\documentclass{article}+ −
\usepackage{../style}+ −
\usepackage{../graphics}+ −
+ − \begin{document}+ −
+ − \section*{Homework 3}+ −
+ − \HEADER+ −
+ − \begin{enumerate}+ −
\item The regular expression matchers in Java, Python and Ruby can be+ −
very slow with some (basic) regular expressions. What is the main+ −
reason for this inefficient computation?+ −
+ −
\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} $r ::=+ −
\ZERO \;|\; \ONE \;|\; c \;|\; r_1 + r_2 \;|\;+ −
r_1 \cdot r_2 \;|\; r^*$ + −
\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}+ −
% $\textit{zeroable(r)} \;\text{if and only if}\; + −
% L(r) = \{\}$+ −
% \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 automaton $A(\varSigma, Q, Q_0, F,+ −
\delta)$, 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 automaton 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 \textbf{(Deleted for 2017, 2018, 2019)}+ −
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?+ −
+ − \item Prove that for all regular expressions $r$ we have+ −
+ −
\begin{center} + −
$\textit{nullable}(r) \quad \text{if and only if} + −
\quad [] \in L(r)$ + −
\end{center}+ −
+ − Write down clearly in each case what you need to prove+ −
and what are the assumptions. + −
+ − + −
\item \POSTSCRIPT + −
\end{enumerate}+ −
+ − \end{document}+ −
+ − %%% Local Variables: + −
%%% mode: latex+ −
%%% TeX-master: t+ −
%%% End: + −