## On bounded truth-table, conjunctive, and randomized reductions to sparse sets.(English)Zbl 0919.03034

Shyamasundar, R. (ed.), Foundations of software technology and theoretical computer science. 12th conference, New Delhi, India, December 18–20, 1992. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 652, 140-151 (1992).
Summary: In this paper we study the consequences of the existence of sparse hard sets for NP and other complexity classes under certain types of deterministic, randomized and nondeterministic reductions. We show that if an NP-complete set is bounded truth-table reducible to some set that conjunctively reduces to a sparse set then P = NP. We next show that if an NP-complete set is bounded truth-table reducible to some set that randomly reduces (via a $$\text{co}$$-$$\text{rp}$$ reduction) to some set that conjunctively reduces to a sparse set then RP = NP. Finally, we prove that if a co-NP-complete set reduces via a nondeterministic polynomial time many-one reduction to a co-sparse set then $$\text{PH} =\Theta^p_2$$. On the other hand, we show that nondeterministic polynomial time many-one reductions to sparse sets are as powerful as nondeterministic Turing reductions to sparse sets.
For the entire collection see [Zbl 0856.00048].

### MSC:

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