A {\em determinisitc finite automata (DFA)} $M$ is a 5-tuple 

$(Q, \Sigma, \delta, s, F)$, where:

\begin{enumerate}

  \item $Q$ is a finite set of {\em states}, also denoted $Q_M$.

  \item $\Sigma$ is a finite set of {\em alphabets}, also denoted $\Sigma_M$.

  \item $\delta$ is a {\em transition function} of type @{text "Q \<times> \<Sigma> \<Rightarrow> Q"} (a total function),

           also denoted $\delta_M$.

  \item @{text "s \<in> Q"} is a state called {\em initial state}, also denoted $s_M$.

  \item @{text "F \<subseteq> Q"} is a set of states named {\em accepting states}, also denoted $F_M$.

\end{enumerate}

Therefore, we have $M = (Q_M, \Sigma_M, \delta_M, s_M, F_M)$. Every DFA $M$ can be interpreted as 

a function assigning states to strings, denoted $\dfa{M}$, the  definition of which is as the following:

\begin{equation}

\begin{aligned}

  \dfa{M}([]) &\equiv s_M \\

   \dfa{M}(xa) &\equiv  \delta_M(\dfa{M}(x), a)

\end{aligned}

\end{equation}

A string @{text "x"} is said to be {\em accepted} (or {\em recognized}) by a DFA $M$ if 

$\dfa{M}(x) \in F_M$. The language recoginzed by DFA $M$, denoted

$L(M)$, is defined as:

\begin{equation}

  L(M) \equiv \{x~|~\dfa{M}(x) \in F_M\}

\end{equation}

The standard way of specifying a laugage $\Lang$ as {\em regular} is by stipulating that:

$\Lang = L(M)$ for some DFA $M$.

For any DFA $M$, the DFA obtained by changing initial state to another $p \in Q_M$ is denoted $M_p$, 

which is defined as:

\begin{equation}

    M_p \ \equiv\ (Q_M, \Sigma_M, \delta_M, p, F_M) 

\end{equation}

Two states $p, q \in Q_M$ are said to be {\em equivalent}, denoted $p \approx_M q$, iff.

\begin{equation}\label{m_eq_def}

  L(M_p) = L(M_q)

\end{equation}

It is obvious that $\approx_M$ is an equivalent relation over $Q_M$. and 

the partition induced by $\approx_M$ has $|Q_M|$ equivalent classes.

By overloading $\approx_M$,  an equivalent relation over strings can be defined:

\begin{equation}

  x \approx_M y ~ ~ \equiv ~ ~ \dfa{M}(x) \approx_M \dfa{M}(y)

\end{equation}

It can be proved that the the partition induced by $\approx_M$ also has $|Q_M|$ equivalent classes.

It is also easy to show that: if $x \approx_M y$, then $x \approx_{L(M)} y$, and this means

$\approx_M$ is a more refined equivalent relation than $\approx_{L(M)}$. Since partition induced by

$\approx_M$ is finite, the one induced by $\approx_{L(M)}$ must also be finite. Now, we get one direction

of \mht:

\begin{Lem}[\mht , Direction one]

  If a language $\Lang$ is regular (i.e. $\Lang = L(M)$ for some DFA $M$), then 

  the partition induced by $\approx_\Lang$ is finite.

\end{Lem}

The other direction is:

\begin{Lem}[\mht , Direction two]\label{auto_mh_d2}

  If  the partition induced by $\approx_\Lang$ is finite, then

  $\Lang$ is regular (i.e. $\Lang = L(M)$ for some DFA $M$).

\end{Lem}

To prove this lemma, a DFA $M_\Lang$ is constructed out of $\approx_\Lang$ with:

\begin{subequations}

\begin{eqnarray}

  Q_{M_\Lang} ~ & \equiv & ~ \{ \cls{x}{\approx_\Lang}~|~ x \in \Sigma^* \}\\

  \Sigma_{M_\Lang} ~ & \equiv & ~ \Sigma_M \\

  \delta_{M_\Lang} ~ & \equiv & ~ \left (\lambda (\cls{x}{\approx_\Lang}, a).  \cls{xa}{\approx_\Lang} \right) \\

  s_{M_\Lang} ~ & \equiv & ~ \cls{[]}{\approx_\Lang} \\

  F_{M_\Lang} ~ & \equiv & ~ \{ \cls{x}{\approx_\Lang}~|~ x \in \Lang \}

\end{eqnarray}

\end{subequations}

From the assumption of lemma \ref{auto_mh_d2}, we have that $\{ \cls{x}{\approx_\Lang}~|~ x \in \Sigma^* \}$

is finite

It can be proved that $\Lang = L(M_\Lang)$.

*}

text {*