Hard Sets are Hard to Find

We investigate the frequency of complete sets for various complexity classes within EXP under several polynomial-time reductions in the sense of resource bounded measure. We show that these sets are scarce: The sets that are complete under 6 p n ?tt-reductions for NP, the levels of the polynomial-time hierarchy, and PSPACE have p 2-measure zero for any constant < 1. The 6 p n c ?T-complete sets for EXP have p 2-measure zero for any constant c. Assuming MA 6 = EXP, the 6 p tt-complete sets for EXP have p-measure zero. A key ingredient is the Small Span Theorem, which states that for any set A in EXP at least one of its lower span (i.e., the sets that reduce to A) or its upper span (i.e., the sets that A reduces to) has p 2-measure zero. Previous to our work, the theorem was only known to hold for 6 p btt-reductions. We establish it for 6 p n o(1) ?tt-reductions.