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Does truth-table of linear norm reduce the one-query tautologies to a random oracle?

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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.

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Authors and Affiliations

  1. Faculty of Liberal Arts, The Open University of Japan, Ōoka 2-31-1, Minami-ku, Yokohama, 232-0061, Japan

    Masahiro Kumabe

  2. Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa, Hachioji, Tokyo, 192-0397, Japan

    Toshio Suzuki

  3. Department of Mathematics, Tohoku University, Sendai, 980-8578, Japan

    Takeshi Yamazaki

Authors
  1. Masahiro Kumabe
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  2. Toshio Suzuki
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  3. Takeshi Yamazaki
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Correspondence to Toshio Suzuki.

Additional information

The authors are partially supported by Grant-in-Aid for Scientific Research (No. 14740082 and No. 17540131), Japan Society for the Promotion of Science.

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Kumabe, M., Suzuki, T. & Yamazaki, T. Does truth-table of linear norm reduce the one-query tautologies to a random oracle?. Arch. Math. Logic 47, 159–180 (2008). https://doi.org/10.1007/s00153-008-0076-4

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  • DOI: https://doi.org/10.1007/s00153-008-0076-4

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