From 2d759ec6cb746e43acca809f162edb12f3c7bcc4 Mon Sep 17 00:00:00 2001 From: robin Date: Fri, 25 Mar 2011 18:21:47 +0000 Subject: [PATCH] Went though J.Howse notes. Need * example concrete diagram * example fmmd analysisa case (the hi fi separates) * make the diagrams consistent * --- logic_diagram/Makefile | 4 +- logic_diagram/logic_diagram.tex | 250 +++++--------------------------- 2 files changed, 40 insertions(+), 214 deletions(-) diff --git a/logic_diagram/Makefile b/logic_diagram/Makefile index 8d29472..26d6001 100644 --- a/logic_diagram/Makefile +++ b/logic_diagram/Makefile @@ -1,4 +1,4 @@ - +PDF_READER = acroread # # Make the propositional logic diagram a paper # @@ -9,7 +9,7 @@ paper: paper.tex logic_diagram_paper.tex #dvipdf paper pdflatex cannot use eps ffs pdflatex paper.tex mv paper.pdf logic_diagram_paper.pdf - okular logic_diagram_paper.pdf + $(PDF_READER) logic_diagram_paper.pdf # Remove the need for referncing graphics in subdirectories diff --git a/logic_diagram/logic_diagram.tex b/logic_diagram/logic_diagram.tex index e63ad49..9ff8361 100644 --- a/logic_diagram/logic_diagram.tex +++ b/logic_diagram/logic_diagram.tex @@ -167,7 +167,7 @@ The concrete definitions for PLD's and Spider Diagrams\cite{howse:sd} share many A concrete {\em Propositional logic diagram} is a set of labelled {\em contours} (closed curves) in the plane. The minimal regions formed by the closed curves -can by occupied by `test cases'. +can by occupied by `test cases' (represented by asterisks). The `test cases' may be joined by joining lines. A group of `test cases' connected by joining lines is defined as a `test case disjunction' or Spider. @@ -228,8 +228,9 @@ $$ \mathbb{R}^{2} - \; \bigcup_{\hat{c} \in \hat{C}(\hat{d})}\hat{c}$$ $$ \hat{z} = \bigcap_{c \in \mathcal{X}} {interior} (\hat{c}) - \; \cup \; - \bigcap_{\hat{c} \in \hat{C}-X} + %\; \cup \; + \cap % J.Howse correction 25MAR2011 + \bigcap_{\hat{c} \in \hat{C}-X} exterior (\hat{c}) $$ @@ -248,7 +249,9 @@ is non empty, then $\hat{z}$ is a concrete zone of $\hat{d}$. A zone is a union } -Each minimal region in the plane may be inhabited by one or more `test cases'. +%%- Each minimal region in the plane may be inhabited by one or more `test cases'. +%%- above statement problematic, esp when having double sim diagrams, should have been zone anyway + % One or more because in software the same logical conditions mean existing in the same % region. For electroincs or mechanical, one test case per region is % mandatory. How to describe ????? @@ -256,7 +259,7 @@ Each test case can be associated with the set of contours that enclose it. %defined the minimal region it inhabits. { -\definition{ $ \mathcal{Z}_{d}:T(d)\rightarrow \mathcal{C}$ is a function +\definition{ $ \mathcal{Z}_{d}:T(d)\rightarrow \mathcal{P}\mathcal{C}$ is a function associating a test-point with a set of contours in the plane. This corresponds to the interior of the contours defining the zone. } } @@ -300,7 +303,7 @@ An $SMG$ can be considered to be a collection of test~cases. %OR consider an $SMG$ as a tree whose nodes are test cases. -Let d be a PLD : An $SMG$ is a collection of test~cases in d where +Let d be a PLD : A Symtomatically Merged Group ($SMG$) is a collection of test~cases in d where the test~cases belong to a graph connected by joining lines. } @@ -343,8 +346,8 @@ To obtain the set of propositions from a PLD, each $SMG$ must be traversed along each joining line. For each test case in the $SMG$ a new section of the equation is exclusive-disjunctively appended to it. % -Let conjunctive logic equation associated with a test case -be determined from the contours that enclose it. +The conjunctive logic term associated with a test case +is determined from the contours that enclose it. i.e. the contours $\mathcal{X}$ from the zone it inhabits. { @@ -369,25 +372,27 @@ $ a \wedge b \wedge c $. { \definition{ -Let $\mathcal{G}$ be a function that returns a logic equation for a given $SMG$ +Let $\mathcal{G}$ be a function that returns a logic term for a given $SMG$ $fmg$ in the diagram, where an SMG is a non empty set of test cases. % $t$ is a `test case' -$\mathcal{G}$ has a domain of SMG and a range of $P$ given as -as +$\mathcal{G}$ has domain of SMG and range of $P$ given as + $$ \mathcal{G}:SMG \rightarrow P. $$ -The logic equation (using $oplus$ to represent exclusive-or) representing an SMG $p_{fmg}$ is given thus; +The logic term (using $oplus$ to represent exclusive-or) representing an SMG $p_{fmg}$ is given thus; $$\mathcal{G}(fmg) = \bigoplus_{t \in fmg} (\; \mathcal{F}_{t} (t) \;) \; .$$ } } -The semantics of the diagram is the set of logic equations representing all its SMGs, +The semantics of the diagram is the set of logic terms representing all its SMGs, along with unused zones (i.e. zones that are not inhabited by SMGs). -Thus the abstract representation of the diagram, becomes a list of logic equations -and unused available zones. -% -% THIS ABOVE COULD BE ANOTHER DEFINITION +Thus the abstract representation of the diagram, becomes a list of logic terms +% and unused available zones +. + + + \section{Context, functional groups, failure modes and symptoms} @@ -446,7 +451,7 @@ with the joining lines defining the its symptom collection. The `symptom collection' represents the functional group at a higher level of failure mode abstraction. That is to say, the contours in the new diagram represent the ways in which the {\fg} considered -as an entity can fail. The new represents the +as an entity can fail. The new diagram represents the {\fg} as a higher level component, or {\textbf{derived component}}, with its own set of failure modes. \ifthenelse {\boolean{paper}} @@ -574,7 +579,7 @@ substituting the test cases for their propositional logic equations gives \paragraph{Failure Analysis Interpretation} Equation \ref{eqn:l_or} would be interpreted to mean that either failure mode a or b occurring, would have the same failure symptom for the circuit/functional~group -under analysis. +under analysis. If $a \wedge b$ occurred this could have a completely different failure mode symptom. @@ -634,8 +639,9 @@ of $ R = b \oplus c $. \vspace{0.3cm} \begin{tabular}{||c|c|l||} \hline \hline {\em $SMG$ } & {\em Failure Mode equation } & {\em comments } \\ \hline - Q & $(a)$ & Symptom Q is active when fault mode `a` is \\ \hline + P & $(b \wedge c)$ & Symptom P is active when `$b \wedge a$' is \\ \hline + Q & $(a)$ & Symptom Q is active when fault mode `a` is \\ \hline R & $(b \oplus c)$ & Symptom R is active when either `b' or `c' is \\ \hline % T & T & T \\ \hline \hline \end{tabular} @@ -736,19 +742,19 @@ is represented by an inhibit gate.\cite{nasafta}[pp41-42],\cite{nucfta}. -The diagram \ref{fig:inhibit} has a test case in the contour $C$. -Contour $C$ is \textbf{enclosed} by contour $A$. This says -that for failure~mode $C$ to occur failure mode $A$ +The diagram \ref{fig:inhibit} has a test case in the contour $c$. +Contour $c$ is \textbf{enclosed} by contour $a$. This says +that for failure~mode $c$ to occur failure mode $a$ must have occurred. A famous example of this is the space shuttle `O' ring failure that caused the 1986 Challenger disaster~\cite{challenger}~\cite{wdycwopt}. For the failure mode to occur, the ambient temperature had to be below a critical value. -If we take the failure mode of the `O' ring to be $C$ -and the temperature below critical to be $A$, we can see that -the low temperature failure~mode $C$ can only occur if $A$ is true. +If we take the failure mode of the `O' ring to be $c$ +and the temperature below critical to be $a$, we can see that +the low temperature failure~mode $c$ can only occur if $a$ is true. The `O' ring could fail in a different way independent of the critical temperature and this is -represented, for the sake of this example, by contour $D$. +represented, for the sake of this example, by contour $d$. In terms of propositional logic, the inhibit gate of FTA\cite{nasafta}[pp 41-42], and the contour enclosure of PLD represent {\em implication}. @@ -775,158 +781,11 @@ $$ R3 = d $$ \paragraph{How this would be interpreted in failure analysis} -In failure analysis, $R2$ is the symptom of either failure~mode $A$ or $C$ -occurring. $R1$ is the symptom of $B$ or $A \wedge C$ occurring. -Note that although $R2$ is a symptom of the functional~group, on its own -it will not lead to a dangerous failure~mode of the subsystem. +In failure analysis, $R2$ is the symptom of either failure~mode $a$ +occurring, this may not necessarily lead to failure mode `c'. $R1$ is the symptom of $a \wedge c$ occurring, +not that because $c$ is enclosed by $a$, $c$ cannot occur unless $a$ has. -% \subsection { Representing Logical Negation } -% -% \begin{figure}[h+] -% %\centering -% %\input{ldor.tex} -% \begin{center} -% \includegraphics[width=200pt,bb=0 0 450 404]{logic_diagram/ldneg.eps} -% % resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404 -% \end{center} -% %\includegraphics[scale=0.60]{ldneg.eps} -% \caption{Logical Negation} -% \label{fig:ld_neg} -% \end{figure} % OR -% -% Diagram \ref{fig:ld_neg} represents the logical equation $$ P = a \wedge b \wedge \neg c $$. -% -% \paragraph{How this would be interpreted in failure analysis} -% In failure analysis this test case represents the scenario where failure modes $a$ and $b$ -% are active but $c$ is not. -% -% -% -% \clearpage -% \subsection { Logical XOR example } -% -% An exclusive or condition is represented by diagram \ref{fig:ld_xor}. -% The Equations represented are as follows. -% -% firstly looking at the test case points -% $$ P = (\neg a \wedge b) $$ -% $$ Q = (\neg b \wedge a) $$ -% -% now joining them with the disjuctive line -% $$ R = P \vee Q $$ -% -% Giving R as a Boolean equation -% $$ R = (\neg a \wedge b) \vee (\neg b \wedge a) $$ -% or taking the symbol $\oplus$ to mean exclusive-or -% $$R = a \oplus b $$ -% -% -% % \begin{figure}[h] %% SOMETHING IS WRONG says latex. very helpful tell me what it fucking is then -% % \centering -% % \caption{Example `XOR' Diagram} -% % \includegraphics[scale=0.80]{ldxor.eps} -% % \label{fig:ld_xor} -% % \end{figure} % XOR -% -% -% \begin{figure}[h+] -% %\centering -% %\input{millivolt_sensor.tex} -% \begin{center} -% % bb= llx lly urx ury; -% \includegraphics[width=200pt,bb=0pt 0pt 800pt 800pt]{logic_diagram/ldxor.eps} -% % resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404 -% \end{center} -% -% %\includegraphics[scale=0.4]{ldxor.eps} -% \caption{Logical XOR} -% \label{fig:ld_xor} -% \end{figure} -% -% \clearpage -% \subsection { Logical IMPLICATION example } -% -% -% An implication $a \rightarrow b$ is represented by diagram \ref{fig:ld_imp}. -% The Equations represented are as follows. -% -% Looking at the conjuctive environment of the test cases -% $$P = (\neg a)$$ -% $$Q = (b)$$ -% From the joining `disjunctive' line R in the diagram. -% $$R = P \vee Q$$ -% Leading to -% $$R = (\neg a) \vee (b)$$ -% which is the standard logic equation for implication. -% -% \begin{figure}[h+] -% %\centering -% %\input{millivolt_sensor.tex} -% \begin{center} -% \includegraphics[width=200pt,bb=0 0 450 404]{logic_diagram/ldimp.eps} -% % resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404 -% \end{center} -% %\includegraphics[scale=0.4]{ldimp.eps} -% \caption{Logical Implication} -% \label{fig:ld_imp} -% \end{figure} -% -% \tiny -% %\vspace{0.3cm} -% \begin{tabular}{||c|c|c|c||} \hline \hline -% -% {\em $a$ } & {\em $b$ } & {implication \em $(\neg a) \vee (b) $ } \\ \hline -% F & F & T \\ \hline -% F & T & T \\ \hline -% T & F & F \\ \hline -% T & T & T \\ \hline \hline: -% \end{tabular} -% %\vspace{0.3cm} -% \normalsize -% -% \clearpage -% \subsection { Diagram representing several Logic Equations Example } -% -% -% bb=0 0 450 404 -% -% -% \begin{figure}[h+] -% %\centering -% %\input{millivolt_sensor.tex} -% \begin{center} -% \includegraphics[width=200pt,bb=0pt 0pt 600pt 600pt]{logic_diagram/ldmeq.eps} -% % resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404 -% \end{center} -% %\includegraphics[scale=0.4]{ldmeq.eps} -% \caption{Several Logical Expressions} -% \label{fig:ld_meq} -% \end{figure} -% -% %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' -% %condition. -% -% Three simple equations are represented in the diagram \ref{fig:ld_dc}. -% -% %The Set of contours $\mho$ represent the `don't care' conditions. -% -% The Equations represented are as follows. -% -% %$$ Q = a \; | \; \mho\{b,c\} $$ -% -% %$$ P = b \wedge c \; | \; \mho\{a\} $$ -% -% $$ Q = a $$ -% $$ P = b \wedge c $$ -% $$ R = b \vee c $$ -% -% % XXXXXX gives annoying impossible to understand syntax messages -% %\small -% %\bibliography{vmgbibliography,mybib} -% %\bibliography{vmgbibliography} -% %\normalsize -% \section{Intended use in FMMD} The intention for these diagrams is that they are used to collect @@ -1010,43 +869,10 @@ The test case AFE represents the condition where all four engines have failed \c \label{fig:allfour} \end{figure} -%\subsection{Example Sub-system} -% -%For instance were a `power supply' being analysed there could be several -%individual component faults or combinations that lead to -%a situation where there is no power. This can be described as a state -%of the powersupply being modeelled as NO\_POWER. -%These can all be collected by DISJUCNTION, i.e. that this this or this -%fault occuring will cause the NO\_POWER fault. Visually this disjuction is -%indicated by the joining lines. -%As far as the user of the `power supply' is concerned, the power supply has failed -%with the failure mode $NO\_POWER$. -%The `power supply' module, after this process will have a defined set of -%fault modes and may be considered as a component at a higher -%level of abstraction. This module can then be combined -%with others at the same abstraction level. -%Note that because this is a fault collection process -%the number of component faults for a module -%must be less than or equal to the sum of the number of component faults. +\section{Conclusion} -%Typeset in \ \ {\huge \LaTeX} \ \ on \ \ \today - -%\begin{verbatim} -%CVS Revision Identity $Id: logic_diagram.tex,v 1.17 2010/01/06 13:41:32 robin Exp $ -%\end{verbatim} -%\ifthenelse {\boolean{paper}} -%{ -% \bibliographystyle{plain} -% \bibliography{../vmgbibliography,../mybib} -% -%} -%{ -%} -%Compiled last \today -%\end{document} - -%\theend +% Elevator Pitch %\pagebreak[4] \clearpage \ No newline at end of file