Counting dependent and independent strings

The paper gives estimations for the sizes of the the following sets: (1) the set of strings that have a given dependency with a fixed string, (2) the set of strings that are pairwise \alpha independent, (3) the set of strings that are mutually \alpha independent. The relevant definitions are as follows: C(x) is the Kolmogorov complexity of the string x. A string y has \alpha -dependency with a string x if C(y) - C(y|x) \geq \alpha. A set of strings {x_1, \ldots, x_t} is pairwise \alpha-independent if for all i different from j, C(x_i) - C(x_i | x_j) \leq \alpha. A tuple of strings (x_1, \ldots, x_t) is mutually \alpha-independent if C(x_{\pi(1)} \ldots x_{\pi(t)}) \geq C(x_1) + \ldots + C(x_t) - \alpha, for every permutation \pi of [t]

Impossibility of independence amplification in Kolmogorov complexity theory

The paper studies randomness extraction from sources with bounded independence and the issue of independence amplification of sources, using the framework of Kolmogorov complexity. The dependency of strings $x$ and $y$ is ${\rm dep}(x,y) = \max\{C(x) - C(x \mid y), C(y) - C(y\mid x)\}$, where $C(\cdot)$ denotes the Kolmogorov complexity. It is shown that there exists a computable Kolmogorov extractor $f$ such that, for any two $n$-bit strings with complexity $s(n)$ and dependency $\alpha(n)$, it outputs a string of length $s(n)$ with complexity $s(n)- \alpha(n)$ conditioned by any one of the input strings. It is proven that the above are the optimal parameters a Kolmogorov extractor can achieve. It is shown that independence amplification cannot be effectively realized. Specifically, if (after excluding a trivial case) there exist computable functions $f_1$ and $f_2$ such that ${\rm dep}(f_1(x,y), f_2(x,y)) \leq \beta(n)$ for all $n$-bit strings $x$ and $y$ with ${\rm dep}(x,y) \leq \alpha(n)$, then $\beta(n) \geq \alpha(n) - O(\log n)$

On double Hurwitz numbers in genus 0

We study double Hurwitz numbers in genus zero counting the number of covers \CP^1\to\CP^1 with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise polynomials in the multiplicities of the preimages of the branching points. We describe the partition of the parameter space into polynomiality domains, called chambers, and provide an expression for the difference of two such polynomials for two neighboring chambers. Besides, we provide an explicit formula for the polynomial in a certain chamber called totally negative, which enables us to calculate double Hurwitz numbers in any given chamber as the polynomial for the totally negative chamber plus the sum of the differences between the neighboring polynomials along a path connecting the totally negative chamber with the given one.Comment: 17 pages, 3 figure

Upper bounds for the number of orbital topological types of planar polynomial vector fields "modulo limit cycles"

The paper deals with planar polynomial vector fields. We aim to estimate the number of orbital topological equivalence classes for the fields of degree n. An evident obstacle for this is the second part of Hilbert's 16th problem. To circumvent this obstacle we introduce the notion of equivalence modulo limit cycles. This paper is the continuation of the author's paper in [Mosc. Math. J. 1 (2001), no. 4] where the lower bound of the form 2^{cn^2} has been obtained. Here we obtain the upper bound of the same form. We also associate an equipped planar graph to every planar polynomial vector field, this graph is a complete invariant for orbital topological classification of such fields.Comment: 23 pages, 5 figure

MDL Convergence Speed for Bernoulli Sequences

The Minimum Description Length principle for online sequence estimation/prediction in a proper learning setup is studied. If the underlying model class is discrete, then the total expected square loss is a particularly interesting performance measure: (a) this quantity is finitely bounded, implying convergence with probability one, and (b) it additionally specifies the convergence speed. For MDL, in general one can only have loss bounds which are finite but exponentially larger than those for Bayes mixtures. We show that this is even the case if the model class contains only Bernoulli distributions. We derive a new upper bound on the prediction error for countable Bernoulli classes. This implies a small bound (comparable to the one for Bayes mixtures) for certain important model classes. We discuss the application to Machine Learning tasks such as classification and hypothesis testing, and generalization to countable classes of i.i.d. models.Comment: 28 page

Noncomputability Arising In Dynamical Triangulation Model Of Four-Dimensional Quantum Gravity

Computations in Dynamical Triangulation Models of Four-Dimensional Quantum Gravity involve weighted averaging over sets of all distinct triangulations of compact four-dimensional manifolds. In order to be able to perform such computations one needs an algorithm which for any given $N$ and a given compact four-dimensional manifold $M$ constructs all possible triangulations of $M$ with $\leq N$ simplices. Our first result is that such algorithm does not exist. Then we discuss recursion-theoretic limitations of any algorithm designed to perform approximate calculations of sums over all possible triangulations of a compact four-dimensional manifold.Comment: 8 Pages, LaTex, PUPT-132

Genus expansion for real Wishart matrices

We present an exact formula for moments and cumulants of several real compound Wishart matrices in terms of an Euler characteristic expansion, similar to the genus expansion for complex random matrices. We consider their asymptotic values in the large matrix limit: as in a genus expansion, the terms which survive in the large matrix limit are those with the greatest Euler characteristic, that is, either spheres or collections of spheres. This topological construction motivates an algebraic expression for the moments and cumulants in terms of the symmetric group. We examine the combinatorial properties distinguishing the leading order terms. By considering higher cumulants, we give a central limit-type theorem for the asymptotic distribution around the expected value

Algorithmic Complexity for Short Binary Strings Applied to Psychology: A Primer

Since human randomness production has been studied and widely used to assess executive functions (especially inhibition), many measures have been suggested to assess the degree to which a sequence is random-like. However, each of them focuses on one feature of randomness, leading authors to have to use multiple measures. Here we describe and advocate for the use of the accepted universal measure for randomness based on algorithmic complexity, by means of a novel previously presented technique using the the definition of algorithmic probability. A re-analysis of the classical Radio Zenith data in the light of the proposed measure and methodology is provided as a study case of an application.Comment: To appear in Behavior Research Method