On hard sets

On P-selective Sets and EXP Hard Sets

Let PSel be the class of all P-selective sets. We show that DTIME(2nk+a) 6 Pnk T(PSel) for all k; a > 0. It implies EXP 6 Pnc T(PSel) for all c > 0. This greatly improves Toda's result that EXP 6 Ptt(PSel) since Ptt(PSel) is equal to PO(logn) T(PSel). We construct an oracle A such that DTIME(2nk)A PAnk+1+a-T(PSel). We also show that the symmetric di erence of a EPm-hard set and a P-selective se...

On Kolmogorov Complexity of Hard Sets

There are two main informal ideas related to informational content of hard sets. The first says that a set of low informational content can not be hard or complete for a great enough complexity class because of lack of information accessible with bounded resources. One of the formalization of this idea is well-known Hartmanis Conjecture: Sparse sets are not Ccomplete if C is great enough comple...

On Sparse Hard Sets for Counting Classes

Hard Sets are Hard to Find

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: The sets that are complete under 6 p n ?tt-reductions for NP, the levels of the polynomial-time hierarchy, and PSPACE have p 2-measure zero for any constant < 1. The 6 p n c ?T-complete sets...

Sparse Hard Sets for

Sparse hard sets for complexity classes has been a central topic for two decades. The area is motivated by the desire to clarify relationships between completeness/hardness and density of languages and studies the existence of sparse complete/hard sets for various complexity classes under various reducibilities. Very recently, we have seen remarkable progress in this area for low-level complexi...