\ifthenelse {\boolean{paper}}
{
\begin{abstract}
In failure mode analysis, it is essential to
know the failure modes of the sub-systems and components used.
This paper outlines a technique for determining the failure modes of a sub-system given 
its components.
%
This paper describes a process for taking a functional~group of components,
applying FMEA analysis on all the component failure modes possible in that functional~group,
and then determining how that functional group can fail.
%
With this information, we can treat the functional group
as a component in its own right.
This new component is a derived component.
For a top down technique this would correspond to a low~level sub-system.
%The technique uses a graphical notation, based on Euler\cite{eulerviz} and Constraint diagrams\cite{constraint} to model failure modes and failure mode common symptom collection.  The technique is designed for making building blocks for a hierarchical fault model.

Once the failure modes have been determined for a sub-system/derived~component,
this derived component can be combined with others to form functional groups
to model 
higher level sub-systems/derived~components. 
In this way a hierarchy to represent the fault behaviour
of a system can be built from the bottom~up. This process can continue
until there is a complete hierarchy representing the failure mode
behaviour of the entire system under analysis.
%FMMD hierarchy
Using the FMMD technique the hierarchy is built from the bottom up to ensure complete failure mode coverage.
Because the process is bottom-up, syntax checking and tracking can ensure that
no component failure mode can be overlooked.
Once a hierarchy is in place it can be converted into a fault data model.
%
From the fault data model, automatic generation
of FTA\cite{nasafta} (Fault Tree Analysis) and mimimal cuts sets\cite{nucfta} are possible. 
Also statistical reliability/probability of failure~on~demand\cite{en61508} and MTTF (Mean Time to Failure) calculations can be produced 
automatically, where component failure mode statistics are available\cite{mil1991}.
%
%
This paper focuses on the process of building the blocks, the symptom extraction or abstraction process, that is key to creating an FMMD hierarchy.
\end{abstract}
}
{
\section{Introduction}

In failure mode analysis, it is essential to
know the failure modes of the sub-systems and components used.
This paper outlines a technique for determining the failure modes of a sub-system given 
its components.
%
This chapter describes a process for taking a functional group of components,
applying FMEA analysis on all the component failure modes possible in that functional~group,
and then determining how that functional group can fail.
%
%
With this information, we can treat the functional group
as a component in its own right.
This new component is a derived component.
For a top down technique this would correspond to a low~level sub-system.
%The technique uses a graphical notation, based on Euler\cite{eulerviz} and Constraint diagrams\cite{constraint} to model failure modes and failure mode common symptom collection.  The technique is designed for making building blocks for a hierarchical fault model.
%
Once the failure modes have been determined for a sub-system/derived~component,
this derived component can be combined with others to form functional groups
to model 
higher level sub-systems/derived~components. 
%
In this way a hierarchy to represent the fault behaviour
of a system can be built from the bottom~up. This process can continue
until there is a complete hierarchy representing the failure mode
behaviour of the entire system under analysis.
%FMMD hierarchy
Using the FMMD technique the hierarchy is built from the bottom up to ensure complete failure mode coverage.
Because the process is bottom-up, syntax checking and tracking can ensure that
no component failure mode can be overlooked.
Once a hierarchy is in place it can be converted into a fault data model.
%
From the fault data model, automatic generation
of FTA\cite{nasafta} (Fault Tree Analysis) and mimimal cuts sets\cite{nucfta} are possible. 
Also statistical reliability/probability of failure~on~demand\cite{en61508} and MTTF (Mean Time to Failure) calculations can be produced 
automatically, where component failure mode statistics are available\cite{mil1991}.
%
This chapter focuses on the process of building the blocks, the symptom extraction or abstraction process, that is key to creating an FMMD hierarchy.


}


%\clearpage

\section{Fault Finding and Failure Mode Analysis}

\subsection{Top Down or natural trouble shooting}
It is interesting here to look at the `natural' trouble shooting process.
Fault finding is intinctively performed from the top-down.
A faulty piece of equipment is examined and will have a 
symptom or specific fault. The suspected area or sub-system within the 
equipment will next be looked into. 
The trouble shooter will look for behaviour that is unusual or incorrect
to determine the next area or sub~system to look into, each time
moving to a more detailed lower level.
Specific measurements
and checks will be made, and finally a component or a low level sub-system
will be found to be faulty.
A natural fault finding process is thus top~down.
\subsection{FMMD - Bottom~up Analysis}
The FMMD technique does not follow the `natural fault finding' or top down approach,
it instead works from the bottom up.
Starting with a collection of base~components that form
a simple functional group, the effect of all component error modes are
examined, as to their effect on the functional group.
The effects on the functional group can then be collected as common symptoms,
and now we may treat the functional group as a component as  it has a known set of failure modes.
By reusing the `components' derived from functional~groups an entire
hierarichal failure mode mode of the system can be built.
By working from the bottom up, we can trace all possible sources
that could cause a particular mode of equipment failure.
This means that at the design stage of a product all component failure
modes must be considered. The aim here is for complete failure mode coverage.
This also means that we can obtain statistical estimates based on the known reliabilities
of the components.
%It also means that every component failure mode must at the very least be considered.

\subsection{Static Analysis}

In the field of safety critical engineering; to comply with 
European Law a product must be certified under the approriate `EN' standard.
Typically environmental stress, EMC, electrical stressing, endurance tests, 
software~inspections and project~management quality reviews are applied\cite{sccs}.

Static testing is also applied. This is theoretical analysis of the design of the product from the safety 
perspective.
Three main techniques are currently used,
Statistical failure models, FMEA (Failure mode Effects Analysis) and FTA (Fault Tree Analysis).
The technique outlined here aims to provide a mathematical frame work
to assist in the production of these three results of static analysis.
From the model created by the FMMD technique, the three  above failure mode
descriptions can be derived.

{
The aims are
\begin{itemize}
 \item To automate the process where possible
 \item To apply a documented trail for each analysis phase (determination of functional groups, and analysis of component failure modes on those groups)
 \item To use a modular approach so that analysed sub-systems can be re-used 
 \item Automatically ensure no failure mode is unhandled
 \item To produce a data model from which FTA, FMEA and statistical failure models may be obtained automatically
\end{itemize}
}


\subsection{Systems, functional groups, sub-systems and failure modes}

It is helpful here to define some terms, `system', `functional~group', `component', `base~component' and `derived~component/sub-system'.
These are listed in table~\ref{tab:symexdef}.

A System, is really any coherent entity that would be sold as a product. % safety critical product.
A sub-system is a system that is part of some larger system.
For instance a stereo amplifier separate is a sub-system. The 
whole Sound System, consists perhaps of the following `sub-systems': 
CD-player, tuner, amplifier~separate, loudspeakers and ipod~interface.
 
%Thinking like this is a top~down analysis approach
%and is the way in which FTA\cite{nucfta} analyses a System
%and breaks it down.

A sub-system will be composed of components, which
may themselves be sub-systems. However each `component'
will have a fault/failure behaviour and it should
always be possible to obtain a set of failure modes
for each `component'.  In FMMD terms a sub-system is a derived component.

If we look at the sound system example, 
the CD~player could fail in several distinct ways, 
and this couldbe due to a large number of
component failure modes.
%no matter what has happened to it or has gone wrong inside it.


Using the reasoning that working from the bottom up forces the consideration of all possible
component failures (which can be missed in a top~down approach)
we are presented with a problem. Which initial collections of base components should we choose ?

For instance in the CD~player example; to start at the bottom; we are presented with
a massive list of base~components, resistors, motors, user~switches, laser~diodes, all sorts !
Clearly, working from the bottom~up, we need to pick small
collections of components that work together in some way.
These are termed `functional~groups'. For instance the  circuitry that powers the laser diode
to illuminate the CD might contain a handful of components, and as such would make a good candidate
to be one of the base level functional~groups.


In choosing the lowest level (base component) sub-systems we would look 
for the smallest `functional~groups' of components within a system. 
We can define a functional~group as a set of components that interact
to perform a specific function.

When we have analysed the fault behaviour of a functional group, we can treat it as a `black box'.
We can now call our functional~group a sub-system or a derived~component. 
The goal here is to know how it will behave under fault conditions !
%Imagine buying one such `sub~system'  from a very honest vendor.
%One of those sir, yes but be warned it may fail in these distinct ways, here
%in the honest data sheet the set of failure modes is listed!


%This type of thinking is starting to become more commonplace in product literature, with the emergence
%of reliability safety standards such as IOC1508\cite{sccs},EN61508\cite{en61508}.
%FIT (Failure in Time - expected number of failures per billion hours of operation) values
%are published for some micro-controllers. A micro~controller
%is a complex sub-system in its self and could be considered a `black~box' with a given reliability.
%\footnote{Microchip sources give an FIT of 4 for their PIC18 series micro~controllers\cite{microchip}, The DOD
%1991 reliability manual\cite{mil1991} applies a FIT of 100 for this generic type of component}

Electrical components have detailed datasheets associated with them. A useful extension of this could
be failure modes of the component, with environmental factors and MTTF statistics.
Currently this sort of failure mode information is generally only  available for generic component types\cite{mil1991}.

 
%At higher levels of analysis, functional~groups are pre-analysed sub-systems that interact to
%erform a given function.

%\vspace{0.3cm}
\begin{table}[h]
\center
\begin{tabular}{||l|l||} \hline \hline
  {\em Definition } & {\em Description}    \\ \hline
System & A product designed to  \\
       & work as a coherent entity  \\  \hline   
Sub-system & A part of a system, \\
-or- derived component           & sub-systems may contain sub-systems. \\     
                                 & derived~components may by derived   \\     
                                 & from derived components   \\    \hline 
Failure mode & A way in which a System, \\
             & Sub-system or component can fail \\     \hline  
Functional Group & A collection of sub-systems and/or \\
                 & components that interact to \\
                 & perform a specific function  \\    \hline  
Base Component & Any bought in component, which \\
               & hopefully has a known set of failure modes  \\    \hline  
 \hline
\end{tabular}
\caption{Symptom Extraction Definitions}
\label{tab:symexdef}
\end{table}
%\vspace{0.3cm}


\section{The Symptom abstraction Process}

% TO DO: separate these two:

\paragraph{Symptom Extraction Described}

The objective of `symptom abstraction' is to analyse the functional~group and find 
how it can fail
when specified components within it fail.
Once we know how functional~group can fail, we can treat it as a component or sub-system
with its own set of failure modes.

\paragraph{FMEA applied to the Functional Group}
As the functional~group is a set of components, the failure~modes
that we have to consider are all the failure modes of its components.
Each failure mode (or combination of) investigated is termed a `test case'.
Each `test case' is analysed.
%
The component failure modes in each test case 
are examined with respect to their effect on the functional~group.
%
The aim of this analysis is to find out how the functional~group reacts
to each of the test case conditions.
The goal of the process is to produce a set of failure modes from the perspective of the functional~group.




\paragraph{Symptom Identification}
When all `test~cases' have been analysed, a second phase is applied.
%
This looks at the results of the `test~cases' as symptoms 
of the sub-system. 
Single component failures (or combinations) within the functional~group may cause unique symptoms. 
However, many failures, when looked at from the perspective of the functional group, will have the same symptoms.
These can be collected as `common symptoms'.
To go back to the CD~player example, a failed
output stage, and a failed internal audio amplifier, 
will both cause the same failure; $no\_sound$ !




\paragraph{Collection of Symptoms}
The common symptoms of failure and lone~component failure~modes are identified and collected.
We can now consider the functional~group as a component and the common symptoms as its failure modes.
Note  that here because the process is bottom up, we can ensure that all failure modes
associated with a functional~group have been handled.
Were failure~modes missed, any failure mode model could be dangerously incomplete. 
It is possible here for an automated system to flag unhandled failure modes.
\ref{requirement at the start}


\section{The Process : To analyse a base level Derived~Component/sub-system}

To sumarise:

\begin{itemize}
 \item Determine a minimal functional group
 \item Obtain the list of components in the functional group
 \item Collect the failure modes for each component
% \item Draw these as contours on a diagram
% \item Where si,ultaneous failures are examined use overlapping contours
% \item For each region on the diagram, make a test case
 \item Examine each failure mode of all the components in the functional~group, and determine their effects on the failure~mode behaviour of the functional group
 \item Collect common symptoms. Imagine you are handed this functional group as a `black box', a `sub-system' to use.
Determine which test cases produce the same fault symptoms {\em from the perspective of the functional~group}.% Join common symptoms with lines connecting them (sometimes termed a `spider').
 \item The lone test cases and the common~symptoms are now the fault mode behaviour of the sub-system/derived~component. 
 \item A new `derived component' can now be created where each common~symptom, or lone test case is a failure~mode of this new component.
\end{itemize}




\section{A general derived Component/Sub-System example}

Consider a functional group $FG$ with components $C_1$, $C_2$ and $C_3$.

$$ FG = \{ C_1 , C_2 , C_3 \} $$

Each component has a set of related fault modes (i.e. ways in which it can fail to operate correctly).
Let us define the following failure modes for each component, defining a function $FM()$ 
that is passed a component and returns the set of failure modes associated with it
\footnote{Base component failure modes are defined, often with 
statistics and evironmental factors in  a variety of sources. \cite{mil1991}
}.

To re-cap from the definitions chapter \ref{chap:definitions}.

Let the set of all possible components  be $\mathcal{C}$
and let the set of all possible failure modes be $\mathcal{F}$.

We can define a function $FM$  

\begin{equation}
FM : \mathcal{C} \mapsto \mathcal{P}\mathcal{F}
\end{equation}

defined by (where $C$ is a component and $F$ is a set of failure modes):

$$  FM ( C ) = F $$

%\\

And for this example:

$$ FM(C_1) =  \{ a_1, a_2, a_3 \} $$
$$ FM(C_2) =  \{ b_1, b_2 \} $$ 
$$ FM(C_3) =  \{ c_1, c_2 \} $$


\paragraph{Finding all failure modes within the functional group}

For FMMD failure mode analysis, we need to consider the failure modes
from all the components in the functional group as a flat set.
This can be found by applying function $FM$ to all the components
in the functional~group and taking the union of them thus:

$$ FunctionalGroupAllFailureModes = \bigcup_{j \in \{1...n\}} FM(C_j) $$

We can actually overload the notation for the function FM
and define it for the set components within a functional group $FG$ (i.e. where $FG \subset \mathcal{C} $) thus:

\begin{equation}
FM : FG \mapsto \mathcal{F}
\end{equation}

Applied to the functional~group $FG$ in the example above:
\begin{equation}
 FM(FG) = \{a_1, a_2, a_3, b_1, b_2, c_1, c_2 \}
\end{equation} 

This can be seen as all the failure modes that can affect the failure mode group $FG$.

\subsection{Analysis of the functional group failure modes}

For this example we shall consider single failure modes.
%For each of the failure modes from $FM(FG)$ we shall
%create a test case ($fgfm_i$). Next each test case is examined/analysed
%and its effect on the functional group determined.

\par
%\vspace{0.3cm}
\begin{table}[h]
\begin{tabular}{||c|c|c|c||} \hline \hline
  {\em Component Failure Mode } & {\em test case} & {\em Functional Group} & {\em Functional Group}    \\ 
  {\em                        } & {\em          } & {\em failure mode}     & {\em Symptom}                \\ \hline
%
$a\_1$ & $fs\_1$ & $fgfm_{1}$ & SP2     \\      \hline
$a\_2$ & $fs\_2$ & $fgfm_{2}$ & SP1 \\    \hline 
$a\_3$ & $fs\_3$ & $fgfm_{3}$ & SP2\\    \hline  
$b\_1$ & $fs\_4$ & $fgfm_{4}$ & SP1 \\    \hline  
$b\_2$ & $fs\_5$ & $fgfm_{5}$ & SP1 \\     \hline  
$c\_1$ & $fs\_6$ & $fgfm_{6}$ & \\    \hline  
$c\_2$ & $fs\_7$ & $fgfm_{7}$ & SP2\\    \hline
%
 \hline
\end{tabular}
\caption{Component to functional group to failure symptoms example}
\label{tab:fexsymptoms}
\end{table}
%\vspace{0.3cm}

Table~\ref{tab:fexsymptoms} shows the analysis process.
As we are only looking at single fault possibilities for this example each failure mode
is represented by a test~case.
The Component failure modes become test cases\footnote{The test case stage is necessary because for more complex analysis we have to consider the effects of combinations of component failure modes}.
The test cases are analysed w.r.t. the functional~group.
These become functional~group~failure~modes ($fgfm$'s).
The functional~group~failure~modes are how the functional group fails for the test~case, rather than how the components failed.

For the sake of example, let us consider the fault symptoms of $\{fgfm_2, fgfm_4, fgfm_5\}$  to be 
identical from the perspective of the functional~group.
That is to say, the way in which functional~group fails if $fgfm_2$, $fgfm_4$ or $fgfm_5$ % failure modes
occur, is going to be the same. 
For example, in our CD player example, this could mean the common symptom `no\_sound'.
No matter which component failure modes, or combinations thereof cause the problem, 
the failure symptom is the same. 
It may be of interest to the manufacturers and designers of the CD player why it failed, but 
as far as we the users are concerned, it has only one symptom, 
`no\_sound'!
We can thus group these component failure modes into a common symptom, $SP1$, thus
$ SP1 = \{fgfm_2, fgfm_4, fgfm_5\}$.
% These can then be joined to form a spider. 
Likewise
let $SP2 = \{fgfm_1, fgfm_3, fgfm_7\}$  be  an identical failure mode {\em from the perspective of the functional~group}.
Let $\{fgfm_6\}$ be a distinct failure mode {\em from the perspective of the functional~group i.e. it cannot be grouped as a common symptom}.


We have now in $SP1$, $SP2$ and $fgfm_6$ as the three ways in which this functional~group can fail.
In other words we have derived failure modes for this functional~group.
We can place these in a set of symptoms, $SP$.
%
$$ SP = \{ SP1, SP2, fgfm_6 \} $$
%
%
These three symptoms can be considered  the set of failure modes for the functional~group,  and
we can treat it as though it were a {\em black box}
or a {\em component} to be used in higher level designs.
%
The next stage of the process could be applied automatically.
Each common symptom becomes a failure mode of 
a newly created derived component. Let $DC$ be the newly derived component.
This is assigned  the failure modes that were derived from the functional~group.
We can thus apply the function $FM$ on this newly derived component thus:

$$ FM(DC) = \{ SP1, SP2, fgfm_6 \} $$

Note that $fgfm_6$, while  %being a failure mode has 
not being grouped as a common symptom
has \textbf{not dissappeared from the analysis process}.
Were the designer to have overlooked this test case, it would appear in the derived component.
This is rather like a child not eating his lunch and being served it cold for dinner\footnote{Although I was only ever threatened with a cold dinner once, my advice to all nine year olds faced with this dilemma, it is best to throw the brussel sprouts out of the dining~room window while the adults are not watching!}!
The process must not allow failure modes to be ignored or forgotten (see project aims in section \ref{requirements}).


\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, we may now
write 

$$
\bowtie : SubSystemComponentFaultModes \mapsto DerivedComponent 
$$
%
%\begin{equation}
%  \bowtie(FG_{cfm}) = DC
%\end{equation} 
%
%or applying the function $FM$ to obtain the $FG_{cfm}$ set 
%
Where DC is a derived component, and FG is a functional group:

\begin{equation}
  \bowtie(FM(FG)) = DC
\end{equation}


%The $SS_{fm}$ set of fault modes can be represented as a diagram with each fault~mode of $SS$ being a contour.
%The derivation of $SS_{fm}$ is represented graphically using the `$\bowtie$' symbol, as in figure \ref{fig:gensubsys4}

% \begin{figure}[h+]
%  \centering
%  \includegraphics[width=3in,height=3in]{./symptom_abstraction4.jpg}
%  % synmptom_abstraction.jpg: 570x601 pixel, 80dpi, 18.10x19.08 cm, bb=0 0 513 541
%  \label{fig:gensubsys3}
%  \caption{Deriving a new diagram}


This sub-system or derived~component $DC$ , with its three error modes, can now be treated as a component (although at a higher level of abstraction)
with known failure modes. 
This process can be repeated using derived~components to build a 
hierarchical 
fault~mode 
model.





%\section{A Formal Algorithmic Description of `Symptom Abstraction'}
\section{Algorithmic Description}

The algorithm for {\em symptom extraction} is described in 
this section
%describes the symptom abstraction process 
using set theory.  

The {\em symptom abstraction process} (given the symbol `$\bowtie$') takes a functional group $FG$
and converts it to a derived~component/sub-system $DC$.
%The sub-system $SS$ is a collection
%of failure~modes of the sub-system.
Note that
$DC$ is a derived component at a higher level of fault analysis abstraction,
than the functional~group it was derived from.
However, it can still be treated  
as a component with a known set of failure modes.
\paragraph{Enumerating abstraction levels}
If $DC$ were to be included in a functional~group,
that functional~group must be considered to be at a higher level of
abstraction than a base level functional~group.
%
In fact, if the abstraction level is enumerated, 
the functional~group must take the abstraction level
of the highest assigned to any of its components.
%
$DC$ can be used as a system building block at a higher
level of fault abstraction. Because the derived components
merge to form functional  groups, a converging hierarchy is
naturally formed with the abstraction level increasing with each tier.


The algorithm, representing the function $\bowtie$, has been broken down into five stages, each following on from the other.
These are described using the Algorithm environment in the next section \ref{algorithms}.
By defining the process and describing it using set theory, constraints and 
verification checks in the process can be stated formally.

\clearpage
\subsection{Algorithmic Description of Symptom Abstraction \\ Determine Failure Modes to examine}

The first stage is to find the failure modes to consider for
analysis.
Let $FG$ be the set of components in the functional group under analysis, and $c$
be components that are members of it.
%%
%% Algorithm 1
%%

\begin{algorithm}[h+]
 ~\label{alg:sympabs1}
\caption{Determine failure modes: $FG \mapsto F$} \label{alg:sympabs11}
\begin{algorithmic}[1]
%\REQUIRE Obtain a list of components for the System $S$ under investigation. 
%ENSURE Decomposition of $S$ into atomic components where each component $c$ has a know set of $fm$ failure modes.


%\STATE Determine functional groups $fg_n \subset S$ of components, where n is an index number and the number of functional groups found. 

\STATE  { Let $FG$ be a set of components } \COMMENT{ The functional group should be chosen to be minimally sized collections of components that perform a specific function}
\STATE { Let $c$ represent a component}

\ENSURE{  Each component $c \in FG $ has a known set of failure modes i.e. $ \forall c \in FG | FM(c) \neq \emptyset$ }
 
\STATE {let $F=FM(FG)$ be a set of all failure modes to consider for the functional~group $FG$}

%\STATE {Collect all failure modes from all the components in FG into the set $FG_{cfm}$}
%\FORALL { $c \in FG $ }
%\STATE { $ FM(c) \in FG_{cfm} $ } \COMMENT {Collect all failure modes from each component into the set $FM_{cfm}$}
%\ENDFOR

%\hline
Algorthim \ref{alg:sympabs11} has taken a functional~group $FG$ and returned a set of failure~modes $F=FM(FG)$ where each component has a known set of failure~modes.
The next task is to formulate `test cases'. These are the combinations of failure~modes that will be used
in the analysis stages.
 

\end{algorithmic}
\end{algorithm}

\clearpage
\subsection{Algorithmic Description of Symptom Abstraction \\ Determine Test Cases}

From the failure modes associated with the functional~group
we now need to determine test cases.
The test cases are collections of failure modes.
These could be formed from single failure modes or failure modes in combination.
Let $TC$ be the set of test cases associated withthe functional group $FG$.

%%
%% Algorithm 2
%%

 
\begin{algorithm}[h+]
 ~\label{alg:sympabs2}
\caption{Determine Test Cases: $F \mapsto TC $} \label{alg:sympabs22}
\begin{algorithmic}[1]

   \REQUIRE {Determine the test cases to be applied}
   
     \STATE { All test cases are now chosen by the investigating engineer(s). Typically all single 
           component failures are investigated
            with some specially selected combination faults}
  
   \STATE   { Let $TC$ be a set of test cases }
   \STATE { Let $tc_j$ be set of component failure modes where $j$ is an index of $J$} 
   \COMMENT { Each set $tc_j$ is a `test case' }
   \STATE   { $ \forall j \in J  |  tc_j \in TC $ } \COMMENT {Ensure the test cases are complete and unique}

   \FORALL { $tc_j \in TC$ }
     %\ENSURE   {$ tc_j  \in \bigcap FG_{cfm} $}
     \ENSURE   {$ tc_j  \in \mathcal{P}(F))$}
     \COMMENT { require that the test case is a member of the powerset of $F$ }
     \ENSURE { $ \forall \; j2 \; \in J ( \forall \; j1 \; \in J | tc_{j1} \neq tc_{j2} \; \wedge \; j1 \neq j2 ) $}
     \COMMENT { Test cases must be unique } 
   \ENDFOR 

 
  
   \STATE { let $f$ represet a component failure mode }
   \REQUIRE { That all failure modes are represented in at least one test case }
   \ENSURE { $ \forall f | (f \in F))    \wedge  (f \in \bigcup TC) $ }
   \COMMENT { This corresponds to checking that at least each failure mode is considered at 
              least once in the analysis; more rigorous cardinality constraint
              checks may be required for some safety standards}

% some european standards
%                imply checking all double fault combinations\cite{en298} }

%\hline
Algorithm \ref{alg:sympabs22} has taken the set of failure modes $ F=FM(FG) $ and returned a set of test cases $TC$.
The next stage is to analyse the effect of each test case on the functional group.

\end{algorithmic}
\end{algorithm}

\clearpage
\subsection{Algorithmic Description of Symptom Abstraction \\ Analyse Test Cases}
%%
%% Algorithm 3
%%
The test cases are now analysed for their impact on the behaviour of the functional~group.
Let $R$ be a set of results indexed by $j$ (the same index used to identify the test cases $tc_{j}$).


\begin{algorithm}[h+]
 ~\label{alg:sympabs3}
\caption{Analyse Test Cases: $ TC \mapsto R $} \label{alg:sympabs33}
\begin{algorithmic}[1]
 \STATE  { let r be a `test case result'}
 \STATE  { Let the function $Analyse : tc \mapsto r $ } \COMMENT { This analysis is a human activity, examining the failure~modes in the test case and determining how the functional~group will fail under those conditions}
 \STATE  { $ R $ is a set of test case results $r_j \in R$ where the index $j$ corresponds to $tc_j \in TC$}
 \FORALL { $tc_j \in TC$ } 
     \STATE { $ rc_j = Analyse(tc_j) $} \COMMENT {this is Fault Mode Effects Analysis (FMEA) applied in the context of the functional group}
     \STATE { $ rc_j \in R $ } \COMMENT{Add $rc_j$ to the set R}
   \ENDFOR 

%\hline
Algorithm \ref{alg:sympabs33} has built the set $R$, the sub-system/functional group results for each test case. 
\end{algorithmic}
\end{algorithm}


\clearpage
\subsection{Algorithmic Description of Symptom Abstraction \\ Find Common Symptoms}
%%
%% Algorithm 4
%%
This stage analyses the results from bottom-up FMEA analysis ($R$), and collects
results that, from the perspective of the functional~group, have the same failure symptom.
Let set $SP$ be the set of symptoms for the functional group $FG$.

\begin{algorithm}[h+]
 ~\label{alg:sympabs4}

\caption{Find Common Symptoms: $ R \mapsto SP $} \label{alg:sympabs44}

\begin{algorithmic}[1]


   %\REQUIRE {All failure modes for the components in $fm_i = FM(fg_i)$}
 \STATE   {Let $sp_l$ be a set of `test cases results' where $l$ is an index set $L$} 
   \STATE   {Let $SP$ be a set whose members are sets $sp_l$} 
   \COMMENT{ $SP$ is the set of `fault symptoms' for the sub-system}
%   
   %\COMMENT{This corresponds to a fault symptom of the functional group $FG$}
    %\COMMENT{where double failure modes are required the cardinality constrained powerset of two must be applied to each failure mode}

   \FORALL { $ r_j \in R$ } 
          \STATE    { $sp_l \in \mathcal{P} R \wedge sp_l \in SP$ } 
          %\STATE   { $sp_l \in \bigcap R \wedge sp_l \in SP$ } 
     \COMMENT{ Collect common symptoms.
 Analyse the sub-system's fault behaviour under the failure modes in $tc_j$ and determine the symptoms $sp_l$ that it
causes in the functional group $FG$}
       %\ENSURE { $ \forall l2 \in L ( \forall l1 \in L | \exists a \in sp_{l1} \neq \exists b \in sp_{l2} \wedge l1 \neq l2 ) $}

       \ENSURE {$ \forall a \in sp_l  |    \forall sp_i \in \bigcap_{i=1..L} SP  ( sp_i = sp_l  \implies a \in sp_i)$} 

      \COMMENT { Ensure that the elements in each $sp_l$ are not present in any other $sp_l$ set }
       	  
   \ENDFOR
    \STATE  { The Set $SP$ can now be considered to be the set of fault modes for the sub-system that $FG$ represents}

%\hline
Algorithm \ref{alg:sympabs44}  raises the failure~mode abstraction level.
The failures have now been considered not from the component level, but from the sub-system or
functional~group level.
We now have a set $SP$ of the symptoms of failure.

\end{algorithmic}
\end{algorithm}

\clearpage
\subsection{Algorithmic Description of Symptom Abstraction \\ Create Derived Component}
%%
%% Algorithm 5
%%
This final stage, is the creation of the derived component.
This derived component may now be used to build
new functional groups at higher levels of fault abstraction.
Let $DC$ be a derived component with its own set of failure~modes.

\begin{algorithm}[h+]
 ~\label{alg:sympabs5}

\caption{Treat the symptoms as failure modes of the Derived~Component/Sub-System: $ SP \mapsto DC $} \label{alg:sympabs55}

\begin{algorithmic}[1]

 \STATE { Let $DC$ be a derived component with failure modes $f$ indexed by $l$ }
 \FORALL { $sp_l \in SP$ }
     \STATE { $ f_l = ConvertToFaultMode(sp_l) $}
     \STATE { $ f_l \in DC $} \COMMENT{ this is saying place $f_l$ into $DC$'s collection of failure modes}
   \ENDFOR 
%\hline

Algorithm \ref{alg:sympabs55} is the final stage in the process. We now have a 
derived~component $DC$, which has its own set of failure~modes. This can now be
used in conjection with other components (or derived~components) 
to form functional~groups at a higher level of failure~mode~abstraction.
Hierarchies of fault abstraction can be built that can model an entire SYSTEM.
\end{algorithmic}
\end{algorithm}


\section{To conclude}

The symptom abstraction technique allows us to take a functional group of components, analyse the failure
mode behaviour and create a new entity, a derived~component, that has its own set of failure modes.
This process naturally takes one step to building a hierarchical failure mode model
from the bottom-up. 

\subsection{Hierarchical Simplification}

Because symptom abstraction collects fault modes, the number of faults to handle decreases
as the hierarchy progresses upwards.
This is seen by casual observation of real life Systems. At the highest levels the number of faults
is significantly less than the sum of its component failure modes. 
A Sound system might have, for instance only four faults at its highest or System level,
\small
$$ SoundSystemFaults = \{TUNER\_FAULT, CD\_FAULT, SOUND\_OUT\_FAULT, IPOD\_FAULT\}$$
\normalsize
The number of causes for any of these faults is very large. 
It does not matter to the user, which combination of causes caused the fault.
But as the hierarchy goes up in abstraction level, the number of faults goes down.

\subsection{Tracable Fault Modes}

Because the fault modes are determined from the bottom-up, the causes
for all high level faults naturally form trees.
Minimal cut sets \cite{nasafta} can be determined from these, and by
analysing the statistical likelyhood of the component failures,
the MTTF and SIL\cite{en61508} levels can be automatically calculated.


\vspace{40pt}
\today