Sizes of Cohen-type forcing condition : Application to randomness and computational complexity

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We improve the main theorem of the following paper. Toshio Suzuki, Archive for Mathematical Logic, vcol.44 (2005) pp. 751-762. The constant norm in the main theorem of the above paper is relaxed to multiple of constant and the norm. Summary of this result may be found in the following research report. M. Kumabe, T Suzuki and T. Yamazaki, "Truth-table reductions and minimum sizes of forcing conditions", Surikaisekikenkyusho-kokyuroku, vol. 1533 (2007), Research Institute of Mathematical Sciences, Kyoto University.We study Boolean functions defined by copies of a Boolean decision tree connected in parallel. An input string is given to leaves, and an output string is given by roots. We investigate which kind of weak randomness property is conserved by this type of Boolean functions. We showed a necessary condition of Martin-Loef randomness is conserved by this type of Boolean functions. The necessary condition is that for every positive integer r, a give oracle is r-generic in the sense of Dowd. We report this result at the following conference talk. "Which weak-randomness property is preserved by Boolean decision trees?", Proof theory 2007 (2007 December 3-4), Tokyo Metropolitan-University.