### Abstract

In this paper, we study one-word-decreasing self-reducible sets which are introduced by Lozano and Torán (1991). These are the usual self-reducible sets with the peculiarity that the self-reducibility machine makes at most one query and this is lexicographically smaller than the input. We show first that for all counting classes defined by a predicate on the number of accepting paths there exist complete sets which are one-word-decreasing self-reducible. Using this fact, we can prove that for any class K chosen from {PP, NP, C ^{=} P, MOD_{2}P, MOD_{3} P,...} it holds that (1) if there is a sparse ≤^{P}
_{btt}-hard set for K then K ⊆ P, and (2) if there is a sparse ≤^{SN}
_{btt}-hard set for K then K ⊆ NP {frown} co-NP. This generalizes the result of Ogiwara and Watanabe (1991) to the mentioned complexity classes.

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

Pages (from-to) | 255-275 |

Number of pages | 21 |

Journal | Theoretical Computer Science |

Volume | 112 |

Issue number | 2 |

DOIs | |

State | Published - May 10 1993 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Computational Theory and Mathematics

### Cite this

*Theoretical Computer Science*,

*112*(2), 255-275. https://doi.org/10.1016/0304-3975(93)90020-T

**On sparse hard sets for counting classes.** / Ogihara, Mitsunori; Lozano, Antoni.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 112, no. 2, pp. 255-275. https://doi.org/10.1016/0304-3975(93)90020-T

}

TY - JOUR

T1 - On sparse hard sets for counting classes

AU - Ogihara, Mitsunori

AU - Lozano, Antoni

PY - 1993/5/10

Y1 - 1993/5/10

N2 - In this paper, we study one-word-decreasing self-reducible sets which are introduced by Lozano and Torán (1991). These are the usual self-reducible sets with the peculiarity that the self-reducibility machine makes at most one query and this is lexicographically smaller than the input. We show first that for all counting classes defined by a predicate on the number of accepting paths there exist complete sets which are one-word-decreasing self-reducible. Using this fact, we can prove that for any class K chosen from {PP, NP, C = P, MOD2P, MOD3 P,...} it holds that (1) if there is a sparse ≤P btt-hard set for K then K ⊆ P, and (2) if there is a sparse ≤SN btt-hard set for K then K ⊆ NP {frown} co-NP. This generalizes the result of Ogiwara and Watanabe (1991) to the mentioned complexity classes.

AB - In this paper, we study one-word-decreasing self-reducible sets which are introduced by Lozano and Torán (1991). These are the usual self-reducible sets with the peculiarity that the self-reducibility machine makes at most one query and this is lexicographically smaller than the input. We show first that for all counting classes defined by a predicate on the number of accepting paths there exist complete sets which are one-word-decreasing self-reducible. Using this fact, we can prove that for any class K chosen from {PP, NP, C = P, MOD2P, MOD3 P,...} it holds that (1) if there is a sparse ≤P btt-hard set for K then K ⊆ P, and (2) if there is a sparse ≤SN btt-hard set for K then K ⊆ NP {frown} co-NP. This generalizes the result of Ogiwara and Watanabe (1991) to the mentioned complexity classes.

UR - http://www.scopus.com/inward/record.url?scp=0027591701&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0027591701&partnerID=8YFLogxK

U2 - 10.1016/0304-3975(93)90020-T

DO - 10.1016/0304-3975(93)90020-T

M3 - Article

AN - SCOPUS:0027591701

VL - 112

SP - 255

EP - 275

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 2

ER -