Abstract
We show the following results for polynomial-time reducibility to R C , the set of Kolmogorov random strings.
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If P ≠ NP, then SAT does not dtt-reduce to R C .
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If PH does not collapse, then SAT does not n α-tt-reduce to R C for any α< 1.
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If PH does not collapse, then SAT does not n α-T-reduce to R C for any \(\alpha < {{1}\over{2}}\).
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There is a problem in E that does not dtt-reduce to R C .
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There is a problem in E that does not n α-tt-reduce to R C , for any α< 1.
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There is a problem in E that does not n α-T-reduce to R C , for any \(\alpha < {{1}\over{2}}\).
These results hold for both the plain and prefix-free variants of Kolmogorov complexity and are also independent of the choice of the universal machine.
This research was supported in part by NSF grant 0652601 and by an NWO travel grant. Part of this research was done while the author was on sabbatical at CWI.
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Hitchcock, J.M. (2010). Lower Bounds for Reducibility to the Kolmogorov Random Strings. In: Ferreira, F., Löwe, B., Mayordomo, E., Mendes Gomes, L. (eds) Programs, Proofs, Processes. CiE 2010. Lecture Notes in Computer Science, vol 6158. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13962-8_22
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