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 complexity class [5]. The conjecture have been established for different classes C under various reducibilities in 70–80 (see [7]). For NP the best known result is for polynomial time bounded truthtable reducibility (≤pbtt) [9]. For polynomial time Turing reducibility ≤ p T the conjuncture is still open. In [8] a partial result was obtained: if NP has sparse ≤pT -complete set then polynomial hierarchy collapses at the level ∆2 (lg n). In [4] it was shown that if P 6= NP then there is no ≤ p T -hard for NP set A such that A ⊆ {222 n : n ∈ ω}. The question if it is possible to replace 22 2n by 22 n in the statement was left as open. In the first part of this paper we show that if P 6= NP then any NP-hard set A has infinitely many initial segments with polynomial time Kolmogorov-Markov complexity greater than double logarithm of their length. This implies an affirmative answer to the question above. The second idea related to the informational content of hard sets says that if some set has very high Kolmogorov complexity then information in it is encoded in that way that it is impossible to extract it by resource-bounded reducibilities. In [9] it was shown that if P 6= NP then no language with high space-bounded Kolmogorov complexity can be ≤pbtt-hard for NP. In the work of Book and Lutz [2] languages in ESPACE are investigated which are ≤pbtt-reducible to sets with very high space-bounded Kolmogorov