A Toom rule that increases the thickness of sets

Toom's north-east-self voting cellular automaton rule R is known to suppress small minorities. A variant which we call R^+ is also known to turn an arbitrary initial configuration into a homogenous one (without changing the ones that were homogenous to start with). Here we show that R^+ always increases a certain property of sets called thickness. This result is intended as a step towards a proof of the fast convergence towards consensus under R^+. The latter is observable experimentally, even in the presence of some noise.Comment: 16 pages, 8 figure

Randomness on computable probability spaces - A dynamical point of view

We extend the notion of randomness (in the version introduced by Schnorr) to computable probability spaces and compare it to a dynamical notion of randomness: typicality. Roughly, a point is typical for some dynamic, if it follows the statistical behavior of the system (Birkhoff’s pointwise ergodic theorem). We prove that a point is Schnorr random if and only if it is typical for every mixing computable dynamics. To prove the result we develop some tools for the theory of computable probability spaces (for example, morphisms) that are expected to have other applications

Inequalities for space-bounded Kolmogorov complexity

There is a parallelism between Shannon information theory and algorithmic information theory. In particular, the same linear inequalities are true for Shannon entropies of tuples of random variables and Kolmogorov complexities of tuples of strings (Hammer et al., 1997), as well as for sizes of subgroups and projections of sets (Chan, Yeung, Romashchenko, Shen, Vereshchagin, 1998--2002). This parallelism started with the Kolmogorov-Levin formula (1968) for the complexity of pairs of strings with logarithmic precision. Longpr\'e (1986) proved a version of this formula for space-bounded complexities. In this paper we prove an improved version of Longpr\'e's result with a tighter space bound, using Sipser's trick (1980). Then, using this space bound, we show that every linear inequality that is true for complexities or entropies, is also true for space-bounded Kolmogorov complexities with a polynomial space overhead.Comment: [Extended version, with full proofs added; some corrections are made

International Trade Issues of the Russian Federation

Trade and capital flows between Russia and the rest of the world are now significant for both partners. The economic reforms introduced in Russia since 1991 have converted an autarkic, highly regulated economy into a relatively open one. The dramatic change followed from the abolition of central planning and complex exchange rate controls as Yeltsin came to power in Russia and the Soviet Union collapsed. Yet the years since 1991 are not simply a record of tearing down trade barriers. Instead Russia's role in the international economy appears to be erratic and inconsistent. Also the transformation of earlier inter-republic deliveries between former republics of the Soviet Union to trade between independent states implied the sometimes controversial establishment of new trade barriers. The country's struggle to develop a viable trade policy provides unique insights into the consequences of the conflicts of economic ideas: free trade versus protectionism; rewards for economic efficiency versus social equity; and macroeconomic stability versus maintaining employment. The clash among policy proposals has been reflected in political struggles, for the decisions on these matters have an impact on the lives of the 179 million Russians. The papers that make up this volume are from a conference held in May 1994 at IIASA, in Laxenburg, Austria. The conference was on Russia's international trade issues, aside from its ties to the republics of the former Soviet Union, a topic of another conference in 1993

The clairvoyant demon has a hard task

ABSTRACT. Consider the integer lattice L = Z 2. For some m ≥ 4, let us color each column of this lattice independently and uniformly into one of m colors. We do the same for the rows, independently from the columns. A point of L will be called blocked if its row and column have the same color. We say that this random configuration percolates if there is a path in L starting at the origin, consisting of rightward and upward unit steps, and avoiding the blocked points. As a problem arising in distributed computing, it has been conjectured that for m ≥ 4, the configuration percolates with positive probability. This has now been proved (in a later paper) for large m. We prove that the probability that there is percolation to distance n but not to infinity is not exponentially small in n. This narrows the range of methods available for proving the conjecture. 1. STATEMENT OF THE RESULT 1.1. Introduction. Let x = (x(0), x(1),...) be an infinite sequence and u = (u(0), u(1),...) be a binary sequence with elements in {0, 1}. Let sn = ∑ n−1 i=0 u(i). We define the delayed version x (u) of x, b

Correction to: Bounds on conditional probabilities with applications

Ahlswede R, Gács P, Körner J. Correction to: Bounds on conditional probabilities with applications. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete. 1977;39(4):353-354

Reliable Cellular Automata With Self-Organization

In a probabilistic cellular automaton in which all local transitions have positive probability, the problem of keeping a bit of information indefinitely is nontrivial, even in an infinite automaton. Still, there is a solution in 2 dimensions, and this solution can be used to construct a simple 3-dimensional discrete-time universal fault-tolerant cellular automaton. This technique does not help muchtosolve the following problems: remembering a bit of information in 1 dimension; computing in dimensions lower than 3; computing in any dimension with non-synchronized transitions. Our mor