Phase transitions and memory effects in the dynamics of Boolean networks

The generating functional method is employed to investigate the synchronous dynamics of Boolean networks, providing an exact result for the system dynamics via a set of macroscopic order parameters. The topology of the networks studied and its constituent Boolean functions represent the system's quenched disorder and are sampled from a given distribution. The framework accommodates a variety of topologies and Boolean function distributions and can be used to study both the noisy and noiseless regimes; it enables one to calculate correlation functions at different times that are inaccessible via commonly used approximations. It is also used to determine conditions for the annealed approximation to be valid, explore phases of the system under different levels of noise and obtain results for models with strong memory effects, where existing approximations break down. Links between Boolean networks and general Boolean formulas are identified and results common to both system types are highlighted. © 2012 Copyright Taylor and Francis Group, LLC.

Public key cryptography and error correcting codes as Ising models

We employ the methods of statistical physics to study the performance of Gallager type error-correcting codes. In this approach, the transmitted codeword comprises Boolean sums of the original message bits selected by two randomly-constructed sparse matrices. We show that a broad range of these codes potentially saturate Shannon's bound but are limited due to the decoding dynamics used. Other codes show sub-optimal performance but are not restricted by the decoding dynamics. We show how these codes may also be employed as a practical public-key cryptosystem and are of competitive performance to modern cyptographical methods.

Noisy random Boolean formulae:a statistical physics perspective

Properties of computing Boolean circuits composed of noisy logical gates are studied using the statistical physics methodology. A formula-growth model that gives rise to random Boolean functions is mapped onto a spin system, which facilitates the study of their typical behavior in the presence of noise. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding macroscopic phase transitions. The framework is employed for deriving results on error-rates at various function-depths and function sensitivity, and their dependence on the gate-type and noise model used. These are difficult to obtain via the traditional methods used in this field.

On the robustness of random Boolean formulae

Random Boolean formulae, generated by a growth process of noisy logical gates are analyzed using the generating functional methodology of statistical physics. We study the type of functions generated for different input distributions, their robustness for a given level of gate error and its dependence on the formulae depth and complexity and the gates used. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding typical-case phase transitions. Results for error-rates, function-depth and sensitivity of the generated functions are obtained for various gate-type and noise models. © 2010 IOP Publishing Ltd.

Computing with noise: phase transitions in Boolean formulas

Computing circuits composed of noisy logical gates and their ability to represent arbitrary Boolean functions with a given level of error are investigated within a statistical mechanics setting. Existing bounds on their performance are straightforwardly retrieved, generalized, and identified as the corresponding typical-case phase transitions. Results on error rates, function depth, and sensitivity, and their dependence on the gate-type and noise model used are also obtained.

Reliability analysis for hazard and operability studies

Fault tree analysis is used as a tool within hazard and operability (Hazop) studies. The present study proposes a new methodology for obtaining the exact TOP event probability of coherent fault trees. The technique uses a top-down approach similar to that of FATRAM. This new Fault Tree Disjoint Reduction Algorithm resolves all the intermediate events in the tree except OR gates with basic event inputs so that a near minimal cut sets expression is obtained. Then Bennetts' disjoint technique is applied and remaining OR gates are resolved. The technique has been found to be appropriate as an alternative to Monte Carlo simulation methods when rare events are countered and exact results are needed. The algorithm has been developed in FORTRAN 77 on the Perq workstation as an addition to the Aston Hazop package. The Perq graphical environment enabled a friendly user interface to be created. The total package takes as its input cause and symptom equations using Lihou's form of coding and produces both drawings of fault trees and the Boolean sum of products expression into which reliability data can be substituted directly.

Dynamics of Boolean networks: an exact solution

The dynamics of Boolean networks (BN) with quenched disorder and thermal noise is studied via the generating functional method. A general formulation, suitable for BN with any distribution of Boolean functions, is developed. It provides exact solutions and insight into the evolution of order parameters and properties of the stationary states, which are inaccessible via existing methodology. We identify cases where the commonly used annealed approximation is valid and others where it breaks down. Broader links between BN and general Boolean formulas are highlighted.

Reliability of vegetation community information derived using decorana ordination and fuzzy c-means clustering

Descriptions of vegetation communities are often based on vague semantic terms describing species presence and dominance. For this reason, some researchers advocate the use of fuzzy sets in the statistical classification of plant species data into communities. In this study, spatially referenced vegetation abundance values collected from Greek phrygana were analysed by ordination (DECORANA), and classified on the resulting axes using fuzzy c-means to yield a point data-set representing local memberships in characteristic plant communities. The fuzzy clusters matched vegetation communities noted in the field, which tended to grade into one another, rather than occupying discrete patches. The fuzzy set representation of the community exploited the strengths of detrended correspondence analysis while retaining richer information than a TWINSPAN classification of the same data. Thus, in the absence of phytosociological benchmarks, meaningful and manageable habitat information could be derived from complex, multivariate species data. We also analysed the influence of the reliability of different surveyors' field observations by multiple sampling at a selected sample location. We show that the impact of surveyor error was more severe in the Boolean than the fuzzy classification. © 2007 Springer.

On reliable computation by noisy random Boolean formulas

We study noisy computation in randomly generated k-ary Boolean formulas. We establish bounds on the noise level above which the results of computation by random formulas are not reliable. This bound is saturated by formulas constructed from a single majority-like gate. We show that these gates can be used to compute any Boolean function reliably below the noise bound.