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Introduction
Many analysts have heard the rule about avoiding, as much as possible, the use of the Boolean operator NOT for constructing fault trees (FTs) in system safety and reliability assessments. As we know, the "NOT" operator applied to an event produces the negated or complementary event (i.e., the negated event probability is one minus the probability of the original event). The existence of negated events in an FT makes its logic "non-coherent," a coined term that is by no means a misnomer. As an anecdotal reference, my personal experience in dealing with non-coherent FTs has been misleading so far (i.e., "giving wrong ideas," as defined in the Oxford English Dictionary).
From a computational perspective, traditional coherent FTs are populated with events (i.e., variables) that are associated with system/component adverse states ("faults" in the broad sense of the word). The use of a negated event in the FT logic implies that a cutset containing the original event (i.e., sequence of independent faults leading to the FT top event) cannot occur if another cutset containing the negated event occurs as well. This not only creates mutually exclusive sequences of events, but its subtlety encourages the analyst to believe that non-coherent FT reduction, solution and interpretation may be carried out following the usual guidelines of coherent diagrams.
Three simple examples are used here to illustrate how effortlessly the manipulation of non-coherent logic produces seemingly reasonable outcomes, but entails flawed assumptions that become evident only after the FT logic is scrutinized in detail. A brief discussion about the use of Binary Decision Diagrams (BDDs) for solving non-coherent FTs is also included.
Example 1: A Visit to the Courthouse
Suppose we visit the courthouse in a certain town, and we observe people who stop at the office of the local justice of the peace to register their marriages. Let A be the independent probability of finding in the office an adult woman who is going to get married (i.e., female spouse), while B is the probability of finding a marrying adult man (i.e., male spouse). We disregard any liaison between staff members who are employed in that office in order to be free of bias in our experiment.
Depending on the law where you live, there is a limiting age below which a young person, legally a minor, needs parental consent and/or a judicial order to be married to an adult. Let  represent the probability of a marrying female minor, and  the probability of a marrying male minor. The bars on the events indicate the negated events after applying the Boolean operator NOT.
If we assume that the event C is the issuing of a marriage consent, then the FT top event S in Figure 1 quantifies the event that a marriage takes place at the office:
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(1)
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where '*' and '+' represent the Boolean operators AND and OR. The Figure 1 FT is identical to the one given by Tang and Dugan in their non-coherent FT analysis of a system with triple modular redundancy [Ref. 1].
Figure 1 — Non-Coherent Fault Tree of Example 1.
In simple words, Equation (1) is the union of the following events: a marrying adult couple, OR-ed with an adult female marrying a male minor with consent, OR-ed with an adult male marrying a female minor with consent. The equation above can be reduced to the following expression, according to the rules of Boolean algebra that are summarized in Table 1:
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(2)
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Table 1 — Basic Boolean Operations [Ref. 1].
A * 1 = A | |
A * = 0 | Null Set |
A * A = A | |
A * B = B * A | Commutative Law |
A + = 1 | Complementation |
A + 0 = A | |
A + 1 = 1 | |
A + A = A | |
A * (B + C) = A * B + A * C | Distributive Law |
A * (B * C) = (A * B) * C | Associative Law |
| De Morgan's Theorem |
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Therefore, the top event probability in Equation (2) is the union of the joint probability of a marrying adult couple, with the probability of consent, intersection with the probability of either an adult female marrying a male minor, or an adult male marrying a female minor.
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