Index Sets and Presentations of Complexity Classes ( revised
نویسنده
چکیده
This paper draws close connections between the ease of presenting a given complexity class C and the position of the index sets IC = f i : L(Mi) 2 C g and JC = f i : Mi is total ^ L(Mi) = 2 C g in the arithmetical hierarchy. For virtually all classes C studied in the literature, the lowest levels attainable are IC 2 P03 and JC 2 Q02; the rst holds i C is 02-presentable, and the second i C is recursively presentable. A general kind of priority argument is formulated, and it is shown that every property enforcible by it is not recursively presentable. It follows that the classes of P-immune and P-biimmune languages in exponential time are not recursively presentable. It is shown that for all C with IC = 2P03, \many" members of C do not provably (in true 2-arithmetic) belong to C. A class H is exhibited such that whether IH 2 P03 is open, and IH = 2 P03 implies that the polynomial hierarchy is in nite.
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