Abstract
The nature of the recursive real μ(UB(x)) in Structural Complexity Theory is discussed, in which μ is Lebesgue measure of random oracles and UB(x) denotes the set of every oracle Z such that the polynomial-time computable oracle-dependent language Bz accepts x. To formalize this argument we introduced the class Lℬ of any language Lm(B) expressing a lower bound of its measure of random oracles on an oracle-dependent language Bz in a relativized class ℬz, and we investigated the nature of this class. The class Cℬ of languages was further introduced to examine the circumstances of Lℬ, each language L of which indicates a lower bound of the cardinality of elements in L by exponentially-many searchings. Originating from the P-hierarchy, the related LP- and CP-hierarchies was obtainable and their structures were also studied. Our best result for the lower and upper bounds of the computational complexity of LP is that PP ≤ pm LP⊆PSPACE. More from an angle of relativization, a natural relativization of each Lℬ and Cℬ is treated, and the probability-one separations of the first levels of the LP- and CP-hierarchies are shown.
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© 1989 Springer-Verlag
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Yamakami, T. (1989). Computational complexity of languages counting random oracles. In: Shinoda, J., Tugué, T., Slaman, T.A. (eds) Mathematical Logic and Applications. Lecture Notes in Mathematics, vol 1388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083671
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DOI: https://doi.org/10.1007/BFb0083671
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