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The complexity and distribution of hard problems

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2 Author(s)
Juedes, D.W. ; Dept. of Comput. Sci., Iowa State Univ., Ames, IA, USA ; Lutz, J.H.

Measure-theoretic aspects of the ⩽mP-reducibility structure of exponential time complexity classes E=DTIME(2linear) and E2=DTIME(2 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 ⩽mP-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 ⩽m 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 ⩽mP-complete languages for E form a measure 0 subset of E (and similarly in E2). This latter fact is seen to be a special case of a more general theorem, namely, that every ⩽mP-degree (e.g. the degree of all ⩽mP-complete languages for NP) has measure 0 in E and in E2

Published in:

Foundations of Computer Science, 1993. Proceedings., 34th Annual Symposium on

Date of Conference:

3-5 Nov 1993