
\abstract{ This chapter looks at current methodologies 
for static analysis of safety critical systems 
and looks at the statistical justifications for their application.
}

 
\section{Introduction}


\section{Current Methods for Safety Critical Analysis}


\subsection{Deterministic Approach}
\paragraph{NOT WRITTEN YET PLEASE IGNORE}
No single component fault may lead to a dangerous condition.
EN298 En230 etc

\subsection{Bayes Theorem}
\paragraph{NOT WRITTEN YET PLEASE IGNORE}
\label{bayes}
Describe application - likely hood of faults being the cause of symptoms -
probablistic approach - no direct causation paths to the higher~abstraction fault mode.
Often for instance a component in a module within a module within a module etc
that has a probability of causing a SYSTEM level fault.

Used in FTA\cite{NASA}\cite{NUK}.
The idea being that probabilities can be assigned to components 
failing, causing system level errors.

 Problems, difficult to get reliable stats 
for probability to cause because of small sample numbers... 

FMMD approach can by traversing down the tree  use known component failure figures
to  get {\em accurate} probabilities and potential causes.
%$$ c1 \cap c2 \eq \emptyset  | c1 \neq c2 \wedge c1,c2 \in C \wedge C \in U  $$

%Thus if the failure~modes are pairwaise mutually exclusive they qualify for inclusion into the 
%unitary~state set family.

\subsection{ Saftey Integrity Level Analysis }
\paragraph{NOT WRITTEN YET PLEASE IGNORE}
\label{sil}
This technique looks at all components in the parts list
and asks what the effect of the component failing will be.
Note that particular failure modes of the compoent are not considered.
The component can fail in any of its failure modes from the perspective of this analysis.
The analyst has to make a choice between four conditions:

\begin{itemize}
\item sd - A safe fault that is detected by an automated system 
\item su - A safe fault that is undetected by an automated system
\item dd - A potentially dangerous fault that is detected by an automated system
\item du - A potentially dangerous fault that is not detected by an automated system
\end{itemize}
Actually this is almost how sil analysis is done, because
the base components are listed
and their failure result as either sd su dd du

A formula is then applied according to the system architecture 1oo1 2oo3 3oo3 etc

What is not done is the probability for all these conditions, the sil analysis
person simple has to decide which it is.
Another fault in this is that it is very difficult to
extract meaning ful stats
for how likely the detection systems are to pick the fault up, or even to introduce a fault of their own.

\subsection{Tests of Hypotheses and Significance}
\paragraph{NOT WRITTEN YET PLEASE IGNORE}
Linked in with Bayes theorem
Accident analysis
plane crashes and faults etc
In high reliability systems the fauls are often logged - strange occurances - 
processors resetting - what are the common factors - P values -
for instance very high voltage spikes can reset micro controllers -
but how do you corrollate that with unshielded suppressed contactors...

Maybe looking at the equipment and seeing if there is a 5\%
level of the error being caused ?
i.e. using it to search for these conditions ?


Actually this could be used to refine the SIL method \ref{sil}
and give probabilities for the four conditions.
