An excursion to the Kolmogorov random strings

AbstractWe study the sets of resource-bounded Kolmogorov random strings:Rt={x|Ct(n)(x)⩾|x|} fort(n)=2nk. We show that the class of sets that Turing reduce toRthas measure 0 inEXPwith respect to the resource-bounded measure introduced by Lutz. From this we conclude thatRtis not Turing-complete forEXP. This contrasts with the resource-unbounded setting. ThereRis Turing-complete forco-RE. We show that the class of sets to whichRtbounded truth-table reduces, hasp2-measure 0 (therefore, measure 0 inEXP). This answers an open question of Lutz, giving a natural example of a language that is not weakly complete forEXPand that reduces to a measure 0 class inEXP. It follows that the sets that are ⩽pbbt-hard forEXPhavep2-measure 0

Instance complexities of hard and weakly hard problems

This thesis investigates the instance complexities of problems that are hard or weakly hard for exponential time under polynomial time, many-one reductions. It is shown that almost every instance of almost every problem in exponential time has essentially maximal instance complexity. It follows that every weakly hard problem has a dense set of such maximally hard instances. This extends the theorem, due to Orponen, Ko, Schöning and Watanabe (1994), that every hard problem for exponential time has a dense set of maximally hard instances. Complementing this, it is shown that every hard problem for exponential time also has a dense set of unusually easy instances

Random Strings Make Hard Instances

Peer reviewe

Random Strings Make Hard Instances

Peer reviewe

Random Strings Make Hard Instances

We establish the truth of the "instance complexity conjecture" in the case of DEXT-complete sets w.r.t. polynomial time computations, and r.e. complete sets w.r.t. recursive computations. Specifically, we obtain for every DEXT-complete set A an exponentially dense subset C such that for every nondecreasing polynomial t(n) = !(n log n), ic t (x : A) K t (x) \Gamma c holds for some constant c and all x 2 C, where ic t and K t are the t-bounded instance complexity and Kolmogorov complexity measures, respectively. For r.e. complete sets A we obtain an infinite set C ` A such that ic 1 (x : A) K 1 (x) \Gamma c holds for some constant c and all x 2 C. The proofs are based on the observation that Kolmogorov random strings are individually hard to recognize by bounded computations