Went though J.Howse notes.

Need
* example concrete diagram
* example fmmd analysisa case (the hi fi separates)
* make the diagrams consistent
*

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

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 @@ $$ \mathbb{R}^{2} - \; \bigcup_{\hat{c} \in \hat{C}(\hat{d})}\hat{c}$$
 $$ \hat{z} =  \bigcap_{c \in \mathcal{X}} 
              {interior} 
              (\hat{c}) 
-             \; \cup \;
+             %\; \cup \;
-              \bigcap_{\hat{c} \in \hat{C}-X} 
+             \cap % J.Howse correction 25MAR2011 
+             \bigcap_{\hat{c} \in \hat{C}-X} 
              exterior (\hat{c}) 
 $$
 
@@ -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
+The conjunctive logic term associated with a test case
-be determined from the contours that enclose it.
+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
+$\mathcal{G}$ has domain of SMG and range of $P$ given as
-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
+Thus the abstract representation of the diagram, becomes a list of logic terms
-and unused available zones.
+% and unused available zones
-%
+.
-% THIS ABOVE COULD BE ANOTHER DEFINITION
+
+
+
 
 \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 $C$.
+The diagram \ref{fig:inhibit} has a test case in the contour $c$.
-Contour $C$ is \textbf{enclosed} by contour $A$. This says
+Contour $c$ is \textbf{enclosed} by contour $a$. This says
-that for failure~mode $C$ to occur failure mode $A$
+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 $C$
+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
+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 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 $C$
+In failure analysis, $R2$ is the symptom of either failure~mode $a$ 
-occurring. $R1$ is the symptom of $B$ or $A \wedge C$ occurring.
+occurring, this may not necessarily lead to failure mode `c'. $R1$ is the symptom of $a \wedge c$ occurring,
-Note that although $R2$ is a symptom of the functional~group, on its own
+not that because $c$ is enclosed by $a$, $c$ cannot occur unless $a$ has.
-it will not lead to a dangerous failure~mode of the subsystem.
 
 
-% \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}
+\section{Conclusion}
-%
-%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.
 
-%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