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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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