Complete Sets under NonAdaptive Reductions are Scarce
Harry Buhrman, Dieter van Melkebeek
We investigate the frequency of complete sets for various complexity classes
within EXP under nonadaptive reductions in the sense of resource bounded
measure. We show that these sets are rare:
* The sets that are complete under <=^p_{n^\alpha-tt}reductions for NP, the
levels of the polynomialtime hierarchy, PSPACE, and EXP have p_2-measure
zero for any constant \alpha < 1.
* Assuming MA \neq EXP, the <=^p_{tt}complete sets for PSPACE and the
\Deltalevels of the polynomialtime hierarchy have pmeasure 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^2measure zero. Previous
to our work, the theorem was only known to hold for <=^p_{k-tt}-reductions for
any constant k. We establish it for <=^p_{n^{o(1)}-tt}-reductions.