CT199703: Buhrman, Harry and van Melkebeek, Dieter (1997) Complete Sets under NonAdaptive Reductions are Scarce. [Report]
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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^\alphatt}reductions for NP, the levels of the polynomialtime hierarchy, PSPACE, and EXP have p_2measure 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_{ktt}reductions for any constant k. We establish it for <=^p_{n^{o(1)}tt}reductions.
Item Type:  Report 

Report Nr:  CT199703 
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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