\documentclass{article}

\usepackage{../style}

\usepackage{../graphics}

\begin{document}

\section*{Homework 3}

\begin{enumerate}

\item What is a regular language?

\item Assume you have an alphabet consisting of the letters

      $a$, $b$ and $c$ only. (1) Find a regular expression

      that recognises the two strings $ab$ and $ac$. (2) Find

      a regular expression that matches all strings

      \emph{except} these two strings. Note, you can only use

      regular expressions of the form 

      \begin{center} $r ::=

      \varnothing \;|\; \epsilon \;|\; c \;|\; r_1 + r_2 \;|\;

      r_1 \cdot r_2 \;|\; r^*$ 

      \end{center}

\item Define the function \textit{zeroable} which takes a

      regular expression as argument and returns a boolean.

      The function should satisfy the following property:

\begin{center}

$\textit{zeroable(r)} \;\text{if and only if}\; L(r) = \varnothing$

\end{center}

\item Given the alphabet $\{a,b\}$. Draw the automaton that has two states, say  $q_0$ and $q_1$.

The starting state is $q_0$ and the final state is $q_1$. The transition

function is given by

\begin{center}

\begin{tabular}{l}

$(q_0, a) \rightarrow q_0$\\

$(q_0, b) \rightarrow q_1$\\

$(q_1, b) \rightarrow q_1$

\end{tabular}

\end{center}

What is the languages recognised by this automaton?

\item Give a non-deterministic finite automaton that can recognise 

the language $L(a\cdot (a + b)^* \cdot c)$. 

\item Given the following deterministic finite automaton over the alphabet $\{0, 1\}$,

find the corresponding minimal automaton. In case states can be merged,

state clearly which states can

be merged.

\begin{center}

\begin{tikzpicture}[scale=3, line width=0.7mm]

  \node[state, initial]        (q0) at ( 0,1) {$q_0$};

  \node[state]                    (q1) at ( 1,1) {$q_1$};

  \node[state, accepting] (q4) at ( 2,1) {$q_4$};

  \node[state]                    (q2) at (0.5,0) {$q_2$};

  \node[state]                    (q3) at (1.5,0) {$q_3$};

  \path[->] (q0) edge node[above] {$0$} (q1)

                  (q0) edge node[right] {$1$} (q2)

                  (q1) edge node[above] {$0$} (q4)

                  (q1) edge node[right] {$1$} (q2)

                  (q2) edge node[above] {$0$} (q3)

                  (q2) edge [loop below] node {$1$} ()

                  (q3) edge node[left] {$0$} (q4)

                  (q3) edge [bend left=95, looseness = 2.2] node [left=2mm] {$1$} (q0)

                  (q4) edge [loop right] node {$0, 1$} ()

                  ;

\end{tikzpicture}

\end{center}

\item Define the language $L(M)$ accepted by a deterministic finite automaton $M$.

\end{enumerate}

\end{document}

%%% Local Variables: 

%%% mode: latex

%%% TeX-master: t

%%% End: