Complete Sets under Non-Adaptive Reductions are Scarce
Harry Buhrman, Dieter van Melkebeek
Abstract:
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.