Qualitative temporal analysis: Towards a full implementation of the Fault Tree Handbook

The Fault tree handbook has become the de facto standard for fault tree analysis (FTA), defining the notation and mathematical foundation of this widely used safety analysis technique. The Handbook recognises that classical combinatorial fault trees employing only Boolean gates cannot capture the potentially critical significance of the temporal ordering of failure events in a system. Although the Handbook proposes two dynamic gates that could remedy this, a Priority-AND and an Exclusive-OR gate, these gates were never accurately defined. This paper proposes extensions to the logical foundation of fault trees that enable use of these dynamic gates in an extended and more powerful FTA. The benefits of this approach are demonstrated on a generic triple-module standby redundant system exhibiting dynamic behaviour

Categorical invariance and structural complexity in human concept learning

An alternative account of human concept learning based on an invariance measure of the categorical\ud stimulus is proposed. The categorical invariance model (CIM) characterizes the degree of structural\ud complexity of a Boolean category as a function of its inherent degree of invariance and its cardinality or\ud size. To do this we introduce a mathematical framework based on the notion of a Boolean differential\ud operator on Boolean categories that generates the degrees of invariance (i.e., logical manifold) of the\ud category in respect to its dimensions. Using this framework, we propose that the structural complexity\ud of a Boolean category is indirectly proportional to its degree of categorical invariance and directly\ud proportional to its cardinality or size. Consequently, complexity and invariance notions are formally\ud unified to account for concept learning difficulty. Beyond developing the above unifying mathematical\ud framework, the CIM is significant in that: (1) it precisely predicts the key learning difficulty ordering of\ud the SHJ [Shepard, R. N., Hovland, C. L.,&Jenkins, H. M. (1961). Learning and memorization of classifications.\ud Psychological Monographs: General and Applied, 75(13), 1-42] Boolean category types consisting of three\ud binary dimensions and four positive examples; (2) it is, in general, a good quantitative predictor of the\ud degree of learning difficulty of a large class of categories (in particular, the 41 category types studied\ud by Feldman [Feldman, J. (2000). Minimization of Boolean complexity in human concept learning. Nature,\ud 407, 630-633]); (3) it is, in general, a good quantitative predictor of parity effects for this large class of\ud categories; (4) it does all of the above without free parameters; and (5) it is cognitively plausible (e.g.,\ud cognitively tractable)

Scaled Boolean Algebras

Scaled Boolean algebras are a category of mathematical objects that arose from attempts to understand why the conventional rules of probability should hold when probabilities are construed, not as frequencies or proportions or the like, but rather as degrees of belief in uncertain propositions. This paper separates the study of these objects from that not-entirely-mathematical problem that motivated them. That motivating problem is explicated in the first section, and the application of scaled Boolean algebras to it is explained in the last section. The intermediate sections deal only with the mathematics. It is hoped that this isolation of the mathematics from the motivating problem makes the mathematics clearer.Comment: 53 pages, 8 Postscript figures, Uses ajour.sty from Academic Press, To appear in Advances in Applied Mathematic

The Boolean Model in the Shannon Regime: Three Thresholds and Related Asymptotics

Consider a family of Boolean models, indexed by integers $n \ge 1$, where the $n$-th model features a Poisson point process in ${\mathbb{R}}^n$ of intensity $e^{n \rho_n}$ with $\rho_n \to \rho$ as $n \to \infty$, and balls of independent and identically distributed radii distributed like $\bar X_n \sqrt{n}$, with $\bar X_n$ satisfying a large deviations principle. It is shown that there exist three deterministic thresholds: $\tau_d$ the degree threshold; $\tau_p$ the percolation threshold; and $\tau_v$ the volume fraction threshold; such that asymptotically as $n$ tends to infinity, in a sense made precise in the paper: (i) for $\rho < \tau_d$, almost every point is isolated, namely its ball intersects no other ball; (ii) for $\tau_d< \rho< \tau_p$, almost every ball intersects an infinite number of balls and nevertheless there is no percolation; (iii) for $\tau_p< \rho< \tau_v$, the volume fraction is 0 and nevertheless percolation occurs; (iv) for $\tau_d< \rho< \tau_v$, almost every ball intersects an infinite number of balls and nevertheless the volume fraction is 0; (v) for $\rho > \tau_v$, the whole space covered. The analysis of this asymptotic regime is motivated by related problems in information theory, and may be of interest in other applications of stochastic geometry