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\section { DFAs}
This week, we will study computational devices called \textit { deterministic finite automata} . \par
A DFA has a simple job: it will either \say { accept} or \say { reject} a string of letters.
\vspace { 2mm}
Consider the automaton $ A $ shown below:
\begin { center}
\begin { tikzpicture}
\begin { scope} [layer = nodes]
\node [main] (a) at (0, 0) { $ a $ } ;
\node [accept] (b) at (2, 0) { $ b $ } ;
\node [main] (c) at (5, 0) { $ c $ } ;
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\node [start] (s) at (-2, 0) { \texttt { start} } ;
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\end { scope}
\draw [->]
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(s) edge (a)
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(a) edge node[label] { $ 1 $ } (b)
(a) edge[loop above] node[label] { $ 0 $ } (a)
(b) edge[bend left] node[label] { $ 0 $ } (c)
(b) edge[loop above] node[label] { $ 1 $ } (b)
(c) edge[bend left] node[label] { $ 0 , 1 $ } (b)
;
\end { tikzpicture}
\end { center}
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$ A $ takes strings of letters in the alphabet $ \{ 0 , 1 \} $ and reads them left to right, one letter at a time. \par
Starting in the state $ a $ , the automaton $ A $ will move between states along the edge marked by each letter. \par
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\vspace { 2mm}
Note that node $ b $ has a \say { double edge} in the diagram above. This means that the state $ b $ is \textit { accepting} . Any string that makes $ A $ end in state $ b $ is \textit { accepted} . Similarly, strings that end in states $ a $ or $ c $ are \textit { rejected} . \par
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\vspace { 2mm}
For example, consider the string \texttt { 1011} . \par
$ A $ will go through the states $ a - b - c - b - b $ while processing this string. \par
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\problem { }
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Which of the following strings are accepted by $ A $ ? \par
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\begin { itemize}
\item \texttt { 1}
\item \texttt { 1010}
\item \texttt { 1110010}
\item \texttt { 1000100}
\end { itemize}
\vfill
\problem { }
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Describe the general form of a string accepted by $ A $ . \par
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\hint { Work backwards from the accepting state, and decide what all the strings must look like at the end in order to be accepted.}
\begin { solution}
$ A $ will accept strings that contain at least one $ 1 $ and end with an even (possibly 0) number of zeroes.
\end { solution}
\vfill
\pagebreak
Now consider the automaton $ B $ , which uses the alphabet $ \{ a, b \} $ . \par
It starts in the state $ s $ and has two accepting states $ a _ 1 $ and $ b _ 1 $ .
\begin { center}
\begin { tikzpicture}
\begin { scope} [layer = nodes]
\node [main] (s) at (0, 0) { $ s $ } ;
\node [accept] (a1) at (-2, -0.5) { $ a _ 1 $ } ;
\node [main] (a2) at (-2, -2.5) { $ a _ 2 $ } ;
\node [accept] (b1) at (2, -0.5) { $ b _ 1 $ } ;
\node [main] (b2) at (2, -2.5) { $ b _ 2 $ } ;
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\node [start] (start) at (0, 1) { \texttt { start} } ;
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\end { scope}
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\clip (-4, -3.5) rectangle (4, 1);
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\draw [->]
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(start) edge (s)
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(s) edge node[label] { \texttt { a} } (a1)
(a1) edge[loop left] node[label] { \texttt { a} } (a1)
(a1) edge[bend left] node[label] { \texttt { b} } (a2)
(a2) edge[bend left] node[label] { \texttt { a} } (a1)
(a2) edge[loop left] node[label] { \texttt { b} } (a2)
(s) edge node[label] { \texttt { b} } (b1)
(b1) edge[loop right] node[label] { \texttt { b} } (b1)
(b1) edge[bend left] node[label] { \texttt { a} } (b2)
(b2) edge[bend left] node[label] { \texttt { b} } (b1)
(b2) edge[loop right] node[label] { \texttt { a} } (b2)
;
\end { tikzpicture}
\end { center}
\problem { }
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Which of the following strings are accepted by $ B $ ?
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\begin { itemize}
\item \texttt { aa}
\item \texttt { abba}
\item \texttt { abbba}
\item \texttt { baabab}
\end { itemize}
\vfill
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\problem { } <SameStartAndEnd>
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Describe the strings accepted by $ B $ .
\begin { solution}
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$ B $ accepts strings that start and end with the same letter.
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\end { solution}
\vfill
\pagebreak
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\problem { } <fibonacci>
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How many strings of length $ n $ are accepted by the automaton $ C $ ?
\begin { center}
\begin { tikzpicture}
\begin { scope} [layer = nodes]
\node [main] (0) at (0, 0) { $ 0 $ } ;
\node [accept] (1) at (3, 0) { $ 1 $ } ;
\node [main] (2) at (5, 0) { $ 2 $ } ;
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\node [start] (s) at (-2, 0) { \texttt { start} } ;
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\end { scope}
\draw [->]
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(s) edge (0)
(0) edge[loop above] node[label] { \texttt { b} } (0)
(0) edge[bend left] node[label] { \texttt { a} } (1)
(1) edge[bend left] node[label] { \texttt { b} } (0)
(1) edge node[label] { \texttt { a} } (2)
(2) edge[loop above] node[label] { \texttt { a,b} } (2)
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;
\end { tikzpicture}
\end { center}
\begin { solution}
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If $ A _ n $ is the number of accepted strings of length $ n $ , then $ A _ n = A _ { n - 1 } + A _ { n - 2 } $ . \par
Computing initial conditions, we see that $ A _ n $ is an $ n + 2 $ -th Fibonacci number.
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\end { solution}
%\begin{remark}
%Note that all the states in our DFAs $A$, $B$ and $C$ from figures 1, 2, 3 have outgoing symbols for each letter of the alphabet. %Do the same for your DFAs.
%\end{remark}
\vfill
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\definition { }
An \textit { alphabet} is a finite set of symbols. \par
\definition { }
A \textit { string} over an alphabet $ Q $ is a finite sequence of symbols from $ Q $ . \par
We denote the empty string $ \varepsilon $ . \par
\vspace { 2mm}
$ Q ^ * $ is the set of all possible strings over $ Q $ . \par
For example, $ \{ \texttt { 0 } , \texttt { 1 } \} ^ * $ is the set $ \{ \varepsilon , \texttt { 0 } , \texttt { 1 } , \texttt { 00 } , \texttt { 01 } , \texttt { 10 } , \texttt { 11 } , \texttt { 000 } ,... \} $ \par
Note that this set contains the empty string.
\definition { }
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A \textit { language} over an alphabet $ Q $ is a subset of $ Q ^ * $ . \par
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For example, the language \say { strings of length 2} over $ \{ \texttt { 0 } , \texttt { 1 } \} $ is $ \{ \texttt { 00 } , \texttt { 01 } , \texttt { 10 } , \texttt { 11 } \} $
\definition { }
We say a language $ L $ is \textit { recognized} by a DFA if that DFA accepts a string $ w $ if and only if $ w \in L $ .
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\pagebreak
\problem { }
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Draw DFAs that recognize the following languages. In all parts, the alphabet is $ \{ 0 , 1 \} $ :
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\begin { itemize}
\item $ \{ w~ | ~w~ \text { begins with a \texttt { 1 } and ends with a \texttt { 0 } } \} $
\item $ \{ w~ | ~w~ \text { contains at least three \texttt { 1 } s } \} $
\item $ \{ w~ | ~w~ \text { contains the substring \texttt { 0101 } ( i.e, $ w = x\texttt { 0101} y$ for some $ x$ and $ y$ ) } \} $
\item $ \{ w~ | ~w~ \text { has length at least three and its third symbol is a \texttt { 0 } } \} $
\item $ \{ w~ | ~w~ \text { starts with \texttt { 0 } and has odd length, or starts with \texttt { 1 } and has even length } \} $
\item $ \{ w~ | ~w~ \text { doesn't contain the substring \texttt { 110 } } \} $
\end { itemize}
\begin { solution}
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$ \{ w~ | ~w~ \text { begins with a \texttt { 1 } and ends with a \texttt { 0 } } \} $
\begin { center}
\begin { tikzpicture}
\begin { scope} [layer = nodes]
\node [accept] (0) at (0, 2) { $ \phantom { 0 } $ } ;
\node [main] (1) at (3, 2) { $ \phantom { 0 } $ } ;
\node [main] (2) at (0, 0) { $ \phantom { 0 } $ } ;
\node [main] (3) at (3, 0) { $ \phantom { 0 } $ } ;
\node [start] (s) at (-2, 0) { \texttt { start} } ;
\end { scope}
\clip (-2, -1) rectangle (4.5, 3);
\draw [->]
(s) edge (2)
(0) edge[loop left] node[label] { \texttt { 1} } (0)
(0) edge[bend left] node[label] { \texttt { 1} } (1)
(1) edge[loop right] node[label] { \texttt { 1} } (1)
(1) edge[bend left] node[label] { \texttt { 0} } (0)
(2) edge[out=90, in=270] node[label] { \texttt { 1} } (1)
(2) edge node[label] { \texttt { 0} } (3)
(3) edge[loop right] node[label] { \texttt { 1,0} } (3)
;
\end { tikzpicture}
\end { center}
\linehack { }
$ \{ w~ | ~w~ \text { contains at least three \texttt { 1 } s } \} $
\begin { center}
\begin { tikzpicture}
\begin { scope} [layer = nodes]
\node [start] (s) at (-2, 0) { \texttt { start} } ;
\node [main] (0) at (0, 0) { $ \phantom { 0 } $ } ;
\node [main] (1) at (2, 0) { $ \phantom { 0 } $ } ;
\node [main] (2) at (4, 0) { $ \phantom { 0 } $ } ;
\node [accept] (3) at (6, 0) { $ \phantom { 0 } $ } ;
\end { scope}
\draw [->]
(s) edge (0)
(0) edge[loop above] node[label] { \texttt { 0} } (0)
(1) edge[loop above] node[label] { \texttt { 0} } (1)
(2) edge[loop above] node[label] { \texttt { 0} } (2)
(3) edge[loop above] node[label] { \texttt { 0,1} } (3)
(0) edge node[label] { \texttt { 1} } (1)
(1) edge node[label] { \texttt { 1} } (2)
(2) edge node[label] { \texttt { 1} } (3)
;
\end { tikzpicture}
\end { center}
\linehack { }
$ \{ w~ | ~w~ \text { contains the substring \texttt { 0101 } } \} $
\begin { center}
\begin { tikzpicture}
\begin { scope} [layer = nodes]
\node [start] (s) at (-2, 0) { \texttt { start} } ;
\node [main] (0) at (0, 0) { $ \phantom { 0 } $ } ;
\node [main] (1) at (2, 1) { $ \phantom { 0 } $ } ;
\node [main] (2) at (4, 1) { $ \phantom { 0 } $ } ;
\node [main] (3) at (0, 3) { $ \phantom { 0 } $ } ;
\node [accept] (4) at (2, 3) { $ \phantom { 0 } $ } ;
\end { scope}
% Tikz includes invisible handles in picture size.
% This crops the image to fix sizing.
\clip (-2, -1.75) rectangle (5, 5.25);
\draw [->]
(s) edge (0)
(0) edge[loop above] node[label] { \texttt { 1} } (0)
(0) edge[bend right] node[label] { \texttt { 0} } (1)
(1) edge[loop above] node[label] { \texttt { 0} } (1)
(1) edge node[label] { \texttt { 1} } (2)
(3) edge[bend right] node[label] { \texttt { 0} } (1)
(3) edge node[label] { \texttt { 1} } (4)
(4) edge[loop above] node[label] { \texttt { 0,1} } (4)
;
\draw [->, rounded corners = 10mm]
(2) to (4, 5) to node[label] { \texttt { 0} } (0, 5) to (3)
;
\draw [->, rounded corners = 10mm]
(2) to (4, -1.5) to node[label] { \texttt { 1} } (0, -1.5) to (0)
;
\end { tikzpicture}
\end { center}
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Notice that after getting two 0's in a row we don't reset to the initial state.
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\pagebreak
$ \{ w~ | ~w~ \text { has length at least three and its third symbol is a \texttt { 0 } } \} $
\begin { center}
\begin { tikzpicture}
\begin { scope} [layer = nodes]
\node [start] (s) at (-2, 0) { \texttt { start} } ;
\node [main] (0) at (0, 0) { $ \phantom { 0 } $ } ;
\node [main] (1) at (2, 0) { $ \phantom { 0 } $ } ;
\node [main] (2) at (4, 0) { $ \phantom { 0 } $ } ;
\node [accept] (3) at (6, 1) { $ \phantom { 0 } $ } ;
\node [accept] (4) at (6, -1) { $ \phantom { 0 } $ } ;
\end { scope}
\clip (-2, -2.5) rectangle (7, 2.5);
\draw [->]
(s) edge (0)
(0) edge node[label] { \texttt { 0,1} } (1)
(1) edge node[label] { \texttt { 0,1} } (2)
(2) edge node[label] { \texttt { 0} } (3)
(2) edge node[label] { \texttt { 1} } (4)
(3) edge[loop above] node[label] { \texttt { 0,1} } (3)
(4) edge[loop below] node[label] { \texttt { 0,1} } (4)
;
\end { tikzpicture}
\end { center}
\linehack { }
$ \{ w~ | ~w~ \text { starts with \texttt { 0 } and has odd length, or starts with \texttt { 1 } and has even length } \} $
\begin { center}
\begin { tikzpicture}
\begin { scope} [layer = nodes]
\node [start] (s) at (-2, 0) { \texttt { start} } ;
\node [main] (0) at (0, 0) { $ \phantom { 0 } $ } ;
\node [accept] (1) at (2, 1) { $ \phantom { 0 } $ } ;
\node [main] (2) at (4, 1) { $ \phantom { 0 } $ } ;
\end { scope}
\draw [->]
(s) edge (0)
(0) edge node[label] { \texttt { 0} } (1)
(1) edge[bend left] node[label] { \texttt { 0,1} } (2)
(2) edge[bend left] node[label] { \texttt { 0,1} } (1)
;
\draw [->, rounded corners = 5mm]
(0) to node[label] { \texttt { 1} } (4, 0) to (2)
;
\end { tikzpicture}
\end { center}
\linehack { }
$ \{ w~ | ~w~ \text { doesn't contain the substring \texttt { 110 } } \} $
\begin { center}
\begin { tikzpicture}
\begin { scope} [layer = nodes]
\node [start] (s) at (-2, 0){ \texttt { start} } ;
\node [accept] (0) at (0, 0) { $ \phantom { 0 } $ } ;
\node [accept] (1) at (2, 0) { $ \phantom { 0 } $ } ;
\node [accept] (2) at (4, 0) { $ \phantom { 0 } $ } ;
\node [main] (3) at (6, 0) { $ \phantom { 0 } $ } ;
\end { scope}
\draw [->]
(s) edge (0)
(0) edge[loop above] node[label] { \texttt { 0} } (0)
(2) edge[loop above] node[label] { \texttt { 1} } (2)
(3) edge[loop above] node[label] { \texttt { 0,1} } (3)
(0) edge[bend left] node[label] { \texttt { 1} } (1)
(1) edge[bend left] node[label] { \texttt { 0} } (0)
(1) edge node[label] { \texttt { 1} } (2)
(2) edge node[label] { \texttt { 0} } (3)
;
\end { tikzpicture}
\end { center}
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Notice that after getting three 1's in a row we don't reset to the initial state.
\end { solution}
\vfill
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\pagebreak
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\problem { }
Draw a DFA over an alphabet $ \{ \texttt { a } , \texttt { b } , \texttt { @ } , \texttt { . } \} $ recognizing the language of strings of the form \texttt { user@website.domain} , where \texttt { user} , \texttt { website} and \texttt { domain} are nonempty strings over $ \{ \texttt { a } , \texttt { b } \} $ and \texttt { domain} has length 2 or 3.
\begin { solution}
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\begin { center}
\begin { tikzpicture}
\begin { scope} [layer = nodes]
\node [start] (start) at (-2, 0) { \texttt { start} } ;
\node [main] (0) at (0, 0) { $ \phantom { 0 } $ } ;
\node [main] (1) at (0, 2) { $ \phantom { 0 } $ } ;
\node [main] (2) at (0, 4) { $ \phantom { 0 } $ } ;
\node [main] (3) at (0, 6) { $ \phantom { 0 } $ } ;
\node [main] (4) at (0, 8) { $ \phantom { 0 } $ } ;
\node [main] (5) at (0, 10) { $ \phantom { 0 } $ } ;
\node [accept] (6) at (0, 12) { $ \phantom { 0 } $ } ;
\node [accept] (7) at (0, 14) { $ \phantom { 0 } $ } ;
\node [main] (8) at (5, 7) { $ \phantom { 0 } $ } ;
\end { scope}
\draw [->]
(start) edge (0)
(0) edge node[label] { \texttt { a,b} } (1)
(1) edge node[label] { \texttt { @} } (2)
(2) edge node[label] { \texttt { a,b} } (3)
(3) edge node[label] { \texttt { .} } (4)
(4) edge node[label] { \texttt { a,b} } (5)
(5) edge node[label] { \texttt { a,b} } (6)
(6) edge node[label] { \texttt { a,b} } (7)
(1) edge[loop left] node[label] { \texttt { a,b} } (1)
(3) edge[loop left] node[label] { \texttt { a,b} } (3)
(0) edge[out=0, in=270] node[label] { \texttt { @,.} } (8)
(1) edge[out=0, in=245] node[label] { \texttt { .} } (8)
(2) edge[out=0, in=220] node[label] { \texttt { @,.} } (8)
(3) edge[out=0, in=195] node[label] { \texttt { @} } (8)
(4) edge[out=0, in=170] node[label] { \texttt { @,.} } (8)
(5) edge[out=0, in=145] node[label] { \texttt { @,.} } (8)
(6) edge[out=0, in=120] node[label] { \texttt { @,.} } (8)
(7) edge[out=0, in=95] node[label] { \texttt { a,b,@,.} } (8)
;
\draw [->, rounded corners = 5mm]
(8) to +(1.5, 1) to node[label] { \texttt { a,b,@,.} } +(1.5, -1) to (8)
;
\end { tikzpicture}
\end { center}
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\end { solution}
\vfill
\pagebreak
\problem { }
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Draw a state diagram for a DFA over an alphabet of your choice that accepts exactly $ f ( n ) $ strings of length $ n $ if \par
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\begin { itemize}
\item $ f ( n ) = n $
\item $ f ( n ) = n + 1 $
\item $ f ( n ) = 3 ^ n $
\item $ f ( n ) = n ^ 2 $
\item $ f ( n ) $ is a Tribonacci number. \par
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Tribonacci numbers are defined by the sequence $ f ( 0 ) = 0 $ , $ f ( 1 ) = 1 $ , $ f ( 2 ) = 1 $ ,
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and $ f ( n ) = f ( n - 1 ) + f ( n - 2 ) + f ( n - 3 ) $ for $ n \ge 3 $ \par
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\hint { Fibonacci numbers are given by the automaton prohibiting two \texttt { `a'} s in a row.}
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\end { itemize}
\begin { solution}
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\textbf { Part 4:} $ f ( n ) = n ^ 2 $ \par
Consider the language of words over $ \{ 0 , 1 , 2 \} $ that have the sum of their digits equal to $ 2 $ . \par
Such words must contain two ones or one two:
\begin { center}
\begin { tikzpicture}
\begin { scope} [layer = nodes]
\node [start] (start) at (-2, 0) { \texttt { start} } ;
\node [main] (0) at (0, 0) { $ \phantom { 0 } $ } ;
\node [accept] (1) at (2, 0) { $ \phantom { 0 } $ } ;
\node [main] (2) at (0, -2) { $ \phantom { 0 } $ } ;
\node [main] (3) at (2, -2) { $ \phantom { 0 } $ } ;
\end { scope}
\clip (-2, 1.5) rectangle (4, -2.75);
\draw [->]
(start) edge (0)
(0) edge[loop above] node[label] { \texttt { 0} } (0)
(1) edge[loop above] node[label] { \texttt { 0} } (1)
(2) edge[loop left] node[label] { \texttt { 0} } (2)
(3) edge[loop right] node[label] { \texttt { 0,1,2} } (3)
(0) edge node[label] { \texttt { 2} } (1)
(0) edge node[label] { \texttt { 1} } (2)
(1) edge node[label] { \texttt { 1,2} } (3)
(2) edge node[label] { \texttt { 1} } (1)
(2) edge node[label] { \texttt { 2} } (3)
;
\end { tikzpicture}
\end { center}
\linehack { }
\textbf { Part 5:} Tribonacci numbers \par
Using the hint, we get the following automaton. \par
It rejects all strings with three \texttt { 'a'} s in a row.
\begin { center}
\begin { tikzpicture}
\begin { scope} [layer = nodes]
\node [start] (start) at (-2, 0) { \texttt { start} } ;
\node [accept] (0) at (0, 0) { $ \phantom { 0 } $ } ;
\node [accept] (1) at (0, 2) { $ \phantom { 0 } $ } ;
\node [accept] (2) at (2, 0) { $ \phantom { 0 } $ } ;
\node [main] (3) at (4, 0) { $ \phantom { 0 } $ } ;
\end { scope}
\draw [->]
(start) edge (0)
(0) edge[loop below] node[label] { \texttt { b} } (0)
(3) edge[loop above] node[label] { \texttt { a,b} } (3)
(0) edge[bend left] node[label] { \texttt { a} } (1)
(1) edge[bend left] node[label] { \texttt { b} } (0)
(1) edge[bend left] node[label] { \texttt { a} } (2)
(2) edge node[label] { \texttt { a} } (3)
(2) edge node[label] { \texttt { b} } (0)
;
\end { tikzpicture}
\end { center}
This automaton rejects all strings with three \texttt { 'a'} s in a row. If we count accepted strings, we get the Tribonacci numbers with an offest: $ f ( 0 ) = 1 $ , $ f ( 1 ) = 2 $ , $ f ( 2 ) = 4 $ , ... \par
\pagebreak
We can fix this by adding a node and changing the start state:
\begin { center}
\begin { tikzpicture}
\begin { scope} [layer = nodes]
\node [start] (start) at (1, -2) { \texttt { start} } ;
\node [accept] (0) at (0, 0) { $ \phantom { 0 } $ } ;
\node [accept] (1) at (0, 2) { $ \phantom { 0 } $ } ;
\node [accept] (2) at (2, 0) { $ \phantom { 0 } $ } ;
\node [main] (3) at (4, 0) { $ \phantom { 0 } $ } ;
\node [main] (4) at (3, -2) { $ \phantom { 0 } $ } ;
\end { scope}
\draw [->]
(start) edge (4)
(2) edge node[label] { \texttt { b} } (0)
(4) edge node[label] { \texttt { b} } (2)
(4) edge node[label] { \texttt { a} } (3)
(0) edge[loop below] node[label] { \texttt { b} } (0)
(3) edge[loop above] node[label] { \texttt { a,b} } (3)
(0) edge[bend left] node[label] { \texttt { a} } (1)
(1) edge[bend left] node[label] { \texttt { b} } (0)
(1) edge[bend left] node[label] { \texttt { a} } (2)
(2) edge node[label] { \texttt { a} } (3)
(2) edge node[label] { \texttt { b} } (0)
;
\end { tikzpicture}
\end { center}
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\end { solution}
\vfill
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\pagebreak
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% \problem{}
% Draw a DFA over an alphabet $\{a, b, c\}$, accepting all the suffixes of the string $abbc$ (including $\varepsilon$) and only them.
\problem { }
Draw a DFA recognizing the language of strings over $ \{ \texttt { 0 } , \texttt { 1 } \} $ in which \texttt { 0} is the third digit from the end. \par
Prove that any such DFA must have at least 8 states.
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\begin { solution}
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\textbf { Part 1:} \par
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Index the states by three-letter suffixes \texttt { 000} , \texttt { 001} , ..., \texttt { 111} . All strings that end with letters $ d _ 1 d _ 2 d _ 3 $ will end up in the state $ d _ 1 d _ 2 d _ 3 $ . We accept all states that start with a \texttt { 0} . \par
Note that we can start at any node if we ignore strings with fewer than three letters.
\begin { center}
\begin { tikzpicture}
\begin { scope} [layer = nodes]
\node [start] (start) at (-2, 0) { \texttt { start} } ;
\node [main] (7) at (0, 0) { \texttt { 111} } ;
\node [accept] (3) at (0, -2) { \texttt { 011} } ;
\node [main] (6) at (2, -2) { \texttt { 110} } ;
\node [main] (4) at (4, -2) { \texttt { 100} } ;
\node [accept] (1) at (-4, -4) { \texttt { 001} } ;
\node [main] (5) at (0, -4) { \texttt { 101} } ;
\node [accept] (2) at (-2, -4) { \texttt { 010} } ;
\node [accept] (0) at (-2, -6) { \texttt { 000} } ;
\end { scope}
\draw [->]
(0) edge[loop left, looseness = 7] node[label] { \texttt { 0} } (0)
(7) edge[loop above, looseness = 7] node[label] { \texttt { 1} } (7)
(start) edge (7)
(0) edge[out=90,in=-90] node[label] { \texttt { 1} } (1)
(1) edge node[label] { \texttt { 0} } (2)
(1) edge[out=45,in=-135] node[label] { \texttt { 1} } (3)
(2) edge[bend left] node[label] { \texttt { 1} } (5)
(3) edge node[label] { \texttt { 0} } (6)
(3) edge node[label] { \texttt { 1} } (7)
(5) edge[bend left] node[label] { \texttt { 0} } (2)
(5) edge node[label] { \texttt { 1} } (3)
(6) edge[bend left] node[label] { \texttt { 0} } (4)
(6) edge[out=-90,in=0] node[label] { \texttt { 1} } (5)
(7) edge[out=0,in=90] node[label] { \texttt { 0} } (6)
;
\draw [->, rounded corners = 10mm]
(4) to (4, 2) to node[label] { \texttt { 1} } (-4, 2) to (1)
;
\draw [->, rounded corners = 10mm]
(4) to (4, -6) to node[label] { \texttt { 0} } (0)
;
\draw [->, rounded corners = 5mm]
(2) to (-2, -5) to node[label] { \texttt { 0} } (3, -5) to (3, -2) to (4)
;
\end { tikzpicture}
\end { center}
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\linehack { }
\textbf { Part 2:} \par
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Strings \texttt { 000} , \texttt { 001} , ..., \texttt { 111} must lead to pairwise different states. \par
\vspace { 2mm}
Assume \texttt { 101} and \texttt { 010} lead to the same state. Append a \texttt { 1} to the end of the string. \par
\texttt { 101} will become \texttt { 011} , and \texttt { 010} will become \texttt { 101} . These must be different states, since we accept \texttt { 011} and reject \texttt { 101} . We now have a contradiction: one edge cannot lead to two states!
\vspace { 2mm}
\texttt { 101} and \texttt { 010} must thus correspond to distinct states. \par
We can repeat this argument for any other pair of strings. \par
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\end { solution}
\vfill
\pagebreak
