The complexity and distribution of hard problems

Measure-theoretic aspects of the ⩽_m^P-reducibility structure of exponential time complexity classes E=DTIME(2^linear) 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 ⩽_m^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 ⩽_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 ⩽_m^P-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 ⩽_m^P-degree (e.g. the degree of all ⩽_m^P-complete languages for NP) has measure 0 in E and in E₂