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\section{Introduction}

This chapter 

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\section{From UML Model to Object Model}

Let us consider a theoretical FMMD model. For the sake of simplicity
consider that all components and functional groups have only two failure modes that
we will label $a$ and $b$.
We can start with some base components, of types C and K  say, $\{ C_1, C_2, C_3, K_4, C_5, C_6, K_7 \}$.
Thus applying the function $fm$ to any of the components
gives error modes identified by a or b.

For the sake of example, let us say that each component has two failure
modes $a$ and $b$. So the function $fm$ applied to
$C_1$ yields $C_{1 a}$ and $C_{1 b}$:
i.e. $fm(C_1) = \{ C_{1 a}, C_{1 b} \}$.

HOW UML OBJECT MODEL OF COMPONENT AND ITS ERROR MODES

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We can organise these into functional groups (where the superscript
represents the FMMD hierarchy level, or $\alpha$ value, thus:
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We can organise these into functional groups (where the superscript 
represents the $\alpha$ value, see section \ref{alpha}), thus:
}

$$ FG^0_1 = \{C_1, C_2\},$$
$$ FG^0_2 = \{C_1, C_3, K_4\},$$
$$ FG^0_3 = \{C_5, C_6, K_7\}.$$

Note that in this model the base~component $C_1$ has been used in
two separate functional groups.
Also that the component type $K$ has been used by 
two different functional groups.

\paragraph{Symptom Extraction.}
A processes of symptom extraction is now applied to the functional groups.
Again for the sake of example, let us say that each functional
group has one or two symptoms again subscripted by $a$ and $b$.

Applying symptom abstraction to $FG^0_1$ i.e. $\bowtie fm ( FG^0_1 ) = \{ FG^0_{1 a}, FG^0_{1 b}  \} $ 
We can now create a new derived component, $DC^1_1$, whose failure
modes are the symptoms of $FG^0_1 $ thus $ fm ( {DC}^1_1 )  = \{ FG^0_{1 a}, FG^0_{1 b}  \} $.

UML OBJECT MODEL OF  DERIVED COMPONENT TOO




\subsection{Using Derived Components in Functional Groups}


HERE should how the hierarchy is built, how the inheritance works etc

HAVE an example. totally theoretical. HAVE Common mode failure detection AND Common dependency detection

\subsection{Directed Acyclic Graph}

Show how the hierarchy can be represented as a DAG

draw a dag

\subsection{Traversing the datamodel}

Show how we can find multiple causes for a SYSTEM level error

\subsubsection{Common mode failure detection}

Describe what a common mode failure is.

show how common mode failures can be detected by using the parts list (same components can all have their
error modes turned on, and the effect can be seen on the system, automatically tracing
common mode failures.


\subsubsection{Common dependency detection}

The same component can be relied on by different functional groups within a system
For instance a power supply spur (i.e. supplying a particular isolated voltage say)
could have many functional groups depending or linked to its failure modes.

Show how FMMD makes this tracable


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\section{Current Static Failure Mode Methodologies}
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