# 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.

## Keywords

Truth-table reduction Computational complexity Random oracle Monotone Boolean formula Forcing complexity## Mathematics Subject Classification (2000)

68Q15 03D15## Preview

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