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On random reductions from sparse sets to tally sets. (English) Zbl 0780.68044
A set \(S\subseteq\{0,1\}^*\) is called sparse if for each natural \(n\) it contains only a polynomial number of words of length \(n\). A set \(T\subseteq \{1\}^*\) is called tally. It is shown that every sparse set \(S\) can be many-one reduced to an appropriate tally set \(T\) by a polynomial-time, randomized reduction. The proof is based on the Chinese remainder theorem. As a consequence of the main result it is proven that there exists a tally set in NP which is complete for all sparse sets in NP under polynomial randomized many-one reduction. This is a partial answer to an open problem posed by Hartmanis and Yesha in 1984.

68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
03D15 Complexity of computation (including implicit computational complexity)
Full Text: DOI
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