CiE'10 Proceedings of the Programs, proofs, process and 6th international conference on Computability in Europe
Generic density and small span theorem
FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
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Measure-theoretic aspects of the /spl les//sub m//sup P/-reducibility structure of exponential time complexity classes E=DTIME(2/sup linear/) and E/sub 2/=DTIME(2/sup polynomial/) are investigated. Particular attention is given to the complexity (measured by the size of complexity cores) and distribution (abundance in the sense of measure) of languages that are /spl les//sub m//sup P/-hard for E and other complexity classes. Tight upper and lower bounds on the size of complexity cores of hard languages are derived. The upper bounds say that the /spl les//sub m//sup P/-hard languages for E are unusually simple in, the sense that they have smaller complexity cores than most languages in E. It follows that the /spl les//sub m//sup P/-complete languages for E form a measure 0 subset of E (and similarly in E/sub 2/). This latter fact is seen to be a special case of a more general theorem, namely, that every /spl les//sub m//sup P/-degree (e.g. the degree of all /spl les//sub m//sup P/-complete languages for NP) has measure 0 in E and in E/sub 2/.