Definition and Existence of Super Complexity Cores

In this paper we deene and study super complexity cores of languages L with respect to classes C with L 6 2 C. A super complexity core S of L can be considered as an innnite set of strings for which the decision problem for L is very hard to solve with respect to the available \resources" xed by C even for algorithms which have to compute the correct result only for all inputs x 2 S. For example let C = P and S be a super complexity core of L. Then S is innnite and all deterministic Turing machines M , which output 1 on input x 2 S \ L and 0 on input x 2 S \ L, need more than polynomially many steps on all but nitely many inputs x 2 S. We prove that for all non-empty, countable classes of languages C which are closed under nite variation, nite union, and under complement and for all languages L 6 2 C it follows that such a super complexity core of L with respect to C exists. Moreover we show: Given a recursively enumerable class C of languages and a recursive language L, if there is a super complexity core of L with respect to C, then there exists a recursive super complexity core, too. Thus for L 6 2 BPP (PP, PSPACE, : : :) there exists a set S such that all BPP Turing machines (PP Turing machines, PSPACE Turing machines, : : :), which compute the characteristic function of L correctly at least on inputs x 2 S, need more than polynomially many steps (tape cells) on almost all inputs x 2 S.