Showing papers on "Randomness published in 1997"

An introduction to Kolmogorov complexity and its applications

Abstract: The book is outstanding and admirable in many respects. ... is necessary reading for all kinds of readers from undergraduate students to top authorities in the field. Journal of Symbolic Logic Written by two experts in the field, this is the only comprehensive and unified treatment of the central ideas and their applications of Kolmogorov complexity. The book presents a thorough treatment of the subject with a wide range of illustrative applications. Such applications include the randomness of finite objects or infinite sequences, Martin-Loef tests for randomness, information theory, computational learning theory, the complexity of algorithms, and the thermodynamics of computing. It will be ideal for advanced undergraduate students, graduate students, and researchers in computer science, mathematics, cognitive sciences, philosophy, artificial intelligence, statistics, and physics. The book is self-contained in that it contains the basic requirements from mathematics and computer science. Included are also numerous problem sets, comments, source references, and hints to solutions of problems. New topics in this edition include Omega numbers, KolmogorovLoveland randomness, universal learning, communication complexity, Kolmogorov's random graphs, time-limited universal distribution, Shannon information and others.

Inference for Non-random Samples

Abstract: Observational data are often analysed as if they had resulted from a controlled study, and yet the tacit assumption of randomness can be crucial for the validity of inference. We take some simple statistical models and supplement them by adding a parameter θ which reflects the degree of non-randomness in the sample. For a randomized study θ is known to be 0. We examine the profile log-likelihood for θ and the sensitivity of inference to small non-zero values of θ. Particular models cover the analysis of survey data with item non-response, the paired comparison t-test and two group comparisons using observational data with covariates. Some practical examples are discussed. Allowing for sampling bias increases the uncertainty of estimation and weakens the significance of treatment effects, sometimes substantially so.

Making Sense of Randomness" Implicit Encoding as a Basis for Judgment

Abstract: People attempting to generate random sequences usually produce more alternations than expected by chance. They also judge overalternating sequences as maximally random. In this article, the authors review findings, implications, and explanatory mechanisms concerning subjective randomness. The authors next present the general approach of the mathematical theory of complexity, which identifies the length of the shortest program for reproducing a sequence with its degree of randomness. They describe three experiments, based on mean group responses, indicating that the perceived randomness of a sequence is better predicted by various measures of its encoding difficulty than by its objective randomness. These results seem to imply that in accordance with the complexity view, judging the extent of a sequence's randomness is based on an attempt to mentally encode it. The experience of randomness may result when this attempt fails. Judging a situation as more or less random is often the key to important cognitions and behaviors. Perceiving a situation as nonchance calls for explanations, and it marks the onset of inductive inference (Lopes, 1982). Lawful environments encourage a coping orientation. One may try to control a situation by predicting its outcome, replicating, changing, or even by avoiding it. In contrast, there seems to be no point in patterning our behavior in a random environment. Although people feel that they know what they mean when speaking of randomness (Kac, 1983) and they communicate in everyday and professional affairs using their shared intuitive understanding of the term, it.is one of the most elusive concepts in mathematics. Randomness resists easy or precise definition, nor is there a decisive test for determining its presence (Ayton, Hunt, & Wright, 1989, 1991; Chaitin, 1975; Falk, 1991; Lopes, 1982; Pollatsek & Konold, 1991; Wagenaar, 1972a, 1991; Zabell, 1992). Attempted definitions of randomness involvo intricate philosophical and mathematical problems (Ayer, 1965;

Estimation and Prediction for a Class of Dynamic Nonlinear Statistical Models

Abstract: A class of nonlinear state-space models, characterized by a single source of randomness, is introduced. A special case, the model underpinning the multiplicative Holt-Winters method of forecasting, is identified. Maximum likelihood estimation based on exponential smoothing instead of a Kalman filter, and with the potential to be applied in contexts involving non-Gaussian disturbances, is considered. A method for computing prediction intervals is proposed and evaluated on both simulated and real data.

Software reliability via run-time result-checking

Abstract: We review the field of result-checking, discussing simple checkers and self-correctors. We argue that such checkers could profitably be incorporated in software as an aid to efficient debugging and enhanced reliability. We consider how to modify traditional checking methodologies to make them more appropriate for use in real-time, real-number computer systems. In particular, we suggest that checkers should be allowed to use stored randomness: that is, that they should be allowed to generate, preprocess, and store random bits prior to run-time, and then to use this information repeatedly in a series of run-time checks. In a case study of checking a general real-number linear transformation (e.g., a Fourier Transform), we present a simple checker which uses stored randomness, and a self-corrector which is particularly efficient if stored randomness is employed.

Unit cells for the simulation of hexagonal ice

Abstract: A number of periodic lattices have historically been used to represent ice-1h in computer simulations. These vary in size, shape, and method of generation, and while they have served their intended purposes, their properties have rarely been documented in detail and their intercompatibility is unknown. We develop a method for generating sets of internally consistent lattices and apply it to determine eight unit cells containing from 96 to 768 water molecules in both near-cubic and slab arrangements. It can easily be applied to generate additional (larger) cells or representations of specific crystal faces. Each unit cell in this set has zero net dipole moment and minimal net quadrupole moment and is optimized using four different criteria to measure the randomness of the hydrogen bonding; if required, these criteria can easily be modified to suit the intended application and alternate sets thus generated. We find that Cota and Hoover’s much used constraint for selecting unit cells with zero dipole moment is too restrictive, not permitting a fully random hydrogen-bonding network; also, unit-cell generation methods based on potential-energy minimization are found to prefer unrepresentative, highly ordered structures.

Not all (possibly) “random” sequences are created equal

Abstract: The need to assess the randomness of a single sequence, especially a finite sequence, is ubiquitous, yet is unaddressed by axiomatic probability theory. Here, we assess randomness via approximate entropy (ApEn), a computable measure of sequential irregularity, applicable to single sequences of both (even very short) finite and infinite length. We indicate the novelty and facility of the multidimensional viewpoint taken by ApEn, in contrast to classical measures. Furthermore and notably, for finite length, finite state sequences, one can identify maximally irregular sequences, and then apply ApEn to quantify the extent to which given sequences differ from maximal irregularity, via a set of deficit (defm) functions. The utility of these defm functions which we show allows one to considerably refine the notions of probabilistic independence and normality, is featured in several studies, including (i) digits of e, π, √2, and √3, both in base 2 and in base 10, and (ii) sequences given by fractional parts of multiples of irrationals. We prove companion analytic results, which also feature in a discussion of the role and validity of the almost sure properties from axiomatic probability theory insofar as they apply to specified sequences and sets of sequences (in the physical world). We conclude by relating the present results and perspective to both previous and subsequent studies.

Computational Physics: Problem Solving with Computers

Abstract: Partial table of contents: GENERALITIES. Computing Software Basics. Errors and Uncertainties in Computations. APPLICATIONS. Data Fitting. Deterministic Randomness. Monte Carlo Applications. Differentiation. Differential Equations and Oscillations. Anharmonic Oscillations. Unusual Dynamics of Nonlinear Systems. Differential Chaos in Phase Space. Matrix Computing and Subroutine Libraries. Bound States in Momentum Space. Computing Hardware Basics: Memory and CPU. High-Performance Computing: Profiling and Tuning. Object-Oriented Programming: Kinematics. Thermodynamic Simulations: The Ising Model. Fractals. PARTIAL DIFFERENTIAL EQUATIONS. Heat Flow. Waves on a String. NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS. Solitons, The KdeV Equation. Confined Electronic Wave Packets. Appendices. Glossary. References. Index.

Probability and random processes for electrical engineers

Abstract: The basic concepts the concept of a random variable a vector random variable introduction to estimation sequence of (IID) random variables random processes the Poisson and Gaussian random processes processing of random processes Markov chains case study - a bus-based switch architecture. Appendices: set theory primer counting methods the historical development of the theory modelling of randomness in engineering systems - a summary.

Brownian motion on a random recursive Sierpinski gasket

Abstract: We introduce a random recursive fractal based on the Sierpinski gasket and construct a diffusion upon the fractal via a Dirichlet form. This form and its symmetrizing measure are determined by the electrical resistance of the fractal. The effective resistance provides a metric with which to discuss the properties of the fractal and the diffusion. The main result is to obtain uniform upper and lower bounds for the transition density of the Brownian motion on the fractal in terms of this metric. The bounds are not tight as there are logarithmic corrections due to the randomness in the environment, and the behavior of the shortest paths in the effective resistance metric is not well understood. The results are deduced from the study of a suitable general branching process.

Real-space renormalization-group study of the two-dimensional Blume-Capel model with a random crystal field

Abstract: The phase diagram of the two-dimensional Blume-Capel model with a random crystal field is investigated within the framework of a real-space renormalization-group approximation. Our results suggest that, for any amount of randomness, the model exhibits a line of Ising-like continuous transitions, as in the pure model, but no first-order transition. At zero temperature the transition is also continuous, but not in the same universality class as the Ising model. In this limit, the attractor (in the renormalization-group sense) is the percolation fixed point of the site diluted spin-1/2 Ising model. The results we found are in qualitative agreement with general predictions made by Berker and Hui on the critical behavior of random models.

Non-Hermitian random matrix models: Free random variable approach

Abstract: Using the standard concepts of free random variables, we show that for a large class of non-Hermitian random matrix models, the support of the eigenvalue distribution follows from their Hermitian analogs using a conformal transformation. We also extend the concepts of free random variables to the class of non-Hermitian matrices, and apply them to the models discussed by Ginibre-Girko (elliptic ensemble) [J. Ginibre, J. Math. Phys. 6, 1440 (1965); V. L. Girko, Spectral Theory of Random Matrices (in Russian) (Nauka, Moscow, 1988)] and Mahaux-Weidenm\"uller (chaotic resonance scattering) [C. Mahaux and H. A. Weidenm\"uller, Shell-model Approach to Nuclear Reactions (North-Holland, Amsterdam, 1969)].

Universal concept of complexity by the dynamic redundance paradigm: Causal randomness, complete wave mechanics, and the ultimate unification of knowledge

Abstract: The fundamental impasses and ruptures in various domains of the canonical, unitary science, or the 'end of science', become the more and more evident. The natural unity of being is recovered within a universal nonperturbative method leading to the dynamic redundance paradigm. It is shown that the unreduced behaviour of any nontrivial (isolated) system includes many equally real, but incompatible dynamic regimes, each of them being roughly equivalent to an ordinary 'complete' solution of the unitary science. Therefore the regimes should 'spontaneously' and randomly replace one another, which provides a universal, purely dynamic origin of randomness. The discovered dynamic redundance leads to the universal, reality-based concept of dynamic complexity and its permanently developing hierarchical structure, alias the world. Its lowest levels give the causally complete mechanics of dynamically quantized elementary fields, extending the double solution found by Louis de Broglie. One obtains a physically complete solution for the 'mysteries' of quantum mechanics, unifyng it with the extended, causal versions of relativity (emergent space and time), quantum and classical gravity (dynamical mass), field theory (electric charge and spin), particle physics, and cosmology. The same key features of the unreduced dynamic complexity determine behaviour of any system at higher levels of complexity described by a case of the single, universal equation in its two related versions, the extended Lagrange-Hamilton (trajectorial) and Schroedinger (distributional) equations. The end of the unitary science opens the renaissance of the ultimately complete and universal understanding initiated by Rene Descartes (but then mechanistically falsified), which is confirmed by many sound, practically important results.

Microstructure functions for a model of statistically inhomogeneous random media

Abstract: We propose a model for statistically inhomogeneous two-phase random media, including functionally graded materials, consisting of inhomogeneous fully penetrable (Poisson distributed) spheres. This model can be constructed for any specified variation of volume fraction and permits one to represent and evaluate certain n-point correlation functions that statistically characterize the microstructure for this model. Unlike the case of statistically homogeneous media, the microstructure functions depend upon the absolute positions of their arguments. However, as with homogeneous random media, this microstructural information will be useful in obtaining rigorous estimates of the effective properties of such systems.

Reducing randomness via irrational numbers

Abstract: We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently com- putable approximations of suitable irrational points. In contrast to the classical technique of DeMillo, Lipton, Schwartz, and Zippel, this methodology can decrease the error probability by increasing the precision of the approximations instead of using more random bits. Consequently, randomized algo- rithms that use the classical technique can generally be improved using the new methodology. To demonstrate the methodology, we discuss two nontrivial applications. The first is to decide whether a graph has a perfect matching in parallel. Our new NC algorithm uses fewer random bits while doing less work than the previously best NC algorithm by Chari, Rohatgi, and Srinivasan. The second application is to test the equality of two multisets of integers. Our new algorithm improves upon the previously best algorithms by Blum and Kannan and can speed up their checking algorithm for sorting programs on a large range of inputs.

Randomness and semigenericity

Abstract: Let L contain only the equality symbol and let L+ be an arbitrary finite symmetric relational language containing L. Suppose probabilities are defined on finite L+ structures with 'edge probability' n-'T. By T', the almost sure theory of random L+-structures we mean the collection of L+-sentences which have limit probability 1. T, denotes the theory of the generic structures for Ka (the collection of finite graphs G with 6,a(G) = IGI-a a edges of G hereditarily nonnegative). 0.1. Theorem. T', the almost sure theory of random L+-structures, is the same as the theory Tc of the Ka,-generic model. This theory is complete, stable, and nearly model complete. Moreover, it has the finite model property and has only infinite models so is not finitely axiomatizable. This paper unites two apparently disparate lines of research. In [8], Shelah and Spencer proved a 0-1-law for first order sentences about random graphs with edge probability na where a is an irrational number between 0 and 1. Answering a question raised by Lynch [5], we extend this result from graphs to hypergraphs (i.e. to arbitrary finite symmetric relational languages). Let T' denote the set of sentences with limit probability 1. The Spencer-Shelah proof proceeded by a process of quantifier elimination which implicitly showed the theories T" were nearly model complete (see below) and complete. Hrushovski in [3] refuted a conjecture of Lachlan by constructing an No-categorical strictly stable pseudoplane. Baldwin and Shi [1] considered a variant on his methods to construct strictly stable (but not to-categorical) theories Ta indexed by irrational a. In this paper we show that for each irrational a, T' = Ta and thus deduce that T, is not finitely axiomatizable and that Ta is stable. Each Ta is the theory of a 'generic' model M, of an amalgamation class Ka of finite structures. Although the Hrushovski examples are easily seen to be nearly model complete this is less clear for the T, since they are not No-categorical. We show that each Ta, is nearly model complete. In the first, purely model theoretic, section of the paper we describe our basic framework and prove a sufficient condition for certain theories, including the Ta, to be nearly model complete. These conditions depend upon a generalization of the notion of genericity of a structure: semigenericity, which is introduced in this paper. In the second section we consider the addition of random relations and Received by the editors September 7, 1994. 1991 Mathematics Subject Classification. Primary 03C10, 05C80.

Source codes as random number generators

Abstract: A random number generator generates fair coin flips by processing deterministically an arbitrary source of nonideal randomness. An optimal random number generator generates asymptotically fair coin flips from a stationary ergodic source at a rate of bits per source symbol equal to the entropy rate of the source. Since optimal noiseless data compression codes produce incompressible outputs, it is natural to investigate their capabilities as optimal random number generators. We show under general conditions that optimal variable-length source codes asymptotically achieve optimal variable-length random bit generation in a rather strong sense. In particular, we show in what sense the Lempel-Ziv (1978) algorithm can be considered an optimal universal random bit generator from arbitrary stationary ergodic random sources with unknown distributions.

Complexity, Logic, and Recursion Theory

Abstract: Resource-bounded measure and randomness degree structures in local degree theory compressibility of infinite binary sequences beyond Godel's theorem - the failure to capture information content progressions of theories of bounded arithmetic on presentations of algebraic structures witness-isomorphic reductions and local search a survey of inductive inference with an emphasis on queries a uniformity of degree structures short course on logic, algebra, and topology the convenience of Tiling. (Part contents).

The Fourth Moment Method

Abstract: Higher moment analysis has typically been used to upper bound certain functions. In this paper, we introduce a new combinatorial method to lower bound the expectation of the absolute value of a random variable X by the expectation of a quartic in X. In the special case where we are looking at the absolute value of a (weighted) sum of {-1,+1} unbiased random variables, we achieve tight bounds, using only a fourth moment, for the total discrepancy of a set system. Because the fourth moment depends only on 4-wise independence, our bounds will hold over polynomially sized distributions, and so these bounds will be directly applicable in removing randomness to obtain NC algorithms. We obtain the first NC algorithms for the problems of total discrepancy, maximum acyclic subgraph, tournament ranking, the Gale--Berlekamp switching game, and edge discrepancy. We show that for most of these applications it is truly necessary to consider a fourth moment by exhibiting a 3-wise independent distribution which does not achieve the required bounds. Our method is strong enough to give a new combinatorial bound on tournament ranking.

Statistical generalizations of the optical cross-section theorem with application to inverse scattering

Abstract: A fundamental result of scattering theory, the so-called optical theorem, applies to situations where the field incident on the scatterer is a monochromatic plane wave and the scatterer is deterministic. We present generalizations of the theorem to situations where either the incident field or the scatterer or both are spatially random. By using these generalizations we demonstrate the possibility of determining the structure of some random scatterers from the knowledge of the power absorbed from two plane waves incident on it.

Chaotic dynamics in a multispecies community

Abstract: It is shown that community dynamics is neither haphazard nor completely directed. This is quite clear from our examination of a concrete example where recovery dynamics in vegetation progressed from an early phase of strong linear determinism to intense randomness with phase transition defined by density. Is it possible to reconstruct the two phase structure in simple mathematical terms? The results show that it is, and that the model is very simple: a discrete-time Markov chain with white noise. Interestingly, the long-term behaviour of the model is complex chaotic and explosive, suggesting that progression from dominant randomness to determinism is a distinctly probable event. And thus a conceptual foundation is laid, through interlinking initial condition, phase structure and explosive chaoticity, for a unifying theory, in which the classical hypotheses of community dynamics appear as special cases.

On the Dealer‘s Randomness Required in Secret Sharing Schemes

Abstract: In this paper we provide upper and lower bounds on the randomness required by the dealer to set up a secret sharing scheme for infinite classes of access structures. Lower bounds are obtained using entropy arguments. Upper bounds derive from a decomposition construction based on combinatorial designs (in particular, t-(v,k,λ) designs). We prove a general result on the randomness needed to construct a scheme for the cycle Cn; when n is odd our bound is tight. We study the access structures on at most four participants and the connected graphs on five vertices, obtaining exact values for the randomness for all them. Also, we analyze the number of random bits required to construct anonymous threshold schemes, giving upper bounds. (Informally, anonymous threshold schemes are schemes in which the secret can be reconstructed without knowledge of which participants hold which shares.)

Renormalizing rectangles and other topics in random matrix theory

Abstract: We consider random Hermitian matrices made of complex or realM×N rectangular blocks, where the blocks are drawn from various ensembles. These matrices haveN pairs of opposite real nonvanishing eigenvalues, as well asM−N zero eigenvalues (forM>N). These zero eigenvalues are “kinematical” in the sense that they are independent of randomness. We study the eigenvalue distribution of these matrices to leading order in the large-N, M limit in which the “rectangularity”r=M/N is held fixed. We apply a variety of methods in our study. We study Gaussian ensembles by a simple diagrammatic method, by the Dyson gas approach, and by a generalization of the Kazakov method. These methods make use of the invariance of such ensembles under the action of symmetry groups. The more complicated Wigner ensemble, which does not enjoy such symmetry properties, is studied by large-N renormalization techniques. In addition to the kinematical δ-function spike in the eigenvalue density which corresponds to zero eigenvalues, we find for both types of ensembles that if |r−1| is held fixed asN→∞, theN nonzero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin. This separation arises because the nonzero eigenvalues are repelled macroscopically from the origin. Finally, we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes, a behavior governed by Airy functions. Asr→1 the lobes come closer, and the Airy oscillatory behavior near the endpoints that are close to zero breaks down. We interpret this breakdown as a signal thatr→1 drives a crossover to the oscillation governed by Bessel functions near the origin for matrices made of square blocks.

Weak Randomness for Large q-State Potts Models in Two Dimensions

Abstract: We have studied the effect of weak randomness on q-state Potts models for q > 4 by measuring the central charges of these models using transfer matrix methods. We obtain a set of new values for the central charges and then show that some of these values are related to one another by a factorization law.

Spectral Curves of Non-Hermitean Hamiltonians

Abstract: Recent analytical and numerical work have shown that the spectrum of the random non-hermitean Hamiltonian on a ring which models the physics of vortex line pinning in superconductors is one dimensional. In the maximally non-hermitean limit, we give a simple "one-line" proof of this feature. We then study the spectral curves for various distributions of the random site energies. We find that a critical transition occurs when the average of the logarithm of the random site energy squared vanishes. For a large class of probability distributions of the site energies, we find that as the randomness increases the energy at which the localization-delocalization transition occurs increases, reaches a maximum, and then decreases. The Cauchy distribution studied previously in the literature does not have this generic behavior. We determine the critical value of the randomness at which "wings" first appear in the energy spectrum. For distributions, such as Cauchy, with infinitely long tails, we show that this critical value is infinitesimally above zero. We determine the density of eigenvalues on the wings for any probability distribution. We show that the localization length on the wings diverges linearly as the energy approaches the energy at which the localization-delocalization transition occurs. These results are all obtained in the maximally non-hermitean limit but for a generic class of probability distributions of the random site energies.