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

Two queries

AbstractWe consider the question whether two queries SAT are as powerful as one query. We show that if PNP[1]=PNP[2] then: Locally either NP=coNP or NP has polynomial-size circuits; PNP=PNP[1]; Σp2⊆Πp2/1; Σp2=UPNP[1]∩RPNP[1]; PH=BPPNP[1]. Moreover, we extend the work of Hemaspaandra, Hemaspaandra, and Hempel to show that if PΣp2[1]=PΣp2[2] then Σp2=Πp2. We also give a relativized world, where PNP[1]=PNP[2], but NP≠coNP

Quantum Pascal's triangle and Sierpinski's carpet

In this paper we consider a quantum version of Pascal's triangle. Pascal's triangle is a well-known triangular array of numbers and when these numbers are plotted modulo 2, a fractal known as the Sierpinski triangle appears. We first prove the appearance of more general fractals when Pascal's triangle is considered modulo prime powers. The numbers in Pascal's triangle can be obtained by scaling the probabilities of the simple symmetric random walk on the line. In this paper we consider a quantum version of Pascal's triangle by replacing the random walk by the quantum walk known as the Hadamard walk. We show that when the amplitudes of the Hadamard walk are scaled to become integers and plotted m