From 16c7f38581aa0c2de1f307a55c92c79e68df29fa Mon Sep 17 00:00:00 2001
From: Robin Clark <r.clark@energytechnologycontrol.com>
Date: Wed, 15 Sep 2010 12:42:13 +0100
Subject: [PATCH] bowtie definition

---
 statistics/statistics.tex      |  2 +-
 symptom_ex_process/process.tex | 39 ++++++++++++++++++++++++++--------
 2 files changed, 31 insertions(+), 10 deletions(-)

diff --git a/statistics/statistics.tex b/statistics/statistics.tex
index 92f692b..f8727bd 100644
--- a/statistics/statistics.tex
+++ b/statistics/statistics.tex
@@ -52,7 +52,7 @@ For the Nuclear power station
 \subsection{Timing And Safety Checking}
 
 \subsubsection{CANopen Timing Definitions}
-
+CAN is a mainstream network and was internationally standardized (ISO 11898–1) in 1993.
 CANopen is a protocol suite based on the hardware of the CANbus\cite{canspec}.
 CANbus is a hardened differential serial communications bus and 
 is arbitration free\footnote{Implemented at the physical and data link layers using DOMINANT and PASSIVE bits, with self monitoring and auto back off
diff --git a/symptom_ex_process/process.tex b/symptom_ex_process/process.tex
index d75a04c..de16042 100644
--- a/symptom_ex_process/process.tex
+++ b/symptom_ex_process/process.tex
@@ -259,14 +259,36 @@ consider DC as being in the set of components i.e. $DC \in \mathcal{C}$
 
 \subsection{Defining the analysis process \\ as a function}
 
-It is useful to define this analysis process as a function.
-Defining the function `$\bowtie$' to represent the {\em symptom abstraction} process,
-and DCFM to represent derived component failure modes, we may now
-write 
+Where $\mathcal{F}$ is the set of all sets of failure modes, and $\mathcal{DC}$ 
+is the set of all derived components, we can define the symptom abstraction process thus:
+$$
+%\bowtie : SubSystemComponentFaultModes \rightarrow DerivedComponent 
+\bowtie : \mathcal{F} \rightarrow \mathcal{DC} 
+$$
+
+\paragraph{Extending $\bowtie$ to {\dcs}}
+
+It is useful to further define the $\bowtie$ function, to
+take the failure modes from derived components (as well as base components)
+and return a new derived component.
+This generalises the function $\bowtie$ and allows us to build
+hierarchical failure mode models.
+
+Where a {\fg} is composed of derived components, for sake of example
+Where $DC_1, DC_2, DC_3 $ are {\dc}'s and $DCFM$ is a set of failure modes thus 
+$FG = \{ DC_1, DC_2, DC_3 \}$ and $DCFM = FM(FG)$
+
+We can apply the symptom abstraction process $\bowtie$
+to the failure mode set $DCFM$.
+
+The case 
+where a {\fg} has been created from {\dcs} 
+using function `$\bowtie$' leads us to
+{\dc}'s at a higher level of fault abstraction.
 
 $$
 %\bowtie : SubSystemComponentFaultModes \rightarrow DerivedComponent 
-\bowtie : DCFM \rightarrow DerivedComponent 
+\bowtie : DCFM \rightarrow DC 
 $$
 %
 %\begin{equation}
@@ -275,10 +297,9 @@ $$
 %
 %or applying the function $fm$ to obtain the $FG_{cfm}$ set 
 %
-To put this in context, where DC is a derived component, and FG is a functional group,
-we may state the process of extracting a set of failure modes from a functional
-group thus:
-
+To put this in context, where FG is a functional group, sourced from base or derived components,
+we may state the process of 
+analysing the failure modes in the {\fg} and returning a {\dc} thus:
 \begin{equation}
   \bowtie(fm(FG)) = DC
 \end{equation}