Abstract
Based on Valiant's class # P of all functions counting the number of accepting computations of nondeterministic polynomial-time Turing machines, the polynomial-time hierarchy of counting functions is introduced. The class PHCF of all functions of this hierarchy and some of its subclasses are characterized by recursion-theoretic means. It turns out that, from the recursion-theoretic point of view, PHCF is an analogue to Kalmár's class E of elementary functions, to the class Pspace of polynomial-space computable functions as well as to the class P of polynomial-time computable functions.
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