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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.

##### MSC:
 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 03D15 Complexity of computation (including implicit computational complexity)
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##### References:
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