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logic diagram must use XOR

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Robin 2010-04-05 13:03:58 +01:00
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146
logic_diagram/logic_diagram.tex
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@ -12,7 +12,7 @@ that control the flow of a computer program. This type of diagram can therefore
integrate logical models from mechanical, electronic and software domains.
Nearly all modern safety critical systems involve these three disiplines.
%
It is intended to be used for analysis of automated safety critical systen
It is intended to be used for analysis of automated safety critical systems.
Many types of safety critical systems now legally
require fault mode effects analysis\cite{FMEA},
but few formal systems exist and wide-spread take-up is
@ -231,12 +231,11 @@ In English:
Test points on the concrete diagram pair-wise connected by a `joining line'
A collection of test points connected by joining lines, is an Fuctionally Merged Group, $FMG$
A collection of test points connected by joining lines, is an Functionally Merged Group, $FMG$
or `test point disjunction'.
An $FMG$ has members which are test points.
{may be merged
and create a
{
\definition{
%A spider is a set of test points where,
%a test point is a member of a spider where it can trace a path connected by joining lines
@ -256,6 +255,7 @@ A singleton test point can be considered a sequence of one test point and is the
% \subsection{Abstract Description of PLD}
%and create a
%
% An Abstract PLD {\em Propositional logic diagram} consists of contours $C$ defining zones $Z$, test points $T$ (which
% are defined by the zone they inhabit) and pair wise connections $W$, which connect test points.
@ -284,11 +284,10 @@ A singleton test point can be considered a sequence of one test point and is the
\item A $FMG$ represents the disjunction of all test points that are members of it.
\end{itemize}
To obtain the set of propositions from a PLD, each $FMG$ must be processed. For each test case
To obtain the set of propositions from a PLD, each $FMG$ must be traversed
along each joining line. For each test case
in the $FMG$ a new section of the equation is disjuctively appended to it.
Let conjunctive logic equation associated with a test point
be determined from the contours that enclose it.
i.e. the contours $\mathcal{X}$ from the zone it inhabits.
@ -397,6 +396,7 @@ In the diagram \ref{fig:ld_and} the area of intersection between the contours $a
represents the conjunction of those conditions. The point $P$ represents the logic equation
$$ P = (a \wedge b) $$
There are no disjunctive joining lines and so this diagram represents one equation only, $ P = (a \wedge b) $.
Note that $P$ is considered to be an $FMG$ with one element, $ (a \wedge b) $
\paragraph{How this would be interpreted in failure analysis}
In failure analysis, this could be considered to be a sub-system with two failure states $a$ and $b$.
@ -430,23 +430,43 @@ $$ P = (a) $$
$$ Q = (b) $$
The two test cases are joined by a the line named $R$.
we thus apply disjunction to the test cases.
$$ R = P \vee Q $$
we thus apply exclusive disjunction to the test cases.
$$ R = P \oplus Q $$
substituting the test cases for their Boolean logic equations gives
$$ R = ((a) \vee (b)) $$.
\begin{equation}
\label{eqn:l_or}
R = ((a) \oplus (b))
\end{equation}
\paragraph{Failure Analysis Interpretation}
Equation \ref{eqn:l_or} would be interpretted to mean that
either failure mode a or b occurring, would have the same failure symptom for the circuit/sub-system
under analysis.
\clearpage
\subsection {Labels and useage}
In diagram \ref{fig:ld_meq} Z and W were labeled but were not necessary for the final expression
of $ R = b \vee c $. The intended use of these diagrams, is that resultant logical conditions be used in a later stage of reasoning.
Test cases joined by disjunction, all become represented in one, resultant equation.
Therefore only test cases not linked by any disjunctive joining lines need be named.
%In diagram \ref{fig:ld_meq} Z and W were labeled but were not necessary for the final expression
%of $ R = b \vee c $. The intended use of these diagrams, is that resultant logical conditions be used in a later stage of reasoning.
%Test cases joined by disjunction, all become represented in one, resultant equation.
%Therefore only test cases not linked by any disjunctive joining lines need be named.
%
%The diagram \ref{fig:ld_meq} can therefore be represented as in diagram \ref{fig:ld_meq2}, with
%two unnamed test cases.
Diagram \ref{fig:ld_meq}
shows three Functionally Merged groups, Q, R and P.
Z and W were labeled but this was not necessary for determination of the final expression
of $ R = b \oplus c $.
%The intended use of these diagrams, is that resultant logical conditions be used in a later stage of reasoning.
%Test cases joined by disjunction, all become represented in one, resultant equation.
%Therefore only test cases not linked by any disjunctive joining lines need be named.
%The diagram \ref{fig:ld_meq} can therefore be represented as in diagram \ref{fig:ld_meq2}, with
%two unnamed test cases.
The diagram \ref{fig:ld_meq} can therefore be represented as in diagram \ref{fig:ld_meq2}, with
two unnamed test cases.
\begin{figure}[h]
\centering
@ -457,7 +477,7 @@ two unnamed test cases.
%
% \begin{figure}[h+]
% %\centering
% %\centeringeragraph
% %\input{millivolt_sensor.tex}
% \begin{center}
% \includegraphics[width=200pt,bb=0pt 0pt 600pt 600pt]{logic_diagram/ldmeq2.jpg}
@ -470,12 +490,20 @@ two unnamed test cases.
\paragraph{How this would be interpreted in failure analysis}
In failure analysis, this could be considered to be a sub-system with two failure states $a$ and $b$.
The proposition $P$ considers the scenario where either failure~mode is active.
Additionally it says that either failure mode $a$ or $b$ being active
will have a resultant effect $R$ on the sub-system. Note that the effect
of $a$ and $b$ both being active is not defined on this diagram.
In failure analysis, this could be considered to be a sub-system with three failure states $a$,$b$ and $c$.
It has three FMG's Q,R and P. Thus there are three ways in which this sub-system can fail.
% \tiny
\vspace{0.3cm}
\begin{tabular}{||c|c|l||} \hline \hline
{\em $FMG$ } & {\em Failure Mode equation } & {\em comments } \\ \hline
Q & $(a)$ & T \\ \hline
P & $(b \oplus c)$ & T \\ \hline
R & $(b \wedge c)$ & F \\ \hline
% T & T & T \\ \hline \hline
\end{tabular}
\vspace{0.3cm}
% \normalsize
\clearpage
@ -488,15 +516,22 @@ Repeated contours are allowed in PLD diagrams.
Logical contradictions or tautologies can be detected automatically by
a software tool which assists in drawing these diagrams.
\begin{figure}[h]
\centering
\includegraphics[bb=0 0 485 206]{logic_diagram/repeated.jpg}
% repeated.jpg: 539x229 pixel, 80dpi, 17.11x7.27 cm, bb=0 0 485 206
\label{fig:repeat}
\includegraphics[width=400pt,bb=0 0 560 195,keepaspectratio=true]{./repeated.jpg}
% repeated.jpg: 560x195 pixel, 72dpi, 19.76x6.88 cm, bb=0 0 560 195
\caption{Contours can appear more than once in a PLD}
\label{fig:repeated}
\end{figure}
%
% \begin{figure}[h]
% \centering
% \includegraphics[bb=0 0 485 206]{logic_diagram/repeated.jpg}
% % repeated.jpg: 539x229 pixel, 80dpi, 17.11x7.27 cm, bb=0 0 485 206
% \label{fig:repeat}
% \end{figure}
% \begin{figure}[h]
% \centering
% \includegraphics[bb=0 0 486 206]{./repeated.jpg}
@ -513,19 +548,22 @@ $$ Q = (a) \wedge (c) $$
The two test cases are joined by a the line named $R1$.
we thus apply disjunction to the test cases.
$$ R1 = P \vee Q $$
$$ R1 = b \vee ( a \wedge c ) $$.
$$ R1 = P \oplus Q $$
$$ R1 = b \oplus ( a \wedge c ) $$.
$R2$ joins two other test cases
$$R2 = a \vee c $$
$$R2 = a \oplus c $$
The test~case residing in the intersection of countours $B$ and $A$
represents the logic equation $R3 = a \wedge b$.
\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.
There is an additional symptom, that of the test case in $A \wedge B$.
In failure analysis, $R2$ is the symptom of either failure~mode $a$ or $c$
occurring. $R1$ is the symptom of $b$ exclusive-or $a \wedge c$ occurring.
The third FMG or symptom shown is test case in $a \wedge b$.
This diagram is incomplete, there is no test case for the fault mode $a$.
The `available region' $a\\b$ has no test case, and this would be considered a `syntax error'
by the FMMD software tool.
@ -537,7 +575,7 @@ There is an additional symptom, that of the test case in $A \wedge B$.
Very often a failure mode can only occurr
given a searate environmental condition.
In Fault Tree Analysis (FTA) this is represented by an inhibit gate.
In Fault Tree Analysis (FTA) this is represented by an inhibit gate.\cite{NASA},\cite{NUK}
\begin{figure}[h]
\centering
@ -559,7 +597,7 @@ that for failure~mode $C$ to occur failure mode $A$
must have occurred.
A well known example of this is the space shuttle `O' ring failure that
caused the 1986 challenger disaster \cite{wdycwopt}.
For the failure mode to occurr the ambiant temperature had to
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
@ -746,24 +784,28 @@ The intention for these diagrams is that they are used to collect
component faults and combinations thereof, into faults that,
at the module level have the same symptoms.
\subsection{Example Sub-system}
The act of collecting common symptoms by joining lines is seen as intuitive.
Syntax checking (looking for contradictions and tautologies), as well as detecting
errors of ommission are automated in the FMMD tool.
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.
%\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.
%Typeset in \ \ {\huge \LaTeX} \ \ on \ \ \today

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logic_diagram/repeated.fig
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@ -1,27 +0,0 @@
#FIG 3.2 Produced by xfig version 3.2.5
Landscape
Center
Metric
A4
100.00
Single
-2
1200 2
5 1 0 1 0 7 50 -1 -1 0.000 0 1 0 0 5173.463 2296.817 3555 2430 5310 3915 6795 2385
5 1 0 1 0 7 50 -1 -1 0.000 0 0 0 0 6772.500 3082.500 5715 2070 6840 1620 7830 2070
1 3 0 1 0 7 50 -1 -1 0.000 1 0.0000 1980 2205 1011 1011 1980 2205 2835 2745
1 3 0 1 0 7 50 -1 -1 0.000 1 0.0000 3150 2205 1028 1028 3150 2205 4095 2610
1 3 0 1 0 7 50 -1 -1 0.000 1 0.0000 6255 2070 1070 1070 6255 2070 7245 2475
1 3 0 1 0 7 50 -1 -1 0.000 1 0.0000 7515 2115 1138 1138 7515 2115 8595 2475
4 0 0 50 -1 0 12 0.0000 4 75 105 2430 2520 *\001
4 0 0 50 -1 0 12 0.0000 4 75 105 6795 2340 *\001
4 0 0 50 -1 0 12 0.0000 4 75 105 7785 2340 *\001
4 0 0 50 -1 0 12 0.0000 4 75 105 5670 2340 *\001
4 0 0 50 -1 0 12 0.0000 4 75 105 3510 2385 *\001
4 0 0 50 -1 0 12 0.0000 4 135 135 1620 1035 A\001
4 0 0 50 -1 0 12 0.0000 4 135 135 2970 945 B\001
4 0 0 50 -1 0 12 0.0000 4 135 135 5985 810 A\001
4 0 0 50 -1 0 12 0.0000 4 135 135 7515 810 C\001
4 0 0 50 -1 0 12 0.0000 4 135 240 4770 3690 R1\001
4 0 0 50 -1 0 12 0.0000 4 135 240 5985 1800 R2\001
4 0 0 50 -1 0 12 0.0000 4 135 240 2340 2160 R3\001

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