## 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:

 [1] Adleman, L.; Manders, K., Reductions that Lie, Proc. 20th ann. IEEE conf. on foundations of computer science, 397-410, (1979) [2] Book, R.V., Tally languages and complexity classes, Inform. and control, 26, 186-193, (1974) · Zbl 0287.68029 [3] Buhrman, H.; Longpré, L.; Spaan, E., Sparse reduces conjunctively to tally, () · Zbl 0830.68042 [4] Chang, R.; Kadin, J.; Rohatgi, P., Connections between the complexity of unique satisfiability and the threshold behavior of randomized reductions, Proc. 6th ann. IEEE conf. on structure in complexity theory, 255-266, (1991) [5] Hartmanis, J., On sparse sets in NP-P, Inform. process. lett., 16, 55-60, (1983) · Zbl 0501.68014 [6] Hartmanis, J.; Yesha, Y., Computation times of NP sets of different densities, Theoret. comput. sci., 34, 17-32, (1984) · Zbl 0985.68515 [7] Niven, I.; Zuckerman, H.S., An introduction to the theory of numbers, (1960), Wiley New York · Zbl 0186.36601 [8] Saluja, S., Relativized limitations of the left set technique and closure classes of sparse sets, Tech. rept. dept. of computer science, (1992), Tata Institute of Fundamental Research Bombay, India [9] Vazirani, U.; Vazirani, V., A natural encoding scheme proved probabilistic polynomial complete, Theoret. comput. sci., 24, 291-300, (1983) · Zbl 0525.68025
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