| author | Christian Urban <urbanc@in.tum.de> |
| Sun, 05 Aug 2018 21:37:35 +0100 | |
| changeset 553 | a05c412b1550 |
| parent 517 | 2ba868eb95cc |
| child 572 | 96af3fbdcd8d |
| permissions | -rw-r--r-- |
| 25 | 1 |
\documentclass[dvipsnames,14pt,t]{beamer}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
2 |
\usepackage{../slides}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
3 |
\usepackage{../graphics}
|
|
215
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
4 |
\usepackage{../langs}
|
|
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
5 |
\usepackage{../data}
|
| 25 | 6 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
7 |
\hfuzz=220pt |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
8 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
9 |
\pgfplotsset{compat=1.11}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
10 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
11 |
\newcommand{\bl}[1]{\textcolor{blue}{#1}}
|
| 25 | 12 |
|
13 |
% beamer stuff |
|
| 445 | 14 |
\renewcommand{\slidecaption}{CFL 03, King's College London}
|
| 25 | 15 |
|
16 |
\begin{document}
|
|
17 |
||
18 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
19 |
\begin{frame}[t]
|
| 25 | 20 |
\frametitle{%
|
21 |
\begin{tabular}{@ {}c@ {}}
|
|
22 |
\\[-3mm] |
|
| 445 | 23 |
\LARGE Compilers and \\[-2mm] |
| 25 | 24 |
\LARGE Formal Languages (3)\\[3mm] |
25 |
\end{tabular}}
|
|
26 |
||
27 |
\normalsize |
|
28 |
\begin{center}
|
|
|
134
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
29 |
\begin{tabular}{lp{8cm}}
|
| 25 | 30 |
Email: & christian.urban at kcl.ac.uk\\ |
| 500 | 31 |
Office: & N7.07 (North Wing, Bush House)\\ |
| 515 | 32 |
Slides: & KEATS (also homework and coursework is there)\\ |
| 25 | 33 |
\end{tabular}
|
34 |
\end{center}
|
|
35 |
||
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
36 |
\end{frame}
|
| 445 | 37 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
38 |
||
39 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
40 |
\begin{frame}[c]
|
|
41 |
\frametitle{Scala Book, Exams}
|
|
42 |
||
43 |
\begin{itemize}
|
|
| 515 | 44 |
\item \small https://nms.kcl.ac.uk/christian.urban/ProgInScala2ed.pdf\normalsize |
45 |
\item homeworks (written exam 80\%) |
|
46 |
\item coursework (20\%)\bigskip |
|
47 |
\item short survey at KEATS; to be answered until Sunday |
|
| 445 | 48 |
\end{itemize}
|
49 |
||
50 |
\end{frame}
|
|
51 |
%%%%%%%%%%% |
|
| 25 | 52 |
|
53 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
54 |
\begin{frame}[c]
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
55 |
\frametitle{Regular Expressions}
|
| 25 | 56 |
|
|
134
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
57 |
In programming languages they are often used to recognise:\medskip |
| 25 | 58 |
|
59 |
\begin{itemize}
|
|
|
133
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
60 |
\item symbols, digits |
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
61 |
\item identifiers |
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
62 |
\item numbers (non-leading zeros) |
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
63 |
\item keywords |
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
64 |
\item comments |
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
65 |
\end{itemize}\bigskip
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
66 |
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
67 |
\mbox{}\hfill\bl{\url{http://www.regexper.com}}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
68 |
\end{frame}
|
|
133
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
69 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
70 |
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
71 |
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
72 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
73 |
\begin{frame}[c]
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
74 |
\frametitle{Last Week}
|
|
133
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
75 |
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
76 |
Last week I showed you a regular expression matcher |
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
77 |
that works provably correct in all cases (we only |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
78 |
started with the proving part though) |
| 25 | 79 |
|
80 |
\begin{center}
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
81 |
\bl{$matches\,s\,r$} \;\;if and only if \;\; \bl{$s \in L(r)$}
|
|
133
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
82 |
\end{center}\bigskip\bigskip
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
83 |
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
84 |
\small |
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
85 |
\textcolor{gray}{\mbox{}\hfill{}by Janusz Brzozowski (1964)}
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
86 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
87 |
\end{frame}
|
| 25 | 88 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
89 |
||
90 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
91 |
\begin{frame}[c]
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
92 |
\frametitle{The Derivative of a Rexp}
|
| 25 | 93 |
|
94 |
\begin{center}
|
|
95 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
|
|
| 445 | 96 |
\bl{$der\, c\, (\ZERO)$} & \bl{$\dn$} & \bl{$\ZERO$} & \\
|
97 |
\bl{$der\, c\, (\ONE)$} & \bl{$\dn$} & \bl{$\ZERO$} & \\
|
|
98 |
\bl{$der\, c\, (d)$} & \bl{$\dn$} & \bl{if $c = d$ then $\ONE$ else $\ZERO$} & \\
|
|
|
133
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
99 |
\bl{$der\, c\, (r_1 + r_2)$} & \bl{$\dn$} & \bl{$der\, c\, r_1 + der\, c\, r_2$} & \\
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
100 |
\bl{$der\, c\, (r_1 \cdot r_2)$} & \bl{$\dn$} & \bl{if $nullable (r_1)$}\\
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
101 |
& & \bl{then $(der\,c\,r_1) \cdot r_2 + der\, c\, r_2$}\\
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
102 |
& & \bl{else $(der\, c\, r_1) \cdot r_2$}\\
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
103 |
\bl{$der\, c\, (r^*)$} & \bl{$\dn$} & \bl{$(der\,c\,r) \cdot (r^*)$} &\medskip\\
|
|
133
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
104 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
105 |
\bl{$\textit{ders}\, []\, r$} & \bl{$\dn$} & \bl{$r$} & \\
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
106 |
\bl{$\textit{ders}\, (c\!::\!s)\, r$} & \bl{$\dn$} & \bl{$\textit{ders}\,s\,(der\,c\,r)$} & \\
|
| 25 | 107 |
\end{tabular}
|
108 |
\end{center}
|
|
109 |
||
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
110 |
\end{frame}
|
| 25 | 111 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
112 |
||
113 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
114 |
\begin{frame}[c]
|
|
| 515 | 115 |
\frametitle{Example}
|
116 |
||
117 |
Given \bl{$r \dn ((a \cdot b) + b)^*$} what is
|
|
118 |
||
119 |
\begin{center}
|
|
120 |
\def\arraystretch{1.5}
|
|
121 |
\begin{tabular}{@{}lcl}
|
|
122 |
\bl{$\der\,a\,((a \cdot b) + b)^*$}
|
|
123 |
& \bl{$\Rightarrow$}& \bl{$\der\,a\, \underline{((a \cdot b) + b)^*}$}\\
|
|
124 |
& \bl{$=$} & \bl{$(der\,a\,(\underline{(a \cdot b) + b})) \cdot r$}\\
|
|
125 |
& \bl{$=$} & \bl{$((der\,a\,(\underline{a \cdot b})) + (der\,a\,b)) \cdot r$}\\
|
|
126 |
& \bl{$=$} & \bl{$(((der\,a\,\underline{a}) \cdot b) + (der\,a\,b)) \cdot r$}\\
|
|
127 |
& \bl{$=$} & \bl{$((\ONE \cdot b) + (der\,a\,\underline{b})) \cdot r$}\\
|
|
128 |
& \bl{$=$} & \bl{$((\ONE \cdot b) + \ZERO) \cdot r$}\\
|
|
129 |
\end{tabular}
|
|
130 |
\end{center}
|
|
131 |
||
132 |
\end{frame}
|
|
133 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
134 |
||
135 |
||
136 |
||
137 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
138 |
\begin{frame}[c]
|
|
| 25 | 139 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
140 |
Input: string \bl{$abc$} and regular expression \bl{$r$}
|
| 25 | 141 |
|
142 |
\begin{enumerate}
|
|
|
133
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
143 |
\item \bl{$der\,a\,r$}
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
144 |
\item \bl{$der\,b\,(der\,a\,r)$}
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
145 |
\item \bl{$der\,c\,(der\,b\,(der\,a\,r))$}\bigskip\pause
|
|
134
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
146 |
\item finally check whether the last regular expression can match the empty string |
| 25 | 147 |
\end{enumerate}
|
148 |
||
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
149 |
\end{frame}
|
| 515 | 150 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
151 |
||
152 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
153 |
\begin{frame}[c]
|
|
154 |
\frametitle{Simplification}
|
|
155 |
||
156 |
Given \bl{$r \dn ((a \cdot b) + b)^*$} what is
|
|
157 |
||
158 |
\begin{center}
|
|
159 |
\def\arraystretch{2}
|
|
160 |
\begin{tabular}{@{}lcl}
|
|
161 |
\bl{$((\ONE \cdot b) + \ZERO) \cdot r$}
|
|
162 |
& \bl{$\Rightarrow$} & \bl{$((\underline{\ONE \cdot b}) + \ZERO) \cdot r$}\\
|
|
163 |
& \bl{$=$} & \bl{$(\underline{b + \ZERO}) \cdot r$}\\
|
|
164 |
& \bl{$=$} & \bl{$b \cdot r$}\\
|
|
165 |
\end{tabular}
|
|
166 |
\end{center}
|
|
167 |
||
168 |
\end{frame}
|
|
| 25 | 169 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
170 |
||
| 515 | 171 |
|
172 |
||
| 25 | 173 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
174 |
\begin{frame}[c]
|
|
175 |
||
| 515 | 176 |
We proved partially |
|
133
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
177 |
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
178 |
\begin{center}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
179 |
\bl{$nullable(r)$} \;if and only if\; \bl{$[] \in L(r)$}
|
|
133
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
180 |
\end{center}
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
181 |
|
| 445 | 182 |
by induction on the regular expression \bl{$r$}.\bigskip\pause
|
|
134
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
183 |
|
|
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
184 |
\begin{center}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
185 |
{\huge\bf\alert{Any Questions?}}
|
|
134
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
186 |
\end{center}
|
|
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
187 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
188 |
\end{frame}
|
|
133
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
189 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
190 |
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
191 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
192 |
\begin{frame}[c]
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
193 |
|
| 25 | 194 |
We need to prove |
195 |
||
196 |
\begin{center}
|
|
|
133
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
197 |
\bl{$L(der\,c\,r) = Der\,c\,(L(r))$}
|
| 25 | 198 |
\end{center}
|
199 |
||
| 445 | 200 |
also by induction on the regular expression \bl{$r$}.
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
201 |
\end{frame}
|
| 25 | 202 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
203 |
||
204 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
205 |
\begin{frame}[c]
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
206 |
\frametitle{Proofs about Rexps}
|
| 25 | 207 |
|
208 |
\begin{itemize}
|
|
| 445 | 209 |
\item \bl{$P$} holds for \bl{$\ZERO$}, \bl{$\ONE$} and \bl{c}\bigskip
|
|
133
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
210 |
\item \bl{$P$} holds for \bl{$r_1 + r_2$} under the assumption that \bl{$P$} already
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
211 |
holds for \bl{$r_1$} and \bl{$r_2$}.\bigskip
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
212 |
\item \bl{$P$} holds for \bl{$r_1 \cdot r_2$} under the assumption that \bl{$P$} already
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
213 |
holds for \bl{$r_1$} and \bl{$r_2$}.\bigskip
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
214 |
\item \bl{$P$} holds for \bl{$r^*$} under the assumption that \bl{$P$} already
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
215 |
holds for \bl{$r$}.
|
| 25 | 216 |
\end{itemize}
|
217 |
||
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
218 |
\end{frame}
|
| 25 | 219 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
220 |
||
221 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
222 |
\begin{frame}[c]
|
|
|
133
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
223 |
\frametitle{\begin{tabular}{c}Proofs about Natural\\[-1mm] Numbers and Strings\end{tabular}}
|
| 25 | 224 |
|
225 |
\begin{itemize}
|
|
226 |
\item \bl{$P$} holds for \bl{$0$} and
|
|
227 |
\item \bl{$P$} holds for \bl{$n + 1$} under the assumption that \bl{$P$} already
|
|
228 |
holds for \bl{$n$}
|
|
229 |
\end{itemize}\bigskip
|
|
230 |
||
231 |
\begin{itemize}
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
232 |
\item \bl{$P$} holds for \bl{$[]$} and
|
| 25 | 233 |
\item \bl{$P$} holds for \bl{$c\!::\!s$} under the assumption that \bl{$P$} already
|
234 |
holds for \bl{$s$}
|
|
235 |
\end{itemize}
|
|
236 |
||
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
237 |
\end{frame}
|
| 25 | 238 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
239 |
||
240 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
241 |
\begin{frame}[t]
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
242 |
\frametitle{Regular Expressions}
|
| 25 | 243 |
|
244 |
\begin{center}
|
|
|
134
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
245 |
\begin{tabular}{@ {}rrl@ {\hspace{13mm}}l}
|
| 515 | 246 |
\bl{$r$} & \bl{$::=$} & \bl{$\ZERO$} & nothing\\
|
| 445 | 247 |
& \bl{$\mid$} & \bl{$\ONE$} & empty string / "" / $[]$\\
|
|
134
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
248 |
& \bl{$\mid$} & \bl{$c$} & character\\
|
|
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
249 |
& \bl{$\mid$} & \bl{$r_1 \cdot r_2$} & sequence\\
|
|
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
250 |
& \bl{$\mid$} & \bl{$r_1 + r_2$} & alternative / choice\\
|
|
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
251 |
& \bl{$\mid$} & \bl{$r^*$} & star (zero or more)\\
|
|
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
252 |
\end{tabular}
|
| 25 | 253 |
\end{center}
|
|
134
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
254 |
|
|
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
255 |
How about ranges \bl{$[a\mbox{-}z]$}, \bl{$r^+$} and \bl{$\sim{}r$}? Do they increase the
|
|
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
256 |
set of languages we can recognise? |
| 25 | 257 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
258 |
\end{frame}
|
| 25 | 259 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
260 |
||
261 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
262 |
\begin{frame}[c]
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
263 |
\frametitle{Negation of Regular Expr's}
|
| 25 | 264 |
|
265 |
\begin{itemize}
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
266 |
\item \bl{$\sim{}r$} \hspace{6mm} (everything that \bl{$r$} cannot recognise)\medskip
|
|
134
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
267 |
\item \bl{$L(\sim{}r) \dn UNIV - L(r)$}\medskip
|
|
136
cc19e6cbcb68
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
135
diff
changeset
|
268 |
\item \bl{$nullable (\sim{}r) \dn not\, (nullable(r))$}\medskip
|
|
134
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
269 |
\item \bl{$der\,c\,(\sim{}r) \dn \;\sim{}(der\,c\,r)$}
|
|
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
270 |
\end{itemize}\bigskip\pause
|
|
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
271 |
|
|
136
cc19e6cbcb68
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
135
diff
changeset
|
272 |
Used often for recognising comments: |
|
134
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
273 |
|
|
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
274 |
\[ |
|
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
275 |
\bl{/ \cdot * \cdot (\sim{}([a\mbox{-}z]^* \cdot * \cdot / \cdot [a\mbox{-}z]^*)) \cdot * \cdot /}
|
|
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
276 |
\] |
|
e3a8cf96f570
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
133
diff
changeset
|
277 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
278 |
\end{frame}
|
| 25 | 279 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
280 |
||
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
281 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
282 |
\begin{frame}[c]
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
283 |
\frametitle{Negation}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
284 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
285 |
Assume you have an alphabet consisting of the letters \bl{$a$}, \bl{$b$} and \bl{$c$} only.
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
286 |
Find a (basic!) regular expression that matches all strings \emph{\alert{except}}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
287 |
\bl{$ab$} and \bl{$ac$}!
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
288 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
289 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
290 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
291 |
|
| 25 | 292 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
293 |
\begin{frame}[c]
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
294 |
\frametitle{Automata}
|
| 25 | 295 |
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
296 |
A \alert{\bf deterministic finite automaton}, DFA, consists of:
|
| 25 | 297 |
|
298 |
\begin{itemize}
|
|
| 515 | 299 |
\item an alphabet \bl{$\varSigma$}
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
300 |
\item a set of states \bl{$Q$}
|
| 515 | 301 |
\item one of these states is the start state \bl{$\mbox{Q}_0$}
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
302 |
\item some states are accepting states \bl{$F$}, and
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
303 |
\item there is transition function \bl{$\delta$}\bigskip
|
| 25 | 304 |
|
305 |
\small |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
306 |
which takes a state as argument and a character and produces a |
| 515 | 307 |
new state; this function might not be everywhere defined $\Rightarrow$ partial function |
|
133
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
308 |
\end{itemize}
|
|
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
309 |
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
310 |
\begin{center}
|
| 515 | 311 |
\bl{$A(\varSigma, \mbox{Q}, \mbox{Q}_0, F, \delta)$}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
312 |
\end{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
313 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
314 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
315 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
316 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
317 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
318 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
319 |
\begin{frame}[c]
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
320 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
321 |
\begin{center}
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
322 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
323 |
every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 324 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
325 |
\node[state] (Q_1) [right=of Q_0] {$\mbox{Q}_1$};
|
|
326 |
\node[state] (Q_2) [below right=of Q_0] {$\mbox{Q}_2$};
|
|
327 |
\node[state] (Q_3) [right=of Q_2] {$\mbox{Q}_3$};
|
|
328 |
\node[state, accepting] (Q_4) [right=of Q_1] {$\mbox{Q}_4$};
|
|
329 |
\path[->] (Q_0) edge node [above] {\alert{$a$}} (Q_1);
|
|
330 |
\path[->] (Q_1) edge node [above] {\alert{$a$}} (Q_4);
|
|
331 |
\path[->] (Q_4) edge [loop right] node {\alert{$a, b$}} ();
|
|
332 |
\path[->] (Q_3) edge node [right] {\alert{$a$}} (Q_4);
|
|
333 |
\path[->] (Q_2) edge node [above] {\alert{$a$}} (Q_3);
|
|
334 |
\path[->] (Q_1) edge node [right] {\alert{$b$}} (Q_2);
|
|
335 |
\path[->] (Q_0) edge node [above] {\alert{$b$}} (Q_2);
|
|
336 |
\path[->] (Q_2) edge [loop left] node {\alert{$b$}} ();
|
|
337 |
\path[->] (Q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (Q_0);
|
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
338 |
\end{tikzpicture}
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
339 |
\end{center}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
340 |
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
341 |
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
342 |
\begin{itemize}
|
|
139
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
343 |
\item the start state can be an accepting state |
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
344 |
\item it is possible that there is no accepting state |
|
139
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
345 |
\item all states might be accepting (but this does not necessarily mean all strings are accepted) |
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
346 |
\end{itemize}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
347 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
348 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
349 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
350 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
351 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
352 |
\begin{frame}[c]
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
353 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
354 |
\begin{center}
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
355 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
356 |
every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 357 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
358 |
\node[state] (Q_1) [right=of Q_0] {$\mbox{Q}_1$};
|
|
359 |
\node[state] (Q_2) [below right=of Q_0] {$\mbox{Q}_2$};
|
|
360 |
\node[state] (Q_3) [right=of Q_2] {$\mbox{Q}_3$};
|
|
361 |
\node[state, accepting] (Q_4) [right=of Q_1] {$\mbox{Q}_4$};
|
|
362 |
\path[->] (Q_0) edge node [above] {\alert{$a$}} (Q_1);
|
|
363 |
\path[->] (Q_1) edge node [above] {\alert{$a$}} (Q_4);
|
|
364 |
\path[->] (Q_4) edge [loop right] node {\alert{$a, b$}} ();
|
|
365 |
\path[->] (Q_3) edge node [right] {\alert{$a$}} (Q_4);
|
|
366 |
\path[->] (Q_2) edge node [above] {\alert{$a$}} (Q_3);
|
|
367 |
\path[->] (Q_1) edge node [right] {\alert{$b$}} (Q_2);
|
|
368 |
\path[->] (Q_0) edge node [above] {\alert{$b$}} (Q_2);
|
|
369 |
\path[->] (Q_2) edge [loop left] node {\alert{$b$}} ();
|
|
370 |
\path[->] (Q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (Q_0);
|
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
371 |
\end{tikzpicture}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
372 |
\end{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
373 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
374 |
for this automaton \bl{$\delta$} is the function\\
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
375 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
376 |
\begin{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
377 |
\begin{tabular}{lll}
|
| 515 | 378 |
\bl{$(\mbox{Q}_0, a) \rightarrow \mbox{Q}_1$} & \bl{$(\mbox{Q}_1, a) \rightarrow \mbox{Q}_4$}
|
379 |
& \bl{$(\mbox{Q}_4, a) \rightarrow \mbox{Q}_4$}\\
|
|
380 |
\bl{$(\mbox{Q}_0, b) \rightarrow \mbox{Q}_2$} & \bl{$(\mbox{Q}_1, b) \rightarrow \mbox{Q}_2$} &
|
|
381 |
\bl{$(\mbox{Q}_4, b) \rightarrow \mbox{Q}_4$}\\
|
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
382 |
\end{tabular}\ldots
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
383 |
\end{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
384 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
385 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
386 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
387 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
388 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
389 |
\begin{frame}[t]
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
390 |
\frametitle{Accepting a String}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
391 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
392 |
Given |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
393 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
394 |
\begin{center}
|
| 515 | 395 |
\bl{$A(\varSigma, \mbox{Q}, \mbox{Q}_0, F, \delta)$}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
396 |
\end{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
397 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
398 |
you can define |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
399 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
400 |
\begin{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
401 |
\begin{tabular}{l}
|
| 515 | 402 |
\bl{$\widehat{\delta}(q, []) \dn q$}\\
|
403 |
\bl{$\widehat{\delta}(q, c::s) \dn \widehat{\delta}(\delta(q, c), s)$}\\
|
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
404 |
\end{tabular}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
405 |
\end{center}\pause
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
406 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
407 |
Whether a string \bl{$s$} is accepted by \bl{$A$}?
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
408 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
409 |
\begin{center}
|
| 517 | 410 |
\hspace{5mm}\bl{$\widehat{\delta}(\mbox{Q}_0, s) \in F$}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
411 |
\end{center}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
412 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
413 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
414 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
415 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
416 |
\begin{frame}[c]
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
417 |
\frametitle{Regular Languages}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
418 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
419 |
A \alert{\bf language} is a set of strings.\bigskip
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
420 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
421 |
A \alert{\bf regular expression} specifies a language.\bigskip
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
422 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
423 |
A language is \alert{\bf regular} iff there exists
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
424 |
a regular expression that recognises all its strings.\bigskip\bigskip\pause |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
425 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
426 |
not all languages are regular, e.g.~\bl{$a^nb^n$} is not
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
427 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
428 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
429 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
430 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
431 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
432 |
\frametitle{Regular Languages (2)}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
433 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
434 |
A language is \alert{\bf regular} iff there exists a regular
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
435 |
expression that recognises all its strings.\bigskip\medskip |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
436 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
437 |
or {\bf equivalently}\bigskip\medskip
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
438 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
439 |
A language is \alert{\bf regular} iff there exists a
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
440 |
deterministic finite automaton that recognises all its |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
441 |
strings. |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
442 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
443 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
444 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
445 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
446 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
447 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
448 |
\begin{frame}[c]
|
|
348
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
449 |
\frametitle{\begin{tabular}{c}
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
450 |
Non-Deterministic\\[-1mm] |
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
451 |
Finite Automata\end{tabular}}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
452 |
|
| 515 | 453 |
A non-deterministic finite automaton (NFA) consists again of: |
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
454 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
455 |
\begin{itemize}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
456 |
\item a finite set of states |
| 515 | 457 |
\item \underline{some} these states are the start states
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
458 |
\item some states are accepting states, and |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
459 |
\item there is transition \alert{relation}\medskip
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
460 |
\end{itemize}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
461 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
462 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
463 |
\begin{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
464 |
\begin{tabular}{c}
|
| 515 | 465 |
\bl{$(\mbox{Q}_1, a) \rightarrow \mbox{Q}_2$}\\
|
466 |
\bl{$(\mbox{Q}_1, a) \rightarrow \mbox{Q}_3$}\\
|
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
467 |
\end{tabular}
|
| 515 | 468 |
\bl{\ldots}
|
469 |
\hspace{10mm}\pause
|
|
470 |
\bl{$(\mbox{Q}_1, a) \rightarrow \{\mbox{Q}_2, \mbox{Q}_3\}$}
|
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
471 |
\end{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
472 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
473 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
474 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
475 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
476 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
477 |
\begin{frame}[c]
|
| 515 | 478 |
\frametitle{An NFA Example}
|
479 |
||
480 |
% A NFA for (ab* + b)*a |
|
481 |
\begin{center}
|
|
482 |
\begin{tikzpicture}[>=stealth',very thick, auto,
|
|
483 |
every state/.style={minimum size=0pt,inner sep=3pt,
|
|
484 |
draw=blue!50,very thick,fill=blue!20},scale=2] |
|
485 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
|
486 |
\node[state] (Q_1) [right=of Q_0] {$\mbox{Q}_1$};
|
|
487 |
\node[state, accepting] (Q_2) [right=of Q_1] {$\mbox{Q}_2$};
|
|
488 |
\path[->] (Q_0) edge [loop above] node {\alert{$b$}} ();
|
|
489 |
\path[<-] (Q_0) edge node [below] {\alert{$b$}} (Q_1);
|
|
490 |
\path[->] (Q_0) edge [bend left] node [above] {\alert{$a$}} (Q_1);
|
|
491 |
\path[->] (Q_0) edge [bend right] node [below] {\alert{$a$}} (Q_2);
|
|
492 |
\path[->] (Q_1) edge [loop above] node {\alert{$a,b$}} ();
|
|
493 |
\path[->] (Q_1) edge node [above] {\alert{$a$}} (Q_2);
|
|
494 |
\end{tikzpicture}
|
|
495 |
\end{center}
|
|
496 |
||
497 |
\end{frame}
|
|
498 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
499 |
||
500 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
501 |
\begin{frame}[c]
|
|
502 |
\frametitle{Two Epsilon NFA Examples}
|
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
503 |
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
504 |
\small |
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
505 |
\begin{center}
|
|
143
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
506 |
\begin{tabular}[t]{c@{\hspace{9mm}}c}
|
|
136
cc19e6cbcb68
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
135
diff
changeset
|
507 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
|
cc19e6cbcb68
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
135
diff
changeset
|
508 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
|
215
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
509 |
\node[state,initial] (q_0) {$q_0$};
|
|
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
510 |
\node[state] (q_1) [above=of q_0] {$q_1$};
|
|
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
511 |
\node[state, accepting] (q_2) [below=of q_0] {$q_2$};
|
|
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
512 |
\path[->] (q_0) edge node [left] {\alert{$\epsilon$}} (q_1);
|
|
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
513 |
\path[->] (q_0) edge node [left] {\alert{$\epsilon$}} (q_2);
|
|
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
514 |
\path[->] (q_0) edge [loop right] node {\alert{$a$}} ();
|
|
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
515 |
\path[->] (q_1) edge [loop above] node {\alert{$a$}} ();
|
|
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
516 |
\path[->] (q_2) edge [loop below] node {\alert{$b$}} ();
|
|
143
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
517 |
\end{tikzpicture} &
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
518 |
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
519 |
\raisebox{20mm}{
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
520 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
521 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
522 |
\node[state,initial] (r_1) {$r_1$};
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
523 |
\node[state] (r_2) [above=of r_1] {$r_2$};
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
524 |
\node[state, accepting] (r_3) [right=of r_1] {$r_3$};
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
525 |
\path[->] (r_1) edge node [below] {\alert{$b$}} (r_3);
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
526 |
\path[->] (r_2) edge [bend left] node [above] {\alert{$a$}} (r_3);
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
527 |
\path[->] (r_1) edge [bend left] node [left] {\alert{$\epsilon$}} (r_2);
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
528 |
\path[->] (r_2) edge [bend left] node [right] {\alert{$a$}} (r_1);
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
529 |
\end{tikzpicture}}
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
530 |
\end{tabular}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
531 |
\end{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
532 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
533 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
534 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
535 |
|
| 515 | 536 |
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
537 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
538 |
\begin{frame}[c]
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
539 |
\frametitle{Rexp to NFA}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
540 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
541 |
\begin{center}
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
542 |
\begin{tabular}[t]{l@{\hspace{10mm}}l}
|
| 445 | 543 |
\raisebox{1mm}{\bl{$\ZERO$}} &
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
544 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 545 |
\node[state, initial] (Q_0) {$\mbox{}$};
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
546 |
\end{tikzpicture}\\\\
|
| 445 | 547 |
\raisebox{1mm}{\bl{$\ONE$}} &
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
548 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 549 |
\node[state, initial, accepting] (Q_0) {$\mbox{}$};
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
550 |
\end{tikzpicture}\\\\
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
551 |
\raisebox{2mm}{\bl{$c$}} &
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
552 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 553 |
\node[state, initial] (Q_0) {$\mbox{}$};
|
554 |
\node[state, accepting] (Q_1) [right=of Q_0] {$\mbox{}$};
|
|
555 |
\path[->] (Q_0) edge node [below] {\alert{$c$}} (Q_1);
|
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
556 |
\end{tikzpicture}\\\\
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
557 |
\end{tabular}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
558 |
\end{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
559 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
560 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
561 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
562 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
563 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
564 |
\begin{frame}[t]
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
565 |
\frametitle{Case $r_1\cdot r_2$}
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
566 |
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
567 |
\mbox{}\bigskip
|
| 515 | 568 |
\onslide<1->{By recursion we are given two automata:\bigskip}
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
569 |
|
| 515 | 570 |
{\centering%
|
571 |
\only<1>{%
|
|
572 |
\begin{tikzpicture}[node distance=3mm,
|
|
573 |
>=stealth',very thick, |
|
574 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20}]
|
|
575 |
\node[state, initial] (Q_0) {$\mbox{}$};
|
|
576 |
\node[state, initial] (Q_01) [below=1mm of Q_0] {$\mbox{}$};
|
|
577 |
\node[state, initial] (Q_02) [above=1mm of Q_0] {$\mbox{}$};
|
|
578 |
\node (R_1) [right=of Q_0] {$\ldots$};
|
|
579 |
\node[state, accepting] (T_1) [right=of R_1] {$\mbox{}$};
|
|
580 |
\node[state, accepting] (T_2) [above=of T_1] {$\mbox{}$};
|
|
581 |
\node[state, accepting] (T_3) [below=of T_1] {$\mbox{}$};
|
|
582 |
||
583 |
\node (A_0) [right=2.5cm of T_1] {$\mbox{}$};
|
|
584 |
\node[state, initial] (A_01) [above=1mm of A_0] {$\mbox{}$};
|
|
585 |
\node[state, initial] (A_02) [below=1mm of A_0] {$\mbox{}$};
|
|
586 |
||
587 |
\node (b_1) [right=of A_0] {$\ldots$};
|
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
588 |
\node[state, accepting] (c_1) [right=of b_1] {$\mbox{}$};
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
589 |
\node[state, accepting] (c_2) [above=of c_1] {$\mbox{}$};
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
590 |
\node[state, accepting] (c_3) [below=of c_1] {$\mbox{}$};
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
591 |
\begin{pgfonlayer}{background}
|
| 515 | 592 |
\node (1) [rounded corners, inner sep=1mm, thick, |
593 |
draw=black!60, fill=black!20, fit= (Q_0) (R_1) (T_1) (T_2) (T_3)] {};
|
|
594 |
\node (2) [rounded corners, inner sep=1mm, thick, |
|
595 |
draw=black!60, fill=black!20, fit= (A_0) (b_1) (c_1) (c_2) (c_3)] {};
|
|
596 |
\node [yshift=2mm] at (1.north) {$r_1$};
|
|
597 |
\node [yshift=2mm] at (2.north) {$r_2$};
|
|
|
143
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
598 |
\end{pgfonlayer}
|
| 515 | 599 |
\end{tikzpicture}}
|
600 |
\only<2>{%
|
|
601 |
\begin{tikzpicture}[node distance=3mm,
|
|
602 |
>=stealth',very thick, |
|
603 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
|
604 |
\node[state, initial] (Q_0) {$\mbox{}$};
|
|
605 |
\node[state, initial] (Q_01) [below=1mm of Q_0] {$\mbox{}$};
|
|
606 |
\node[state, initial] (Q_02) [above=1mm of Q_0] {$\mbox{}$};
|
|
607 |
\node (r_1) [right=of Q_0] {$\ldots$};
|
|
608 |
\node[state] (t_1) [right=of r_1] {$\mbox{}$};
|
|
609 |
\node[state] (t_2) [above=of t_1] {$\mbox{}$};
|
|
610 |
\node[state] (t_3) [below=of t_1] {$\mbox{}$};
|
|
611 |
||
612 |
\node (A_0) [right=2.5cm of t_1] {$\mbox{}$};
|
|
613 |
\node[state] (A_01) [above=1mm of A_0] {$\mbox{}$};
|
|
614 |
\node[state] (A_02) [below=1mm of A_0] {$\mbox{}$};
|
|
615 |
||
616 |
\node (b_1) [right=of A_0] {$\ldots$};
|
|
617 |
\node[state, accepting] (c_1) [right=of b_1] {$\mbox{}$};
|
|
618 |
\node[state, accepting] (c_2) [above=of c_1] {$\mbox{}$};
|
|
619 |
\node[state, accepting] (c_3) [below=of c_1] {$\mbox{}$};
|
|
620 |
\path[->] (t_1) edge (A_01); |
|
621 |
\path[->] (t_2) edge node [above] {$\epsilon$s} (A_01);
|
|
622 |
\path[->] (t_3) edge (A_01); |
|
623 |
\path[->] (t_1) edge (A_02); |
|
624 |
\path[->] (t_2) edge (A_02); |
|
625 |
\path[->] (t_3) edge node [below] {$\epsilon$s} (A_02);
|
|
626 |
||
627 |
\begin{pgfonlayer}{background}
|
|
628 |
\node (3) [rounded corners, inner sep=1mm, thick, |
|
629 |
draw=black!60, fill=black!20, fit= (Q_0) (c_1) (c_2) (c_3)] {};
|
|
630 |
\node [yshift=2mm] at (3.north) {$r_1\cdot r_2$};
|
|
631 |
\end{pgfonlayer}
|
|
632 |
\end{tikzpicture}}
|
|
633 |
% |
|
634 |
}\bigskip\bigskip |
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
635 |
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
636 |
\small |
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
637 |
We need to (1) change the accepting nodes of the first automaton into non-accepting nodes, and (2) connect them |
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
638 |
via $\epsilon$-transitions to the starting state of the second automaton. |
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
639 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
640 |
\end{frame}
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
641 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
642 |
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
643 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
644 |
\begin{frame}[t]
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
645 |
\frametitle{Case $r_1+ r_2$}
|
| 515 | 646 |
\mbox{}\\[-8mm]
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
647 |
|
| 515 | 648 |
\onslide<1->{By recursion we are given two automata:\smallskip}
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
649 |
|
| 515 | 650 |
\hspace{2cm}
|
651 |
\only<1>{%
|
|
652 |
\begin{tikzpicture}[node distance=3mm,
|
|
653 |
>=stealth',very thick, |
|
654 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},
|
|
655 |
baseline=(current bounding box.center)] |
|
656 |
\node at (0,0) (1) {$\mbox{}$};
|
|
657 |
\node (2) [above=14mm of 1] {};
|
|
658 |
\node[state, initial] (4) [above=1mm of 2] {$\mbox{}$};
|
|
659 |
\node[state, initial] (5) [below=1mm of 2] {$\mbox{}$};
|
|
660 |
\node[state, initial] (3) [below=10mm of 1] {$\mbox{}$};
|
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
661 |
|
| 515 | 662 |
\node (a) [right=of 2] {$\ldots\,$};
|
663 |
\node (a1) [right=of a] {$$};
|
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
664 |
\node[state, accepting] (a2) [above=of a1] {$\mbox{}$};
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
665 |
\node[state, accepting] (a3) [below=of a1] {$\mbox{}$};
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
666 |
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
667 |
\node (b) [right=of 3] {$\ldots$};
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
668 |
\node[state, accepting] (b1) [right=of b] {$\mbox{}$};
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
669 |
\node[state, accepting] (b2) [above=of b1] {$\mbox{}$};
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
670 |
\node[state, accepting] (b3) [below=of b1] {$\mbox{}$};
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
671 |
\begin{pgfonlayer}{background}
|
| 515 | 672 |
\node (1) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (2) (a1) (a2) (a3)] {};
|
673 |
\node (2) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (3) (b1) (b2) (b3)] {};
|
|
674 |
\node [yshift=3mm] at (1.north) {$r_1$};
|
|
675 |
\node [yshift=3mm] at (2.north) {$r_2$};
|
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
676 |
\end{pgfonlayer}
|
| 515 | 677 |
\end{tikzpicture}}
|
678 |
\only<2>{%
|
|
679 |
\begin{tikzpicture}[node distance=3mm,
|
|
680 |
>=stealth',very thick, |
|
681 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},
|
|
682 |
baseline=(current bounding box.center)] |
|
683 |
\node at (0,0) (1) {$\mbox{}$};
|
|
684 |
\node (2) [above=14mm of 1] {$$};
|
|
685 |
\node[state, initial] (4) [above=1mm of 2] {$\mbox{}$};
|
|
686 |
\node[state, initial] (5) [below=1mm of 2] {$\mbox{}$};
|
|
687 |
\node[state, initial] (3) [below=10mm of 1] {$\mbox{}$};
|
|
688 |
||
689 |
\node (a) [right=of 2] {$\ldots\,$};
|
|
690 |
\node (a1) [right=of a] {$$};
|
|
691 |
\node[state, accepting] (a2) [above=of a1] {$\mbox{}$};
|
|
692 |
\node[state, accepting] (a3) [below=of a1] {$\mbox{}$};
|
|
693 |
||
694 |
\node (b) [right=of 3] {$\ldots$};
|
|
695 |
\node[state, accepting] (b1) [right=of b] {$\mbox{}$};
|
|
696 |
\node[state, accepting] (b2) [above=of b1] {$\mbox{}$};
|
|
697 |
\node[state, accepting] (b3) [below=of b1] {$\mbox{}$};
|
|
698 |
||
699 |
%\path[->] (1) edge node [above] {$\epsilon$} (2);
|
|
700 |
%\path[->] (1) edge node [below] {$\epsilon$} (3);
|
|
701 |
||
702 |
\begin{pgfonlayer}{background}
|
|
703 |
\node (3) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (1) (a2) (a3) (b2) (b3)] {};
|
|
704 |
\node [yshift=3mm] at (3.north) {$r_1+ r_2$};
|
|
705 |
\end{pgfonlayer}
|
|
706 |
\end{tikzpicture}}
|
|
707 |
% |
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
708 |
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
709 |
\small |
| 515 | 710 |
\mbox{}\\\medskip
|
| 517 | 711 |
We can just put both automata together. |
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
712 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
713 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
714 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
715 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
716 |
\begin{frame}[c]
|
|
138
b98a0a49c432
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
137
diff
changeset
|
717 |
\frametitle{Case $r^*$}
|
|
b98a0a49c432
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
137
diff
changeset
|
718 |
|
| 515 | 719 |
\onslide<1->{By recursion we are given an automaton for $r$:\smallskip}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
720 |
|
| 515 | 721 |
\hspace{2cm}
|
722 |
\only<1>{\begin{tikzpicture}[node distance=3mm,
|
|
723 |
>=stealth',very thick, |
|
724 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},
|
|
725 |
baseline=(current bounding box.north)] |
|
726 |
\node (2) {$\mbox{}$};
|
|
727 |
\node[state, initial] (4) [above=1mm of 2] {$\mbox{}$};
|
|
728 |
\node[state, initial] (5) [below=1mm of 2] {$\mbox{}$};
|
|
|
138
b98a0a49c432
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
137
diff
changeset
|
729 |
\node (a) [right=of 2] {$\ldots$};
|
|
b98a0a49c432
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
137
diff
changeset
|
730 |
\node[state, accepting] (a1) [right=of a] {$\mbox{}$};
|
|
b98a0a49c432
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
137
diff
changeset
|
731 |
\node[state, accepting] (a2) [above=of a1] {$\mbox{}$};
|
| 515 | 732 |
\node[state, accepting] (a3) [below=of a1] {$\mbox{}$};
|
733 |
\begin{pgfonlayer}{background}
|
|
734 |
\node (1) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (2) (a1) (a2) (a3)] {};
|
|
735 |
\node [yshift=3mm] at (1.north) {$r$};
|
|
736 |
\end{pgfonlayer}
|
|
737 |
\end{tikzpicture}}
|
|
738 |
\only<2->{\begin{tikzpicture}[node distance=3mm,
|
|
739 |
>=stealth',very thick, |
|
740 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},
|
|
741 |
baseline=(current bounding box.north)] |
|
742 |
\node at (0,0) [state, initial,accepting] (1) {$\mbox{}$};
|
|
743 |
\node (2) [right=16mm of 1] {$\mbox{}$};
|
|
744 |
\node[state] (4) [above=1mm of 2] {$\mbox{}$};
|
|
745 |
\node[state] (5) [below=1mm of 2] {$\mbox{}$};
|
|
746 |
\node (a) [right=of 2] {$\ldots$};
|
|
|
138
b98a0a49c432
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
137
diff
changeset
|
747 |
\node[state] (a1) [right=of a] {$\mbox{}$};
|
|
b98a0a49c432
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
137
diff
changeset
|
748 |
\node[state] (a2) [above=of a1] {$\mbox{}$};
|
| 515 | 749 |
\node[state] (a3) [below=of a1] {$\mbox{}$};
|
750 |
\path[->] (1) edge node [below] {$\epsilon$} (4);
|
|
751 |
\path[->] (1) edge node [below] {$\epsilon$} (5);
|
|
752 |
\path[->] (a1) edge [bend left=45] node [below] {$\epsilon$} (1);
|
|
753 |
\path[->] (a2) edge [bend right] node [below] {$\epsilon$} (1);
|
|
754 |
\path[->] (a3) edge [bend left=45] node [below] {$\epsilon$} (1);
|
|
|
138
b98a0a49c432
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
137
diff
changeset
|
755 |
\begin{pgfonlayer}{background}
|
| 515 | 756 |
\node (2) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (1) (a2) (a3)] {};
|
757 |
\node [yshift=3mm] at (2.north) {$r^*$};
|
|
|
138
b98a0a49c432
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
137
diff
changeset
|
758 |
\end{pgfonlayer}
|
| 515 | 759 |
\end{tikzpicture}}
|
760 |
\bigskip |
|
|
138
b98a0a49c432
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
137
diff
changeset
|
761 |
|
|
139
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
762 |
\onslide<3->{
|
|
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
763 |
Why can't we just have an epsilon transition from the accepting states to the starting state?} |
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
764 |
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
765 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
766 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
767 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
768 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
769 |
\begin{frame}[c]
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
770 |
\frametitle{Subset Construction}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
771 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
772 |
|
|
136
cc19e6cbcb68
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
135
diff
changeset
|
773 |
\begin{textblock}{5}(1,1)
|
|
cc19e6cbcb68
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
135
diff
changeset
|
774 |
\begin{center}
|
| 515 | 775 |
\begin{tikzpicture}[node distance=6mm,scale=0.7,>=stealth',very thick,
|
|
136
cc19e6cbcb68
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
135
diff
changeset
|
776 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 777 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
778 |
\node[state] (Q_1) [above=of Q_0] {$\mbox{Q}_1$};
|
|
779 |
\node[state, accepting] (Q_2) [below=of Q_0] {$\mbox{Q}_2$};
|
|
780 |
\path[->] (Q_0) edge node [left] {\alert{$\epsilon$}} (Q_1);
|
|
781 |
\path[->] (Q_0) edge node [left] {\alert{$\epsilon$}} (Q_2);
|
|
782 |
\path[->] (Q_0) edge [loop right] node {\alert{$a$}} ();
|
|
783 |
\path[->] (Q_1) edge [loop above] node {\alert{$a$}} ();
|
|
784 |
\path[->] (Q_2) edge [loop below] node {\alert{$b$}} ();
|
|
|
136
cc19e6cbcb68
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
135
diff
changeset
|
785 |
\end{tikzpicture}
|
|
cc19e6cbcb68
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
135
diff
changeset
|
786 |
\end{center}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
787 |
\end{textblock}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
788 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
789 |
\begin{textblock}{11}(6.5,4.5)
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
790 |
\begin{tabular}{r|cl}
|
|
139
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
791 |
nodes \textcolor{white}{*} & $a$ & $b$\\
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
792 |
\hline |
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
793 |
$\{\}$ \textcolor{white}{*} & \onslide<2->{$\{\}$} & \onslide<2->{$\{\}$}\\
|
|
139
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
794 |
$\{0\}$ \textcolor{white}{*} & \onslide<3->{$\{0,1,2\}$} & \onslide<3->{$\{2\}$}\\
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
795 |
$\{1\}$ \textcolor{white}{*} & \onslide<3->{$\{1\}$} & \onslide<3->{$\{\}$}\\
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
796 |
$\{2\}$ \onslide<5->{*} & \onslide<3->{$\{\}$} & \onslide<3->{$\{2\}$}\\
|
|
139
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
797 |
$\{0,1\}$ \textcolor{white}{*} &\onslide<4->{$\{0,1,2\}$} & \onslide<4->{$\{2\}$}\\
|
|
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
798 |
$\{0,2\}$ \onslide<5->{*} &\onslide<4->{$\{0,1,2\}$} & \onslide<4->{$\{2\}$}\\
|
|
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
799 |
$\{1,2\}$ \onslide<5->{*} & \onslide<4->{$\{1\}$} & \onslide<4->{$\{2\}$}\\
|
|
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
800 |
\onslide<5->{s:} $\{0,1,2\}$ \onslide<5->{*}& \onslide<4->{$\{0,1,2\}$} & \onslide<4->{$\{2\}$}\\
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
801 |
\end{tabular}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
802 |
\end{textblock}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
803 |
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
804 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
805 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
806 |
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
807 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
808 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
809 |
\frametitle{The Result}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
810 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
811 |
\footnotesize |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
812 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
813 |
\begin{tikzpicture}[scale=0.5,>=stealth',very thick,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
814 |
every state/.style={minimum size=0pt,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
815 |
draw=blue!50,very thick,fill=blue!20}, |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
816 |
baseline=0mm] |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
817 |
\node[state,initial,accepting] (q012) {$\{0,1,2\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
818 |
\node[state,accepting] (q02) [right=of q012] {$\{0,2\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
819 |
\node[state] (q01) [above=of q02] {$\{0,1\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
820 |
\node[state,accepting] (q12) [below=of q02] {$\{1,2\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
821 |
\node[state] (q0) [right=2cm of q01] {$\{0\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
822 |
\node[state] (q1) [right=2.5cm of q02] {$\{1\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
823 |
\node[state,accepting] (q2) [right=1.5cm of q12] {$\{2\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
824 |
\node[state] (qn) [right=of q1] {$\{\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
825 |
|
|
348
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
826 |
\path[->] (q012) edge [loop below] node {\alert{$a$}} ();
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
827 |
\path[->] (q012) edge node [above] {\alert{$b$}} (q2);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
828 |
\path[->] (q12) edge [bend left] node [below,pos=0.4] {\alert{$a$}} (q1);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
829 |
\path[->] (q12) edge node [below] {\alert{$b$}} (q2);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
830 |
\path[->] (q02) edge node [above] {\alert{$a$}} (q012);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
831 |
\path[->] (q02) edge [bend left] node [above, pos=0.8] {\alert{$b$}} (q2);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
832 |
\path[->] (q01) edge node [below] {\alert{$a$}} (q012);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
833 |
\path[->] (q01) edge [bend left] node [above] {\alert{$b$}} (q2);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
834 |
\path[->] (q0) edge node [below] {\alert{$a$}} (q012);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
835 |
\path[->] (q0) edge node [right, pos=0.2] {\alert{$b$}} (q2);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
836 |
\path[->] (q1) edge [loop above] node {\alert{$a$}} ();
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
837 |
\path[->] (q1) edge node [above] {\alert{$b$}} (qn);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
838 |
\path[->] (q2) edge [loop right] node {\alert{$b$}} ();
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
839 |
\path[->] (q2) edge node [below] {\alert{$a$}} (qn);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
840 |
\path[->] (qn) edge [loop above] node {$\alert{a},\alert{b}$} ();
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
841 |
\end{tikzpicture}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
842 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
843 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
844 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
845 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
846 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
847 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
848 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
849 |
\begin{frame}[c]
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
850 |
\frametitle{Removing Dead States}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
851 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
852 |
\footnotesize |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
853 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
854 |
\begin{tabular}{ll}
|
| 445 | 855 |
DFA: & (original) NFA:\\ |
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
856 |
\raisebox{10mm}{%
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
857 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
858 |
every state/.style={minimum size=0pt,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
859 |
draw=blue!50,very thick,fill=blue!20}] |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
860 |
\node[state,initial,accepting] (q012) {$\{0,1,2\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
861 |
\node[state,accepting] (q2) [right=of q012] {$\{2\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
862 |
\node[state] (qn) [right=of q2] {$\{\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
863 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
864 |
\path[->] (q012) edge [loop below] node {\alert{$a$}} ();
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
865 |
\path[->] (q012) edge node [above] {\alert{$b$}} (q2);
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
866 |
\path[->] (q2) edge [loop below] node {\alert{$b$}} ();
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
867 |
\path[->] (q2) edge node [below] {\alert{$a$}} (qn);
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
868 |
\path[->] (qn) edge [loop above] node |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
869 |
{$\alert{a},\alert{b}$} ();
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
870 |
\end{tikzpicture}}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
871 |
& |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
872 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
873 |
every state/.style={minimum size=0pt,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
874 |
draw=blue!50,very thick,fill=blue!20},] |
| 515 | 875 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
876 |
\node[state] (Q_1) [above=of Q_0] {$\mbox{Q}_1$};
|
|
877 |
\node[state, accepting] (Q_2) [below=of Q_0] {$\mbox{Q}_2$};
|
|
878 |
\path[->] (Q_0) edge node [left] {\alert{$\epsilon$}} (Q_1);
|
|
879 |
\path[->] (Q_0) edge node [left] {\alert{$\epsilon$}} (Q_2);
|
|
880 |
\path[->] (Q_0) edge [loop right] node {\alert{$a$}} ();
|
|
881 |
\path[->] (Q_1) edge [loop above] node {\alert{$a$}} ();
|
|
882 |
\path[->] (Q_2) edge [loop below] node {\alert{$b$}} ();
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
883 |
\end{tikzpicture}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
884 |
\end{tabular}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
885 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
886 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
887 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
888 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
889 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
890 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
891 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
892 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
893 |
\begin{frame}[c]
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
894 |
\frametitle{Regexps and Automata}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
895 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
896 |
\begin{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
897 |
\begin{tikzpicture}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
898 |
\node (rexp) {\bl{\bf Regexps}};
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
899 |
\node (nfa) [right=of rexp] {\bl{\bf NFAs}};
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
900 |
\node (dfa) [right=of nfa] {\bl{\bf DFAs}};
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
901 |
\onslide<2->{\node (mdfa) [right=of dfa] {\bl{\bf \begin{tabular}{c}minimal\\ DFAs\end{tabular}}};}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
902 |
\path[->, red, line width=2mm] (rexp) edge node [above=4mm, black] {\begin{tabular}{c@{\hspace{9mm}}}Thompson's\\[-1mm] construction\end{tabular}} (nfa);
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
903 |
\path[->, red, line width=2mm] (nfa) edge node [above=4mm, black] {\begin{tabular}{c}subset\\[-1mm] construction\end{tabular}}(dfa);
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
904 |
\onslide<2->{\path[->, red, line width=2mm] (dfa) edge node [below=9mm, black] {minimisation} (mdfa);}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
905 |
\end{tikzpicture}\\
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
906 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
907 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
908 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
909 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
910 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
911 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
912 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
913 |
\frametitle{DFA Minimisation}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
914 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
915 |
\begin{enumerate}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
916 |
\item Take all pairs \bl{$(q, p)$} with \bl{$q \not= p$}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
917 |
\item Mark all pairs that accepting and non-accepting states |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
918 |
\item For all unmarked pairs \bl{$(q, p)$} and all characters \bl{$c$} test whether
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
919 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
920 |
\bl{$(\delta(q, c), \delta(p,c))$}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
921 |
\end{center}
|
|
348
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
922 |
are marked. If yes in at least one case, then also mark \bl{$(q, p)$}.
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
923 |
\item Repeat last step until no change. |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
924 |
\item All unmarked pairs can be merged. |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
925 |
\end{enumerate}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
926 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
927 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
928 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
929 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
930 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
931 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
932 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
933 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
934 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
935 |
every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 936 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
937 |
\node[state] (Q_1) [right=of Q_0] {$\mbox{Q}_1$};
|
|
938 |
\node[state] (Q_2) [below right=of Q_0] {$\mbox{Q}_2$};
|
|
939 |
\node[state] (Q_3) [right=of Q_2] {$\mbox{Q}_3$};
|
|
940 |
\node[state, accepting] (Q_4) [right=of Q_1] {$\mbox{Q}_4$};
|
|
941 |
\path[->] (Q_0) edge node [above] {\alert{$a$}} (Q_1);
|
|
942 |
\path[->] (Q_1) edge node [above] {\alert{$a$}} (Q_4);
|
|
943 |
\path[->] (Q_4) edge [loop right] node {\alert{$a, b$}} ();
|
|
944 |
\path[->] (Q_3) edge node [right] {\alert{$a$}} (Q_4);
|
|
945 |
\path[->] (Q_2) edge node [above] {\alert{$a$}} (Q_3);
|
|
946 |
\path[->] (Q_1) edge node [right] {\alert{$b$}} (Q_2);
|
|
947 |
\path[->] (Q_0) edge node [above] {\alert{$b$}} (Q_2);
|
|
948 |
\path[->] (Q_2) edge [loop left] node {\alert{$b$}} ();
|
|
949 |
\path[->] (Q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (Q_0);
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
950 |
\end{tikzpicture}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
951 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
952 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
953 |
\mbox{}\\[-20mm]\mbox{}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
954 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
955 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
956 |
\begin{tikzpicture}[scale=0.8,line width=0.8mm]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
957 |
\draw (0,0) -- (4,0); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
958 |
\draw (0,1) -- (4,1); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
959 |
\draw (0,2) -- (3,2); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
960 |
\draw (0,3) -- (2,3); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
961 |
\draw (0,4) -- (1,4); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
962 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
963 |
\draw (0,0) -- (0, 4); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
964 |
\draw (1,0) -- (1, 4); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
965 |
\draw (2,0) -- (2, 3); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
966 |
\draw (3,0) -- (3, 2); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
967 |
\draw (4,0) -- (4, 1); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
968 |
|
| 515 | 969 |
\draw (0.5,-0.5) node {$\mbox{Q}_0$};
|
970 |
\draw (1.5,-0.5) node {$\mbox{Q}_1$};
|
|
971 |
\draw (2.5,-0.5) node {$\mbox{Q}_2$};
|
|
972 |
\draw (3.5,-0.5) node {$\mbox{Q}_3$};
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
973 |
|
| 515 | 974 |
\draw (-0.5, 3.5) node {$\mbox{Q}_1$};
|
975 |
\draw (-0.5, 2.5) node {$\mbox{Q}_2$};
|
|
976 |
\draw (-0.5, 1.5) node {$\mbox{Q}_3$};
|
|
977 |
\draw (-0.5, 0.5) node {$\mbox{Q}_4$};
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
978 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
979 |
\draw (0.5,0.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
980 |
\draw (1.5,0.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
981 |
\draw (2.5,0.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
982 |
\draw (3.5,0.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
983 |
\end{tikzpicture}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
984 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
985 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
986 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
987 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
988 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
989 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
990 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
991 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
992 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
993 |
\begin{tabular}{@{\hspace{-8mm}}cc@{}}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
994 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
995 |
every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 996 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
997 |
\node[state] (Q_1) [right=of Q_0] {$\mbox{Q}_1$};
|
|
998 |
\node[state] (Q_2) [below right=of Q_0] {$\mbox{Q}_2$};
|
|
999 |
\node[state] (Q_3) [right=of Q_2] {$\mbox{Q}_3$};
|
|
1000 |
\node[state, accepting] (Q_4) [right=of Q_1] {$\mbox{Q}_4$};
|
|
1001 |
\path[->] (Q_0) edge node [above] {\alert{$a$}} (Q_1);
|
|
1002 |
\path[->] (Q_1) edge node [above] {\alert{$a$}} (Q_4);
|
|
1003 |
\path[->] (Q_4) edge [loop right] node {\alert{$a, b$}} ();
|
|
1004 |
\path[->] (Q_3) edge node [right] {\alert{$a$}} (Q_4);
|
|
1005 |
\path[->] (Q_2) edge node [above] {\alert{$a$}} (Q_3);
|
|
1006 |
\path[->] (Q_1) edge node [right] {\alert{$b$}} (Q_2);
|
|
1007 |
\path[->] (Q_0) edge node [above] {\alert{$b$}} (Q_2);
|
|
1008 |
\path[->] (Q_2) edge [loop left] node {\alert{$b$}} ();
|
|
1009 |
\path[->] (Q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (Q_0);
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1010 |
\end{tikzpicture}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1011 |
& |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1012 |
\raisebox{9mm}{\begin{tikzpicture}[scale=0.6,line width=0.8mm]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1013 |
\draw (0,0) -- (4,0); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1014 |
\draw (0,1) -- (4,1); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1015 |
\draw (0,2) -- (3,2); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1016 |
\draw (0,3) -- (2,3); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1017 |
\draw (0,4) -- (1,4); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1018 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1019 |
\draw (0,0) -- (0, 4); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1020 |
\draw (1,0) -- (1, 4); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1021 |
\draw (2,0) -- (2, 3); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1022 |
\draw (3,0) -- (3, 2); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1023 |
\draw (4,0) -- (4, 1); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1024 |
|
| 515 | 1025 |
\draw (0.5,-0.5) node {$\mbox{Q}_0$};
|
1026 |
\draw (1.5,-0.5) node {$\mbox{Q}_1$};
|
|
1027 |
\draw (2.5,-0.5) node {$\mbox{Q}_2$};
|
|
1028 |
\draw (3.5,-0.5) node {$\mbox{Q}_3$};
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1029 |
|
| 515 | 1030 |
\draw (-0.5, 3.5) node {$\mbox{Q}_1$};
|
1031 |
\draw (-0.5, 2.5) node {$\mbox{Q}_2$};
|
|
1032 |
\draw (-0.5, 1.5) node {$\mbox{Q}_3$};
|
|
1033 |
\draw (-0.5, 0.5) node {$\mbox{Q}_4$};
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1034 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1035 |
\draw (0.5,0.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1036 |
\draw (1.5,0.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1037 |
\draw (2.5,0.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1038 |
\draw (3.5,0.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1039 |
\draw (0.5,1.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1040 |
\draw (2.5,1.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1041 |
\draw (0.5,3.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1042 |
\draw (1.5,2.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1043 |
\end{tikzpicture}}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1044 |
\end{tabular}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1045 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1046 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1047 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1048 |
\mbox{}\\[-20mm]\mbox{}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1049 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1050 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1051 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1052 |
every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 1053 |
\node[state,initial] (Q_02) {$\mbox{Q}_{0, 2}$};
|
1054 |
\node[state] (Q_13) [right=of Q_02] {$\mbox{Q}_{1, 3}$};
|
|
1055 |
\node[state, accepting] (Q_4) [right=of Q_13] {$\mbox{Q}_{4\phantom{,0}}$};
|
|
1056 |
\path[->] (Q_02) edge [bend left] node [above] {\alert{$a$}} (Q_13);
|
|
1057 |
\path[->] (Q_13) edge [bend left] node [below] {\alert{$b$}} (Q_02);
|
|
1058 |
\path[->] (Q_02) edge [loop below] node {\alert{$b$}} ();
|
|
1059 |
\path[->] (Q_13) edge node [above] {\alert{$a$}} (Q_4);
|
|
1060 |
\path[->] (Q_4) edge [loop above] node {\alert{$a, b$}} ();
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1061 |
\end{tikzpicture}\\
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1062 |
minimal automaton |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1063 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1064 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1065 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1066 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1067 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1068 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1069 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1070 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1071 |
\frametitle{Alternatives}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1072 |
\mbox{}\\[-17mm]\mbox{}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1073 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1074 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1075 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1076 |
every state/.style={minimum size=0pt,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1077 |
inner sep=2pt,draw=blue!50,very thick,fill=blue!20}] |
| 515 | 1078 |
\only<1>{\node[state,initial] (Q_0) {$\mbox{Q}_0$};}
|
1079 |
\only<2->{\node[state,accepting] (Q_0) {$\mbox{Q}_0$};}
|
|
1080 |
\node[state] (Q_1) [right=of Q_0] {$\mbox{Q}_1$};
|
|
1081 |
\node[state] (Q_2) [below right=of Q_0] {$\mbox{Q}_2$};
|
|
1082 |
\node[state] (Q_3) [right=of Q_2] {$\mbox{Q}_3$};
|
|
1083 |
\only<1>{\node[state, accepting] (Q_4) [right=of Q_1] {$\mbox{Q}_4$};}
|
|
1084 |
\only<2->{\node[state, initial right] (Q_4) [right=of Q_1] {$\mbox{Q}_4$};}
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1085 |
\only<1-2>{
|
| 515 | 1086 |
\path[->] (Q_0) edge node [above] {\alert{$a$}} (Q_1);
|
1087 |
\path[->] (Q_1) edge node [above] {\alert{$a$}} (Q_4);
|
|
1088 |
\path[->] (Q_4) edge [loop above] node {\alert{$a, b$}} ();
|
|
1089 |
\path[->] (Q_3) edge node [right] {\alert{$a$}} (Q_4);
|
|
1090 |
\path[->] (Q_2) edge node [above] {\alert{$a$}} (Q_3);
|
|
1091 |
\path[->] (Q_1) edge node [right] {\alert{$b$}} (Q_2);
|
|
1092 |
\path[->] (Q_0) edge node [above] {\alert{$b$}} (Q_2);
|
|
1093 |
\path[->] (Q_2) edge [loop left] node {\alert{$b$}} ();
|
|
1094 |
\path[->] (Q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (Q_0);}
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1095 |
\only<3->{
|
| 515 | 1096 |
\path[<-] (Q_0) edge node [above] {\alert{$a$}} (Q_1);
|
1097 |
\path[<-] (Q_1) edge node [above] {\alert{$a$}} (Q_4);
|
|
1098 |
\path[<-] (Q_4) edge [loop above] node {\alert{$a, b$}} ();
|
|
1099 |
\path[<-] (Q_3) edge node [right] {\alert{$a$}} (Q_4);
|
|
1100 |
\path[<-] (Q_2) edge node [above] {\alert{$a$}} (Q_3);
|
|
1101 |
\path[<-] (Q_1) edge node [right] {\alert{$b$}} (Q_2);
|
|
1102 |
\path[<-] (Q_0) edge node [above] {\alert{$b$}} (Q_2);
|
|
1103 |
\path[<-] (Q_2) edge [loop left] node {\alert{$b$}} ();
|
|
1104 |
\path[<-] (Q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (Q_0);}
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1105 |
\end{tikzpicture}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1106 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1107 |
\mbox{}\\[-18mm]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1108 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1109 |
\small |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1110 |
\begin{itemize}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1111 |
\item<2-> exchange initial / accepting states\\[-2mm] |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1112 |
\item<3-> reverse all edges\\[-2mm] |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1113 |
\item<4-> subset construction $\Rightarrow$ DFA\\[-2mm] |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1114 |
\item<5-> remove dead states\\[-2mm] |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1115 |
\item<6-> repeat once more |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1116 |
\onslide<6->{$\Rightarrow$ minimal DFA}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1117 |
\end{itemize}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1118 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1119 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1120 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1121 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1122 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1123 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1124 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1125 |
\frametitle{Regexps and Automata}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1126 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1127 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1128 |
\begin{tikzpicture}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1129 |
\node (rexp) {\bl{\bf Regexps}};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1130 |
\node (nfa) [right=of rexp] {\bl{\bf NFAs}};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1131 |
\node (dfa) [right=of nfa] {\bl{\bf DFAs}};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1132 |
\onslide<1->{\node (mdfa) [right=of dfa] {\bl{\bf \begin{tabular}{c}minimal\\ DFAs\end{tabular}}};}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1133 |
\path[->, red, line width=2mm] (rexp) edge node [above=4mm, black] {\begin{tabular}{c@{\hspace{9mm}}}Thompson's\\[-1mm] construction\end{tabular}} (nfa);
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1134 |
\path[->, red, line width=2mm] (nfa) edge node [above=4mm, black] {\begin{tabular}{c}subset\\[-1mm] construction\end{tabular}}(dfa);
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1135 |
\onslide<1->{\path[->, red, line width=2mm] (dfa) edge node [below=9mm, black] {minimisation} (mdfa);}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1136 |
\onslide<2->{\path[->, red, line width=2mm] (dfa) edge [bend left=45] (rexp);}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1137 |
\end{tikzpicture}\\
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1138 |
\end{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1139 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1140 |
\end{frame}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1141 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1142 |
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1143 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1144 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1145 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1146 |
\begin{frame}<2>[c]
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1147 |
\frametitle{DFA to Rexp}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1148 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1149 |
\begin{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1150 |
\begin{tikzpicture}[scale=2,>=stealth',very thick,
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1151 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 1152 |
\only<1>{\node[state, initial] (q0) at ( 0,1) {$\mbox{Q}_0$};}
|
1153 |
\only<2->{\node[state, initial,accepting] (q0) at ( 0,1) {$\mbox{Q}_0$};}
|
|
1154 |
\only<1>{\node[state] (q1) at ( 1,1) {$\mbox{Q}_1$};}
|
|
1155 |
\only<2->{\node[state,accepting] (q1) at ( 1,1) {$\mbox{Q}_1$};}
|
|
1156 |
\only<1>{\node[state, accepting] (q2) at ( 2,1) {$\mbox{Q}_2$};}
|
|
1157 |
\only<2->{\node[state] (q2) at ( 2,1) {$\mbox{Q}_2$};}
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1158 |
\path[->] (q0) edge[bend left] node[above] {\alert{$a$}} (q1)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1159 |
(q1) edge[bend left] node[above] {\alert{$b$}} (q0)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1160 |
(q2) edge[bend left=50] node[below] {\alert{$b$}} (q0)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1161 |
(q1) edge node[above] {\alert{$a$}} (q2)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1162 |
(q2) edge [loop right] node {\alert{$a$}} ()
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1163 |
(q0) edge [loop below] node {\alert{$b$}} ()
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1164 |
; |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1165 |
\end{tikzpicture}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1166 |
\end{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1167 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1168 |
\onslide<3>{How to get from a DFA to a regular expression?}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1169 |
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1170 |
\end{frame}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1171 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1172 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1173 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1174 |
\begin{frame}[c]
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1175 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1176 |
\begin{center}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1177 |
\begin{tikzpicture}[scale=2,>=stealth',very thick,
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1178 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 1179 |
\only<1->{\node[state, initial] (q0) at ( 0,1) {$\mbox{Q}_0$};}
|
1180 |
\only<1->{\node[state] (q1) at ( 1,1) {$\mbox{Q}_1$};}
|
|
1181 |
\only<1->{\node[state] (q2) at ( 2,1) {$\mbox{Q}_2$};}
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1182 |
\path[->] (q0) edge[bend left] node[above] {\alert{$a$}} (q1)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1183 |
(q1) edge[bend left] node[above] {\alert{$b$}} (q0)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1184 |
(q2) edge[bend left=50] node[below] {\alert{$b$}} (q0)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1185 |
(q1) edge node[above] {\alert{$a$}} (q2)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1186 |
(q2) edge [loop right] node {\alert{$a$}} ()
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1187 |
(q0) edge [loop below] node {\alert{$b$}} ()
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1188 |
; |
|
139
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
1189 |
\end{tikzpicture}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1190 |
\end{center}\pause\bigskip
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1191 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1192 |
\onslide<2->{
|
| 445 | 1193 |
You know how to solve since school days, no? |
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1194 |
\begin{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1195 |
\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
|
| 515 | 1196 |
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$2\, \mbox{Q}_0 + 3 \,\mbox{Q}_1 + 4\, \mbox{Q}_2$}\\
|
1197 |
\bl{$\mbox{Q}_1$} & \bl{$=$} & \bl{$2 \,\mbox{Q}_0 + 3\, \mbox{Q}_1 + 1\, \mbox{Q}_2$}\\
|
|
1198 |
\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$1\, \mbox{Q}_0 + 5\, \mbox{Q}_1 + 2\, \mbox{Q}_2$}\\
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1199 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1200 |
\end{tabular}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1201 |
\end{center}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1202 |
} |
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1203 |
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1204 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1205 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1206 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1207 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1208 |
\begin{frame}[c]
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1209 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1210 |
\begin{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1211 |
\begin{tikzpicture}[scale=2,>=stealth',very thick,
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1212 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 1213 |
\only<1->{\node[state, initial] (q0) at ( 0,1) {$\mbox{Q}_0$};}
|
1214 |
\only<1->{\node[state] (q1) at ( 1,1) {$\mbox{Q}_1$};}
|
|
1215 |
\only<1->{\node[state] (q2) at ( 2,1) {$\mbox{Q}_2$};}
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1216 |
\path[->] (q0) edge[bend left] node[above] {\alert{$a$}} (q1)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1217 |
(q1) edge[bend left] node[above] {\alert{$b$}} (q0)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1218 |
(q2) edge[bend left=50] node[below] {\alert{$b$}} (q0)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1219 |
(q1) edge node[above] {\alert{$a$}} (q2)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1220 |
(q2) edge [loop right] node {\alert{$a$}} ()
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1221 |
(q0) edge [loop below] node {\alert{$b$}} ()
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1222 |
; |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1223 |
\end{tikzpicture}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1224 |
\end{center}\bigskip
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1225 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1226 |
\onslide<2->{
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1227 |
\begin{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1228 |
\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
|
| 515 | 1229 |
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\ONE + \mbox{Q}_0\,b + \mbox{Q}_1\,b + \mbox{Q}_2\,b$}\\
|
1230 |
\bl{$\mbox{Q}_1$} & \bl{$=$} & \bl{$\mbox{Q}_0\,a$}\\
|
|
1231 |
\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$\mbox{Q}_1\,a + \mbox{Q}_2\,a$}\\
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1232 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1233 |
\end{tabular}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1234 |
\end{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1235 |
} |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1236 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1237 |
\onslide<3->{
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1238 |
Arden's Lemma: |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1239 |
\begin{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1240 |
If \bl{$q = q\,r + s$}\; then\; \bl{$q = s\, r^*$}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1241 |
\end{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1242 |
} |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1243 |
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1244 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1245 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1246 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1247 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1248 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1249 |
\frametitle{Regexps and Automata}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1250 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1251 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1252 |
\begin{tikzpicture}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1253 |
\node (rexp) {\bl{\bf Regexps}};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1254 |
\node (nfa) [right=of rexp] {\bl{\bf NFAs}};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1255 |
\node (dfa) [right=of nfa] {\bl{\bf DFAs}};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1256 |
\onslide<1->{\node (mdfa) [right=of dfa] {\bl{\bf \begin{tabular}{c}minimal\\ DFAs\end{tabular}}};}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1257 |
\path[->, red, line width=2mm] (rexp) edge node [above=4mm, black] {\begin{tabular}{c@{\hspace{9mm}}}Thompson's\\[-1mm] construction\end{tabular}} (nfa);
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1258 |
\path[->, red, line width=2mm] (nfa) edge node [above=4mm, black] {\begin{tabular}{c}subset\\[-1mm] construction\end{tabular}}(dfa);
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1259 |
\onslide<1->{\path[->, red, line width=2mm] (dfa) edge node [below=9mm, black] {minimisation} (mdfa);}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1260 |
\onslide<1->{\path[->, red, line width=2mm] (dfa) edge [bend left=45] (rexp);}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1261 |
\end{tikzpicture}\\
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1262 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1263 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1264 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1265 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1266 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1267 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1268 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1269 |
\frametitle{Regular Languages (3)}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1270 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1271 |
A language is \alert{\bf regular} iff there exists a regular
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1272 |
expression that recognises all its strings.\bigskip\medskip |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1273 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1274 |
or {\bf equivalently}\bigskip\medskip
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1275 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1276 |
A language is \alert{\bf regular} iff there exists a
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1277 |
deterministic finite automaton that recognises all its |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1278 |
strings.\bigskip\medskip\pause |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1279 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1280 |
Why is every finite set of strings a regular language? |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1281 |
\end{frame}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1282 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1283 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1284 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1285 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1286 |
\begin{frame}[c]
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1287 |
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1288 |
Given the function |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1289 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1290 |
\begin{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1291 |
\bl{\begin{tabular}{r@{\hspace{1mm}}c@{\hspace{1mm}}l}
|
| 445 | 1292 |
$rev(\ZERO)$ & $\dn$ & $\ZERO$\\ |
1293 |
$rev(\ONE)$ & $\dn$ & $\ONE$\\ |
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1294 |
$rev(c)$ & $\dn$ & $c$\\ |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1295 |
$rev(r_1 + r_2)$ & $\dn$ & $rev(r_1) + rev(r_2)$\\ |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1296 |
$rev(r_1 \cdot r_2)$ & $\dn$ & $rev(r_2) \cdot rev(r_1)$\\ |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1297 |
$rev(r^*)$ & $\dn$ & $rev(r)^*$\\ |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1298 |
\end{tabular}}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1299 |
\end{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1300 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1301 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1302 |
and the set |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1303 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1304 |
\begin{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1305 |
\bl{$Rev\,A \dn \{s^{-1} \;|\; s \in A\}$}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1306 |
\end{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1307 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1308 |
prove whether |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1309 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1310 |
\begin{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1311 |
\bl{$L(rev(r)) = Rev (L(r))$}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1312 |
\end{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1313 |
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1314 |
\end{frame}
|
|
133
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
1315 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 25 | 1316 |
|
1317 |
||
1318 |
\end{document}
|
|
1319 |
||
1320 |
%%% Local Variables: |
|
1321 |
%%% mode: latex |
|
1322 |
%%% TeX-master: t |
|
1323 |
%%% End: |
|
1324 |