Does truth-table of linear norm reduce the one-query tautologies to a random oracle?
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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.
KeywordsTruth-table reduction Computational complexity Random oracle Monotone Boolean formula Forcing complexity
Mathematics Subject Classification (2000)68Q15 03D15
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- 2.Ambos-Spies K.: Randomness, relativizations, and polynomial reducibilities. In: Selman, A.L. (eds) Structure in Complexity Theory. Lecture Notes in Computer Sciences, vol. 223, pp. 23–34. Springer, Berlin (1986)Google Scholar
- 4.Ambos-Spies K., Mayordomo E.: Resource-bounded measure and randomness. In: Sorbi, A. (eds) Complexity, Logic, and Recursion Theory. Lecture Notes in Pure and Applied Mathematics, vol. 187, pp. 1–47. Marcel Dekker, New York (1997)Google Scholar
- 10.Jockusch, C.G.: Reducibilities in recursive function theory, Ph.D. thesis, MIT Press, Cambridge (1966)Google Scholar
- 12.Kumabe, M., Suzuki, T., Yamazaki, T.: Logarithmic truth-table reductions and minimum sizes of forcing conditions (preliminary draft). In: Proof Theory and Computation Theory, Kyoto, 2005, Sūrikaisekikenkyusho Kōkyuroku, no. 1442, pp. 42–47 (2005)Google Scholar
- 13.Kumabe, M., Suzuki, T., Yamazaki, T., Kumabe, M., Suzuki, T., Yamazaki, T.: Truth-table reductions and minimum sizes of forcing conditions (preliminary draft). In: Sūrikaisekikenkyusho Kōkyuroku, no. 1533, pp. 9–14 (2007)Google Scholar
- 16.Rogers, H. Jr.: Theory of recursive functions and effective computability. Massachusetts Institute of Technology (1987) (Original edition: MacGraw-Hill, New York, 1967)Google Scholar
- 17.Sacks G.E.: Degrees of Unsolvability, Annals of Mathematics Studies, vol. 55. Princeton university press, Princeton (1963)Google Scholar
- 19.Suzuki, T.: Computational complexity of Boolean formulas with query symbols. Doctoral dissertation, Institute of Mathematics, University of Tsukuba, Tsukuba-City, Japan (1999)Google Scholar