## Computing functions with parallel queries to NP.(English)Zbl 0873.68058

Summary: The class $$\Theta ^{p}_{2}$$ of languages polynomial-time truth-table reducible to sets in NP has a wide range of different characterizations. We consider several functional versions of $$\Theta ^{p}_{2}$$ based on these characterizations. We show that in this way the three function classes FL$$^{\text{NP}}_{\log}$$, FP$$^{\text{NP}}_{\log}$$, and FP$$^{\text{NP}}_{\parallel}$$ are obtained. In contrast to the language case the function classes seem to all be different. We give evidence in support of this fact by showing that FL$$^{\text{NP}}_{\log}$$ coincides with any of the other classes then L=P, and that the equality of the classes FP$$^{\text{NP}}_{\log}$$ and FP$$^{\text{NP}}_{\parallel}$$ would imply that the number of nondeterministic bits needed for the computation of any problem in NP can be reduced by a polylogarithmic factor, and that the problem can be computed deterministically with a subexponential time bound of order $$2^{n^{O(1/\log \log n)}}.$$

### MSC:

 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 68Q45 Formal languages and automata
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### References:

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