The Complexity and Distribution of Hard Problems

Measure-theoretic aspects of the <:-reducibility structure of the exponential time complexity classes E=DTIME(21inea‘) and E2 = DTIME(2Po’ynomia’) are investigated. Particular attention is given to the complexity (measured b y the size of complexity cores) and distribution (abundance in the sense of measure) of languages that are <:-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 <:-hard languages for E are unusuaIly simple, i n the sense that they have smaller complexity cores than most languages in E. It follows that the <:-complete languages for E form a measure 0 subset of E (and similarly in Ez). This latter fact is seen to be a special case of a more general theorem, namely, that every <:-degree (e.g., the degree of all <:-complete languages for N P ) has measure 0 in E and in E2.