Limits on the computational power of random strings

2010

We show the following results for polynomial-time reducibility to R C , the set of Kolmogorov random strings. If P ≠ NP, then SAT does not dtt-reduce to R C . If PH does not collapse, then SAT does not n α -tt-reduce to R C for any α If PH does not collapse, then SAT does not n α -T-reduce to R C for any \(\alpha . There is a problem in E that does not dtt-reduce to R C . There is a problem in E that does not n α -tt-reduce to R C , for any α There is a problem in E that does not n α -T-reduce to R C , for any \(\alpha . These results hold for both the plain and prefix-free variants of Kolmogorov complexity and are also independent of the choice of the universal machine.

Annals of Pure and Applied Logic, 2006

We investigate the question of whether one can characterize complexity classes (such as PSPACE or NEXP) in terms of efficient reducibility to the set of Kolmogorovrandom strings R C. This question arises because PSPACE ⊆ P RC and NEXP ⊆ NP RC , and no larger complexity classes are known to be reducible to R C in this way. We show that this question cannot be posed without explicitly dealing with issues raised by the choice of universal machine in the definition of Kolmogorov complexity. What follows is a list of some of our main results.

Bulletin of Symbolic Logic, 2013

We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability.

1995

We study the set of resource bounded Kolmogorov random strings: R t = fx j K t (x) jxjg for t a time constructible function such that t(n) 2 n 2 and t(n) 2 2 n O(1). We show that the class of sets that Turing reduce to R t has measure 0 in EXP with respect to the resource-bounded measure introduced by Lutz Lut92]. From this we conclude that R t is not Turing-complete for EXP. This contrasts the resource unbounded setting. There R is Turing-complete for coRE. We show that the class of sets to which R t bounded truth-table reduces, has p 2-measure 0 (therefore, measure 0 in EXP). This answers an open question of Lutz, giving a natural example of a language that is not weakly-complete for EXP and that reduces to a measure 0 class in EXP. It follows that the sets that are p btt-hard for EXP have p 2-measure 0. The measure in EXP just de ned is known to be nontrivial because of the Measure Conservation Theorem Lut92], stating that EXP does not have p 2-measure 0. Similarly, p-measure and measure in E are de ned as follows De nition 7 A class X f0; 1g 1 has p-measure 0 (and we denote it p (X) = 0) i there exists a martingale d 2 p such that, X S d]. A set X f0; 1g 1 has p-measure 1 (and we denote it p (X) = 1) i X c has p-measure 0.

Logical Methods in Computer Science, 2014

We investigate the truth-table degrees of (co-)c.e.\ sets, in particular, sets of random strings. It is known that the set of random strings with respect to any universal prefix-free machine is Turing complete, but that truth-table completeness depends on the choice of universal machine. We show that for such sets of random strings, any finite set of their truth-table degrees do not meet to the degree~0, even within the c.e. truth-table degrees, but when taking the meet over all such truth-table degrees, the infinite meet is indeed~0. The latter result proves a conjecture of Allender, Friedman and Gasarch. We also show that there are two Turing complete c.e. sets whose truth-table degrees form a minimal pair.

Journal of Computer and System Sciences, 1996

We establish the truth of the \instance complexity conjecture" in the case of DEXT-complete sets over polynomial time computations, and r.e. complete sets over recursive computations. Speci cally, we obtain for every DEXT-complete set A an exponentially dense subset C and a constant k such that for every nondecreasing polynomial t(n) = (n k ), ic t (x : A) K t (x) ? 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 any r.e. complete set A we obtain an in nite set C A such that ic(x : A) K(x) ? c holds for some constant c and all x 2 C, where ic and K denote the time-unbounded versions of instance and Kolmogorov complexities, respectively. The proofs are based on the observation that Kolmogorov random strings are individually hard to recognize by bounded computations.

SIAM Journal on Computing, 2006

We show that sets consisting of strings of high Kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and non-uniform reductions. These sets are provably not complete under the usual many-one reductions. Let R C , R Kt , R KS , R KT be the sets of strings x having complexity at least |x|/2, according to the usual Kolmogorov complexity measure C, Levin's time-bounded Kolmogorov complexity Kt [Lev84], a space-bounded Kolmogorov measure KS, and a new time-bounded Kolmogorov complexity measure KT, respectively. Our main results are:

The starting point of this work is the basic question of whether there exists a formal and meaningful way to limit the computational power that a time bounded randomized Turing Machine can employ on its randomness. We attack this question using a fascinating connection between space and time bounded machines given by Cook [4]: a Turing Machine S running in space s with access to an unbounded stack is equivalent to a Turing Machine T running in time 2 O(s). We extend S with access to a read-only tape containing 2 O(s) uniform random bits, and a usual error regime: one-sided or two-sided, and bounded or unbounded. We study the effect of placing a bound p on the number of passes S is allowed on its random tape. It follows from Cook's results that: • If p = 1 (one-way access) and the error is one-sided unbounded, S is equivalent to deterministic T. • If p = ∞ (unrestricted access), S is equivalent to randomized T (with the same error). As our first two contributions, we completely resolve the case of unbounded error. We show that we cannot meaningfully interpolate between deterministic and randomized T by increasing p: • If p = 1 and the error is two-sided unbounded, S is still equivalent to deterministic T. • If p = 2 and the error is unbounded, S is already equivalent to randomized T (with the same error). In the bounded error case, we consider a logarithmic space Stack Machine S that is allowed p passes over its randomness. Of particular interest is the case p = 2 (log n) i , where n is the input length, and i is a positive integer. Intuitively, we show that S performs polynomial time computation on its input and parallel (preprocessing plus NC i) computation on its randomness. Formally, we introduce Randomness Compilers. In this model, a polynomial time Turing Machine gets an input x and outputs a (polynomial size, bounded fan-in) circuit C x that takes random inputs. Acceptance of x is determined by the acceptance probability of C x. We say that the randomness compiler has depth d if C x has depth d(|x|). As our third contribution, we show that: • S simulates, and is in turn simulated by, a randomness compiler with depth O (log n) i , and O (log n) i+1 , respectively. Randomness Compilers are a formal refinement of polynomial time randomized Turing Machines that might elicit independent interest.

SIAM Journal on Computing, 2000

We study the set of incompressible strings for various resource bounded versions of Kolmogorov complexity. The resource bounded versions of Kolmogorov complexity we study are: polynomial time CD complexity de ned by Sipser, the nondeterministic variant due to Buhrman and Fortnow, and the polynomial space bounded Kolmogorov complexity, CS introduced by Hartmanis. For all of these measures we de ne the set of random strings R CD t , R CND t , and R CS s as the set

Springer eBooks, 2004

We study constructive and resource-bounded scaled dimension as an information content measure and obtain several results that parallel previous work on unscaled dimension. Scaled dimension for finite strings is developed and shown to be closely related to Kolmogorov complexity. The scaled dimension of an infinite sequence is characterized by the scaled dimensions of its prefixes. We obtain an exact Kolmogorov complexity characterization of scaled dimension. Juedes and Lutz (1996) established a small span theorem for P/poly-Turing reductions which asserts that for any problem A in ESPACE, either the class of problems reducible to A (the lower span) or the class of problems to which A is reducible (the upper span) has measure 0 in ESPACE. We apply our Kolmogorov complexity characterization to improve this to (−3) rdorder scaled dimension 0 in ESPACE. As a consequence we obtain a new upper bound on the Kolmogorov complexity of Turing-hard sets for ESPACE.