\part{Sequence alignment}

\section{Simililarity between sequences}

A function $d$ is a distance between two sequences $x$ and $y$ in an alphabet $\Sigma$ if
\begin{itemize}
  \item $x, y \in \Sigma^{*}, d(x, x) = 0$
  \item $\forall x, y \in \Sigma^{*}$ $d(x,y) = d(y,x)$
  \item $\forall x, y, z \in \Sigma^{*}$ $d(x, z) \leq d(x, y) + d(x, z)$
\end{itemize}

Here we are interested by the distance that is able to represent the transformation of $x$ to $y$ using three types of basic operations:
\begin{itemize}
  \item Substition
  \item Insertion
  \item Deletion
\end{itemize}

Example:
\begin{itemize}
\item  $sub(a, b) = \begin{cases} 0 & \text{if} a = b \\ 1 &\text{otherwise} \end{cases}$.
\item $del(a) = 1$
\item $ins(a) = 1$
\end{itemize}
