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logic_diagram/logic_diagram.tex
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@ -3,7 +3,7 @@
{
\begin{abstract}
%This chapter describes using diagrams to represent propositional logic.
Propositial Logic Diagrams (PLD) have been designed to provide an intuitive method for visualising and manipulating
Propositional Logic Diagrams (PLD) have been designed to provide an intuitive method for visualising and manipulating
a specific sub-set of logic equations, to express fault modes in Mechanical and Electronic Systems.
PLDs are a variant of constraint diagrams. Contours used to express
sets represent failure modes and the Symptomatically merged groups
@ -15,7 +15,7 @@ PLD Diagrams can also be used to model the structure of software
and the flow of data through 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.
Nearly all modern safety critical systems involve these three disciplines.
%
It is intended to be used for analysis of automated safety critical systems.
Many types of safety critical systems now legally
@ -35,7 +35,7 @@ for the analysis of safety critical software and hardware systems.
}
{
\section{Intrduction}
Propositial Logic Diagrams (PLD) have been designed to provide an intuitive method for visualising and manipulating
Propositional Logic Diagrams (PLD) have been designed to provide an intuitive method for visualising and manipulating
a specific sub-set of logic equations, to express fault modes in Mechanical and Electronic Systems.
PLDs are a variant of constraint diagrams. Contours used to express
sets represent failure modes and the Symptomatically merged groups
@ -47,7 +47,7 @@ PLD Diagrams can also be used to model the structure of software
and the flow of data through 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.
Nearly all modern safety critical systems involve these three disciplines.
%
It is intended to be used for analysis of automated safety critical systems.
Many types of safety critical systems now legally
@ -86,7 +86,7 @@ for the analysis of safety critical software and hardware systems.
}
Propositional Logic Diagrams (PLDs) have been created
to collect and simplfy fault~modes in safety critical systems undergoing
to collect and simplify fault~modes in safety critical systems undergoing
static analysis.%\cite{sccs}\cite{en61508}.
%
This type of analysis treats failure modes within a system as logical
@ -96,7 +96,7 @@ within these systems, and aids the collection of
common failure symptoms.
%
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.
in a text editor or spreadsheet, a visual method is perceived as being more intuitive.
%Traditional set theory is often represented by Euler\cite{euler} or Spider\cite{spider}
@ -166,7 +166,7 @@ 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.
Spiders may be labelled.
%To differentiate these from common Euler diagram notation (normally used to represent set theory)
%the curves are drawn using dotted and dashed lines.
@ -218,7 +218,7 @@ $$ \mathbb{R}^{2} - \; \bigcup_{\hat{c} \in \hat{C}(\hat{d})}\hat{c}$$
{
\definition
{
Let d be a PLD and $ \mathcal{X} \subseteq \hat{C}(\hat{d})$ a set of countours.
Let d be a PLD and $ \mathcal{X} \subseteq \hat{C}(\hat{d})$ a set of contours.
If the set
$$ \hat{z} = \bigcap_{c \in \mathcal{X}}
{interior}
@ -252,7 +252,7 @@ Each test point can be associated with the set of contours that enclose it.
{
\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.
associating a test-point with a set of contours in the plane. This corresponds to the interior of the contours defining the zone.
}
}
@ -834,7 +834,7 @@ at the module level have the same symptoms.
The act of collecting common symptoms by joining lines is seen as intuitive.
Syntax checking (looking for contradictions and tautologies), as well as detecting
errors of ommission are automated in the FMMD tool.
errors of omission are automated in the FMMD tool.
\section{Double Simultaneous Fault Modelling}
@ -845,7 +845,7 @@ that not only single component failure modes must be considered in
analysis, but that the possibility of two component failing
simultaneously must be considered.
EN298 states that if a burner controller is in `lock out' (i.e. has detected a fault
and has ordered a shutdown) a secondary fault cannot be allowed to put the equipement under control (the burner) into a dangerous state.
and has ordered a shutdown) a secondary fault cannot be allowed to put the equipment under control (the burner) into a dangerous state.
To cover this rigorously, we are bound to consider more than one fault being active at a time.
\paragraph{Covering Double faults in a PLD Diagram}
Because we are allowed to repeat contours in a PLD diagram,