### 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) |
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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 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Theoretical Computer Science*,

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