Sparse Oracles, Lowness, and Highness
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چکیده
The polynomial-time hierarchy has been studied extensively since it lies between the class P of languages accepted deterministically in polynomial time and the class PSPACE of languages accepted (deterministically or nondeterministically) in polynomial space. Since the P =? NP problem is still unsolved, it is not known whether the hierarchy is nontrivial, although it is known that there is a relativization which allows the hierarchy to exist to at least three levels [3]. The purpose of the present paper is to study the role that structural notions such as being sparse, having polynomial-size circuits, etc., have in determining the underlying structure of complexity classes and in particular the underlying structure of the polynomial-time hierarchy. Sch~ning [15] has considered a decomposition of the class NP which depends on the number of distinct levels in the polynomial-time hierarchy. Call a set A in NP "low" if for some n, zP (A) C zP and n n call a set B in NP "high" if for some n, zP C zP(B). Thus, if n 1 -n A is low, then with respect to the operator zP(÷) A does not encode n the power of a quantifier, but if A is high, then with respect to the operator Z~(), A does encode the power of a quantifier. It is easy to see that if a set in NP is both high and low, then the polynomial-
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تاریخ انتشار 1984