CT-1997-03: Complete Sets under Non-Adaptive Reductions are Scarce

CT-1997-03: Buhrman, Harry and van Melkebeek, Dieter (1997) Complete Sets under Non-Adaptive Reductions are Scarce. [Report]

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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.

Item Type: Report
Report Nr: CT-1997-03
Series Name: Computation and Complexity Theory (CT)
Year: 1997
Date Deposited: 12 Oct 2016 14:39
Last Modified: 12 Oct 2016 14:39
URI: https://eprints.illc.uva.nl/id/eprint/1074

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