\documentclass{article}+ −
\usepackage{../style}+ −
\usepackage{../graphics}+ −
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\begin{document}+ −
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\section*{Homework 3}+ −
+ −
\HEADER+ −
+ −
\begin{enumerate}+ −
\item What is a regular language? Are there alternative ways to define this+ −
notion? If yes, give an explanation why they define the same notion.+ −
+ −
\item Why is every finite set of strings 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 language 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 a deterministic finite automata $A(Q, q_0, F, \delta)$,+ −
define which language is recognised by this automaton. Can you + −
define also the language defined by a non-deterministic automaton?+ −
+ −
\item Given the following deterministic finite automata over the+ −
alphabet $\{a, b\}$, find an automaton that recognises the+ −
complement language. (Hint: Recall that for the algorithm from the+ −
lectures, the automaton needs to be in completed form, that is have+ −
a transition for every letter from the alphabet.)+ −
+ −
\begin{center}+ −
\begin{tikzpicture}[>=stealth',very thick,auto,+ −
every state/.style={minimum size=0pt,+ −
inner sep=2pt,draw=blue!50,very thick,+ −
fill=blue!20},scale=2]+ −
\node[state, initial] (q0) at ( 0,1) {$q_0$};+ −
\node[state, accepting] (q1) at ( 1,1) {$q_1$};+ −
\path[->] (q0) edge node[above] {$a$} (q1)+ −
(q1) edge [loop right] node {$b$} ();+ −
\end{tikzpicture}+ −
\end{center}+ −
+ −
+ −
+ −
%\item Given the following deterministic finite automaton+ −
%+ −
%\begin{center}+ −
%\begin{tikzpicture}[scale=3, line width=0.7mm]+ −
% \node[state, initial] (q0) at ( 0,1) {$q_0$};+ −
% \node[state,accepting] (q1) at ( 1,1) {$q_1$};+ −
% \node[state, accepting] (q2) at ( 2,1) {$q_2$};+ −
% \path[->] (q0) edge node[above] {$b$} (q1)+ −
% (q1) edge [loop above] node[above] {$a$} ()+ −
% (q2) edge [loop above] node[above] {$a, b$} ()+ −
% (q1) edge node[above] {$b$} (q2)+ −
% (q0) edge[bend right] node[below] {$a$} (q2)+ −
% ;+ −
%\end{tikzpicture}+ −
%\end{center}+ −
%find the corresponding minimal automaton. State clearly which nodes+ −
%can be merged.+ −
+ −
\item Given the following non-deterministic finite automaton over the+ −
alphabet $\{a, b\}$, find a deterministic finite automaton that+ −
recognises the same language:+ −
+ −
\begin{center}+ −
\begin{tikzpicture}[>=stealth',very thick,auto,+ −
every state/.style={minimum size=0pt,+ −
inner sep=2pt,draw=blue!50,very thick,+ −
fill=blue!20},scale=2]+ −
\node[state, initial] (q0) at ( 0,1) {$q_0$};+ −
\node[state] (q1) at ( 1,1) {$q_1$};+ −
\node[state, accepting] (q2) at ( 2,1) {$q_2$};+ −
\path[->] (q0) edge node[above] {$a$} (q1)+ −
(q0) edge [loop above] node[above] {$b$} ()+ −
(q0) edge [loop below] node[below] {$a$} ()+ −
(q1) edge node[above] {$a$} (q2);+ −
\end{tikzpicture}+ −
\end{center}+ −
+ −
\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}[>=stealth',very thick,auto,+ −
every state/.style={minimum size=0pt,+ −
inner sep=2pt,draw=blue!50,very thick,+ −
fill=blue!20},scale=2]+ −
\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 Given the following finite deterministic automaton over the alphabet $\{a, b\}$:+ −
+ −
\begin{center}+ −
\begin{tikzpicture}[scale=2,>=stealth',very thick,auto,+ −
every state/.style={minimum size=0pt,+ −
inner sep=2pt,draw=blue!50,very thick,+ −
fill=blue!20}]+ −
\node[state, initial, accepting] (q0) at ( 0,1) {$q_0$};+ −
\node[state, accepting] (q1) at ( 1,1) {$q_1$};+ −
\node[state] (q2) at ( 2,1) {$q_2$};+ −
\path[->] (q0) edge[bend left] node[above] {$a$} (q1)+ −
(q1) edge[bend left] node[above] {$b$} (q0)+ −
(q2) edge[bend left=50] node[below] {$b$} (q0)+ −
(q1) edge node[above] {$a$} (q2)+ −
(q2) edge [loop right] node {$a$} ()+ −
(q0) edge [loop below] node {$b$} ()+ −
;+ −
\end{tikzpicture}+ −
\end{center}+ −
+ −
Give a regular expression that can recognise the same language as+ −
this automaton. (Hint: If you use Brzozwski's method, you can assume+ −
Arden's lemma which states that an equation of the form $q = q\cdot r + s$+ −
has the unique solution $q = s \cdot r^*$.)+ −
+ −
\item If a non-deterministic finite automaton (NFA) has+ −
$n$ states. How many states does a deterministic + −
automaton (DFA) that can recognise the same language+ −
as the NFA maximal need?+ −
+ −
\end{enumerate}+ −
+ −
\end{document}+ −
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