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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 89
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: M. Papadrakakis and B.H.V. Topping
Paper 74
Generalized Top and Bottom Binary n-Tuples L. González
Department of Mathematics, University of Las Palmas de Gran Canaria, Spain , "Generalized Top and Bottom Binary n-Tuples", in M. Papadrakakis, B.H.V. Topping, (Editors), "Proceedings of the Sixth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 74, 2008. doi:10.4203/ccp.89.74
Keywords: complex stochastic Boolean systems, reliability engineering, intrinsic order, top binary n-tuples, bottom binary n-tuples, generalized top binary n-tuples, generalized bottom binary n-tuples.
Summary
A complex stochastic Boolean system (CSBS) is a system depending on an arbitrary number n of random Boolean variables x1,...,xn. Each one of the 2n possible situations associated to such a system is given by a binary n-tuple u in {0,1}n of 0s and 1s, and it has its own occurrence probability Pr{u}.
The intrinsic order criterion [1,2] leads us to establish a partition of the set {0,1}n of binary n-tuples into three kinds of binary strings: top, bottom and jumping binary n-tuples. A binary n-tuple u is called top (bottom, respectively) if its occurrence probability Pr{u} is always among the 2n largest (smallest, respectively) ones. A binary n-tuple u is called jumping if it is neither top nor bottom. Top and bottom n-tuples has been characterized in [3]. In this paper, we generalize these three kinds of binary strings by defining the k-top, k-bottom and k-jumping binary n-tuples (1 <= k <= 2n). A binary n-tuple u is called k-top (k-bottom, respectively) if its occurrence probability Pr{u} is always among the k largest (smallest, respectively) ones. A binary n-tuple u is called k-jumping if it is neither k-top nor k-bottom. We present a symmetry property that reduces the study of the k-bottom strings to the study of the k-top strings simply by changing 0s to 1s and 1s to 0s in the binary n-tuples (i.e, interchanging the role of 0s and 1s). We establish some properties and necessary conditions for both the k-top and k-bottom binary n-tuples. Next, we present a necessary and sufficient condition for the k-top binary n-tuples. To state this characterization theorem, we first determine the cardinality of the set Cu of all binary binary n-tuples that are intrinsically less than or equal to a given n-tuple u. The set Cu has been widely studied in [4]. For characterizing the k-bottom binary n-tuples it is enough to use the above mentioned symmetry property. For identifying the k-jumping binary n-tuples it is enough to use the fact that they are neither k-top nor k-bottom. These theoretical results may be applied to the reliability analysis of technical systems (or, in general, CSBSs arising from many scientific or engineering areas) described by a stochastic Boolean function. References
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