140
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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|
1 |
\documentclass{article}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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|
2 |
\usepackage{../style}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
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|
3 |
\usepackage{../langs}
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268
Christian Urban <christian dot urban at kcl dot ac dot uk>
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|
4 |
\usepackage{../graphics}
|
140
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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|
5 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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|
6 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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|
7 |
\begin{document}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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|
8 |
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
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|
9 |
\section*{Handout 3 (Automata)}
|
140
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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|
10 |
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
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|
11 |
Every formal language course I know of bombards you first with
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
12 |
automata and then to a much, much smaller extend with regular
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
13 |
expressions. As you can see, this course is turned upside
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
14 |
down: regular expressions come first. The reason is that
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
15 |
regular expressions are easier to reason about and the notion
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
16 |
of derivatives, although already quite old, only became more
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
17 |
widely known rather recently. Still let us in this lecture
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
18 |
have a closer look at automata and their relation to regular
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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|
19 |
expressions. This will help us with understanding why the
|
251
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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|
20 |
regular expression matchers in Python and Ruby are so slow
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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|
21 |
with certain regular expressions. The central definition
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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|
22 |
is:\medskip
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142
Christian Urban <christian dot urban at kcl dot ac dot uk>
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|
23 |
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
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|
24 |
\noindent
|
251
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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|
25 |
A \emph{deterministic finite automaton} (DFA), say $A$, is
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
26 |
defined by a four-tuple written $A(Q, q_0, F, \delta)$ where
|
142
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
27 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
28 |
\begin{itemize}
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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|
29 |
\item $Q$ is a finite set of states,
|
142
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
30 |
\item $q_0 \in Q$ is the start state,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
31 |
\item $F \subseteq Q$ are the accepting states, and
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
32 |
\item $\delta$ is the transition function.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
33 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
34 |
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251
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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|
35 |
\noindent The transition function determines how to
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
36 |
``transition'' from one state to the next state with respect
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
37 |
to a character. We have the assumption that these transition
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
38 |
functions do not need to be defined everywhere: so it can be
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
39 |
the case that given a character there is no next state, in
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
40 |
which case we need to raise a kind of ``failure exception''. A
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
41 |
typical example of a DFA is
|
142
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
42 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
43 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
44 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
45 |
every state/.style={minimum size=0pt,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
46 |
inner sep=2pt,draw=blue!50,very thick,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
47 |
fill=blue!20},scale=2]
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
48 |
\node[state,initial] (q_0) {$q_0$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
49 |
\node[state] (q_1) [right=of q_0] {$q_1$};
|
142
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
50 |
\node[state] (q_2) [below right=of q_0] {$q_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
51 |
\node[state] (q_3) [right=of q_2] {$q_3$};
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
52 |
\node[state, accepting] (q_4) [right=of q_1] {$q_4$};
|
142
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
53 |
\path[->] (q_0) edge node [above] {$a$} (q_1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
54 |
\path[->] (q_1) edge node [above] {$a$} (q_4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
55 |
\path[->] (q_4) edge [loop right] node {$a, b$} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
56 |
\path[->] (q_3) edge node [right] {$a$} (q_4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
57 |
\path[->] (q_2) edge node [above] {$a$} (q_3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
58 |
\path[->] (q_1) edge node [right] {$b$} (q_2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
59 |
\path[->] (q_0) edge node [above] {$b$} (q_2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
60 |
\path[->] (q_2) edge [loop left] node {$b$} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
61 |
\path[->] (q_3) edge [bend left=95, looseness=1.3] node [below] {$b$} (q_0);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
62 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
63 |
\end{center}
|
140
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
64 |
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
65 |
\noindent In this graphical notation, the accepting state
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
66 |
$q_4$ is indicated with double circles. Note that there can be
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
67 |
more than one accepting state. It is also possible that a DFA
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
68 |
has no accepting states at all, or that the starting state is
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
69 |
also an accepting state. In the case above the transition
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
70 |
function is defined everywhere and can be given as a table as
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
71 |
follows:
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
72 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
73 |
\[
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
74 |
\begin{array}{lcl}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
75 |
(q_0, a) &\rightarrow& q_1\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
76 |
(q_0, b) &\rightarrow& q_2\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
77 |
(q_1, a) &\rightarrow& q_4\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
78 |
(q_1, b) &\rightarrow& q_2\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
79 |
(q_2, a) &\rightarrow& q_3\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
80 |
(q_2, b) &\rightarrow& q_2\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
81 |
(q_3, a) &\rightarrow& q_4\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
82 |
(q_3, b) &\rightarrow& q_0\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
83 |
(q_4, a) &\rightarrow& q_4\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
84 |
(q_4, b) &\rightarrow& q_4\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
85 |
\end{array}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
86 |
\]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
87 |
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
88 |
We need to define the notion of what language is accepted by
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
89 |
an automaton. For this we lift the transition function
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
90 |
$\delta$ from characters to strings as follows:
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
91 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
92 |
\[
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
93 |
\begin{array}{lcl}
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
94 |
\hat{\delta}(q, []) & \dn & q\\
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
95 |
\hat{\delta}(q, c\!::\!s) & \dn & \hat{\delta}(\delta(q, c), s)\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
96 |
\end{array}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
97 |
\]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
98 |
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
99 |
\noindent This lifted transition function is often called
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
100 |
``delta-hat''. Given a string, we start in the starting state
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
101 |
and take the first character of the string, follow to the next
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
102 |
sate, then take the second character and so on. Once the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
103 |
string is exhausted and we end up in an accepting state, then
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
104 |
this string is accepted by the automaton. Otherwise it is not
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
105 |
accepted. So $s$ is in the \emph{language accepted by the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
106 |
automaton} $A(Q, q_0, F, \delta)$ iff
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
107 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
108 |
\[
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
109 |
\hat{\delta}(q_0, s) \in F
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
110 |
\]
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
111 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
112 |
\noindent I let you think about a definition that describes
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
113 |
the set of strings accepted by an automaton.
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
114 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
115 |
|
292
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
116 |
While with DFAs it will always be clear that given a character
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
117 |
what the next state is (potentially none), it will be useful
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
118 |
to relax this restriction. That means we have several
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
119 |
potential successor states. We even allow ``silent
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
120 |
transitions'', also called epsilon-transitions. They allow us
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
121 |
to go from one state to the next without having a character
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
122 |
consumed. We label such silent transition with the letter
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
123 |
$\epsilon$. The resulting construction is called a
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
124 |
\emph{non-deterministic finite automaton} (NFA) given also as
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
125 |
a four-tuple $A(Q, q_0, F, \rho)$ where
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
126 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
127 |
\begin{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
128 |
\item $Q$ is a finite set of states
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
129 |
\item $q_0$ is a start state
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
130 |
\item $F$ are some accepting states with $F \subseteq Q$, and
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
131 |
\item $\rho$ is a transition relation.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
132 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
133 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
134 |
\noindent
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
135 |
Two typical examples of NFAs are
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
136 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
137 |
\begin{tabular}[t]{c@{\hspace{9mm}}c}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
138 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
139 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
140 |
\node[state,initial] (q_0) {$q_0$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
141 |
\node[state] (q_1) [above=of q_0] {$q_1$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
142 |
\node[state, accepting] (q_2) [below=of q_0] {$q_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
143 |
\path[->] (q_0) edge node [left] {$\epsilon$} (q_1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
144 |
\path[->] (q_0) edge node [left] {$\epsilon$} (q_2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
145 |
\path[->] (q_0) edge [loop right] node {$a$} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
146 |
\path[->] (q_1) edge [loop above] node {$a$} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
147 |
\path[->] (q_2) edge [loop below] node {$b$} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
148 |
\end{tikzpicture} &
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
149 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
150 |
\raisebox{20mm}{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
151 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
152 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
153 |
\node[state,initial] (r_1) {$r_1$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
154 |
\node[state] (r_2) [above=of r_1] {$r_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
155 |
\node[state, accepting] (r_3) [right=of r_1] {$r_3$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
156 |
\path[->] (r_1) edge node [below] {$b$} (r_3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
157 |
\path[->] (r_2) edge [bend left] node [above] {$a$} (r_3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
158 |
\path[->] (r_1) edge [bend left] node [left] {$\epsilon$} (r_2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
159 |
\path[->] (r_2) edge [bend left] node [right] {$a$} (r_1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
160 |
\end{tikzpicture}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
161 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
162 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
163 |
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
164 |
\noindent There are, however, a number of points you should
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
165 |
note. Every DFA is a NFA, but not vice versa. The $\rho$ in
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
166 |
NFAs is a transition \emph{relation} (DFAs have a transition
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
167 |
function). The difference between a function and a relation is
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
168 |
that a function has always a single output, while a relation
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
169 |
gives, roughly speaking, several outputs. Look at the NFA on
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
170 |
the right-hand side above: if you are currently in the state
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
171 |
$r_2$ and you read a character $a$, then you can transition to
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
172 |
either $r_1$ \emph{or} $r_3$. Which route you take is not
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
173 |
determined. This means if we need to decide whether a string
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
174 |
is accepted by a NFA, we might have to explore all
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
175 |
possibilities. Also there is the special silent transition in
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
176 |
NFAs. As mentioned already this transition means you do not
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
177 |
have to ``consume'' any part of the input string, but
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
178 |
``silently'' change to a different state. In the left picture,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
179 |
for example, if you are in the starting state, you can
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
180 |
silently move either to $q_1$ or $q_2$.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
181 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
182 |
|
269
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
183 |
\subsubsection*{Thompson Construction}
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
184 |
|
251
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
185 |
The reason for introducing NFAs is that there is a relatively
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
186 |
simple (recursive) translation of regular expressions into
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
187 |
NFAs. Consider the simple regular expressions $\varnothing$,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
188 |
$\epsilon$ and $c$. They can be translated as follows:
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
189 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
190 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
191 |
\begin{tabular}[t]{l@{\hspace{10mm}}l}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
192 |
\raisebox{1mm}{$\varnothing$} &
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
193 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
194 |
\node[state, initial] (q_0) {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
195 |
\end{tikzpicture}\\\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
196 |
\raisebox{1mm}{$\epsilon$} &
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
197 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
198 |
\node[state, initial, accepting] (q_0) {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
199 |
\end{tikzpicture}\\\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
200 |
\raisebox{2mm}{$c$} &
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
201 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
202 |
\node[state, initial] (q_0) {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
203 |
\node[state, accepting] (q_1) [right=of q_0] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
204 |
\path[->] (q_0) edge node [below] {$c$} (q_1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
205 |
\end{tikzpicture}\\\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
206 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
207 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
208 |
|
251
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
209 |
\noindent The case for the sequence regular expression $r_1
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
210 |
\cdot r_2$ is as follows: We are given by recursion two
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
211 |
automata representing $r_1$ and $r_2$ respectively.
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
212 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
213 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
214 |
\begin{tikzpicture}[node distance=3mm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
215 |
>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
216 |
\node[state, initial] (q_0) {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
217 |
\node (r_1) [right=of q_0] {$\ldots$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
218 |
\node[state, accepting] (t_1) [right=of r_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
219 |
\node[state, accepting] (t_2) [above=of t_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
220 |
\node[state, accepting] (t_3) [below=of t_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
221 |
\node[state, initial] (a_0) [right=2.5cm of t_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
222 |
\node (b_1) [right=of a_0] {$\ldots$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
223 |
\node[state, accepting] (c_1) [right=of b_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
224 |
\node[state, accepting] (c_2) [above=of c_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
225 |
\node[state, accepting] (c_3) [below=of c_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
226 |
\begin{pgfonlayer}{background}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
227 |
\node (1) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (q_0) (r_1) (t_1) (t_2) (t_3)] {};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
228 |
\node (2) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (a_0) (b_1) (c_1) (c_2) (c_3)] {};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
229 |
\node [yshift=2mm] at (1.north) {$r_1$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
230 |
\node [yshift=2mm] at (2.north) {$r_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
231 |
\end{pgfonlayer}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
232 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
233 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
234 |
|
251
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
235 |
\noindent The first automaton has some accepting states. We
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
236 |
obtain an automaton for $r_1\cdot r_2$ by connecting these
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
237 |
accepting states with $\epsilon$-transitions to the starting
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
238 |
state of the second automaton. By doing so we make them
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
239 |
non-accepting like so:
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
240 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
241 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
242 |
\begin{tikzpicture}[node distance=3mm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
243 |
>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
244 |
\node[state, initial] (q_0) {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
245 |
\node (r_1) [right=of q_0] {$\ldots$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
246 |
\node[state] (t_1) [right=of r_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
247 |
\node[state] (t_2) [above=of t_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
248 |
\node[state] (t_3) [below=of t_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
249 |
\node[state] (a_0) [right=2.5cm of t_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
250 |
\node (b_1) [right=of a_0] {$\ldots$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
251 |
\node[state, accepting] (c_1) [right=of b_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
252 |
\node[state, accepting] (c_2) [above=of c_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
253 |
\node[state, accepting] (c_3) [below=of c_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
254 |
\path[->] (t_1) edge node [above, pos=0.3] {$\epsilon$} (a_0);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
255 |
\path[->] (t_2) edge node [above] {$\epsilon$} (a_0);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
256 |
\path[->] (t_3) edge node [below] {$\epsilon$} (a_0);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
257 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
258 |
\begin{pgfonlayer}{background}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
259 |
\node (3) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (q_0) (c_1) (c_2) (c_3)] {};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
260 |
\node [yshift=2mm] at (3.north) {$r_1\cdot r_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
261 |
\end{pgfonlayer}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
262 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
263 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
264 |
|
251
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
265 |
\noindent The case for the choice regular expression $r_1 +
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
266 |
r_2$ is slightly different: We are given by recursion two
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
267 |
automata representing $r_1$ and $r_2$ respectively.
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
268 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
269 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
270 |
\begin{tikzpicture}[node distance=3mm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
271 |
>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
272 |
\node at (0,0) (1) {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
273 |
\node[state, initial] (2) [above right=16mm of 1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
274 |
\node[state, initial] (3) [below right=16mm of 1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
275 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
276 |
\node (a) [right=of 2] {$\ldots$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
277 |
\node[state, accepting] (a1) [right=of a] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
278 |
\node[state, accepting] (a2) [above=of a1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
279 |
\node[state, accepting] (a3) [below=of a1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
280 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
281 |
\node (b) [right=of 3] {$\ldots$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
282 |
\node[state, accepting] (b1) [right=of b] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
283 |
\node[state, accepting] (b2) [above=of b1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
284 |
\node[state, accepting] (b3) [below=of b1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
285 |
\begin{pgfonlayer}{background}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
286 |
\node (1) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (2) (a1) (a2) (a3)] {};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
287 |
\node (2) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (3) (b1) (b2) (b3)] {};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
288 |
\node [yshift=3mm] at (1.north) {$r_1$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
289 |
\node [yshift=3mm] at (2.north) {$r_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
290 |
\end{pgfonlayer}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
291 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
292 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
293 |
|
251
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
294 |
\noindent Each automaton has a single start state and
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
295 |
potentially several accepting states. We obtain a NFA for the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
296 |
regular expression $r_1 + r_2$ by introducing a new starting
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
297 |
state and connecting it with an $\epsilon$-transition to the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
298 |
two starting states above, like so
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
299 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
300 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
301 |
\hspace{2cm}\begin{tikzpicture}[node distance=3mm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
302 |
>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
303 |
\node at (0,0) [state, initial] (1) {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
304 |
\node[state] (2) [above right=16mm of 1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
305 |
\node[state] (3) [below right=16mm of 1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
306 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
307 |
\node (a) [right=of 2] {$\ldots$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
308 |
\node[state, accepting] (a1) [right=of a] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
309 |
\node[state, accepting] (a2) [above=of a1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
310 |
\node[state, accepting] (a3) [below=of a1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
311 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
312 |
\node (b) [right=of 3] {$\ldots$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
313 |
\node[state, accepting] (b1) [right=of b] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
314 |
\node[state, accepting] (b2) [above=of b1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
315 |
\node[state, accepting] (b3) [below=of b1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
316 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
317 |
\path[->] (1) edge node [above] {$\epsilon$} (2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
318 |
\path[->] (1) edge node [below] {$\epsilon$} (3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
319 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
320 |
\begin{pgfonlayer}{background}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
321 |
\node (3) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (1) (a2) (a3) (b2) (b3)] {};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
322 |
\node [yshift=3mm] at (3.north) {$r_1+ r_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
323 |
\end{pgfonlayer}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
324 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
325 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
326 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
327 |
\noindent
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
328 |
Finally for the $*$-case we have an automaton for $r$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
329 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
330 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
331 |
\begin{tikzpicture}[node distance=3mm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
332 |
>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
333 |
\node at (0,0) (1) {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
334 |
\node[state, initial] (2) [right=16mm of 1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
335 |
\node (a) [right=of 2] {$\ldots$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
336 |
\node[state, accepting] (a1) [right=of a] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
337 |
\node[state, accepting] (a2) [above=of a1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
338 |
\node[state, accepting] (a3) [below=of a1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
339 |
\begin{pgfonlayer}{background}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
340 |
\node (1) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (2) (a1) (a2) (a3)] {};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
341 |
\node [yshift=3mm] at (1.north) {$r$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
342 |
\end{pgfonlayer}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
343 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
344 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
345 |
|
251
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
346 |
\noindent and connect its accepting states to a new starting
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
347 |
state via $\epsilon$-transitions. This new starting state is
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
348 |
also an accepting state, because $r^*$ can recognise the
|
251
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
349 |
empty string. This gives the following automaton for $r^*$:
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
350 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
351 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
352 |
\begin{tikzpicture}[node distance=3mm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
353 |
>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
354 |
\node at (0,0) [state, initial,accepting] (1) {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
355 |
\node[state] (2) [right=16mm of 1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
356 |
\node (a) [right=of 2] {$\ldots$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
357 |
\node[state] (a1) [right=of a] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
358 |
\node[state] (a2) [above=of a1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
359 |
\node[state] (a3) [below=of a1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
360 |
\path[->] (1) edge node [above] {$\epsilon$} (2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
361 |
\path[->] (a1) edge [bend left=45] node [above] {$\epsilon$} (1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
362 |
\path[->] (a2) edge [bend right] node [below] {$\epsilon$} (1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
363 |
\path[->] (a3) edge [bend left=45] node [below] {$\epsilon$} (1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
364 |
\begin{pgfonlayer}{background}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
365 |
\node (2) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (1) (a2) (a3)] {};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
366 |
\node [yshift=3mm] at (2.north) {$r^*$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
367 |
\end{pgfonlayer}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
368 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
369 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
370 |
|
251
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
371 |
\noindent This construction of a NFA from a regular expression
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
372 |
was invented by Ken Thompson in 1968.
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
373 |
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
374 |
|
269
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
375 |
\subsubsection*{Subset Construction}
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
376 |
|
292
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
377 |
What is interesting is that for every NFA we can find a DFA
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
378 |
which recognises the same language. This can, for example, be
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
379 |
done by the \emph{subset construction}. Consider again the NFA
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
380 |
on the left, consisting of nodes labeled $0$, $1$ and $2$.
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
381 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
382 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
383 |
\begin{tabular}{c@{\hspace{10mm}}c}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
384 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
385 |
every state/.style={minimum size=0pt,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
386 |
draw=blue!50,very thick,fill=blue!20},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
387 |
baseline=0mm]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
388 |
\node[state,initial] (q_0) {$0$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
389 |
\node[state] (q_1) [above=of q_0] {$1$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
390 |
\node[state, accepting] (q_2) [below=of q_0] {$2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
391 |
\path[->] (q_0) edge node [left] {$\epsilon$} (q_1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
392 |
\path[->] (q_0) edge node [left] {$\epsilon$} (q_2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
393 |
\path[->] (q_0) edge [loop right] node {$a$} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
394 |
\path[->] (q_1) edge [loop above] node {$a$} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
395 |
\path[->] (q_2) edge [loop below] node {$b$} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
396 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
397 |
&
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
398 |
\begin{tabular}{r|cl}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
399 |
nodes & $a$ & $b$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
400 |
\hline
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
401 |
$\varnothing\phantom{\star}$ & $\varnothing$ & $\varnothing$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
402 |
$\{0\}\phantom{\star}$ & $\{0,1,2\}$ & $\{2\}$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
403 |
$\{1\}\phantom{\star}$ & $\{1\}$ & $\varnothing$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
404 |
$\{2\}\star$ & $\varnothing$ & $\{2\}$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
405 |
$\{0,1\}\phantom{\star}$ & $\{0,1,2\}$ & $\{2\}$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
406 |
$\{0,2\}\star$ & $\{0,1,2\}$ & $\{2\}$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
407 |
$\{1,2\}\star$ & $\{1\}$ & $\{2\}$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
408 |
s: $\{0,1,2\}\star$ & $\{0,1,2\}$ & $\{2\}$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
409 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
410 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
411 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
412 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
413 |
\noindent The nodes of the DFA are given by calculating all
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
414 |
subsets of the set of nodes of the NFA (seen in the nodes
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
415 |
column on the right). The table shows the transition function
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
416 |
for the DFA. The first row states that $\varnothing$ is the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
417 |
sink node which has transitions for $a$ and $b$ to itself.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
418 |
The next three lines are calculated as follows:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
419 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
420 |
\begin{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
421 |
\item suppose you calculate the entry for the transition for
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
422 |
$a$ and the node $\{0\}$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
423 |
\item start from the node $0$ in the NFA
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
424 |
\item do as many $\epsilon$-transition as you can obtaining a
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
425 |
set of nodes, in this case $\{0,1,2\}$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
426 |
\item filter out all notes that do not allow an $a$-transition
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
427 |
from this set, this excludes $2$ which does not permit a
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
428 |
$a$-transition
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
429 |
\item from the remaining set, do as many $\epsilon$-transition
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
430 |
as you can, this yields $\{0,1,2\}$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
431 |
\item the resulting set specifies the transition from $\{0\}$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
432 |
when given an $a$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
433 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
434 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
435 |
\noindent Similarly for the other entries in the rows for
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
436 |
$\{0\}$, $\{1\}$ and $\{2\}$. The other rows are calculated by
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
437 |
just taking the union of the single node entries. For example
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
438 |
for $a$ and $\{0,1\}$ we need to take the union of $\{0,1,2\}$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
439 |
(for $0$) and $\{1\}$ (for $1$). The starting state of the DFA
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
440 |
can be calculated from the starting state of the NFA, that is
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
441 |
$0$, and then do as many $\epsilon$-transitions as possible.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
442 |
This gives $\{0,1,2\}$ which is the starting state of the DFA.
|
292
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
443 |
The terminal states in the DFA are given by all sets that
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
444 |
contain a $2$, which is the terminal state of the NFA. This
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
445 |
completes the subset construction.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
446 |
|
292
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
447 |
There are two points to note: One is that very often the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
448 |
resulting DFA contains a number of ``dead'' nodes that are
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
449 |
never reachable from the starting state (that is that the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
450 |
calculated DFA in this example is not a minimal DFA). Such
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
451 |
dead nodes can be safely removed without changing the language
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
452 |
that is recognised by the DFA. Another point is that in some
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
453 |
cases, however, the subset construction produces a DFA that
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
454 |
does \emph{not} contain any dead nodes\ldots{}that means it
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
455 |
calculates a minimal DFA. Which in turn means that in some
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
456 |
cases the number of nodes by going from NFAs to DFAs
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
457 |
exponentially increases, namely by $2^n$ (which is the number
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
458 |
of subsets you can form for $n$ nodes).
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
459 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
460 |
|
269
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
461 |
\subsubsection*{Brzozowski's Method}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
462 |
|
292
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
463 |
As said before, we can also go into the other direction---from
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
464 |
DFAs to regular expressions. Brzozowski's method calculates
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
465 |
a regular expression using familiar transformations for
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
466 |
solving equational systems. Consider the DFA:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
467 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
468 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
469 |
\begin{tikzpicture}[scale=1.5,>=stealth',very thick,auto,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
470 |
every state/.style={minimum size=0pt,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
471 |
inner sep=2pt,draw=blue!50,very thick,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
472 |
fill=blue!20}]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
473 |
\node[state, initial] (q0) at ( 0,1) {$q_0$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
474 |
\node[state] (q1) at ( 1,1) {$q_1$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
475 |
\node[state, accepting] (q2) at ( 2,1) {$q_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
476 |
\path[->] (q0) edge[bend left] node[above] {$a$} (q1)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
477 |
(q1) edge[bend left] node[above] {$b$} (q0)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
478 |
(q2) edge[bend left=50] node[below] {$b$} (q0)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
479 |
(q1) edge node[above] {$a$} (q2)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
480 |
(q2) edge [loop right] node {$a$} ()
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
481 |
(q0) edge [loop below] node {$b$} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
482 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
483 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
484 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
485 |
\noindent for which we can set up the following equational
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
486 |
system
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
487 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
488 |
\begin{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
489 |
q_0 & = & \epsilon + q_0\,b + q_1\,b + q_2\,b\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
490 |
q_1 & = & q_0\,a\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
491 |
q_2 & = & q_1\,a + q_2\,a
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
492 |
\end{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
493 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
494 |
\noindent There is an equation for each node in the DFA. Let
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
495 |
us have a look how the right-hand sides of the equations are
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
496 |
constructed. First have a look at the second equation: the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
497 |
left-hand side is $q_1$ and the right-hand side $q_0\,a$. The
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
498 |
right-hand side is essentially all possible ways how to end up
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
499 |
in $q_1$. There is only one incoming edge from $q_0$ consuming
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
500 |
an $a$. Therefore we say: if we are in $q_0$ consuming an $a$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
501 |
then we end up in $q_1$. Therefore the right hand side is
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
502 |
state followed by character---in this case $q_0\,a$. Now lets
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
503 |
have a look at the third equation: there are two incoming
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
504 |
edges. Therefore we have two terms, namely $q_1\,a$ and
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
505 |
$q_2\,a$. These terms are separated by $+$. The first states
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
506 |
that if in state $q_1$ consuming an $a$ will bring you to
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
507 |
$q_2$, and the secont that being in $q_2$ and consuming an $a$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
508 |
will make you stay in $q_2$. The right-hand side of the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
509 |
first equation is constructed similarly: there are three
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
510 |
incoming edges, therefore there are three terms. There is
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
511 |
one exception in that we also ``add'' $\epsilon$ to the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
512 |
first equation, because it corresponds to the starting state
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
513 |
in the DFA.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
514 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
515 |
Having constructed the equational system, the question is
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
516 |
how to solve it? Remarkably the rules are very similar to
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
517 |
solving usual linear equational systems. For example the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
518 |
second equation does not contain the variable $q_1$ on the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
519 |
right-hand side of the equation. We can therefore eliminate
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
520 |
$q_1$ from the system by just substituting this equation
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
521 |
into the other two. This gives
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
522 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
523 |
\begin{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
524 |
q_0 & = & \epsilon + q_0\,b + q_0\,a\,b + q_2\,b\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
525 |
q_2 & = & q_0\,a\,a + q_2\,a
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
526 |
\end{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
527 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
528 |
\noindent where in Equation (4) we have two occurences
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
529 |
of $q_0$. Like the laws about $+$ and $\cdot$, we can simplify
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
530 |
Equation (4) to obtain the following two equations:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
531 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
532 |
\begin{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
533 |
q_0 & = & \epsilon + q_0\,(b + a\,b) + q_2\,b\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
534 |
q_2 & = & q_0\,a\,a + q_2\,a
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
535 |
\end{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
536 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
537 |
\noindent Unfortunately we cannot make any more progress with
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
538 |
substituting equations, because both (6) and (7) contain the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
539 |
variable on the left-hand side also on the right-hand side.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
540 |
Here we need to now use a law that is different from the usual
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
541 |
laws. It is called \emph{Arden's rule}. It states that
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
542 |
if an equation is of the form $q = q\,r + s$ then it can be
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
543 |
transformed to $q = s\, r^*$. Since we can assume $+$ is
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
544 |
symmetric, equation (7) is of that form: $s$ is $q_0\,a\,a$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
545 |
and $r$ is $a$. That means we can transform Equation (7)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
546 |
to obtain the two new equations
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
547 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
548 |
\begin{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
549 |
q_0 & = & \epsilon + q_0\,(b + a\,b) + q_2\,b\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
550 |
q_2 & = & q_0\,a\,a\,(a^*)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
551 |
\end{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
552 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
553 |
\noindent Now again we can substitute the second equation into
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
554 |
the first in order to eliminate the variable $q_2$.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
555 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
556 |
\begin{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
557 |
q_0 & = & \epsilon + q_0\,(b + a\,b) + q_0\,a\,a\,(a^*)\,b
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
558 |
\end{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
559 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
560 |
\noindent Pulling $q_0$ out as a single factor gives:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
561 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
562 |
\begin{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
563 |
q_0 & = & \epsilon + q_0\,(b + a\,b + a\,a\,(a^*)\,b)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
564 |
\end{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
565 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
566 |
\noindent This equation is again of the form so that we can
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
567 |
apply Arden's rule ($r$ is $b + a\,b + a\,a\,(a^*)\,b$ and $s$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
568 |
is $\epsilon$). This gives as solution for $q_0$ the following
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
569 |
regular expression:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
570 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
571 |
\begin{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
572 |
q_0 & = & \epsilon\,(b + a\,b + a\,a\,(a^*)\,b)^*
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
573 |
\end{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
574 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
575 |
\noindent SInce this is a regular expression, we can simplify
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
576 |
away the $\epsilon$ to obtain the slightly simpler regular
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
577 |
expression
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
578 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
579 |
\begin{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
580 |
q_0 & = & (b + a\,b + a\,a\,(a^*)\,b)^*
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
581 |
\end{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
582 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
583 |
\noindent
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
584 |
Now we can unwind this process and obtain the solutions
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
585 |
for the other equations. This gives:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
586 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
587 |
\begin{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
588 |
q_0 & = & (b + a\,b + a\,a\,(a^*)\,b)^*\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
589 |
q_1 & = & (b + a\,b + a\,a\,(a^*)\,b)^*\,a\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
590 |
q_2 & = & (b + a\,b + a\,a\,(a^*)\,b)^*\,a\,a\,(a)^*
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
591 |
\end{eqnarray}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
592 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
593 |
\noindent Finally, we only need to ``add'' up the equations
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
594 |
which correspond to a terminal state. In our running example,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
595 |
this is just $q_2$. Consequently, a regular expression
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
596 |
that recognises the same language as the automaton is
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
597 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
598 |
\[
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
599 |
(b + a\,b + a\,a\,(a^*)\,b)^*\,a\,a\,(a)^*
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
600 |
\]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
601 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
602 |
\noindent You can somewhat crosscheck your solution
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
603 |
by taking a string the regular expression can match and
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
604 |
and see whether it can be matched by the automaton.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
605 |
One string for example is $aaa$ and \emph{voila} this
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
606 |
string is also matched by the automaton.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
607 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
608 |
We should prove that Brzozowski's method really produces
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
609 |
an equivalent regular expression for the automaton. But
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
610 |
for the purposes of this module, we omit this.
|
269
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
611 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
612 |
\subsubsection*{Automata Minimization}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
613 |
|
270
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
614 |
As seen in the subset construction, the translation
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
615 |
of an NFA to a DFA can result in a rather ``inefficient''
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
616 |
DFA. Meaning there are states that are not needed. A
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
617 |
DFA can be \emph{minimised} by the following algorithm:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
618 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
619 |
\begin{enumerate}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
620 |
\item Take all pairs $(q, p)$ with $q \not= p$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
621 |
\item Mark all pairs that accepting and non-accepting states
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
622 |
\item For all unmarked pairs $(q, p)$ and all characters $c$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
623 |
test whether
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
624 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
625 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
626 |
$(\delta(q, c), \delta(p,c))$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
627 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
628 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
629 |
are marked. If there is one, then also mark $(q, p)$.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
630 |
\item Repeat last step until no change.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
631 |
\item All unmarked pairs can be merged.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
632 |
\end{enumerate}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
633 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
634 |
\noindent To illustrate this algorithm, consider the following
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
635 |
DFA.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
636 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
637 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
638 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
639 |
every state/.style={minimum size=0pt,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
640 |
inner sep=2pt,draw=blue!50,very thick,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
641 |
fill=blue!20}]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
642 |
\node[state,initial] (q_0) {$q_0$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
643 |
\node[state] (q_1) [right=of q_0] {$q_1$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
644 |
\node[state] (q_2) [below right=of q_0] {$q_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
645 |
\node[state] (q_3) [right=of q_2] {$q_3$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
646 |
\node[state, accepting] (q_4) [right=of q_1] {$q_4$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
647 |
\path[->] (q_0) edge node [above] {$a$} (q_1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
648 |
\path[->] (q_1) edge node [above] {$a$} (q_4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
649 |
\path[->] (q_4) edge [loop right] node {$a, b$} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
650 |
\path[->] (q_3) edge node [right] {$a$} (q_4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
651 |
\path[->] (q_2) edge node [above] {$a$} (q_3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
652 |
\path[->] (q_1) edge node [right] {$b$} (q_2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
653 |
\path[->] (q_0) edge node [above] {$b$} (q_2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
654 |
\path[->] (q_2) edge [loop left] node {$b$} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
655 |
\path[->] (q_3) edge [bend left=95, looseness=1.3] node
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
656 |
[below] {$b$} (q_0);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
657 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
658 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
659 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
660 |
\noindent In Step 1 and 2 we consider essentially a triangle
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
661 |
of the form
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
662 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
663 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
664 |
\begin{tikzpicture}[scale=0.6,line width=0.8mm]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
665 |
\draw (0,0) -- (4,0);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
666 |
\draw (0,1) -- (4,1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
667 |
\draw (0,2) -- (3,2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
668 |
\draw (0,3) -- (2,3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
669 |
\draw (0,4) -- (1,4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
670 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
671 |
\draw (0,0) -- (0, 4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
672 |
\draw (1,0) -- (1, 4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
673 |
\draw (2,0) -- (2, 3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
674 |
\draw (3,0) -- (3, 2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
675 |
\draw (4,0) -- (4, 1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
676 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
677 |
\draw (0.5,-0.5) node {$q_0$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
678 |
\draw (1.5,-0.5) node {$q_1$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
679 |
\draw (2.5,-0.5) node {$q_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
680 |
\draw (3.5,-0.5) node {$q_3$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
681 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
682 |
\draw (-0.5, 3.5) node {$q_1$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
683 |
\draw (-0.5, 2.5) node {$q_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
684 |
\draw (-0.5, 1.5) node {$q_3$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
685 |
\draw (-0.5, 0.5) node {$q_4$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
686 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
687 |
\draw (0.5,0.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
688 |
\draw (1.5,0.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
689 |
\draw (2.5,0.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
690 |
\draw (3.5,0.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
691 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
692 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
693 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
694 |
\noindent where the lower row is filled with stars, because in
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
695 |
the corresponding pairs there is always one state that is
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
696 |
accepting ($q_4$) and a state that is non-accepting (the other
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
697 |
states).
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
698 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
699 |
Now in Step 3 we need to fill in more stars according whether
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
700 |
one of the next-state pairs are marked. We have to do this
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
701 |
for every unmarked field until there is no change anymore.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
702 |
This gives the triangle
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
703 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
704 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
705 |
\begin{tikzpicture}[scale=0.6,line width=0.8mm]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
706 |
\draw (0,0) -- (4,0);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
707 |
\draw (0,1) -- (4,1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
708 |
\draw (0,2) -- (3,2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
709 |
\draw (0,3) -- (2,3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
710 |
\draw (0,4) -- (1,4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
711 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
712 |
\draw (0,0) -- (0, 4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
713 |
\draw (1,0) -- (1, 4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
714 |
\draw (2,0) -- (2, 3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
715 |
\draw (3,0) -- (3, 2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
716 |
\draw (4,0) -- (4, 1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
717 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
718 |
\draw (0.5,-0.5) node {$q_0$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
719 |
\draw (1.5,-0.5) node {$q_1$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
720 |
\draw (2.5,-0.5) node {$q_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
721 |
\draw (3.5,-0.5) node {$q_3$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
722 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
723 |
\draw (-0.5, 3.5) node {$q_1$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
724 |
\draw (-0.5, 2.5) node {$q_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
725 |
\draw (-0.5, 1.5) node {$q_3$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
726 |
\draw (-0.5, 0.5) node {$q_4$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
727 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
728 |
\draw (0.5,0.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
729 |
\draw (1.5,0.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
730 |
\draw (2.5,0.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
731 |
\draw (3.5,0.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
732 |
\draw (0.5,1.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
733 |
\draw (2.5,1.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
734 |
\draw (0.5,3.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
735 |
\draw (1.5,2.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
736 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
737 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
738 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
739 |
\noindent which means states $q_0$ and $q_2$, as well as $q_1$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
740 |
and $q_3$ can be merged. This gives the following minimal DFA
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
741 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
742 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
743 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
744 |
every state/.style={minimum size=0pt,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
745 |
inner sep=2pt,draw=blue!50,very thick,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
746 |
fill=blue!20}]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
747 |
\node[state,initial] (q_02) {$q_{0, 2}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
748 |
\node[state] (q_13) [right=of q_02] {$q_{1, 3}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
749 |
\node[state, accepting] (q_4) [right=of q_13]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
750 |
{$q_{4\phantom{,0}}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
751 |
\path[->] (q_02) edge [bend left] node [above] {$a$} (q_13);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
752 |
\path[->] (q_13) edge [bend left] node [below] {$b$} (q_02);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
753 |
\path[->] (q_02) edge [loop below] node {$b$} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
754 |
\path[->] (q_13) edge node [above] {$a$} (q_4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
755 |
\path[->] (q_4) edge [loop above] node {$a, b$} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
756 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
757 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
758 |
|
269
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
759 |
\subsubsection*{Regular Languages}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
760 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
761 |
Given the constructions in the previous sections we obtain
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
762 |
the following picture:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
763 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
764 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
765 |
\begin{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
766 |
\node (rexp) {\bf Regexps};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
767 |
\node (nfa) [right=of rexp] {\bf NFAs};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
768 |
\node (dfa) [right=of nfa] {\bf DFAs};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
769 |
\node (mdfa) [right=of dfa] {\bf\begin{tabular}{c}minimal\\ DFAs\end{tabular}};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
770 |
\path[->,line width=1mm] (rexp) edge node [above=4mm, black] {\begin{tabular}{c@{\hspace{9mm}}}Thompson's\\[-1mm] construction\end{tabular}} (nfa);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
771 |
\path[->,line width=1mm] (nfa) edge node [above=4mm, black] {\begin{tabular}{c}subset\\[-1mm] construction\end{tabular}}(dfa);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
772 |
\path[->,line width=1mm] (dfa) edge node [below=5mm, black] {minimisation} (mdfa);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
773 |
\path[->,line width=1mm] (dfa) edge [bend left=45] (rexp);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
774 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
775 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
776 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
777 |
\noindent By going from regular expressions over NFAs to DFAs,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
778 |
we can always ensure that for every regular expression there
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
779 |
exists a NFA and DFA that can recognise the same language.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
780 |
Although we did not prove this fact. Similarly by going from
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
781 |
DFAs to regular expressions, we can make sure for every DFA
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
782 |
there exists a regular expression that can recognise the same
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
783 |
language. Again we did not prove this fact.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
784 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
785 |
The interesting conclusion is that automata and regular
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
786 |
expressions can recognise the same set of languages:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
787 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
788 |
\begin{quote} A language is \emph{regular} iff there exists a
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
789 |
regular expression that recognises all its strings.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
790 |
\end{quote}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
791 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
792 |
\noindent or equivalently
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
793 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
794 |
\begin{quote} A language is \emph{regular} iff there exists an
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
795 |
automaton that recognises all its strings.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
796 |
\end{quote}
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
797 |
|
269
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
798 |
\noindent So for deciding whether a string is recognised by a
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
799 |
regular expression, we could use our algorithm based on
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
800 |
derivatives or NFAs or DFAs. But let us quickly look at what
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
801 |
the differences mean in computational terms. Translating a
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
802 |
regular expression into a NFA gives us an automaton that has
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
803 |
$O(n)$ nodes---that means the size of the NFA grows linearly
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
804 |
with the size of the regular expression. The problem with NFAs
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
805 |
is that the problem of deciding whether a string is accepted
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
806 |
is computationally not cheap. Remember with NFAs we have
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
807 |
potentially many next states even for the same input and also
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
808 |
have the silent $\epsilon$-transitions. If we want to find a
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
809 |
path from the starting state of an NFA to an accepting state,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
810 |
we need to consider all possibilities. In Ruby and Python this
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
811 |
is done by a depth-first search, which in turn means that if a
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
812 |
``wrong'' choice is made, the algorithm has to backtrack and
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
813 |
thus explore all potential candidates. This is exactly the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
814 |
reason why Ruby and Python are so slow for evil regular
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
815 |
expressions. The alternative is to explore the search space
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
816 |
in a breadth-first fashion, but this might incur a memory
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
817 |
penalty.
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
818 |
|
269
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
819 |
To avoid the problems with NFAs, we can translate them
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
820 |
into DFAs. With DFAs the problem of deciding whether a
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
821 |
string is recognised or not is much simpler, because in
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
822 |
each state it is completely determined what the next
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
823 |
state will be for a given input. So no search is needed.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
824 |
The problem with this is that the translation to DFAs
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
825 |
can explode exponentially the number of states. Therefore when
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
826 |
this route is taken, we definitely need to minimise the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
827 |
resulting DFAs in order to have an acceptable memory
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
828 |
and runtime behaviour.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
829 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
830 |
But this does not mean that everything is bad with automata.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
831 |
Recall the problem of finding a regular expressions for the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
832 |
language that is \emph{not} recognised by a regular
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
833 |
expression. In our implementation we added explicitly such a
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
834 |
regular expressions because they are useful for recognising
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
835 |
comments. But in principle we did not need to. The argument
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
836 |
for this is as follows: take a regular expression, translate
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
837 |
it into a NFA and DFA that recognise the same language. Once
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
838 |
you have the DFA it is very easy to construct the automaton
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
839 |
for the language not recognised by an DFA. If the DAF is
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
840 |
completed (this is important!), then you just need to exchange
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
841 |
the accepting and non-accepting states. You can then translate
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
842 |
this DFA back into a regular expression.
|
268
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
843 |
|
292
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
844 |
Not all languages are regular. The most well-known example
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
845 |
of a language that is not regular consists of all the strings
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
846 |
of the form
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
847 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
848 |
\[a^n\,b^n\]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
849 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
850 |
\noindent meaning strings that have the same number of $a$s
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
851 |
and $b$s. You can try, but you cannot find a regular
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
852 |
expression for this language and also not an automaton. One
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
853 |
can actually prove that there is no regular expression nor
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
854 |
automaton for this language, but again that would lead us too
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
855 |
far afield for what we want to do in this module.
|
270
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
856 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
857 |
|
140
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
858 |
\end{document}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
859 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
860 |
%%% Local Variables:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
861 |
%%% mode: latex
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
862 |
%%% TeX-master: t
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
863 |
%%% End:
|