This editorial contains Vladimir V’yugin’s short biography and a selective review of his contributions to several areas of mathematics, computer science, and their applications.

Abstract Eroders are monotonic cellular automata with a linearly ordered state set that eventually wipe out any finite island of nonzero states. One-dimensional eroders were studied by Gal’perin in the 1970s, who presented a simple combinatorial characterization of the class. The multi-dimensional case has been studied by Toom and others, but no such characterization has been found. We prove a similar characterization for those one-dimensional monotonic cellular automata that are eroders even in the presence of random noise.

We extend the notion of randomness (in the version introduced by Schnorr) to computable Probability Spaces and compare it to a \emph{dynamical} notion of randomness: typicality. Roughly, a point is \emph{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 \emph{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.

Two infinite walks on the same finite graph are called compatible if it is possible to introduce delays into them in such a way that they never collide. About 10 years ago, Peter Winkler asked the question: for which graphs are two independent walks compatible with positive probability. Up to now, no such graphs were found. We show in this paper that large complete graphs have this property. The question is equivalent to a certain dependent percolation with a power-law behavior: the probability that the origin is blocked at distance n but not closer decreases only polynomially fast and not, as usual, exponentially.

Extends algorithmic information theory to quantum mechanics, taking a universal semi-computable density matrix ("universal probability") as a starting point, and defines complexity (an operator) as its negative logarithm. A number of properties of Kolmogorov complexity extend naturally to the new domain. Approximately, a quantum state is simple if it is within a small distance from a low-dimensional subspace of low Kolmogorov complexity. The von-Neumann entropy of a computable density matrix is within an additive constant from the average complexity. Some of the theory of randomness translates to the new domain. We explore the relations of the new quantity to the quantum Kolmogorov complexity defined by P.M.B. Vita/spl acute/nyi (1999) (we show that the latter is sometimes as large as 2n-2 log n) and the qubit complexity defined by A. Berthiaume et al. (2000). The "cloning" properties of our complexity measure are similar to those of qubit complexity.

Two infinite 0-1 sequences are called compatible when it is possible to cast out 0's from both in such a way that they become complementary to each other. Answering a question of Peter Winkler, we show that if the two 0-1-sequences are random i.i.d. and independent from each other, with probability p of 1's, then if p is sufficiently small they are compatible with positive probability. The question is equivalent to a certain dependent percolation with a power-law behavior: the probability that the origin is blocked at distance n but not closer decreases only polynomially fast and not, as usual, exponentially.

While Kolmogorov (1965, 1983) complexity is the accepted absolute measure of information content of an individual finite object, a similarly absolute notion is needed for the relation between an individual data sample and an individual model summarizing the information in the data, for example, a finite set (or probability distribution) where the data sample typically came from. The statistical theory based on such relations between individual objects can be called algorithmic statistics, in contrast to classical statistical theory that deals with relations between probabilistic ensembles. We develop the algorithmic theory of statistic, sufficient statistic, and minimal sufficient statistic. This theory is based on two-part codes consisting of the code for the statistic (the model summarizing the regularity, the meaningful information, in the data) and the model-to-data code. In contrast to the situation in probabilistic statistical theory, the algorithmic relation of (minimal) sufficiency is an absolute relation between the individual model and the individual data sample. We distinguish implicit and explicit descriptions of the models. We give characterizations of algorithmic (Kolmogorov) minimal sufficient statistic for all data samples for both description modes-in the explicit mode under some constraints. We also strengthen and elaborate on earlier results for the "Kolmogorov structure function" and "absolutely nonstochastic objects"-those objects for which the simplest models that summarize their relevant information (minimal sufficient statistics) are at least as complex as the objects themselves. We demonstrate a close relation between the probabilistic notions and the algorithmic ones: (i) in both cases there is an "information non-increase" law; (ii) it is shown that a function is a probabilistic sufficient statistic iff it is with high probability (in an appropriate sense) an algorithmic sufficient statistic.

While Kolmogorov (1965) complexity is the accepted absolute measure of information content in an individual finite object, a similarly absolute notion is needed for the information distance between two individual objects, for example, two pictures. We give several natural definitions of a universal information metric, based on length of shortest programs for either ordinary computations or reversible (dissipationless) computations. It turns out that these definitions are equivalent up to an additive logarithmic term. We show that the information distance is a universal cognitive similarity distance. We investigate the maximal correlation of the shortest programs involved, the maximal uncorrelation of programs (a generalization of the Slepian-Wolf theorem of classical information theory), and the density properties of the discrete metric spaces induced by the information distances. A related distance measures the amount of nonreversibility of a computation. Using the physical theory of reversible computation, we give an appropriate (universal, antisymmetric, and transitive) measure of the thermodynamic work required to transform one object in another object by the most efficient process. Information distance between individual objects is needed in pattern recognition where one wants to express effective notions of "pattern similarity" or "cognitive similarity" between individual objects and in thermodynamics of computation where one wants to analyze the energy dissipation of a computation from a particular input to a particular output.

In a noisy cellular automaton, even if it is infinite, it is non-trivial to keep a bit of information for more than a constant number of steps. A clever solution in 2 dimensions has been applied to a simple 3-dimensional fault-tolerant cellular automaton. This technique did not solve the following problems: remembering a bit of information in 1 dimension; computing in dimensions lower than 3, or with non-synchronized transitions. With a more complex technique using a hierarchy of simulations, we construct an asynchronous one-dimensional reliable cellular automaton, which is also "self-organizing". This means that if the input information has constant size, the initial configuration can be homogenous: the hierarchy organizes itself. An application to information storage in positive-temperature Gibbs states is also given.

Proves that the reliable computation of any Boolean function with sensitivity s requires /spl Omega/(s log s) gates if the gates fail independently with a fixed positive probability. This theorem was stated by Dobrushin and Ortyukov (1977), but their proof was found by Pippenger, Stamoulis, and Tsitsiklis (1991) to contain some errors.>

Applying the tools of algorithmic information theory, we compare several candidates for an asymptotically machine-independent. absolute measure of the informational or ``cognitive`` distance between discrete objects x and y. The maximum of the conditional Kolmogorov complexities max{l_brace}K(y{vert_bar}z) K(m{vert_bar}y){r_brace}, is shown to be optimal, in the sense of being minimal within an additive constant among semicomputable, symmetric, positive semidefinite functions of z and y satisfying a reasonable normalization condition and obeying the triangle intequality. The optimal metric, in turn, differs by at most an additive logarithmic term from the size of the smallest program for a universal reversible computer to transform x into y. This program functions in a `catalytic`` capacity, being retained in the computer before, during, and after the computation. Similarly, the sum of the conditional complexities. K(y{vert_bar}x) + K(x{vert_bar}y), is shown to be equal within a logarithmic term to the minimal amount Of information flowing out and in during a reversible computation in which the program is not retained. Finally. using the physical theory of reversible computation, it is shown that the simple difference K(x) - K(y) is an appropriate (ie universal, antisymmetric, and transitive) measure of the amount of thermodynamic work required to transform string x into string y by the most efficient process.

We build a three-dimensional array of unreliable cellular automata that can simulate a universal Turing machine (more generally, a one-dimensional universal iterative array) reliably. This is the first reliable real-time simulation. The encoding is simple repetition, and no decoding is needed. The construction is based on Toom's work.

We construct a one-dimensional array of cellular automata on which arbitrarily large computations can be implemented reliably, even though each automaton at each step makes an error with some constant probability. In statistical physics, this construction leads to the refutation of the “positive probability conjecture,” which states that any one-dimensional infinite particle system with positive transition probabilities is ergodic. Our approach takes its origin from Kurdyumov's ideas for this refutation. To compute reliability with unreliable components, von Neumann proposed Boolean circuits whose intricate interconnection pattern (arising from the error-correcting organization) he had to assume to be immune to errors. In a uniform cellular medium, the error-correcting organization exists only in “software,” therefore errors threaten to disable it. The real technical novelty of the paper is therefore the construction of a self-repairing organization.

We build a three-dimensional array of unreliable cellular automata that can simulate a universal Turing machine (more generally, a one-dimensional universal iterative array) reliably. This is the first reliable real-time simulation. The encoding is simple repetition, and no decoding is needed. The construction is based on Toom's work.

We construct a one-dimensional array of cellular automata on which arbitrarily large computations can be implemented reliably, even though each automaton at each step makes an error with some constant probability. To compute reliably with unreliable components, von Neumann proposed Boolean circuits whose intricate interconnection pattern (arising from the error-correcting organization) he had to assume to be immune to errors. In a uniform cellular medium, the error-correcting organization exists only in “software”, therefore errors threaten to disable it. The real technical novelty of the paper is therefore the construction of a self-repairing organization.

Several results in Algorithmic Information Theory establish upper bounds on the difference between descriptional complexity and the logarithm of "apriori probability". It was conjectured that these two quantities coincide to within an additive constant. Here, we disprove this conjecture and show that the known overall upper bound on the difference is exact. The proof uses a memory-allocation game between two players called User and Server. User sends incremental requests of memory space for certain structured items, Server allocates this space in a write-once memory. For each item, some of the allocated space is required to be in one piece, in order to live a short address. We also present some related results.