### Abstract

In our former works, for a given concept of reduction, we study the following hypothesis: "For a random oracle A, with probability one, the degree of the one-query tautologies with respect to A is strictly higher than the degree of A." In our former works (Suzuki in Kobe J. Math. 15, 91-102, 1998; in Inf. Comput. 176, 66-87, 2002; in Arch. Math. Logic 44, 751-762), the following three results are shown: The hypothesis for p-T (polynomial-time Turing) reduction is equivalent to the assertion that the probabilistic complexity class R is not equal to NP; The hypothesis for p-tt (polynomial-time truth-table) reduction implies that P is not NP; The hypothesis holds for each of the following: disjunctive reduction, conjunctive reduction, and p-btt (polynomial-time bounded-truth-table) reduction. In this paper, we show the following three results: (1) Let c be a positive real number. We consider a concept of truth-table reduction whose norm is at most c times size of input, where for a relativized propositional formula F, the size of F denotes the total number of occurrences of propositional variables, constants and propositional connectives. Then, our main result is that the hypothesis holds for such tt-reduction, provided that c is small enough. How small c can we take so that the above holds? It depends on our syntactic convention on one-query tautologies. In our setting, the statement holds for all c < 1. (2) The hypothesis holds for monotone truth-table reduction (also called positive reduction). (3) Dowd (in Inf. Comput. 96, 65-76, 1992) shows a polynomial upper bound for the minimum sizes of forcing conditions associated with a random oracle. We apply the above result (1), and get a linear lower bound for the sizes.

Original language | English |
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Pages (from-to) | 159-180 |

Number of pages | 22 |

Journal | Archive for Mathematical Logic |

Volume | 47 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2008 Jul 1 |

### Keywords

- Computational complexity
- Forcing complexity
- Monotone Boolean formula
- Random oracle
- Truth-table reduction

### ASJC Scopus subject areas

- Philosophy
- Logic

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## Cite this

*Archive for Mathematical Logic*,

*47*(2), 159-180. https://doi.org/10.1007/s00153-008-0076-4