision for L, search does not nonadaptively reduce to decision for L and L is not self-reducible, (ii) If UE ≠ E, then there exists a language L ∈ UP - P such that search nonadaptively reduces to decision for L, but L is not self-reducible, (iii) If UE ∩ co-UE ≠ E, then there is a disjunctive self-reducible language L ∈ UP - P for which search does not nonadaptively reduce to decision. We prove that if NE ⊈ BPE, then there is a language L ∈ NP - BPP such that L is randomly self-reducible, not nonadaptively randomly self-reducible, and not self-reducible. We obtain results concerning trade-offs in multiprover interactive proof systems and results that distinguish checkable languages from those that are nonadaptively checkable. Many of our results are proven by constructing p-selective sets. We obtain a p-selective set that is not ≤ P tt-equivalent to any tally language, and we show that if P = PP, then every p-selective set is ≤ P T-equivalent to a tally language. Similarly, if P = NP, then every cheatable set is ≤ P m-equivalent to a tally language. We construct a recursive p-selective tally set that is not cheatable.

AB - We distinguish self-reducibility of a language L with the question of whether search reduces to decision for L. Results include: (i) If NE ≠ E, then there exists a set t in NP - P such that search reduces to decision for L, search does not nonadaptively reduce to decision for L and L is not self-reducible, (ii) If UE ≠ E, then there exists a language L ∈ UP - P such that search nonadaptively reduces to decision for L, but L is not self-reducible, (iii) If UE ∩ co-UE ≠ E, then there is a disjunctive self-reducible language L ∈ UP - P for which search does not nonadaptively reduce to decision. We prove that if NE ⊈ BPE, then there is a language L ∈ NP - BPP such that L is randomly self-reducible, not nonadaptively randomly self-reducible, and not self-reducible. We obtain results concerning trade-offs in multiprover interactive proof systems and results that distinguish checkable languages from those that are nonadaptively checkable. Many of our results are proven by constructing p-selective sets. We obtain a p-selective set that is not ≤ P tt-equivalent to any tally language, and we show that if P = PP, then every p-selective set is ≤ P T-equivalent to a tally language. Similarly, if P = NP, then every cheatable set is ≤ P m-equivalent to a tally language. We construct a recursive p-selective tally set that is not cheatable.

UR - http://www.scopus.com/inward/record.url?scp=0030263710&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030263710&partnerID=8YFLogxK

U2 - 10.1006/jcss.1996.0061

DO - 10.1006/jcss.1996.0061

M3 - Article

AN - SCOPUS:0030263710

VL - 53

SP - 194

EP - 209

JO - Journal of Computer and System Sciences

JF - Journal of Computer and System Sciences

SN - 0022-0000

IS - 2

ER -