@MISC{Regan94indexsets, author = {Kenneth W. Regan}, title = {Index Sets and Presentations of Complexity Classes}, year = {1994} }

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Abstract

This paper draws close connections between the ease of presenting a given complexity class C and the position of the index sets I C = f i : L(M i ) 2 C g and J C = f i : M i is total L(M i ) = 2 C g in the arithmetical hierarchy. For virtually all classes C studied in the literature, the lowest levels attainable are I C 2 P 0 3 and J C 2 Q 0 2 ; the first holds iff C is \Delta 0 2 -presentable, and the second iff 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 I C = 2 P 0 3 , "many" members of C do not provably (in true \Pi 2 -arithmetic) belong to C. A class H is exhibited such that whether I H 2 P 0 3 is open, and I H = 2 P 0 3 implies that the polynomial hierarchy is infinite. 1 Introduction This paper extend...