Hard Sets are Hard to Find Download as PS


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:

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 p2-measure zero. Previous to our work, the best published theorem along these lines held for reductions with a constant number of queries. We establish it for tt-reductions that make no(1) queries.