Filling it out.
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logic_diagram/CVS/Entries
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/fmmd_hierarchy_cimg5040.eps/1.1/Sun Jul 13 17:13:12 2008//
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/ldmeq2.dia/1.1/Sun Jul 13 17:13:14 2008//
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/ldmeq2.eps/1.1/Sun Jul 13 17:13:14 2008//
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/ldneg.dia/1.1/Sun Jul 13 17:13:14 2008//
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/Makefile/1.2/Wed Nov 25 17:14:59 2009//
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/paper.tex/1.4/Sat Nov 28 20:05:52 2009//
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/inhibit.eps/1.1/Fri Jan 1 14:11:12 2010//
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/repeated.eps/1.1/Fri Jan 1 14:20:31 2010//
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/repeated.fig/1.1/Fri Jan 1 14:20:23 2010//
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/logic_diagram.tex/1.17/Wed Jan 6 13:41:32 2010//
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fmd_docs/phd_docs/logic_diagram
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robin@192.168.2.44:/export/home/cvs/
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logic_diagram/Makefile
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#
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# Make the propositional logic diagram a paper
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#
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paper: paper.tex logic_diagram_paper.tex
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#latex paper.tex
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#dvipdf paper pdflatex cannot use eps ffs
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pdflatex paper.tex
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okular paper.pdf
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# Remove the need for referncing graphics in subdirectories
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#
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logic_diagram_paper.tex: logic_diagram.tex
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cat logic_diagram.tex | sed 's/logic_diagram\///' > logic_diagram_paper.tex
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%EOF
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#FIG 3.2 Produced by xfig version 3.2.5
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Landscape
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11497
logic_diagram/logic_diagram.log
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11497
logic_diagram/logic_diagram.log
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749
logic_diagram/logic_diagram.tex
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logic_diagram/logic_diagram.tex
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@ -0,0 +1,749 @@
|
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||||||
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\begin{verbatim}
|
||||||
|
CVS Revision Identity $Id: logic_diagram.tex,v 1.17 2010/01/06 13:41:32 robin Exp $
|
||||||
|
\end{verbatim}
|
||||||
|
|
||||||
|
\begin{abstract}
|
||||||
|
%This chapter describes using diagrams to represent propositional logic.
|
||||||
|
Propositial Logic Diagrams have been designed to provide an intuitive method for visualising and manipulating
|
||||||
|
logic equations, to express fault modes in Mechanical and Electronic Systems.
|
||||||
|
%To aid hierarchical stages of fault analysis, it has been specifically developed for the purpose of
|
||||||
|
%joining conjunctive conditions with disjuctive conditions
|
||||||
|
%to group the effects of failure modes.
|
||||||
|
Diagrams of this type can also be used to model the logical conditions
|
||||||
|
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
|
||||||
|
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
|
||||||
|
not yet the norm.\cite{takeup}.
|
||||||
|
%
|
||||||
|
Because of its visual nature, it is easy to manipulate and model
|
||||||
|
complicated conditions that can lead to dangerous failures in
|
||||||
|
automated systems.
|
||||||
|
|
||||||
|
% No need to talk about abstraction yet, just define PLD PROPERLY
|
||||||
|
|
||||||
|
The Diagrams described here form the mathematical basis for a new visual and formal system
|
||||||
|
for the analysis of safety critical software and hardware systems.
|
||||||
|
\end{abstract}
|
||||||
|
|
||||||
|
%\title{Propositional Logic Diagrams}
|
||||||
|
%\begin{keyword}
|
||||||
|
% fault~tree fault~mode EN298 EN61508 EN12067 EN230 UL1998 safety~critical logic euler venn propositional
|
||||||
|
%\end{keyword}
|
||||||
|
%\end{frontmatter}
|
||||||
|
|
||||||
|
% In software looking at one condition means lots of dont care situations
|
||||||
|
% in static analysis we look at given sub-sets of faults and assume the other faults
|
||||||
|
% are not active.
|
||||||
|
% This is a major cultural difference !
|
||||||
|
% it deserves a whole chapter.
|
||||||
|
|
||||||
|
|
||||||
|
\section{Introduction}
|
||||||
|
|
||||||
|
Propositional Logic Diagrams (PLDs) have been devised
|
||||||
|
to collect and simplfy fault~modes in safety critical systems undergoing
|
||||||
|
static analysis\cite{FMEA}\cite{SIL}.
|
||||||
|
|
||||||
|
This type of analysis treats failure modes within a system as logical
|
||||||
|
states. PLD provides a visual method for modelling failure~mode analysis
|
||||||
|
within these systems, and specifically
|
||||||
|
identifying common failure symptoms in a user friendly way.
|
||||||
|
Contrasting this to looking at many propositional logic equations directly
|
||||||
|
in a text editor or spreadsheet, a visual method is percieved as being more intuitive.
|
||||||
|
|
||||||
|
|
||||||
|
%Traditional set theory is often represented by Euler\cite{euler} or Spider\cite{spider}
|
||||||
|
%diagrams. These use contours to describe inclusion in a set.may be merged and create a
|
||||||
|
%Propositional Logic Diagrams (PLDs) use named contours represent a logical conditions.
|
||||||
|
%Where an Euler diagram would use
|
||||||
|
%overlapping contours to represent inclusion in sets,
|
||||||
|
%PLDs use these to represent conjunction of the conditions.
|
||||||
|
|
||||||
|
%Named reference points may be placed onto the diagram,
|
||||||
|
%these represent test cases for conjunction.
|
||||||
|
%These can be joined by lines to apply disjunction.
|
||||||
|
%In a spider diagram the lines would represent that the object represented coul;d belong to either set.
|
||||||
|
%in a PLD it means that the logical conditions represent disjuction; a boolean OR condition.
|
||||||
|
%these points may be joined.
|
||||||
|
|
||||||
|
PLDs use three visual features that
|
||||||
|
can be combined to represent logic equations. Closed contours (using dashed lines), test cases, and lines that
|
||||||
|
link test cases.
|
||||||
|
All features may be labelled, and the labels must be unique within a diagram, however contours may be repeated in the diagram.
|
||||||
|
%Aditionally a label begining with the `$\neg$' character, applied only to a contour, represents negation.
|
||||||
|
|
||||||
|
|
||||||
|
%Regions defined by contours are used to represent given conjunctive logical conditions.
|
||||||
|
|
||||||
|
Test cases are marked by asterisks. These are used as a visual `anchor'
|
||||||
|
to mark a logical condition, the logical condition being defined by the contours
|
||||||
|
that enclose the region on which the test~case has been placed.
|
||||||
|
The contours that enclose represent conjuction.
|
||||||
|
Test~cases may be connected by joining lines. These lines represent disjunction (Boolean `OR') of
|
||||||
|
test~cases.
|
||||||
|
|
||||||
|
With these three visual syntax elements, we have the basic building blocks for all logic equations possible.
|
||||||
|
\begin{description}
|
||||||
|
\item Test cases - Points on the plane indicating a logical condition.
|
||||||
|
\item Conjunction - Overlapping contours
|
||||||
|
\item Disjunction - Joining of named test cases.
|
||||||
|
%\item Negation - Countours negatively named
|
||||||
|
\end{description}
|
||||||
|
|
||||||
|
% Because of this
|
||||||
|
% we have the complete suite of logical primitives here, conjunction, disjuction and negation.
|
||||||
|
% Form these complex logic equations can be respresented in 2D.
|
||||||
|
|
||||||
|
% Another advantage of this is being able to describe `don't care' conditions.
|
||||||
|
% Very often in digital hardware design, or in a computer program
|
||||||
|
% many logical conditions are `don't care'.
|
||||||
|
% These are difficult to specify in set theory.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\section{Formal Description of PLD}
|
||||||
|
|
||||||
|
Definitions of conrete and abstract PLD's follow.
|
||||||
|
Well-formedness conditions for PLD's are separated from this definition, because of
|
||||||
|
practical differences between the way they are used to represent software as opposed to
|
||||||
|
representing electronics and mechanical systems.
|
||||||
|
|
||||||
|
\subsection{Concrete PLD Definition}
|
||||||
|
|
||||||
|
A concrete {\em Propositional logic diagram} is a set of labeled {\em contours}
|
||||||
|
(closed curves) in the plane. The minimal regions formed by the closed curves
|
||||||
|
can by occupied by `test points'.
|
||||||
|
The `test points' may be joined by joining lines.
|
||||||
|
A group of `test points' connected by joining lines
|
||||||
|
is defined as a `test point disjunction' or Spider.
|
||||||
|
Spiders may be labeled.
|
||||||
|
|
||||||
|
To differentiate these from common Euler diagram notation (normally used to represent set theory)
|
||||||
|
the curves are drawn using dotted and dashed lines.
|
||||||
|
|
||||||
|
\subsection{ PLD Definition}
|
||||||
|
In English:
|
||||||
|
The elements that can be found in a PLD diagram are a number of contours,
|
||||||
|
a number of test points and joining lines that connect
|
||||||
|
test points.
|
||||||
|
{
|
||||||
|
\definition{A concrete PLD $d$ is a set comprising of a set of
|
||||||
|
closed curves $C=C(d)$, a set of test points $T=T(d)$ and
|
||||||
|
a set of test point joining lines $J=J(d)$.
|
||||||
|
$$d=\{C,T,J\}$$
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
In English:
|
||||||
|
Each element of the diagram has a unique label within the diagram.
|
||||||
|
%Thus the set of labels found in a diagram is
|
||||||
|
%a subset of the powerset of characters that can be present in a label.
|
||||||
|
|
||||||
|
%{
|
||||||
|
%\definition{ $ \mathcal{F}_{d}:C(d) \rightarrow \mathcal{P}\Lambda$ is a
|
||||||
|
%function associating a label drawn from an infinite
|
||||||
|
%set of labels $\Lambda$.
|
||||||
|
%}
|
||||||
|
%}
|
||||||
|
|
||||||
|
%In English:
|
||||||
|
%A minimal region of a PLD diagram d is a
|
||||||
|
%region bounded by curves.
|
||||||
|
%connected component of $\mathbb{R}^{2} - \; \bigcup_{c \in C(d)} c$
|
||||||
|
|
||||||
|
%That is to say the complement of all other regions is subtracted from the plane.
|
||||||
|
%- Or in another way- that smallest area defined by the curves that enclose it
|
||||||
|
|
||||||
|
|
||||||
|
%% \hat is used to indicate CONCRETE
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
A minimal region of concrete PLD diagram d is a connected component of
|
||||||
|
|
||||||
|
$$ \mathbb{R}^{2} - \; \bigcup_{\hat{c} \in \hat{C}(\hat{d})}\hat{c}$$
|
||||||
|
|
||||||
|
% I.e. The contours break the connectivity
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition
|
||||||
|
{
|
||||||
|
Let d be a PLD and $ \mathcal{X} \subseteq \hat{C}(\hat{d})$ a set of countours.
|
||||||
|
If the set
|
||||||
|
$$ \hat{z} = \bigcap_{c \in \mathcal{X}}
|
||||||
|
{interior}
|
||||||
|
(\hat{c})
|
||||||
|
\; \cup \;
|
||||||
|
\bigcap_{\hat{c} \in \hat{C}-X}
|
||||||
|
exterior (\hat{c})
|
||||||
|
$$
|
||||||
|
|
||||||
|
is non empty, then $\hat{z}$ is a concrete zone of $\hat{d}$. A zone is a union of minimal regions. The set of all concrete zones of $\hat{d}$
|
||||||
|
is denoted $ \hat{\mathcal{Z}} $.
|
||||||
|
|
||||||
|
% NOT interested in labelling the zones.
|
||||||
|
% but am interested in
|
||||||
|
%The set of labels associated with the contours in $\mathcal{X}$ is the zone label set $\hat{\mathcal{Z}}(\hat{z})$
|
||||||
|
%of $\hat{z}$.
|
||||||
|
%$$ \hat{\mathcal{L}}(\hat{z}) = \bigcup_{\hat{c} \in \mathcal{X}} \mathcal{F}_{d}(\hat{c}) $$
|
||||||
|
%
|
||||||
|
% So Z is the set of all available for use ZONES; great !
|
||||||
|
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
|
Each minimal region in the plane may be inhabited by one or more `test points'.
|
||||||
|
% One or more because in software the same logical conditions mean existing in the same
|
||||||
|
% region. For electroincs or mechanical, one test point per region is
|
||||||
|
% mandatory. How to describe ?????
|
||||||
|
Each test point 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
|
||||||
|
associating a testpoint with a set of contours in the plane. This corresponds to the interior of the contours defining the zone.
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
Pairs of test points may be joined by joining lines.
|
||||||
|
The operator $\stackrel{join}{\leftrightarrow}$ is used to
|
||||||
|
show that two points are joined by a line in the concrete diagram.
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
|
||||||
|
$ \mathcal{F}_{j}$ is a function
|
||||||
|
associating a joining line with a pair of test points. The Join t1,t2 is defined as
|
||||||
|
|
||||||
|
%$$ \mathcal{F}_{d}:J(d)\rightarrow \{t1,t2\ | t1 \in T(d) \wedge t2 \in T(d) \wedge t1 \neq t2 %\wedge t1 \stackrel{join}{\leftrightarrow} t2\} $$
|
||||||
|
$$ \mathcal{F}_{d}:J(d)\rightarrow \{t1,t2\ | t1 \in T(d) \wedge t2 \in T(d) \wedge t1 \neq t2 \} $$
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
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, $FMG$
|
||||||
|
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
|
||||||
|
%to another member of the spider. A singleton test point can be considered a spider.
|
||||||
|
Let d be a PLD : An $FMG$ is a maximal set of test points in d where
|
||||||
|
the test points belong to a sequence connected by joining lines such that:
|
||||||
|
|
||||||
|
$$ t_i \stackrel{join}{\leftrightarrow} t_n, for \; i = 1, ..., n $$
|
||||||
|
|
||||||
|
|
||||||
|
OR consider an $FMG$ as a tree whose nodes are test points.
|
||||||
|
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
A singleton test point can be considered a sequence of one test point and is therefore also an $FMG$.
|
||||||
|
|
||||||
|
|
||||||
|
% \subsection{Abstract Description of PLD}
|
||||||
|
%
|
||||||
|
% 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.
|
||||||
|
% Collections of test points, linked by shared conecting lines, form a set of test point groups $G$.
|
||||||
|
%
|
||||||
|
% A Zone defined by the contours that enclose it in the concrete diagram.
|
||||||
|
%
|
||||||
|
% $$ Z \subseteq C $$
|
||||||
|
%
|
||||||
|
% A test point $t \in T$ in habits a zone on the diagram.
|
||||||
|
%
|
||||||
|
% $$ \eta(t) = Z $$
|
||||||
|
%
|
||||||
|
% A joining line $$ w \in W $$ joins test points.
|
||||||
|
%
|
||||||
|
% $$ w = t1 \stackrel{join}{\rightarrow} t2 | t1 \neq t2 \wedge t1 \in T \wedge t2 \in T $$
|
||||||
|
%
|
||||||
|
% A test point group $g \in G$ is defined by test points linked by shared connecting lines.
|
||||||
|
|
||||||
|
|
||||||
|
\subsection{Semantics of PLD}
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item A closed curve in a PLD represents a condition (logical state) being modelled.
|
||||||
|
\item A test point represents the conjunction of the conditions represented by the curves that enclose it.
|
||||||
|
\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
|
||||||
|
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.
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
|
||||||
|
Let $\mathcal{F}_{t}$ be a function mapping a test point to a proposition / logical equation $p \in P$.
|
||||||
|
The test point inhabits the zone $\mathcal{Z}$ which is a collection of contours (the contours that enclose the test point.
|
||||||
|
|
||||||
|
$$ \mathcal{F}:T \rightarrow P $$
|
||||||
|
|
||||||
|
%$$ \mathcal{F}(t): p = \bigwedge_{c \in \mathcal{Z}} \Lambda c $$
|
||||||
|
|
||||||
|
$$ \mathcal{F}(t): p = \bigwedge_{c \in \mathcal{Z}} c $$
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
In English:
|
||||||
|
Thus a `test point' enclosed by contours labelled $a,b,c$ would be represented by the logic equation
|
||||||
|
$ a \wedge b \wedge c $.
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
Let $\mathcal{G}_{fmg}$ be a function that returns a logic equation for a given $FMG$
|
||||||
|
$fmg$ in the diagram, where an FMG is a non empty set of test points
|
||||||
|
% $t$ is a `test point'
|
||||||
|
|
||||||
|
$$ \mathcal{G}:FMG \rightarrow P_{fmg} $$
|
||||||
|
|
||||||
|
The logic equation representing an FMG $p_{fmg}$ can be determined thus.
|
||||||
|
|
||||||
|
$$\mathcal{G}_{fmg}(fmg) = \bigvee_{t \in fmg} (\; \mathcal{F}_{t} (t) \;) $$
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
The abstract PLD diagram is a set of logic equations representing all FMGs,
|
||||||
|
along with unused zones (i.e. zones that are not inhabited by FMGs).
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
\label{FMGderivation}
|
||||||
|
A diagram can be reduced to a collection of $FMG$s.
|
||||||
|
A new diagram can be derived from this, replacing a contour for each FMG.
|
||||||
|
This diagram is at one higher level of abstraction then the diagram that
|
||||||
|
it was produced from.
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
|
\section{Example Diagrams}
|
||||||
|
|
||||||
|
|
||||||
|
\subsection {How to read a PLD diagram}
|
||||||
|
|
||||||
|
PLD diagrams are read by first looking at the test case points.
|
||||||
|
The test case asterisk will be enclosed by one or more contours.
|
||||||
|
These contours are collected and form the logical conjunction
|
||||||
|
equation for the test case.
|
||||||
|
These test case points thus represent the conjunctive aspects
|
||||||
|
of an equation defined in a PLD. Where these test cases are joined by lines;
|
||||||
|
these represent disjunction of the conjunctive aspects defined by the test cases.
|
||||||
|
Joining lines thus represent dis-junction in a PLD.
|
||||||
|
|
||||||
|
%Where negation and assertion of a logical condition is required in the same diagram, a separate contour can be created, which is
|
||||||
|
%..assigned the same name as its positive counter part, but preceded by a negation `$\neg$' sign.
|
||||||
|
%Obviously were a drawing to show conjunction of a contour and its complement
|
||||||
|
%this would result in a contradiction for any test case placed on it, and would be a visual `syntax error'.
|
||||||
|
|
||||||
|
%Note that negation is handled explicitly. This is to allow `don't care'
|
||||||
|
%conditions. Should a test case be outside a contour, that contour is a `don't care' condition.
|
||||||
|
%In a PLD, contours may be represented in complement, to provide
|
||||||
|
%logical negation. Here the contour name is begun with the negation symbol `$\neg$'.
|
||||||
|
%%To represent conjunction of logical conditions (Boolean `AND'), contours may be overlapped.
|
||||||
|
|
||||||
|
%Providing explicit negation, in addition to disjunction and conjunction
|
||||||
|
%allows us to represent `don't care' or `tri-state' logical conditions. Simply by
|
||||||
|
%not using the conditions we are not interested in they are `don't care'.
|
||||||
|
%
|
||||||
|
|
||||||
|
%\section{Example Logic Diagrams}
|
||||||
|
|
||||||
|
\subsection{ Logical AND example }
|
||||||
|
|
||||||
|
|
||||||
|
\begin{figure}[h+]
|
||||||
|
\begin{center}
|
||||||
|
\includegraphics[width=200pt,bb=0 0 450 404]{logic_diagram/ldand.eps}
|
||||||
|
% resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404
|
||||||
|
\end{center}
|
||||||
|
|
||||||
|
%\includegraphics[scale=0.6]{ldand.eps}
|
||||||
|
\caption{Logical AND}
|
||||||
|
\label{fig:ld_and}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
In the diagram \ref{fig:ld_and} the area of intersection between the contours $a$ and $b$
|
||||||
|
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) $.
|
||||||
|
|
||||||
|
\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 both failure~modes are active.
|
||||||
|
|
||||||
|
\clearpage
|
||||||
|
\subsection { Logical OR example }
|
||||||
|
|
||||||
|
\begin{figure}[h+]
|
||||||
|
%\centering
|
||||||
|
%\input{ldor.tex}
|
||||||
|
\begin{center}
|
||||||
|
\includegraphics[width=200pt,bb=0 0 450 404]{logic_diagram/ldor.eps}
|
||||||
|
% resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404
|
||||||
|
\end{center}
|
||||||
|
%\includegraphics[scale=0.60]{ldor.eps}
|
||||||
|
\caption{Logical OR}
|
||||||
|
\label{fig:ld_or}
|
||||||
|
\end{figure} % OR
|
||||||
|
|
||||||
|
|
||||||
|
The diagram \ref{fig:ld_or} is converted to Boolean logic by first looking at the test cases, and
|
||||||
|
the contours they are placed on.
|
||||||
|
$$ P = (a) $$
|
||||||
|
$$ Q = (b) $$
|
||||||
|
|
||||||
|
The two test cases are joined by a the line named $R$.
|
||||||
|
we thus apply disjunction to the test cases.
|
||||||
|
$$ R = P \vee Q $$
|
||||||
|
substituting the test cases for their Boolean logic equations gives
|
||||||
|
$$ R = ((a) \vee (b)) $$.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\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.
|
||||||
|
|
||||||
|
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
|
||||||
|
%\input{millivolt_sensor.tex}
|
||||||
|
\begin{center}
|
||||||
|
\includegraphics[width=200pt,bb=0pt 0pt 600pt 600pt]{logic_diagram/ldmeq2.eps}
|
||||||
|
% resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404
|
||||||
|
\end{center}
|
||||||
|
%\includegraphics[scale=0.4]{ldmeq2.eps}
|
||||||
|
\caption{Several Logical Expressions with unamed test cases}
|
||||||
|
\label{fig:ld_meq2}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
\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.
|
||||||
|
|
||||||
|
|
||||||
|
\clearpage
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\subsection { Repeated Contour example }
|
||||||
|
|
||||||
|
|
||||||
|
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 0 486 206]{./repeated.eps}
|
||||||
|
% repeated.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 486 206
|
||||||
|
\label{fig:repeated}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
The diagram \ref{fig:repeat} is converted to Boolean logic by first looking at the test cases, and
|
||||||
|
the contours they are placed on.
|
||||||
|
$$ P = (b) $$
|
||||||
|
$$ Q = (a) \wedge (c) $$
|
||||||
|
|
||||||
|
The two test cases are joined by a the line named $R1$.
|
||||||
|
we thus apply disjunction to the test cases.
|
||||||
|
$$ R1 = P \vee Q $$
|
||||||
|
$$ R1 = b \vee ( a \wedge c ) $$.
|
||||||
|
|
||||||
|
$R2$ joins two other test cases
|
||||||
|
$$R2 = a \vee c $$
|
||||||
|
|
||||||
|
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 $A \wedge B$.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\clearpage
|
||||||
|
|
||||||
|
|
||||||
|
\subsection { Inhibit Failure }
|
||||||
|
|
||||||
|
|
||||||
|
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.
|
||||||
|
|
||||||
|
|
||||||
|
\begin{figure}[h]
|
||||||
|
\centering
|
||||||
|
\includegraphics[bb=0 0 364 228]{./inhibit.eps}
|
||||||
|
% inhibit.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 364 228
|
||||||
|
\label{fig:inhibit}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
The diagram \ref{fig:inhibit} has a test case in the contour $C$.
|
||||||
|
Contour $C$ is enclosed by contour $A$. This says
|
||||||
|
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
|
||||||
|
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 occurr if $A$ is true.
|
||||||
|
The `O' ring could fail in a different way independant of the critical temperature and this is
|
||||||
|
represented, for the sake of this example, by contour $D$.
|
||||||
|
|
||||||
|
In terms of propositional logic, the inhibit gate of FTA, and the countour enclosure
|
||||||
|
of PLD represent {\em implication}.
|
||||||
|
\\
|
||||||
|
\tiny
|
||||||
|
\vspace{0.3cm}
|
||||||
|
\begin{tabular}{||c|c|c|c||} \hline \hline
|
||||||
|
{\em $c$ } & {\em $a$ } & {\em $R1$ } \\ \hline
|
||||||
|
F & F & T \\ \hline
|
||||||
|
F & T & T \\ \hline
|
||||||
|
T & F & F \\ \hline
|
||||||
|
T & T & T \\ \hline \hline
|
||||||
|
\end{tabular}
|
||||||
|
\vspace{0.3cm}
|
||||||
|
\normalsize
|
||||||
|
|
||||||
|
$$ R1 = c \implies a $$
|
||||||
|
$$ R2 = a $$
|
||||||
|
$$ R3 = d $$
|
||||||
|
|
||||||
|
|
||||||
|
\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 sub-system, on its own
|
||||||
|
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
|
||||||
|
component faults and combinations thereof, into faults that,
|
||||||
|
at the module level have the same symptoms.
|
||||||
|
|
||||||
|
\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
|
||||||
|
|
||||||
|
\begin{verbatim}
|
||||||
|
CVS Revision Identity $Id: logic_diagram.tex,v 1.17 2010/01/06 13:41:32 robin Exp $
|
||||||
|
\end{verbatim}
|
||||||
|
Compiled last \today
|
||||||
|
%\end{document}
|
||||||
|
|
||||||
|
%\theend
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
723
logic_diagram/logic_diagram.tex.backup
Normal file
723
logic_diagram/logic_diagram.tex.backup
Normal file
@ -0,0 +1,723 @@
|
|||||||
|
|
||||||
|
\begin{verbatim}
|
||||||
|
CVS Revision Identity $Id: logic_diagram.tex,v 1.15 2009/02/09 07:33:27 robin Exp $
|
||||||
|
\end{verbatim}
|
||||||
|
|
||||||
|
\begin{abstract}
|
||||||
|
%This chapter describes using diagrams to represent propositional logic.
|
||||||
|
Propositial Logic Diagrams have been designed to provide an intuitive method for visualising and manipulating
|
||||||
|
logic equations, to express fault modes in Mechanical and Electronic Systems.
|
||||||
|
%To aid hierarchical stages of fault analysis, it has been specifically developed for the purpose of
|
||||||
|
%joining conjunctive conditions with disjuctive conditions
|
||||||
|
%to group the effects of failure modes.
|
||||||
|
Diagrams of this type can also be used to model the logical conditions
|
||||||
|
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
|
||||||
|
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
|
||||||
|
not yet the norm.\cite{takeup}.
|
||||||
|
%
|
||||||
|
Because of its visual nature, it is easy to manipulate and model
|
||||||
|
complicated conditions that can lead to dangerous failures in
|
||||||
|
automated systems.
|
||||||
|
|
||||||
|
% No need to talk about abstraction yet, just define PLD PROPERLY
|
||||||
|
|
||||||
|
The Diagrams described here form the mathematical basis for a new visual and formal system
|
||||||
|
for the analysis of safety critical software and hardware systems.
|
||||||
|
\end{abstract}
|
||||||
|
|
||||||
|
%\title{Propositional Logic Diagrams}
|
||||||
|
%\begin{keyword}
|
||||||
|
% fault~tree fault~mode EN298 EN61508 EN12067 EN230 UL1998 safety~critical logic euler venn propositional
|
||||||
|
%\end{keyword}
|
||||||
|
%\end{frontmatter}
|
||||||
|
|
||||||
|
|
||||||
|
\section{Introduction}
|
||||||
|
|
||||||
|
Propositional Logic Diagrams (PLDs) have been devised
|
||||||
|
to collect and simplfy fault~modes in safety critical systems undergoing
|
||||||
|
static analysis\cite{FMEA}\cite{SIL}.
|
||||||
|
|
||||||
|
This type of analysis treats failure modes within a system as logical
|
||||||
|
states. PLD provides a visual method for modelling failure~mode analysis
|
||||||
|
within these systems, and specifically
|
||||||
|
identifying common failure symptoms in a user friendly way.
|
||||||
|
Contrasting this to looking at many propositional logic equations directly
|
||||||
|
in a text editor or spreadsheet, a visual method is prefferred.
|
||||||
|
|
||||||
|
|
||||||
|
%Traditional set theory is often represented by Euler\cite{euler} or Spider\cite{spider}
|
||||||
|
%diagrams. These use contours to describe inclusion in a set.may be merged and create a
|
||||||
|
%Propositional Logic Diagrams (PLDs) use named contours represent a logical conditions.
|
||||||
|
%Where an Euler diagram would use
|
||||||
|
%overlapping contours to represent inclusion in sets,
|
||||||
|
%PLDs use these to represent conjunction of the conditions.
|
||||||
|
|
||||||
|
%Named reference points may be placed onto the diagram,
|
||||||
|
%these represent test cases for conjunction.
|
||||||
|
%These can be joined by lines to apply disjunction.
|
||||||
|
%In a spider diagram the lines would represent that the object represented coul;d belong to either set.
|
||||||
|
%in a PLD it means that the logical conditions represent disjuction; a boolean OR condition.
|
||||||
|
%these points may be joined.
|
||||||
|
|
||||||
|
PLDs use three visual features that
|
||||||
|
can be combined to represent logic equations. Closed contours (using dashed lines), test cases, and joining lines.
|
||||||
|
All features may be labelled, and the labels must be unique within a diagram, however contours may be repeated in the diagram.
|
||||||
|
%Aditionally a label begining with the `$\neg$' character, applied only to a contour, represents negation.
|
||||||
|
|
||||||
|
|
||||||
|
%Regions defined by contours are used to represent given conjunctive logical conditions.
|
||||||
|
|
||||||
|
Test cases are marked by asterisks. These are used as a visual `anchor'
|
||||||
|
to mark a logical condition, the logical condition being defined by the countours
|
||||||
|
that enclose the region on which the test case has been placed.
|
||||||
|
Test cases may be pair-wise connected by named lines representing disjunction (Boolean `OR') of
|
||||||
|
the conditions defined by the placement of the test case markers.
|
||||||
|
|
||||||
|
With these three visual syntax elements, we have the basic building blocks for all logic equations possible.
|
||||||
|
\begin{description}
|
||||||
|
\item Test cases - Points on the plane indicating a logical condition.
|
||||||
|
\item Conjunction - Overlapping contours
|
||||||
|
\item Disjunction - Joining of named test cases.
|
||||||
|
%\item Negation - Countours negatively named
|
||||||
|
\end{description}
|
||||||
|
|
||||||
|
% Because of this
|
||||||
|
% we have the complete suite of logical primitives here, conjunction, disjuction and negation.
|
||||||
|
% Form these complex logic equations can be respresented in 2D.
|
||||||
|
|
||||||
|
% Another advantage of this is being able to describe `don't care' conditions.
|
||||||
|
% Very often in digital hardware design, or in a computer program
|
||||||
|
% many logical conditions are `don't care'.
|
||||||
|
% These are difficult to specify in set theory.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\section{Formal Description of PLD}
|
||||||
|
|
||||||
|
Definitions of conrete and abstract PLD's follow.
|
||||||
|
Well-formedness conditions for PLD's are separated from this definition, because of
|
||||||
|
practical differences between the way they are used to represent software as opposed to
|
||||||
|
representing electronics and mechanical systems.
|
||||||
|
|
||||||
|
\subsection{Concrete PLD Definition}
|
||||||
|
|
||||||
|
A concrete {\em Propositional logic diagram} is a set of labeled {\em contours}
|
||||||
|
(closed curves) in the plane. The minimal regions formed by the closed curves
|
||||||
|
can by occupied by `test points'.
|
||||||
|
The `test points' may be joined by joining lines.
|
||||||
|
A group of `test points' connected by joining lines
|
||||||
|
is defined as a `test point disjunction' or Spider.
|
||||||
|
Spiders may be labeled.
|
||||||
|
|
||||||
|
To differentiate these from common Euler diagram notation (normally used to represent set theory)
|
||||||
|
the curves are drawn using dotted and dashed lines.
|
||||||
|
|
||||||
|
\subsection{ PLD Definition}
|
||||||
|
In English:
|
||||||
|
The elements that can be found in a PLD diagram are a number of contours,
|
||||||
|
a number of test points and joining lines that connect
|
||||||
|
test points.
|
||||||
|
{
|
||||||
|
\definition{A concrete PLD $d$ is a set comprising of a set of
|
||||||
|
closed curves $C=C(d)$, a set of test points $T=T(d)$ and
|
||||||
|
a set of test point joining lines $J=J(d)$.
|
||||||
|
$$d=\{C,T,J\}$$
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
In English:
|
||||||
|
Each element of the diagram has a unique label within the diagram.
|
||||||
|
%Thus the set of labels found in a diagram is
|
||||||
|
%a subset of the powerset of characters that can be present in a label.
|
||||||
|
|
||||||
|
%{
|
||||||
|
%\definition{ $ \mathcal{F}_{d}:C(d) \rightarrow \mathcal{P}\Lambda$ is a
|
||||||
|
%function associating a label drawn from an infinite
|
||||||
|
%set of labels $\Lambda$.
|
||||||
|
%}
|
||||||
|
%}
|
||||||
|
|
||||||
|
%In English:
|
||||||
|
%A minimal region of a PLD diagram d is a
|
||||||
|
%region bounded by curves.
|
||||||
|
%connected component of $\mathbb{R}^{2} - \; \bigcup_{c \in C(d)} c$
|
||||||
|
|
||||||
|
%That is to say the complement of all other regions is subtracted from the plane.
|
||||||
|
%- Or in another way- that smallest area defined by the curves that enclose it
|
||||||
|
|
||||||
|
|
||||||
|
%% \hat is used to indicate CONCRETE
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
A minimal region of concrete PLD diagram d is a connected component of
|
||||||
|
|
||||||
|
$$ \mathbb{R}^{2} - \; \bigcup_{\hat{c} \in \hat{C}(\hat{d})}\hat{c}$$
|
||||||
|
|
||||||
|
% I.e. The contours break the connectivity
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition
|
||||||
|
{
|
||||||
|
Let d be a PLD and $ \mathcal{X} \subseteq \hat{C}(\hat{d})$ a set of countours.
|
||||||
|
If the set
|
||||||
|
$$ \hat{z} = \bigcap_{c \in \mathcal{X}}
|
||||||
|
{interior}
|
||||||
|
(\hat{c})
|
||||||
|
\; \cup \;
|
||||||
|
\bigcap_{\hat{c} \in \hat{C}-X}
|
||||||
|
exterior (\hat{c})
|
||||||
|
$$
|
||||||
|
|
||||||
|
is non empty, then $\hat{z}$ is a concrete zone of $\hat{d}$. A zone is a union of minimal regions. The set of all concrete zones of $\hat{d}$
|
||||||
|
is denoted $ \hat{\mathcal{Z}} $.
|
||||||
|
|
||||||
|
% NOT interested in labelling the zones.
|
||||||
|
% but am interested in
|
||||||
|
%The set of labels associated with the contours in $\mathcal{X}$ is the zone label set $\hat{\mathcal{Z}}(\hat{z})$
|
||||||
|
%of $\hat{z}$.
|
||||||
|
%$$ \hat{\mathcal{L}}(\hat{z}) = \bigcup_{\hat{c} \in \mathcal{X}} \mathcal{F}_{d}(\hat{c}) $$
|
||||||
|
%
|
||||||
|
% So Z is the set of all available for use ZONES; great !
|
||||||
|
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
|
Each minimal region in the plane may be inhabited by one or more `test points'.
|
||||||
|
% One or more because in software the same logical conditions mean existing in the same
|
||||||
|
% region. For electroincs or mechanical, one test point per region is
|
||||||
|
% mandatory. How to describe ?????
|
||||||
|
Each test point 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
|
||||||
|
associating a testpoint with a set of contours in the plane. This corresponds to the interior of the contours defining the zone.
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
Pairs of test points may be joined by joining lines.
|
||||||
|
The operator $\stackrel{join}{\leftrightarrow}$ is used to
|
||||||
|
show that two points are joined by a line in the concrete diagram.
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
|
||||||
|
$ \mathcal{F}_{j}$ is a function
|
||||||
|
associating a joining line with a pair of test points. The Join t1,t2 is defined as
|
||||||
|
|
||||||
|
%$$ \mathcal{F}_{d}:J(d)\rightarrow \{t1,t2\ | t1 \in T(d) \wedge t2 \in T(d) \wedge t1 \neq t2 %\wedge t1 \stackrel{join}{\leftrightarrow} t2\} $$
|
||||||
|
$$ \mathcal{F}_{d}:J(d)\rightarrow \{t1,t2\ | t1 \in T(d) \wedge t2 \in T(d) \wedge t1 \neq t2 \} $$
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
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, $FMG$
|
||||||
|
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
|
||||||
|
%to another member of the spider. A singleton test point can be considered a spider.
|
||||||
|
Let d be a PLD : An $FMG$ is a maximal set of test points in d where
|
||||||
|
the test points belong to a sequence connected by joining lines such that:
|
||||||
|
|
||||||
|
$$ t_i \stackrel{join}{\leftrightarrow} t_n, for \; i = 1, ..., n $$
|
||||||
|
|
||||||
|
|
||||||
|
OR consider an $FMG$ as a tree whose nodes are test points.
|
||||||
|
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
A singleton test point can be considered a sequence of one test point and is therefore also an $FMG$.
|
||||||
|
|
||||||
|
|
||||||
|
% \subsection{Abstract Description of PLD}
|
||||||
|
%
|
||||||
|
% 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.
|
||||||
|
% Collections of test points, linked by shared conecting lines, form a set of test point groups $G$.
|
||||||
|
%
|
||||||
|
% A Zone defined by the contours that enclose it in the concrete diagram.
|
||||||
|
%
|
||||||
|
% $$ Z \subseteq C $$
|
||||||
|
%
|
||||||
|
% A test point $t \in T$ in habits a zone on the diagram.
|
||||||
|
%
|
||||||
|
% $$ \eta(t) = Z $$
|
||||||
|
%
|
||||||
|
% A joining line $$ w \in W $$ joins test points.
|
||||||
|
%
|
||||||
|
% $$ w = t1 \stackrel{join}{\rightarrow} t2 | t1 \neq t2 \wedge t1 \in T \wedge t2 \in T $$
|
||||||
|
%
|
||||||
|
% A test point group $g \in G$ is defined by test points linked by shared connecting lines.
|
||||||
|
|
||||||
|
|
||||||
|
\subsection{Semantics of PLD}
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item A closed curve in a PLD represents a condition (logical state) being modelled.
|
||||||
|
\item A test point represents the conjunction of the conditions represented by the curves that enclose it.
|
||||||
|
\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
|
||||||
|
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.
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
|
||||||
|
Let $\mathcal{F}_{t}$ be a function mapping a test point to a proposition / logical equation $p \in P$.
|
||||||
|
The test point inhabits the zone $\mathcal{Z}$ which is a collection of contours (the contours that enclose the test point.
|
||||||
|
|
||||||
|
$$ \mathcal{F}:T \rightarrow P $$
|
||||||
|
|
||||||
|
%$$ \mathcal{F}(t): p = \bigwedge_{c \in \mathcal{Z}} \Lambda c $$
|
||||||
|
|
||||||
|
$$ \mathcal{F}(t): p = \bigwedge_{c \in \mathcal{Z}} c $$
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
In English:
|
||||||
|
Thus a `test point' enclosed by contours labelled $a,b,c$ would be represented by the logic equation
|
||||||
|
$ a \wedge b \wedge c $.
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
Let $\mathcal{G}_{fmg}$ be a function that returns a logic equation for a given $FMG$
|
||||||
|
$fmg$ in the diagram, where an FMG is a non empty set of test points
|
||||||
|
% $t$ is a `test point'
|
||||||
|
|
||||||
|
$$ \mathcal{G}:FMG \rightarrow P_{fmg} $$
|
||||||
|
|
||||||
|
The logic equation representing an FMG $p_{fmg}$ can be determined thus.
|
||||||
|
|
||||||
|
$$\mathcal{G}_{fmg}(fmg) = \bigvee_{t \in fmg} (\; \mathcal{F}_{t} (t) \;) $$
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
The abstract PLD diagram is a set of logic equations representing all FMGs,
|
||||||
|
along with unused zones (i.e. zones that are not inhabited by FMGs).
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
\label{FMGderivation}
|
||||||
|
A diagram can be reduced to a collection of $FMG$s.
|
||||||
|
A new diagram can be derived from this, replacing a contour for each FMG.
|
||||||
|
This diagram is at one higher level of abstraction then the diagram that
|
||||||
|
it was produced from.
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
|
\section{Example Diagrams}
|
||||||
|
|
||||||
|
|
||||||
|
\subsection {How to read a PLD diagram}
|
||||||
|
|
||||||
|
PLD diagrams are read by first looking at the test case points.
|
||||||
|
The test case asterisk will be enclosed by one or more contours.
|
||||||
|
These contours are collected and form the logical conjunction
|
||||||
|
equation for the test case.
|
||||||
|
These test case points thus represent the conjunctive aspects
|
||||||
|
of an equation defined in a PLD. Where these test cases are joined by lines;
|
||||||
|
these represent disjunction of the conjunctive aspects defined by the test cases.
|
||||||
|
Joining lines thus represent dis-junction in a PLD.
|
||||||
|
|
||||||
|
%Where negation and assertion of a logical condition is required in the same diagram, a separate contour can be created, which is
|
||||||
|
%..assigned the same name as its positive counter part, but preceded by a negation `$\neg$' sign.
|
||||||
|
%Obviously were a drawing to show conjunction of a contour and its complement
|
||||||
|
%this would result in a contradiction for any test case placed on it, and would be a visual `syntax error'.
|
||||||
|
|
||||||
|
%Note that negation is handled explicitly. This is to allow `don't care'
|
||||||
|
%conditions. Should a test case be outside a contour, that contour is a `don't care' condition.
|
||||||
|
%In a PLD, contours may be represented in complement, to provide
|
||||||
|
%logical negation. Here the contour name is begun with the negation symbol `$\neg$'.
|
||||||
|
%%To represent conjunction of logical conditions (Boolean `AND'), contours may be overlapped.
|
||||||
|
|
||||||
|
%Providing explicit negation, in addition to disjunction and conjunction
|
||||||
|
%allows us to represent `don't care' or `tri-state' logical conditions. Simply by
|
||||||
|
%not using the conditions we are not interested in they are `don't care'.
|
||||||
|
%
|
||||||
|
|
||||||
|
%\section{Example Logic Diagrams}
|
||||||
|
|
||||||
|
\subsection{ Logical AND example }
|
||||||
|
|
||||||
|
|
||||||
|
\begin{figure}[h+]
|
||||||
|
\begin{center}
|
||||||
|
\includegraphics[width=200pt,bb=0 0 450 404]{logic_diagram/ldand.eps}
|
||||||
|
% resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404
|
||||||
|
\end{center}
|
||||||
|
|
||||||
|
%\includegraphics[scale=0.6]{ldand.eps}
|
||||||
|
\caption{Logical AND}
|
||||||
|
\label{fig:ld_and}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
In the diagram \ref{fig:ld_and} the area of intersection between the contours $a$ and $b$
|
||||||
|
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) $.
|
||||||
|
|
||||||
|
\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 both failure~modes are active.
|
||||||
|
|
||||||
|
\clearpage
|
||||||
|
\subsection { Logical OR example }
|
||||||
|
|
||||||
|
\begin{figure}[h+]
|
||||||
|
%\centering
|
||||||
|
%\input{ldor.tex}
|
||||||
|
\begin{center}
|
||||||
|
\includegraphics[width=200pt,bb=0 0 450 404]{logic_diagram/ldor.eps}
|
||||||
|
% resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404
|
||||||
|
\end{center}
|
||||||
|
%\includegraphics[scale=0.60]{ldor.eps}
|
||||||
|
\caption{Logical OR}
|
||||||
|
\label{fig:ld_or}
|
||||||
|
\end{figure} % OR
|
||||||
|
|
||||||
|
|
||||||
|
The diagram \ref{fig:ld_or} is converted to Boolean logic by first looking at the test cases, and
|
||||||
|
the contours they are placed on.
|
||||||
|
$$ P = (a) $$
|
||||||
|
$$ Q = (b) $$
|
||||||
|
|
||||||
|
The two test cases are joined by a the line named $R$.
|
||||||
|
we thus apply disjunction to the test cases.
|
||||||
|
$$ R = P \vee Q $$
|
||||||
|
substituting the test cases for their Boolean logic equations gives
|
||||||
|
$$ R = ((a) \vee (b)) $$.
|
||||||
|
|
||||||
|
|
||||||
|
\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 $b$.
|
||||||
|
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.
|
||||||
|
|
||||||
|
|
||||||
|
\clearpage
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\subsection { Repeated Contour example }
|
||||||
|
|
||||||
|
|
||||||
|
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 0 640 480]{./repeated.jpg}
|
||||||
|
% repeated.jpg: 640x480 pixel, 72dpi, 22.58x16.93 cm, bb=0 0 640 480
|
||||||
|
\caption{Repeated Contours}
|
||||||
|
\label{fig:repeat}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
The diagram \ref{fig:repeat} is converted to Boolean logic by first looking at the test cases, and
|
||||||
|
the contours they are placed on.
|
||||||
|
$$ P = (a) $$
|
||||||
|
$$ Q = (b) \wedge (c) $$
|
||||||
|
|
||||||
|
The two test cases are joined by a the line named $R1$.
|
||||||
|
we thus apply disjunction to the test cases.
|
||||||
|
$$ R1 = P \vee Q $$
|
||||||
|
$$ R1 = b \wedge ( a \vee c ) $$.
|
||||||
|
|
||||||
|
$R2$ joins two other test cases
|
||||||
|
$$R2 = a \vee c $$
|
||||||
|
|
||||||
|
The test~case residing in the intersection of countours $B$ and $A$
|
||||||
|
represents the logic equation $R = a \wedge b$.
|
||||||
|
|
||||||
|
\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 $A \wedge B$.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\clearpage
|
||||||
|
|
||||||
|
|
||||||
|
\subsection { Inhibit Failure }
|
||||||
|
|
||||||
|
|
||||||
|
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.
|
||||||
|
|
||||||
|
\begin{figure}[h]
|
||||||
|
\centering
|
||||||
|
\includegraphics[bb=0 0 640 480]{./inhibit.jpg}
|
||||||
|
% repeated.jpg: 640x480 pixel, 72dpi, 22.58x16.93 cm, bb=0 0 640 480
|
||||||
|
\caption{Inhibit Contours}
|
||||||
|
\label{fig:inhibit}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
The diagram \ref{fig:inhibit} has a test case in the contour $C$.
|
||||||
|
Contour $C$ is enclosed by contour $A$. This says
|
||||||
|
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
|
||||||
|
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 occurr if $A$ is true.
|
||||||
|
The `O' ring could fail in a different way above the critical temperature and this is
|
||||||
|
represented, for the sake of this example, by contour $D$
|
||||||
|
|
||||||
|
|
||||||
|
\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.
|
||||||
|
There is an additional symptom, that of the test case in $A \wedge B$.
|
||||||
|
|
||||||
|
|
||||||
|
% \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
|
||||||
|
%
|
||||||
|
|
||||||
|
\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.
|
||||||
|
|
||||||
|
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
|
||||||
|
%\input{millivolt_sensor.tex}
|
||||||
|
\begin{center}
|
||||||
|
\includegraphics[width=200pt,bb=0pt 0pt 600pt 600pt]{logic_diagram/ldmeq2.eps}
|
||||||
|
% resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404
|
||||||
|
\end{center}
|
||||||
|
%\includegraphics[scale=0.4]{ldmeq2.eps}
|
||||||
|
\caption{Several Logical Expressions with unamed test cases}
|
||||||
|
\label{fig:ld_meq2}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
\section{Intended use in FMMD}
|
||||||
|
|
||||||
|
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}
|
||||||
|
|
||||||
|
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 1.15 2009/02/09 07:33:27 robin Exp $
|
||||||
|
\end{verbatim}
|
||||||
|
Compiled last \today
|
||||||
|
%\end{document}
|
||||||
|
|
||||||
|
%\theend
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
742
logic_diagram/logic_diagram.tex~
Normal file
742
logic_diagram/logic_diagram.tex~
Normal file
@ -0,0 +1,742 @@
|
|||||||
|
|
||||||
|
\begin{verbatim}
|
||||||
|
CVS Revision Identity $Id: logic_diagram.tex,v 1.15 2009/02/09 07:33:27 robin Exp $
|
||||||
|
\end{verbatim}
|
||||||
|
|
||||||
|
\begin{abstract}
|
||||||
|
%This chapter describes using diagrams to represent propositional logic.
|
||||||
|
Propositial Logic Diagrams have been designed to provide an intuitive method for visualising and manipulating
|
||||||
|
logic equations, to express fault modes in Mechanical and Electronic Systems.
|
||||||
|
%To aid hierarchical stages of fault analysis, it has been specifically developed for the purpose of
|
||||||
|
%joining conjunctive conditions with disjuctive conditions
|
||||||
|
%to group the effects of failure modes.
|
||||||
|
Diagrams of this type can also be used to model the logical conditions
|
||||||
|
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
|
||||||
|
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
|
||||||
|
not yet the norm.\cite{takeup}.
|
||||||
|
%
|
||||||
|
Because of its visual nature, it is easy to manipulate and model
|
||||||
|
complicated conditions that can lead to dangerous failures in
|
||||||
|
automated systems.
|
||||||
|
|
||||||
|
% No need to talk about abstraction yet, just define PLD PROPERLY
|
||||||
|
|
||||||
|
The Diagrams described here form the mathematical basis for a new visual and formal system
|
||||||
|
for the analysis of safety critical software and hardware systems.
|
||||||
|
\end{abstract}
|
||||||
|
|
||||||
|
%\title{Propositional Logic Diagrams}
|
||||||
|
%\begin{keyword}
|
||||||
|
% fault~tree fault~mode EN298 EN61508 EN12067 EN230 UL1998 safety~critical logic euler venn propositional
|
||||||
|
%\end{keyword}
|
||||||
|
%\end{frontmatter}
|
||||||
|
|
||||||
|
|
||||||
|
\section{Introduction}
|
||||||
|
|
||||||
|
Propositional Logic Diagrams (PLDs) have been devised
|
||||||
|
to collect and simplfy fault~modes in safety critical systems undergoing
|
||||||
|
static analysis\cite{FMEA}\cite{SIL}.
|
||||||
|
|
||||||
|
This type of analysis treats failure modes within a system as logical
|
||||||
|
states. PLD provides a visual method for modelling failure~mode analysis
|
||||||
|
within these systems, and specifically
|
||||||
|
identifying common failure symptoms in a user friendly way.
|
||||||
|
Contrasting this to looking at many propositional logic equations directly
|
||||||
|
in a text editor or spreadsheet, a visual method is prefferred.
|
||||||
|
|
||||||
|
|
||||||
|
%Traditional set theory is often represented by Euler\cite{euler} or Spider\cite{spider}
|
||||||
|
%diagrams. These use contours to describe inclusion in a set.may be merged and create a
|
||||||
|
%Propositional Logic Diagrams (PLDs) use named contours represent a logical conditions.
|
||||||
|
%Where an Euler diagram would use
|
||||||
|
%overlapping contours to represent inclusion in sets,
|
||||||
|
%PLDs use these to represent conjunction of the conditions.
|
||||||
|
|
||||||
|
%Named reference points may be placed onto the diagram,
|
||||||
|
%these represent test cases for conjunction.
|
||||||
|
%These can be joined by lines to apply disjunction.
|
||||||
|
%In a spider diagram the lines would represent that the object represented coul;d belong to either set.
|
||||||
|
%in a PLD it means that the logical conditions represent disjuction; a boolean OR condition.
|
||||||
|
%these points may be joined.
|
||||||
|
|
||||||
|
PLDs use three visual features that
|
||||||
|
can be combined to represent logic equations. Closed contours (using dashed lines), test cases, and joining lines.
|
||||||
|
All features may be labelled, and the labels must be unique within a diagram, however contours may be repeated in the diagram.
|
||||||
|
%Aditionally a label begining with the `$\neg$' character, applied only to a contour, represents negation.
|
||||||
|
|
||||||
|
|
||||||
|
%Regions defined by contours are used to represent given conjunctive logical conditions.
|
||||||
|
|
||||||
|
Test cases are marked by asterisks. These are used as a visual `anchor'
|
||||||
|
to mark a logical condition, the logical condition being defined by the countours
|
||||||
|
that enclose the region on which the test case has been placed.
|
||||||
|
Test cases may be pair-wise connected by named lines representing disjunction (Boolean `OR') of
|
||||||
|
the conditions defined by the placement of the test case markers.
|
||||||
|
|
||||||
|
With these three visual syntax elements, we have the basic building blocks for all logic equations possible.
|
||||||
|
\begin{description}
|
||||||
|
\item Test cases - Points on the plane indicating a logical condition.
|
||||||
|
\item Conjunction - Overlapping contours
|
||||||
|
\item Disjunction - Joining of named test cases.
|
||||||
|
%\item Negation - Countours negatively named
|
||||||
|
\end{description}
|
||||||
|
|
||||||
|
% Because of this
|
||||||
|
% we have the complete suite of logical primitives here, conjunction, disjuction and negation.
|
||||||
|
% Form these complex logic equations can be respresented in 2D.
|
||||||
|
|
||||||
|
% Another advantage of this is being able to describe `don't care' conditions.
|
||||||
|
% Very often in digital hardware design, or in a computer program
|
||||||
|
% many logical conditions are `don't care'.
|
||||||
|
% These are difficult to specify in set theory.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\section{Formal Description of PLD}
|
||||||
|
|
||||||
|
Definitions of conrete and abstract PLD's follow.
|
||||||
|
Well-formedness conditions for PLD's are separated from this definition, because of
|
||||||
|
practical differences between the way they are used to represent software as opposed to
|
||||||
|
representing electronics and mechanical systems.
|
||||||
|
|
||||||
|
\subsection{Concrete PLD Definition}
|
||||||
|
|
||||||
|
A concrete {\em Propositional logic diagram} is a set of labeled {\em contours}
|
||||||
|
(closed curves) in the plane. The minimal regions formed by the closed curves
|
||||||
|
can by occupied by `test points'.
|
||||||
|
The `test points' may be joined by joining lines.
|
||||||
|
A group of `test points' connected by joining lines
|
||||||
|
is defined as a `test point disjunction' or Spider.
|
||||||
|
Spiders may be labeled.
|
||||||
|
|
||||||
|
To differentiate these from common Euler diagram notation (normally used to represent set theory)
|
||||||
|
the curves are drawn using dotted and dashed lines.
|
||||||
|
|
||||||
|
\subsection{ PLD Definition}
|
||||||
|
In English:
|
||||||
|
The elements that can be found in a PLD diagram are a number of contours,
|
||||||
|
a number of test points and joining lines that connect
|
||||||
|
test points.
|
||||||
|
{
|
||||||
|
\definition{A concrete PLD $d$ is a set comprising of a set of
|
||||||
|
closed curves $C=C(d)$, a set of test points $T=T(d)$ and
|
||||||
|
a set of test point joining lines $J=J(d)$.
|
||||||
|
$$d=\{C,T,J\}$$
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
In English:
|
||||||
|
Each element of the diagram has a unique label within the diagram.
|
||||||
|
%Thus the set of labels found in a diagram is
|
||||||
|
%a subset of the powerset of characters that can be present in a label.
|
||||||
|
|
||||||
|
%{
|
||||||
|
%\definition{ $ \mathcal{F}_{d}:C(d) \rightarrow \mathcal{P}\Lambda$ is a
|
||||||
|
%function associating a label drawn from an infinite
|
||||||
|
%set of labels $\Lambda$.
|
||||||
|
%}
|
||||||
|
%}
|
||||||
|
|
||||||
|
%In English:
|
||||||
|
%A minimal region of a PLD diagram d is a
|
||||||
|
%region bounded by curves.
|
||||||
|
%connected component of $\mathbb{R}^{2} - \; \bigcup_{c \in C(d)} c$
|
||||||
|
|
||||||
|
%That is to say the complement of all other regions is subtracted from the plane.
|
||||||
|
%- Or in another way- that smallest area defined by the curves that enclose it
|
||||||
|
|
||||||
|
|
||||||
|
%% \hat is used to indicate CONCRETE
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
A minimal region of concrete PLD diagram d is a connected component of
|
||||||
|
|
||||||
|
$$ \mathbb{R}^{2} - \; \bigcup_{\hat{c} \in \hat{C}(\hat{d})}\hat{c}$$
|
||||||
|
|
||||||
|
% I.e. The contours break the connectivity
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition
|
||||||
|
{
|
||||||
|
Let d be a PLD and $ \mathcal{X} \subseteq \hat{C}(\hat{d})$ a set of countours.
|
||||||
|
If the set
|
||||||
|
$$ \hat{z} = \bigcap_{c \in \mathcal{X}}
|
||||||
|
{interior}
|
||||||
|
(\hat{c})
|
||||||
|
\; \cup \;
|
||||||
|
\bigcap_{\hat{c} \in \hat{C}-X}
|
||||||
|
exterior (\hat{c})
|
||||||
|
$$
|
||||||
|
|
||||||
|
is non empty, then $\hat{z}$ is a concrete zone of $\hat{d}$. A zone is a union of minimal regions. The set of all concrete zones of $\hat{d}$
|
||||||
|
is denoted $ \hat{\mathcal{Z}} $.
|
||||||
|
|
||||||
|
% NOT interested in labelling the zones.
|
||||||
|
% but am interested in
|
||||||
|
%The set of labels associated with the contours in $\mathcal{X}$ is the zone label set $\hat{\mathcal{Z}}(\hat{z})$
|
||||||
|
%of $\hat{z}$.
|
||||||
|
%$$ \hat{\mathcal{L}}(\hat{z}) = \bigcup_{\hat{c} \in \mathcal{X}} \mathcal{F}_{d}(\hat{c}) $$
|
||||||
|
%
|
||||||
|
% So Z is the set of all available for use ZONES; great !
|
||||||
|
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
|
Each minimal region in the plane may be inhabited by one or more `test points'.
|
||||||
|
% One or more because in software the same logical conditions mean existing in the same
|
||||||
|
% region. For electroincs or mechanical, one test point per region is
|
||||||
|
% mandatory. How to describe ?????
|
||||||
|
Each test point 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
|
||||||
|
associating a testpoint with a set of contours in the plane. This corresponds to the interior of the contours defining the zone.
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
Pairs of test points may be joined by joining lines.
|
||||||
|
The operator $\stackrel{join}{\leftrightarrow}$ is used to
|
||||||
|
show that two points are joined by a line in the concrete diagram.
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
|
||||||
|
$ \mathcal{F}_{j}$ is a function
|
||||||
|
associating a joining line with a pair of test points. The Join t1,t2 is defined as
|
||||||
|
|
||||||
|
%$$ \mathcal{F}_{d}:J(d)\rightarrow \{t1,t2\ | t1 \in T(d) \wedge t2 \in T(d) \wedge t1 \neq t2 %\wedge t1 \stackrel{join}{\leftrightarrow} t2\} $$
|
||||||
|
$$ \mathcal{F}_{d}:J(d)\rightarrow \{t1,t2\ | t1 \in T(d) \wedge t2 \in T(d) \wedge t1 \neq t2 \} $$
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
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, $FMG$
|
||||||
|
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
|
||||||
|
%to another member of the spider. A singleton test point can be considered a spider.
|
||||||
|
Let d be a PLD : An $FMG$ is a maximal set of test points in d where
|
||||||
|
the test points belong to a sequence connected by joining lines such that:
|
||||||
|
|
||||||
|
$$ t_i \stackrel{join}{\leftrightarrow} t_n, for \; i = 1, ..., n $$
|
||||||
|
|
||||||
|
|
||||||
|
OR consider an $FMG$ as a tree whose nodes are test points.
|
||||||
|
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
A singleton test point can be considered a sequence of one test point and is therefore also an $FMG$.
|
||||||
|
|
||||||
|
|
||||||
|
% \subsection{Abstract Description of PLD}
|
||||||
|
%
|
||||||
|
% 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.
|
||||||
|
% Collections of test points, linked by shared conecting lines, form a set of test point groups $G$.
|
||||||
|
%
|
||||||
|
% A Zone defined by the contours that enclose it in the concrete diagram.
|
||||||
|
%
|
||||||
|
% $$ Z \subseteq C $$
|
||||||
|
%
|
||||||
|
% A test point $t \in T$ in habits a zone on the diagram.
|
||||||
|
%
|
||||||
|
% $$ \eta(t) = Z $$
|
||||||
|
%
|
||||||
|
% A joining line $$ w \in W $$ joins test points.
|
||||||
|
%
|
||||||
|
% $$ w = t1 \stackrel{join}{\rightarrow} t2 | t1 \neq t2 \wedge t1 \in T \wedge t2 \in T $$
|
||||||
|
%
|
||||||
|
% A test point group $g \in G$ is defined by test points linked by shared connecting lines.
|
||||||
|
|
||||||
|
|
||||||
|
\subsection{Semantics of PLD}
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item A closed curve in a PLD represents a condition (logical state) being modelled.
|
||||||
|
\item A test point represents the conjunction of the conditions represented by the curves that enclose it.
|
||||||
|
\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
|
||||||
|
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.
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
|
||||||
|
Let $\mathcal{F}_{t}$ be a function mapping a test point to a proposition / logical equation $p \in P$.
|
||||||
|
The test point inhabits the zone $\mathcal{Z}$ which is a collection of contours (the contours that enclose the test point.
|
||||||
|
|
||||||
|
$$ \mathcal{F}:T \rightarrow P $$
|
||||||
|
|
||||||
|
%$$ \mathcal{F}(t): p = \bigwedge_{c \in \mathcal{Z}} \Lambda c $$
|
||||||
|
|
||||||
|
$$ \mathcal{F}(t): p = \bigwedge_{c \in \mathcal{Z}} c $$
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
In English:
|
||||||
|
Thus a `test point' enclosed by contours labelled $a,b,c$ would be represented by the logic equation
|
||||||
|
$ a \wedge b \wedge c $.
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
Let $\mathcal{G}_{fmg}$ be a function that returns a logic equation for a given $FMG$
|
||||||
|
$fmg$ in the diagram, where an FMG is a non empty set of test points
|
||||||
|
% $t$ is a `test point'
|
||||||
|
|
||||||
|
$$ \mathcal{G}:FMG \rightarrow P_{fmg} $$
|
||||||
|
|
||||||
|
The logic equation representing an FMG $p_{fmg}$ can be determined thus.
|
||||||
|
|
||||||
|
$$\mathcal{G}_{fmg}(fmg) = \bigvee_{t \in fmg} (\; \mathcal{F}_{t} (t) \;) $$
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
The abstract PLD diagram is a set of logic equations representing all FMGs,
|
||||||
|
along with unused zones (i.e. zones that are not inhabited by FMGs).
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
\label{FMGderivation}
|
||||||
|
A diagram can be reduced to a collection of $FMG$s.
|
||||||
|
A new diagram can be derived from this, replacing a contour for each FMG.
|
||||||
|
This diagram is at one higher level of abstraction then the diagram that
|
||||||
|
it was produced from.
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
|
\section{Example Diagrams}
|
||||||
|
|
||||||
|
|
||||||
|
\subsection {How to read a PLD diagram}
|
||||||
|
|
||||||
|
PLD diagrams are read by first looking at the test case points.
|
||||||
|
The test case asterisk will be enclosed by one or more contours.
|
||||||
|
These contours are collected and form the logical conjunction
|
||||||
|
equation for the test case.
|
||||||
|
These test case points thus represent the conjunctive aspects
|
||||||
|
of an equation defined in a PLD. Where these test cases are joined by lines;
|
||||||
|
these represent disjunction of the conjunctive aspects defined by the test cases.
|
||||||
|
Joining lines thus represent dis-junction in a PLD.
|
||||||
|
|
||||||
|
%Where negation and assertion of a logical condition is required in the same diagram, a separate contour can be created, which is
|
||||||
|
%..assigned the same name as its positive counter part, but preceded by a negation `$\neg$' sign.
|
||||||
|
%Obviously were a drawing to show conjunction of a contour and its complement
|
||||||
|
%this would result in a contradiction for any test case placed on it, and would be a visual `syntax error'.
|
||||||
|
|
||||||
|
%Note that negation is handled explicitly. This is to allow `don't care'
|
||||||
|
%conditions. Should a test case be outside a contour, that contour is a `don't care' condition.
|
||||||
|
%In a PLD, contours may be represented in complement, to provide
|
||||||
|
%logical negation. Here the contour name is begun with the negation symbol `$\neg$'.
|
||||||
|
%%To represent conjunction of logical conditions (Boolean `AND'), contours may be overlapped.
|
||||||
|
|
||||||
|
%Providing explicit negation, in addition to disjunction and conjunction
|
||||||
|
%allows us to represent `don't care' or `tri-state' logical conditions. Simply by
|
||||||
|
%not using the conditions we are not interested in they are `don't care'.
|
||||||
|
%
|
||||||
|
|
||||||
|
%\section{Example Logic Diagrams}
|
||||||
|
|
||||||
|
\subsection{ Logical AND example }
|
||||||
|
|
||||||
|
|
||||||
|
\begin{figure}[h+]
|
||||||
|
\begin{center}
|
||||||
|
\includegraphics[width=200pt,bb=0 0 450 404]{logic_diagram/ldand.eps}
|
||||||
|
% resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404
|
||||||
|
\end{center}
|
||||||
|
|
||||||
|
%\includegraphics[scale=0.6]{ldand.eps}
|
||||||
|
\caption{Logical AND}
|
||||||
|
\label{fig:ld_and}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
In the diagram \ref{fig:ld_and} the area of intersection between the contours $a$ and $b$
|
||||||
|
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) $.
|
||||||
|
|
||||||
|
\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 both failure~modes are active.
|
||||||
|
|
||||||
|
\clearpage
|
||||||
|
\subsection { Logical OR example }
|
||||||
|
|
||||||
|
\begin{figure}[h+]
|
||||||
|
%\centering
|
||||||
|
%\input{ldor.tex}
|
||||||
|
\begin{center}
|
||||||
|
\includegraphics[width=200pt,bb=0 0 450 404]{logic_diagram/ldor.eps}
|
||||||
|
% resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404
|
||||||
|
\end{center}
|
||||||
|
%\includegraphics[scale=0.60]{ldor.eps}
|
||||||
|
\caption{Logical OR}
|
||||||
|
\label{fig:ld_or}
|
||||||
|
\end{figure} % OR
|
||||||
|
|
||||||
|
|
||||||
|
The diagram \ref{fig:ld_or} is converted to Boolean logic by first looking at the test cases, and
|
||||||
|
the contours they are placed on.
|
||||||
|
$$ P = (a) $$
|
||||||
|
$$ Q = (b) $$
|
||||||
|
|
||||||
|
The two test cases are joined by a the line named $R$.
|
||||||
|
we thus apply disjunction to the test cases.
|
||||||
|
$$ R = P \vee Q $$
|
||||||
|
substituting the test cases for their Boolean logic equations gives
|
||||||
|
$$ R = ((a) \vee (b)) $$.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\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.
|
||||||
|
|
||||||
|
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
|
||||||
|
%\input{millivolt_sensor.tex}
|
||||||
|
\begin{center}
|
||||||
|
\includegraphics[width=200pt,bb=0pt 0pt 600pt 600pt]{logic_diagram/ldmeq2.eps}
|
||||||
|
% resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404
|
||||||
|
\end{center}
|
||||||
|
%\includegraphics[scale=0.4]{ldmeq2.eps}
|
||||||
|
\caption{Several Logical Expressions with unamed test cases}
|
||||||
|
\label{fig:ld_meq2}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
\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.
|
||||||
|
|
||||||
|
|
||||||
|
\clearpage
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\subsection { Repeated Contour example }
|
||||||
|
|
||||||
|
|
||||||
|
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 0 550 250]{./repeated.eps}
|
||||||
|
% repeated.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 550 250
|
||||||
|
\label{fig:repeat}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
The diagram \ref{fig:repeat} is converted to Boolean logic by first looking at the test cases, and
|
||||||
|
the contours they are placed on.
|
||||||
|
$$ P = (b) $$
|
||||||
|
$$ Q = (a) \wedge (c) $$
|
||||||
|
|
||||||
|
The two test cases are joined by a the line named $R1$.
|
||||||
|
we thus apply disjunction to the test cases.
|
||||||
|
$$ R1 = P \vee Q $$
|
||||||
|
$$ R1 = b \vee ( a \wedge c ) $$.
|
||||||
|
|
||||||
|
$R2$ joins two other test cases
|
||||||
|
$$R2 = a \vee c $$
|
||||||
|
|
||||||
|
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 $A \wedge B$.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\clearpage
|
||||||
|
|
||||||
|
|
||||||
|
\subsection { Inhibit Failure }
|
||||||
|
|
||||||
|
|
||||||
|
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.
|
||||||
|
|
||||||
|
|
||||||
|
\begin{figure}[h]
|
||||||
|
\centering
|
||||||
|
\includegraphics[bb=0 0 364 228]{./inhibit.eps}
|
||||||
|
% inhibit.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 364 228
|
||||||
|
\label{fig:inhibit}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
The diagram \ref{fig:inhibit} has a test case in the contour $C$.
|
||||||
|
Contour $C$ is enclosed by contour $A$. This says
|
||||||
|
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
|
||||||
|
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 occurr if $A$ is true.
|
||||||
|
The `O' ring could fail in a different way independant of the critical temperature and this is
|
||||||
|
represented, for the sake of this example, by contour $D$.
|
||||||
|
|
||||||
|
In terms of propositional logic, the inhibit gate of FTA, and the countour enclosure
|
||||||
|
of PLD represent {\em implication}.
|
||||||
|
\\
|
||||||
|
\tiny
|
||||||
|
\vspace{0.3cm}
|
||||||
|
\begin{tabular}{||c|c|c|c||} \hline \hline
|
||||||
|
{\em $c$ } & {\em $a$ } & {\em $R1$ } \\ \hline
|
||||||
|
F & F & T \\ \hline
|
||||||
|
F & T & T \\ \hline
|
||||||
|
T & F & F \\ \hline
|
||||||
|
T & T & T \\ \hline \hline
|
||||||
|
\end{tabular}
|
||||||
|
\vspace{0.3cm}
|
||||||
|
\normalsize
|
||||||
|
|
||||||
|
$$ R1 = c \implies a $$
|
||||||
|
$$ R2 = a $$
|
||||||
|
$$ R3 = d $$
|
||||||
|
|
||||||
|
|
||||||
|
\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 sub-system, on its own
|
||||||
|
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
|
||||||
|
component faults and combinations thereof, into faults that,
|
||||||
|
at the module level have the same symptoms.
|
||||||
|
|
||||||
|
\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
|
||||||
|
|
||||||
|
\begin{verbatim}
|
||||||
|
CVS Revision Identity $Id: logic_diagram.tex,v 1.15 2009/02/09 07:33:27 robin Exp $
|
||||||
|
\end{verbatim}
|
||||||
|
Compiled last \today
|
||||||
|
%\end{document}
|
||||||
|
|
||||||
|
%\theend
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
743
logic_diagram/logic_diagram_paper.tex
Normal file
743
logic_diagram/logic_diagram_paper.tex
Normal file
@ -0,0 +1,743 @@
|
|||||||
|
|
||||||
|
\begin{verbatim}
|
||||||
|
CVS Revision Identity $Id: logic_diagram.tex,v 1.16 2010/01/01 14:09:02 robin Exp $
|
||||||
|
\end{verbatim}
|
||||||
|
|
||||||
|
\begin{abstract}
|
||||||
|
%This chapter describes using diagrams to represent propositional logic.
|
||||||
|
Propositial Logic Diagrams have been designed to provide an intuitive method for visualising and manipulating
|
||||||
|
logic equations, to express fault modes in Mechanical and Electronic Systems.
|
||||||
|
%To aid hierarchical stages of fault analysis, it has been specifically developed for the purpose of
|
||||||
|
%joining conjunctive conditions with disjuctive conditions
|
||||||
|
%to group the effects of failure modes.
|
||||||
|
Diagrams of this type can also be used to model the logical conditions
|
||||||
|
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
|
||||||
|
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
|
||||||
|
not yet the norm.\cite{takeup}.
|
||||||
|
%
|
||||||
|
Because of its visual nature, it is easy to manipulate and model
|
||||||
|
complicated conditions that can lead to dangerous failures in
|
||||||
|
automated systems.
|
||||||
|
|
||||||
|
% No need to talk about abstraction yet, just define PLD PROPERLY
|
||||||
|
|
||||||
|
The Diagrams described here form the mathematical basis for a new visual and formal system
|
||||||
|
for the analysis of safety critical software and hardware systems.
|
||||||
|
\end{abstract}
|
||||||
|
|
||||||
|
%\title{Propositional Logic Diagrams}
|
||||||
|
%\begin{keyword}
|
||||||
|
% fault~tree fault~mode EN298 EN61508 EN12067 EN230 UL1998 safety~critical logic euler venn propositional
|
||||||
|
%\end{keyword}
|
||||||
|
%\end{frontmatter}
|
||||||
|
|
||||||
|
|
||||||
|
\section{Introduction}
|
||||||
|
|
||||||
|
Propositional Logic Diagrams (PLDs) have been devised
|
||||||
|
to collect and simplfy fault~modes in safety critical systems undergoing
|
||||||
|
static analysis\cite{FMEA}\cite{SIL}.
|
||||||
|
|
||||||
|
This type of analysis treats failure modes within a system as logical
|
||||||
|
states. PLD provides a visual method for modelling failure~mode analysis
|
||||||
|
within these systems, and specifically
|
||||||
|
identifying common failure symptoms in a user friendly way.
|
||||||
|
Contrasting this to looking at many propositional logic equations directly
|
||||||
|
in a text editor or spreadsheet, a visual method is percieved as being more intuitive.
|
||||||
|
|
||||||
|
|
||||||
|
%Traditional set theory is often represented by Euler\cite{euler} or Spider\cite{spider}
|
||||||
|
%diagrams. These use contours to describe inclusion in a set.may be merged and create a
|
||||||
|
%Propositional Logic Diagrams (PLDs) use named contours represent a logical conditions.
|
||||||
|
%Where an Euler diagram would use
|
||||||
|
%overlapping contours to represent inclusion in sets,
|
||||||
|
%PLDs use these to represent conjunction of the conditions.
|
||||||
|
|
||||||
|
%Named reference points may be placed onto the diagram,
|
||||||
|
%these represent test cases for conjunction.
|
||||||
|
%These can be joined by lines to apply disjunction.
|
||||||
|
%In a spider diagram the lines would represent that the object represented coul;d belong to either set.
|
||||||
|
%in a PLD it means that the logical conditions represent disjuction; a boolean OR condition.
|
||||||
|
%these points may be joined.
|
||||||
|
|
||||||
|
PLDs use three visual features that
|
||||||
|
can be combined to represent logic equations. Closed contours (using dashed lines), test cases, and lines that
|
||||||
|
link test cases.
|
||||||
|
All features may be labelled, and the labels must be unique within a diagram, however contours may be repeated in the diagram.
|
||||||
|
Aditionally a label begining with the `$\neg$' character, applied only to a contour, represents negation.
|
||||||
|
|
||||||
|
|
||||||
|
%Regions defined by contours are used to represent given conjunctive logical conditions.
|
||||||
|
|
||||||
|
Test cases are marked by asterisks. These are used as a visual `anchor'
|
||||||
|
to mark a logical condition, the logical condition being defined by the contours
|
||||||
|
that enclose the region on which the test~case has been placed.
|
||||||
|
The contours that enclose represent conjuction.
|
||||||
|
Test~cases may be connected by joining lines. These lines represent disjunction (Boolean `OR') of
|
||||||
|
test~cases.
|
||||||
|
|
||||||
|
With these three visual syntax elements, we have the basic building blocks for all logic equations possible.
|
||||||
|
\begin{description}
|
||||||
|
\item Test cases - Points on the plane indicating a logical condition.
|
||||||
|
\item Conjunction - Overlapping contours
|
||||||
|
\item Disjunction - Joining of named test cases.
|
||||||
|
%\item Negation - Countours negatively named
|
||||||
|
\end{description}
|
||||||
|
|
||||||
|
% Because of this
|
||||||
|
% we have the complete suite of logical primitives here, conjunction, disjuction and negation.
|
||||||
|
% Form these complex logic equations can be respresented in 2D.
|
||||||
|
|
||||||
|
% Another advantage of this is being able to describe `don't care' conditions.
|
||||||
|
% Very often in digital hardware design, or in a computer program
|
||||||
|
% many logical conditions are `don't care'.
|
||||||
|
% These are difficult to specify in set theory.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\section{Formal Description of PLD}
|
||||||
|
|
||||||
|
Definitions of conrete and abstract PLD's follow.
|
||||||
|
Well-formedness conditions for PLD's are separated from this definition, because of
|
||||||
|
practical differences between the way they are used to represent software as opposed to
|
||||||
|
representing electronics and mechanical systems.
|
||||||
|
|
||||||
|
\subsection{Concrete PLD Definition}
|
||||||
|
|
||||||
|
A concrete {\em Propositional logic diagram} is a set of labeled {\em contours}
|
||||||
|
(closed curves) in the plane. The minimal regions formed by the closed curves
|
||||||
|
can by occupied by `test points'.
|
||||||
|
The `test points' may be joined by joining lines.
|
||||||
|
A group of `test points' connected by joining lines
|
||||||
|
is defined as a `test point disjunction' or Spider.
|
||||||
|
Spiders may be labeled.
|
||||||
|
|
||||||
|
To differentiate these from common Euler diagram notation (normally used to represent set theory)
|
||||||
|
the curves are drawn using dotted and dashed lines.
|
||||||
|
|
||||||
|
\subsection{ PLD Definition}
|
||||||
|
In English:
|
||||||
|
The elements that can be found in a PLD diagram are a number of contours,
|
||||||
|
a number of test points and joining lines that connect
|
||||||
|
test points.
|
||||||
|
{
|
||||||
|
\definition{A concrete PLD $d$ is a set comprising of a set of
|
||||||
|
closed curves $C=C(d)$, a set of test points $T=T(d)$ and
|
||||||
|
a set of test point joining lines $J=J(d)$.
|
||||||
|
$$d=\{C,T,J\}$$
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
In English:
|
||||||
|
Each element of the diagram has a unique label within the diagram.
|
||||||
|
%Thus the set of labels found in a diagram is
|
||||||
|
%a subset of the powerset of characters that can be present in a label.
|
||||||
|
|
||||||
|
%{
|
||||||
|
%\definition{ $ \mathcal{F}_{d}:C(d) \rightarrow \mathcal{P}\Lambda$ is a
|
||||||
|
%function associating a label drawn from an infinite
|
||||||
|
%set of labels $\Lambda$.
|
||||||
|
%}
|
||||||
|
%}
|
||||||
|
|
||||||
|
%In English:
|
||||||
|
%A minimal region of a PLD diagram d is a
|
||||||
|
%region bounded by curves.
|
||||||
|
%connected component of $\mathbb{R}^{2} - \; \bigcup_{c \in C(d)} c$
|
||||||
|
|
||||||
|
%That is to say the complement of all other regions is subtracted from the plane.
|
||||||
|
%- Or in another way- that smallest area defined by the curves that enclose it
|
||||||
|
|
||||||
|
|
||||||
|
%% \hat is used to indicate CONCRETE
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
A minimal region of concrete PLD diagram d is a connected component of
|
||||||
|
|
||||||
|
$$ \mathbb{R}^{2} - \; \bigcup_{\hat{c} \in \hat{C}(\hat{d})}\hat{c}$$
|
||||||
|
|
||||||
|
% I.e. The contours break the connectivity
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition
|
||||||
|
{
|
||||||
|
Let d be a PLD and $ \mathcal{X} \subseteq \hat{C}(\hat{d})$ a set of countours.
|
||||||
|
If the set
|
||||||
|
$$ \hat{z} = \bigcap_{c \in \mathcal{X}}
|
||||||
|
{interior}
|
||||||
|
(\hat{c})
|
||||||
|
\; \cup \;
|
||||||
|
\bigcap_{\hat{c} \in \hat{C}-X}
|
||||||
|
exterior (\hat{c})
|
||||||
|
$$
|
||||||
|
|
||||||
|
is non empty, then $\hat{z}$ is a concrete zone of $\hat{d}$. A zone is a union of minimal regions. The set of all concrete zones of $\hat{d}$
|
||||||
|
is denoted $ \hat{\mathcal{Z}} $.
|
||||||
|
|
||||||
|
% NOT interested in labelling the zones.
|
||||||
|
% but am interested in
|
||||||
|
%The set of labels associated with the contours in $\mathcal{X}$ is the zone label set $\hat{\mathcal{Z}}(\hat{z})$
|
||||||
|
%of $\hat{z}$.
|
||||||
|
%$$ \hat{\mathcal{L}}(\hat{z}) = \bigcup_{\hat{c} \in \mathcal{X}} \mathcal{F}_{d}(\hat{c}) $$
|
||||||
|
%
|
||||||
|
% So Z is the set of all available for use ZONES; great !
|
||||||
|
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
|
Each minimal region in the plane may be inhabited by one or more `test points'.
|
||||||
|
% One or more because in software the same logical conditions mean existing in the same
|
||||||
|
% region. For electroincs or mechanical, one test point per region is
|
||||||
|
% mandatory. How to describe ?????
|
||||||
|
Each test point 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
|
||||||
|
associating a testpoint with a set of contours in the plane. This corresponds to the interior of the contours defining the zone.
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
Pairs of test points may be joined by joining lines.
|
||||||
|
The operator $\stackrel{join}{\leftrightarrow}$ is used to
|
||||||
|
show that two points are joined by a line in the concrete diagram.
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
|
||||||
|
$ \mathcal{F}_{j}$ is a function
|
||||||
|
associating a joining line with a pair of test points. The Join t1,t2 is defined as
|
||||||
|
|
||||||
|
%$$ \mathcal{F}_{d}:J(d)\rightarrow \{t1,t2\ | t1 \in T(d) \wedge t2 \in T(d) \wedge t1 \neq t2 %\wedge t1 \stackrel{join}{\leftrightarrow} t2\} $$
|
||||||
|
$$ \mathcal{F}_{d}:J(d)\rightarrow \{t1,t2\ | t1 \in T(d) \wedge t2 \in T(d) \wedge t1 \neq t2 \} $$
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
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, $FMG$
|
||||||
|
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
|
||||||
|
%to another member of the spider. A singleton test point can be considered a spider.
|
||||||
|
Let d be a PLD : An $FMG$ is a maximal set of test points in d where
|
||||||
|
the test points belong to a sequence connected by joining lines such that:
|
||||||
|
|
||||||
|
$$ t_i \stackrel{join}{\leftrightarrow} t_n, for \; i = 1, ..., n $$
|
||||||
|
|
||||||
|
|
||||||
|
OR consider an $FMG$ as a tree whose nodes are test points.
|
||||||
|
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
A singleton test point can be considered a sequence of one test point and is therefore also an $FMG$.
|
||||||
|
|
||||||
|
|
||||||
|
% \subsection{Abstract Description of PLD}
|
||||||
|
%
|
||||||
|
% 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.
|
||||||
|
% Collections of test points, linked by shared conecting lines, form a set of test point groups $G$.
|
||||||
|
%
|
||||||
|
% A Zone defined by the contours that enclose it in the concrete diagram.
|
||||||
|
%
|
||||||
|
% $$ Z \subseteq C $$
|
||||||
|
%
|
||||||
|
% A test point $t \in T$ in habits a zone on the diagram.
|
||||||
|
%
|
||||||
|
% $$ \eta(t) = Z $$
|
||||||
|
%
|
||||||
|
% A joining line $$ w \in W $$ joins test points.
|
||||||
|
%
|
||||||
|
% $$ w = t1 \stackrel{join}{\rightarrow} t2 | t1 \neq t2 \wedge t1 \in T \wedge t2 \in T $$
|
||||||
|
%
|
||||||
|
% A test point group $g \in G$ is defined by test points linked by shared connecting lines.
|
||||||
|
|
||||||
|
|
||||||
|
\subsection{Semantics of PLD}
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item A closed curve in a PLD represents a condition (logical state) being modelled.
|
||||||
|
\item A test point represents the conjunction of the conditions represented by the curves that enclose it.
|
||||||
|
\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
|
||||||
|
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.
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
|
||||||
|
Let $\mathcal{F}_{t}$ be a function mapping a test point to a proposition / logical equation $p \in P$.
|
||||||
|
The test point inhabits the zone $\mathcal{Z}$ which is a collection of contours (the contours that enclose the test point.
|
||||||
|
|
||||||
|
$$ \mathcal{F}:T \rightarrow P $$
|
||||||
|
|
||||||
|
%$$ \mathcal{F}(t): p = \bigwedge_{c \in \mathcal{Z}} \Lambda c $$
|
||||||
|
|
||||||
|
$$ \mathcal{F}(t): p = \bigwedge_{c \in \mathcal{Z}} c $$
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
In English:
|
||||||
|
Thus a `test point' enclosed by contours labelled $a,b,c$ would be represented by the logic equation
|
||||||
|
$ a \wedge b \wedge c $.
|
||||||
|
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
Let $\mathcal{G}_{fmg}$ be a function that returns a logic equation for a given $FMG$
|
||||||
|
$fmg$ in the diagram, where an FMG is a non empty set of test points
|
||||||
|
% $t$ is a `test point'
|
||||||
|
|
||||||
|
$$ \mathcal{G}:FMG \rightarrow P_{fmg} $$
|
||||||
|
|
||||||
|
The logic equation representing an FMG $p_{fmg}$ can be determined thus.
|
||||||
|
|
||||||
|
$$\mathcal{G}_{fmg}(fmg) = \bigvee_{t \in fmg} (\; \mathcal{F}_{t} (t) \;) $$
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
The abstract PLD diagram is a set of logic equations representing all FMGs,
|
||||||
|
along with unused zones (i.e. zones that are not inhabited by FMGs).
|
||||||
|
{
|
||||||
|
\definition{
|
||||||
|
\label{FMGderivation}
|
||||||
|
A diagram can be reduced to a collection of $FMG$s.
|
||||||
|
A new diagram can be derived from this, replacing a contour for each FMG.
|
||||||
|
This diagram is at one higher level of abstraction then the diagram that
|
||||||
|
it was produced from.
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
|
\section{Example Diagrams}
|
||||||
|
|
||||||
|
|
||||||
|
\subsection {How to read a PLD diagram}
|
||||||
|
|
||||||
|
PLD diagrams are read by first looking at the test case points.
|
||||||
|
The test case asterisk will be enclosed by one or more contours.
|
||||||
|
These contours are collected and form the logical conjunction
|
||||||
|
equation for the test case.
|
||||||
|
These test case points thus represent the conjunctive aspects
|
||||||
|
of an equation defined in a PLD. Where these test cases are joined by lines;
|
||||||
|
these represent disjunction of the conjunctive aspects defined by the test cases.
|
||||||
|
Joining lines thus represent dis-junction in a PLD.
|
||||||
|
|
||||||
|
%Where negation and assertion of a logical condition is required in the same diagram, a separate contour can be created, which is
|
||||||
|
%..assigned the same name as its positive counter part, but preceded by a negation `$\neg$' sign.
|
||||||
|
%Obviously were a drawing to show conjunction of a contour and its complement
|
||||||
|
%this would result in a contradiction for any test case placed on it, and would be a visual `syntax error'.
|
||||||
|
|
||||||
|
%Note that negation is handled explicitly. This is to allow `don't care'
|
||||||
|
%conditions. Should a test case be outside a contour, that contour is a `don't care' condition.
|
||||||
|
%In a PLD, contours may be represented in complement, to provide
|
||||||
|
%logical negation. Here the contour name is begun with the negation symbol `$\neg$'.
|
||||||
|
%%To represent conjunction of logical conditions (Boolean `AND'), contours may be overlapped.
|
||||||
|
|
||||||
|
%Providing explicit negation, in addition to disjunction and conjunction
|
||||||
|
%allows us to represent `don't care' or `tri-state' logical conditions. Simply by
|
||||||
|
%not using the conditions we are not interested in they are `don't care'.
|
||||||
|
%
|
||||||
|
|
||||||
|
%\section{Example Logic Diagrams}
|
||||||
|
|
||||||
|
\subsection{ Logical AND example }
|
||||||
|
|
||||||
|
|
||||||
|
\begin{figure}[h+]
|
||||||
|
\begin{center}
|
||||||
|
\includegraphics[width=200pt,bb=0 0 450 404]{ldand.eps}
|
||||||
|
% resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404
|
||||||
|
\end{center}
|
||||||
|
|
||||||
|
%\includegraphics[scale=0.6]{ldand.eps}
|
||||||
|
\caption{Logical AND}
|
||||||
|
\label{fig:ld_and}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
In the diagram \ref{fig:ld_and} the area of intersection between the contours $a$ and $b$
|
||||||
|
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) $.
|
||||||
|
|
||||||
|
\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 both failure~modes are active.
|
||||||
|
|
||||||
|
\clearpage
|
||||||
|
\subsection { Logical OR example }
|
||||||
|
|
||||||
|
\begin{figure}[h+]
|
||||||
|
%\centering
|
||||||
|
%\input{ldor.tex}
|
||||||
|
\begin{center}
|
||||||
|
\includegraphics[width=200pt,bb=0 0 450 404]{ldor.eps}
|
||||||
|
% resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404
|
||||||
|
\end{center}
|
||||||
|
%\includegraphics[scale=0.60]{ldor.eps}
|
||||||
|
\caption{Logical OR}
|
||||||
|
\label{fig:ld_or}
|
||||||
|
\end{figure} % OR
|
||||||
|
|
||||||
|
|
||||||
|
The diagram \ref{fig:ld_or} is converted to Boolean logic by first looking at the test cases, and
|
||||||
|
the contours they are placed on.
|
||||||
|
$$ P = (a) $$
|
||||||
|
$$ Q = (b) $$
|
||||||
|
|
||||||
|
The two test cases are joined by a the line named $R$.
|
||||||
|
we thus apply disjunction to the test cases.
|
||||||
|
$$ R = P \vee Q $$
|
||||||
|
substituting the test cases for their Boolean logic equations gives
|
||||||
|
$$ R = ((a) \vee (b)) $$.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\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.
|
||||||
|
|
||||||
|
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
|
||||||
|
%\input{millivolt_sensor.tex}
|
||||||
|
\begin{center}
|
||||||
|
\includegraphics[width=200pt,bb=0pt 0pt 600pt 600pt]{ldmeq2.eps}
|
||||||
|
% resistor_pld.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 450 404
|
||||||
|
\end{center}
|
||||||
|
%\includegraphics[scale=0.4]{ldmeq2.eps}
|
||||||
|
\caption{Several Logical Expressions with unamed test cases}
|
||||||
|
\label{fig:ld_meq2}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
\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.
|
||||||
|
|
||||||
|
|
||||||
|
\clearpage
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\subsection { Repeated Contour example }
|
||||||
|
|
||||||
|
|
||||||
|
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 0 486 206]{./repeated.eps}
|
||||||
|
% repeated.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 486 206
|
||||||
|
\label{fig:repeated}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
The diagram \ref{fig:repeat} is converted to Boolean logic by first looking at the test cases, and
|
||||||
|
the contours they are placed on.
|
||||||
|
$$ P = (b) $$
|
||||||
|
$$ Q = (a) \wedge (c) $$
|
||||||
|
|
||||||
|
The two test cases are joined by a the line named $R1$.
|
||||||
|
we thus apply disjunction to the test cases.
|
||||||
|
$$ R1 = P \vee Q $$
|
||||||
|
$$ R1 = b \vee ( a \wedge c ) $$.
|
||||||
|
|
||||||
|
$R2$ joins two other test cases
|
||||||
|
$$R2 = a \vee c $$
|
||||||
|
|
||||||
|
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 $A \wedge B$.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\clearpage
|
||||||
|
|
||||||
|
|
||||||
|
\subsection { Inhibit Failure }
|
||||||
|
|
||||||
|
|
||||||
|
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.
|
||||||
|
|
||||||
|
|
||||||
|
\begin{figure}[h]
|
||||||
|
\centering
|
||||||
|
\includegraphics[bb=0 0 364 228]{./inhibit.eps}
|
||||||
|
% inhibit.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=0 0 364 228
|
||||||
|
\label{fig:inhibit}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
The diagram \ref{fig:inhibit} has a test case in the contour $C$.
|
||||||
|
Contour $C$ is enclosed by contour $A$. This says
|
||||||
|
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
|
||||||
|
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 occurr if $A$ is true.
|
||||||
|
The `O' ring could fail in a different way independant of the critical temperature and this is
|
||||||
|
represented, for the sake of this example, by contour $D$.
|
||||||
|
|
||||||
|
In terms of propositional logic, the inhibit gate of FTA, and the countour enclosure
|
||||||
|
of PLD represent {\em implication}.
|
||||||
|
\\
|
||||||
|
\tiny
|
||||||
|
\vspace{0.3cm}
|
||||||
|
\begin{tabular}{||c|c|c|c||} \hline \hline
|
||||||
|
{\em $c$ } & {\em $a$ } & {\em $R1$ } \\ \hline
|
||||||
|
F & F & T \\ \hline
|
||||||
|
F & T & T \\ \hline
|
||||||
|
T & F & F \\ \hline
|
||||||
|
T & T & T \\ \hline \hline
|
||||||
|
\end{tabular}
|
||||||
|
\vspace{0.3cm}
|
||||||
|
\normalsize
|
||||||
|
|
||||||
|
$$ R1 = c \implies a $$
|
||||||
|
$$ R2 = a $$
|
||||||
|
$$ R3 = d $$
|
||||||
|
|
||||||
|
|
||||||
|
\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 sub-system, on its own
|
||||||
|
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]{ldneg.eps}
|
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|
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|
||||||
|
% \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
|
||||||
|
% $$R = a \oplus b $$
|
||||||
|
%
|
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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
|
||||||
|
% % \caption{Example `XOR' Diagram}
|
||||||
|
% % \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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|
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|
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|
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|
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|
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|
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|
% \includegraphics[width=200pt,bb=0pt 0pt 800pt 800pt]{ldxor.eps}
|
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|
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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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|
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|
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|
% \end{figure}
|
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%
|
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|
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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}.
|
||||||
|
% The Equations represented are as follows.
|
||||||
|
%
|
||||||
|
% Looking at the conjuctive environment of the test cases
|
||||||
|
% $$P = (\neg a)$$
|
||||||
|
% $$Q = (b)$$
|
||||||
|
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|
||||||
|
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|
||||||
|
% Leading to
|
||||||
|
% $$R = (\neg a) \vee (b)$$
|
||||||
|
% which is the standard logic equation for implication.
|
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|
%
|
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|
% \begin{figure}[h+]
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
% \end{center}
|
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|
% %\includegraphics[scale=0.4]{ldimp.eps}
|
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|
% \caption{Logical Implication}
|
||||||
|
% \label{fig:ld_imp}
|
||||||
|
% \end{figure}
|
||||||
|
%
|
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|
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|
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|
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|
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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
|
||||||
|
% F & T & T \\ \hline
|
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|
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|
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|
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|
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|
% \end{tabular}
|
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|
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|
% \normalsize
|
||||||
|
%
|
||||||
|
% \clearpage
|
||||||
|
% \subsection { Diagram representing several Logic Equations Example }
|
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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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|
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|
% %\centering
|
||||||
|
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|
||||||
|
% \begin{center}
|
||||||
|
% \includegraphics[width=200pt,bb=0pt 0pt 600pt 600pt]{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}
|
||||||
|
% \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.
|
||||||
|
%
|
||||||
|
% %$$ Q = a \; | \; \mho\{b,c\} $$
|
||||||
|
%
|
||||||
|
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|
||||||
|
%
|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
|
%
|
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|
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|
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|
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|
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|
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|
\section{Intended use in FMMD}
|
||||||
|
|
||||||
|
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}
|
||||||
|
|
||||||
|
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
|
||||||
|
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|
||||||
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{\lp@linefill}}\newcommand{\linefootfill}{\def\lp@footfill{\lp@linefill}}
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||||||
|
\newcommand{\blankheadfill}{\def\lp@headfill{\hfill}}\newcommand
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||||||
|
{\blankfootfill}{\def\lp@footfill{\hfill}}\newcommand{\documentnumber}[1]
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||||||
|
{\def\lp@docno{#1}\outerhead{\lp@docno}}\def\lp@docno{}\def\@maketitlet
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||||||
|
{\newpage\null\vskip -14ex\hbox{}\hfill\lp@docno\vskip 13ex\begin{center}
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||||||
|
{\LARGE\@title\par}\vskip 1.5em{\large\lineskip .5em\begin{tabular}[t]{c}
|
||||||
|
\@author\end{tabular}\par}\vskip 1em{\large\@date}\end{center}\par\vskip 3em}
|
||||||
|
\def\abstract{\if@twocolumn\section*{Abstract}\else\small\begin{center}
|
||||||
|
{\bf Abstract\vspace{-.5em}\vspace{0pt}}\end{center}\quotation\fi}\def
|
||||||
|
\endabstract{\if@twocolumn\else\endquotation\fi}\ps@asyoulikeit\catcode`\@=12
|
||||||
|
%
|
||||||
|
%=========== End of {asyoulikeit} page style definition ====================*
|
||||||
|
|
||||||
|
\DeclareSymbolFont{AMSb}{U}{msb}{m}{n}
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||||||
|
\DeclareMathSymbol{\N}{\mathbin}{AMSb}{"4E}
|
||||||
|
\DeclareMathSymbol{\Z}{\mathbin}{AMSb}{"5A}
|
||||||
|
\DeclareMathSymbol{\R}{\mathbin}{AMSb}{"52}
|
||||||
|
\DeclareMathSymbol{\Q}{\mathbin}{AMSb}{"51}
|
||||||
|
\DeclareMathSymbol{\I}{\mathbin}{AMSb}{"49}
|
||||||
|
\DeclareMathSymbol{\C}{\mathbin}{AMSb}{"43}
|
||||||
|
|
||||||
|
|
||||||
|
% Page layout definitions to suit A4 paper
|
||||||
|
\setcounter{secnumdepth}{3} \setcounter{tocdepth}{4}
|
||||||
|
\setlength{\topmargin}{0mm}
|
||||||
|
\setlength{\textwidth}{160mm} \setlength{\textheight}{220mm}
|
||||||
|
\setlength{\oddsidemargin}{0mm} \setlength{\evensidemargin}{0mm}
|
||||||
|
%
|
||||||
|
% Local definitions
|
||||||
|
% -----------------
|
||||||
|
\newcommand{\eg}{{\it e.g.}}
|
||||||
|
\newcommand{\etc}{{\it etc.}}
|
||||||
|
\newcommand{\ie}{{\it i.e.}}
|
||||||
|
\newcommand{\qv}{{\it q.v.}}
|
||||||
|
\newcommand{\viz}{{\it viz.}}
|
||||||
|
\newcommand{\degs}[1]{$#1^\circ$} % Degrees symbol
|
||||||
|
\newcommand{\mins}[1]{$#1^{\scriptsize\prime}$} % Minutes symbol
|
||||||
|
\newcommand{\secs}[1]{$#1^{\scriptsize\prime\prime}$} % Seconds symbol
|
||||||
|
\newcommand{\key}[1]{\fbox{\sc#1}} % Box for keys
|
||||||
|
\newcommand{\?}{\_\hspace{0.115em}} % Proper spacing for
|
||||||
|
% underscore
|
||||||
|
%----- Display example text (#1) in typewriter font
|
||||||
|
|
||||||
|
%\newcommand{\example}[1]{\\ \smallskip\hspace{1in}{\tt #1}\hfil\\
|
||||||
|
% \smallskip\noindent}
|
||||||
|
%
|
||||||
|
%----- Enclose text (#2) in ruled box of given thickness (#1)
|
||||||
|
|
||||||
|
\def\boxit#1#2{\vbox{\hrule height #1pt\hbox{\vrule width #1pt\hskip 5pt
|
||||||
|
\vbox{\vskip 5pt #2 \vskip 5pt}\hskip 5pt
|
||||||
|
\vrule width #1pt}\hrule height #1pt}}
|
||||||
|
|
||||||
|
%----- Display boxed warning text (#1)
|
||||||
|
|
||||||
|
\def\warning#1{\bigskip
|
||||||
|
\setbox1=\vbox{\tolerance=5000\parfillskip=0pt
|
||||||
|
\hsize=3in\noindent#1}
|
||||||
|
\centerline{\boxit{1.0}{\box1}}
|
||||||
|
\bigskip}
|
||||||
|
|
||||||
|
%----- Definitions to aid display of help text
|
||||||
|
% (modelled on \item and \itemitem)
|
||||||
|
|
||||||
|
\def\helpindent#1{\setbox2=\hbox to\parindent{{\it #1}\hfil}
|
||||||
|
\indent\llap{\box2}\ignorespaces}
|
||||||
|
\def\helpitem{\parindent=70pt\par\hang\helpindent}
|
||||||
|
\def\helpitemitem{\parindent=70pt\par\indent \parindent=80pt
|
||||||
|
\hangindent2\parindent \helpindent}
|
||||||
|
|
||||||
|
%----- Tables and footnotes to tables
|
||||||
|
%
|
||||||
|
\newcommand{\spacerA}{\rule{0mm}{4mm}}
|
||||||
|
\newcommand{\spacerB}{\rule[-2mm]{0mm}{5mm}}
|
||||||
|
\footnotesep=5mm
|
||||||
|
\renewcommand{\footnoterule}{{\small Notes:}}
|
||||||
|
|
||||||
|
%% Robin 01AUG2008
|
||||||
|
%%
|
||||||
|
|
||||||
|
\newcounter{examplec}
|
||||||
|
\newcounter{definitionc}
|
||||||
|
\newcounter{summaryc}
|
||||||
|
|
||||||
|
%\@addtoreset{examplec}{chapter}\renewcommand\theexamplec{\thechapter.arabic{examplec}}
|
||||||
|
%\@addtoreset{definitionc}{chapter}
|
||||||
|
%\@addtoreset{summaryc}{chapter}
|
||||||
|
|
||||||
|
%\renewcommand\examplec{\arabic{examplec}}
|
||||||
|
|
||||||
|
%\newenvironment{example}
|
||||||
|
%{
|
||||||
|
% \stepcounter{examplec} \vspace{10pt} \normalfont\bfseries Example:\normalfont\[{\arabic{chapter}.\arabic{examplec}}\]
|
||||||
|
% \normalfont \begin{quote}}{\end{quote}\par}
|
||||||
|
%\newenvironment{definition}
|
||||||
|
%\newenvironment{example}
|
||||||
|
%{
|
||||||
|
% \stepcounter{examplec} \vspace{10pt} \normalfont\bfseries Example:\normalfont\[{\arabic{chapter}.\arabic{examplec}}\]
|
||||||
|
% \normalfont \begin{quote}}{\end{quote}\par}
|
||||||
|
\usepackage{amsthm}
|
||||||
|
|
||||||
|
\newtheorem{example}{Example:}
|
||||||
|
\newtheorem{definition}{Definition:}
|
||||||
|
\newtheorem*{summary}{Summary:}
|
||||||
|
|
||||||
|
%\newenvironment{example}
|
||||||
|
%{ \stepcounter{examplec} \vspace{10pt} \normalfont\bfseries Example:(\arabic{chapter}.\arabic{examplec})
|
||||||
|
% \normalfont \begin{quote}}{\end{quote}\par}
|
||||||
|
|
||||||
|
%
|
||||||
|
%\newenvironment{definition}
|
||||||
|
%{ \stepcounter{definitionc} \vspace{10pt} \normalfont\bfseries Definition:(\arabic{chapter}.\arabic{definitionc})
|
||||||
|
% \normalfont \begin{quote}}{\end{quote}\par}
|
||||||
|
%
|
||||||
|
%\newenvironment{summary}
|
||||||
|
%{ \vspace{10pt} \normalfont\bfseries Summary:
|
||||||
|
% \normalfont \begin{quote}}{\end{quote}\par}
|
||||||
|
%
|
||||||
Loading…
Reference in New Issue
Block a user