Went though J.Howse notes.
Need * example concrete diagram * example fmmd analysisa case (the hi fi separates) * make the diagrams consistent *
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robin
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@ -1,4 +1,4 @@
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PDF_READER = acroread
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#
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# Make the propositional logic diagram a paper
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#
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@ -9,7 +9,7 @@ paper: paper.tex logic_diagram_paper.tex
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#dvipdf paper pdflatex cannot use eps ffs
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pdflatex paper.tex
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mv paper.pdf logic_diagram_paper.pdf
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okular logic_diagram_paper.pdf
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$(PDF_READER) logic_diagram_paper.pdf
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# Remove the need for referncing graphics in subdirectories
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@ -167,7 +167,7 @@ The concrete definitions for PLD's and Spider Diagrams\cite{howse:sd} share many
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A concrete {\em Propositional logic diagram} is a set of labelled {\em contours}
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(closed curves) in the plane. The minimal regions formed by the closed curves
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can by occupied by `test cases'.
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can by occupied by `test cases' (represented by asterisks).
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The `test cases' may be joined by joining lines.
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A group of `test cases' connected by joining lines
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is defined as a `test case disjunction' or Spider.
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@ -228,8 +228,9 @@ $$ \mathbb{R}^{2} - \; \bigcup_{\hat{c} \in \hat{C}(\hat{d})}\hat{c}$$
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$$ \hat{z} = \bigcap_{c \in \mathcal{X}}
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{interior}
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(\hat{c})
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\; \cup \;
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\bigcap_{\hat{c} \in \hat{C}-X}
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%\; \cup \;
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\cap % J.Howse correction 25MAR2011
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\bigcap_{\hat{c} \in \hat{C}-X}
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exterior (\hat{c})
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$$
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@ -248,7 +249,9 @@ is non empty, then $\hat{z}$ is a concrete zone of $\hat{d}$. A zone is a union
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}
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Each minimal region in the plane may be inhabited by one or more `test cases'.
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%%- Each minimal region in the plane may be inhabited by one or more `test cases'.
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%%- above statement problematic, esp when having double sim diagrams, should have been zone anyway
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% One or more because in software the same logical conditions mean existing in the same
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% region. For electroincs or mechanical, one test case per region is
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% mandatory. How to describe ?????
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@ -256,7 +259,7 @@ Each test case can be associated with the set of contours that enclose it.
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%defined the minimal region it inhabits.
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{
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\definition{ $ \mathcal{Z}_{d}:T(d)\rightarrow \mathcal{C}$ is a function
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\definition{ $ \mathcal{Z}_{d}:T(d)\rightarrow \mathcal{P}\mathcal{C}$ is a function
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associating a test-point with a set of contours in the plane. This corresponds to the interior of the contours defining the zone.
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}
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}
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@ -300,7 +303,7 @@ An $SMG$ can be considered to be a collection of test~cases.
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%OR consider an $SMG$ as a tree whose nodes are test cases.
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Let d be a PLD : An $SMG$ is a collection of test~cases in d where
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Let d be a PLD : A Symtomatically Merged Group ($SMG$) is a collection of test~cases in d where
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the test~cases belong to a graph connected by joining lines.
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}
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@ -343,8 +346,8 @@ To obtain the set of propositions from a PLD, each $SMG$ must be traversed
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along each joining line. For each test case
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in the $SMG$ a new section of the equation is exclusive-disjunctively appended to it.
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%
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Let conjunctive logic equation associated with a test case
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be determined from the contours that enclose it.
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The conjunctive logic term associated with a test case
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is determined from the contours that enclose it.
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i.e. the contours $\mathcal{X}$ from the zone it inhabits.
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{
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@ -369,25 +372,27 @@ $ a \wedge b \wedge c $.
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{
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\definition{
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Let $\mathcal{G}$ be a function that returns a logic equation for a given $SMG$
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Let $\mathcal{G}$ be a function that returns a logic term for a given $SMG$
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$fmg$ in the diagram, where an SMG is a non empty set of test cases.
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% $t$ is a `test case'
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$\mathcal{G}$ has a domain of SMG and a range of $P$ given as
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as
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$\mathcal{G}$ has domain of SMG and range of $P$ given as
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$$ \mathcal{G}:SMG \rightarrow P. $$
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The logic equation (using $oplus$ to represent exclusive-or) representing an SMG $p_{fmg}$ is given thus;
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The logic term (using $oplus$ to represent exclusive-or) representing an SMG $p_{fmg}$ is given thus;
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$$\mathcal{G}(fmg) = \bigoplus_{t \in fmg} (\; \mathcal{F}_{t} (t) \;) \; .$$
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}
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}
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The semantics of the diagram is the set of logic equations representing all its SMGs,
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The semantics of the diagram is the set of logic terms representing all its SMGs,
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along with unused zones (i.e. zones that are not inhabited by SMGs).
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Thus the abstract representation of the diagram, becomes a list of logic equations
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and unused available zones.
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%
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% THIS ABOVE COULD BE ANOTHER DEFINITION
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Thus the abstract representation of the diagram, becomes a list of logic terms
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% and unused available zones
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.
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\section{Context, functional groups, failure modes and symptoms}
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@ -446,7 +451,7 @@ with the joining lines defining the its symptom collection.
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The `symptom collection' represents the functional group
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at a higher level of failure mode abstraction. That is to say,
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the contours in the new diagram represent the ways in which the {\fg} considered
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as an entity can fail. The new represents the
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as an entity can fail. The new diagram represents the
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{\fg} as a higher level component, or {\textbf{derived component}}, with its own set of failure modes.
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\ifthenelse {\boolean{paper}}
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@ -574,7 +579,7 @@ substituting the test cases for their propositional logic equations gives
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\paragraph{Failure Analysis Interpretation}
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Equation \ref{eqn:l_or} would be interpreted to mean that
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either failure mode a or b occurring, would have the same failure symptom for the circuit/functional~group
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under analysis.
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under analysis. If $a \wedge b$ occurred this could have a completely different failure mode symptom.
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@ -634,8 +639,9 @@ of $ R = b \oplus c $.
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\vspace{0.3cm}
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\begin{tabular}{||c|c|l||} \hline \hline
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{\em $SMG$ } & {\em Failure Mode equation } & {\em comments } \\ \hline
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Q & $(a)$ & Symptom Q is active when fault mode `a` is \\ \hline
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P & $(b \wedge c)$ & Symptom P is active when `$b \wedge a$' is \\ \hline
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Q & $(a)$ & Symptom Q is active when fault mode `a` is \\ \hline
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R & $(b \oplus c)$ & Symptom R is active when either `b' or `c' is \\ \hline
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% T & T & T \\ \hline \hline
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\end{tabular}
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@ -736,19 +742,19 @@ is represented by an inhibit gate.\cite{nasafta}[pp41-42],\cite{nucfta}.
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The diagram \ref{fig:inhibit} has a test case in the contour $C$.
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Contour $C$ is \textbf{enclosed} by contour $A$. This says
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that for failure~mode $C$ to occur failure mode $A$
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The diagram \ref{fig:inhibit} has a test case in the contour $c$.
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Contour $c$ is \textbf{enclosed} by contour $a$. This says
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that for failure~mode $c$ to occur failure mode $a$
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must have occurred.
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A famous example of this is the space shuttle `O' ring failure that
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caused the 1986 Challenger disaster~\cite{challenger}~\cite{wdycwopt}.
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For the failure mode to occur, the ambient temperature had to
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be below a critical value.
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If we take the failure mode of the `O' ring to be $C$
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and the temperature below critical to be $A$, we can see that
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the low temperature failure~mode $C$ can only occur if $A$ is true.
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If we take the failure mode of the `O' ring to be $c$
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and the temperature below critical to be $a$, we can see that
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the low temperature failure~mode $c$ can only occur if $a$ is true.
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The `O' ring could fail in a different way independent of the critical temperature and this is
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represented, for the sake of this example, by contour $D$.
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represented, for the sake of this example, by contour $d$.
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In terms of propositional logic, the inhibit gate of FTA\cite{nasafta}[pp 41-42], and the contour enclosure
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of PLD represent {\em implication}.
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@ -775,158 +781,11 @@ $$ R3 = d $$
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\paragraph{How this would be interpreted in failure analysis}
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In failure analysis, $R2$ is the symptom of either failure~mode $A$ or $C$
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occurring. $R1$ is the symptom of $B$ or $A \wedge C$ occurring.
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Note that although $R2$ is a symptom of the functional~group, on its own
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it will not lead to a dangerous failure~mode of the subsystem.
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In failure analysis, $R2$ is the symptom of either failure~mode $a$
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occurring, this may not necessarily lead to failure mode `c'. $R1$ is the symptom of $a \wedge c$ occurring,
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not that because $c$ is enclosed by $a$, $c$ cannot occur unless $a$ has.
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% \subsection { Representing Logical Negation }
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%
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% \begin{figure}[h+]
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% %\centering
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% %\input{ldor.tex}
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% \begin{center}
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% \includegraphics[width=200pt,bb=0 0 450 404]{logic_diagram/ldneg.eps}
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% % resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404
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% \end{center}
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% %\includegraphics[scale=0.60]{ldneg.eps}
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% \caption{Logical Negation}
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% \label{fig:ld_neg}
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% \end{figure} % OR
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%
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% Diagram \ref{fig:ld_neg} represents the logical equation $$ P = a \wedge b \wedge \neg c $$.
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%
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% \paragraph{How this would be interpreted in failure analysis}
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% In failure analysis this test case represents the scenario where failure modes $a$ and $b$
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% are active but $c$ is not.
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%
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%
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%
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% \clearpage
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% \subsection { Logical XOR example }
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%
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% An exclusive or condition is represented by diagram \ref{fig:ld_xor}.
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% The Equations represented are as follows.
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%
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% firstly looking at the test case points
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% $$ P = (\neg a \wedge b) $$
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% $$ Q = (\neg b \wedge a) $$
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%
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% now joining them with the disjuctive line
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% $$ R = P \vee Q $$
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%
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% Giving R as a Boolean equation
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% $$ R = (\neg a \wedge b) \vee (\neg b \wedge a) $$
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% or taking the symbol $\oplus$ to mean exclusive-or
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% $$R = a \oplus b $$
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%
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%
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% % \begin{figure}[h] %% SOMETHING IS WRONG says latex. very helpful tell me what it fucking is then
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% % \centering
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% % \caption{Example `XOR' Diagram}
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% % \includegraphics[scale=0.80]{ldxor.eps}
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% % \label{fig:ld_xor}
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% % \end{figure} % XOR
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%
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%
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% \begin{figure}[h+]
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% %\centering
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% %\input{millivolt_sensor.tex}
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% \begin{center}
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% % bb= llx lly urx ury;
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% \includegraphics[width=200pt,bb=0pt 0pt 800pt 800pt]{logic_diagram/ldxor.eps}
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% % resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404
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% \end{center}
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%
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% %\includegraphics[scale=0.4]{ldxor.eps}
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% \caption{Logical XOR}
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% \label{fig:ld_xor}
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% \end{figure}
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%
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% \clearpage
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% \subsection { Logical IMPLICATION example }
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%
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%
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% An implication $a \rightarrow b$ is represented by diagram \ref{fig:ld_imp}.
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% The Equations represented are as follows.
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%
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% Looking at the conjuctive environment of the test cases
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% $$P = (\neg a)$$
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% $$Q = (b)$$
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% From the joining `disjunctive' line R in the diagram.
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% $$R = P \vee Q$$
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% Leading to
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% $$R = (\neg a) \vee (b)$$
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% which is the standard logic equation for implication.
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%
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% \begin{figure}[h+]
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% %\centering
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% %\input{millivolt_sensor.tex}
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% \begin{center}
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% \includegraphics[width=200pt,bb=0 0 450 404]{logic_diagram/ldimp.eps}
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% % resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404
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% \end{center}
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% %\includegraphics[scale=0.4]{ldimp.eps}
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% \caption{Logical Implication}
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% \label{fig:ld_imp}
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% \end{figure}
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%
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% \tiny
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% %\vspace{0.3cm}
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% \begin{tabular}{||c|c|c|c||} \hline \hline
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%
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% {\em $a$ } & {\em $b$ } & {implication \em $(\neg a) \vee (b) $ } \\ \hline
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% F & F & T \\ \hline
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% F & T & T \\ \hline
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% T & F & F \\ \hline
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% T & T & T \\ \hline \hline:
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% \end{tabular}
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% %\vspace{0.3cm}
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% \normalsize
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%
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% \clearpage
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% \subsection { Diagram representing several Logic Equations Example }
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%
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%
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% bb=0 0 450 404
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%
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%
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% \begin{figure}[h+]
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% %\centering
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% %\input{millivolt_sensor.tex}
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% \begin{center}
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% \includegraphics[width=200pt,bb=0pt 0pt 600pt 600pt]{logic_diagram/ldmeq.eps}
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% % resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404
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% \end{center}
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% %\includegraphics[scale=0.4]{ldmeq.eps}
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% \caption{Several Logical Expressions}
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% \label{fig:ld_meq}
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% \end{figure}
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%
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% %The effect of using explicit negation, means that a test case being outside a given contour does not imply negation, it implies a `don't care'
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% %condition.
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%
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% Three simple equations are represented in the diagram \ref{fig:ld_dc}.
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%
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% %The Set of contours $\mho$ represent the `don't care' conditions.
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%
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% The Equations represented are as follows.
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%
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% %$$ Q = a \; | \; \mho\{b,c\} $$
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%
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% %$$ P = b \wedge c \; | \; \mho\{a\} $$
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%
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% $$ Q = a $$
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% $$ P = b \wedge c $$
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% $$ R = b \vee c $$
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%
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% % XXXXXX gives annoying impossible to understand syntax messages
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% %\small
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% %\bibliography{vmgbibliography,mybib}
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% %\bibliography{vmgbibliography}
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% %\normalsize
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%
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\section{Intended use in FMMD}
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The intention for these diagrams is that they are used to collect
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@ -1010,43 +869,10 @@ The test case AFE represents the condition where all four engines have failed \c
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\label{fig:allfour}
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\end{figure}
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%\subsection{Example Sub-system}
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%
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%For instance were a `power supply' being analysed there could be several
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%individual component faults or combinations that lead to
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%a situation where there is no power. This can be described as a state
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%of the powersupply being modeelled as NO\_POWER.
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%These can all be collected by DISJUCNTION, i.e. that this this or this
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%fault occuring will cause the NO\_POWER fault. Visually this disjuction is
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%indicated by the joining lines.
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%As far as the user of the `power supply' is concerned, the power supply has failed
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%with the failure mode $NO\_POWER$.
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%The `power supply' module, after this process will have a defined set of
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%fault modes and may be considered as a component at a higher
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%level of abstraction. This module can then be combined
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%with others at the same abstraction level.
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%Note that because this is a fault collection process
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%the number of component faults for a module
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%must be less than or equal to the sum of the number of component faults.
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\section{Conclusion}
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%Typeset in \ \ {\huge \LaTeX} \ \ on \ \ \today
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%\begin{verbatim}
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%CVS Revision Identity $Id: logic_diagram.tex,v 1.17 2010/01/06 13:41:32 robin Exp $
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%\end{verbatim}
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%\ifthenelse {\boolean{paper}}
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%{
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% \bibliographystyle{plain}
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% \bibliography{../vmgbibliography,../mybib}
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%
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%}
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%{
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%}
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%Compiled last \today
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%\end{document}
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%\theend
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% Elevator Pitch
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%\pagebreak[4]
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\clearpage
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