# Boolean Normal Forms, Shellability, and Reliability Computations

##### Citations

60 citations

### Cites background from "Boolean Normal Forms, Shellability,..."

...By a fundamental result of Boolean logics [Boros et al (2000)]: = M_ i=1 Oi (24) Since is a Bernoulli variable, then E[ ] = P ( = 1) = E[O1 _O2 _ ::: _OM ] = P ([Mi=1Oi = 1) (25) There follows that P ( = 1) = MX s=0 P (Os = 1) X i s P ((Oi = 1) \ (Os = 1)) + :::+ ( 1)MP (\Mi=1(Oi = 1)) (26) Then,…...

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...In these instances, multilinear functions play a crucial role and many reliability results are a consequence of the properties of Boolean functions [Birnbaum (1969), Agrawal and Barlow (1984), Ball and Provan (1988), Fishman (1989), Boros et al (2000), Khachiyan et al (2007).]...

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...Core of the theory is the representation of a system as a set of N components, each of which can be in two possible states [Birnbaum (1969), Ball and Provan (1988), Boros et al (2000), Giglio and Wynn (2004), Khachiyan et al (2007).]...

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39 citations

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### Cites background from "Boolean Normal Forms, Shellability,..."

...Corollary 1 cannot be extended to non-coherent systems because the structure function of some non-coherent systems can be expressed by more than one structure function [64]....

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##### References

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### "Boolean Normal Forms, Shellability,..." refers background in this paper

...In the context of reliability theory, the prime implicants of f represent the minimal cutsets of the system under study, namely, the minimal subsets of elements whose failure causes the whole system to fail (see [3, 25])....

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...One of the fundamental issues in reliability is to compute the probability that a positive Boolean function (describing the state—operating or failed—of a complex system) take value 1 when each variable (representing the state of individual components) takes value 0 or 1 randomly and independently of the value of the other variables (see, for instance, [3, 25])....

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