MA0301/exercise12/main.tex

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\documentclass[12pt]{article}
\usepackage{ntnu}
\usepackage{ntnu-math}
\author{Øystein Tveit}
\title{MA0301 Exercise 12}
\usetikzlibrary{automata, positioning, arrows.meta}
\begin{document}
\ntnuTitle{}
\break{}
\begin{excs}
\exc{}
\[ r = \{a,b\}^* a \{a,b\}^* a \{a,b\}^* a \{a,b\}^* \]
\exc{}
\begin{subexcs}
\subexc{}
\[ \{ab\} \{ab\}^* \]
\subexc{}
\[ a (a | \lambda) b (b | \lambda)\]
\end{subexcs}
\exc{}
\[ M = (Q, \Sigma, \delta, s, F) \]
\begin{align*}
Q &= \{ s_0, s_1, s_2 \} \\
\Sigma &= \{a, b\} \\
\delta &= \begin{Bmatrix}
s_0 \xrightarrow{b} s_1, \\
s_0 \xrightarrow{a} s_2, \\
s_1 \xrightarrow{a,b} s_1, \\
s_2 \xrightarrow{a} s_1, \\
s_2 \xrightarrow{b} s_2
\end{Bmatrix} \\
s &= s_0 \\
F &= \{ s_2 \}
\end{align*}
\includeDiagram[scale=1.6, width=10cm]{diagrams/ex3.tex}
\exc{}
The words $L$ can be described by the regular expression $r$ where
\[ r = a^* b b^* a \{a,b\}^* \]
\exc{}
The words in $L$ can be described by the regular expression $r$ where
\[ r = (a^* b)^3 \{ (a^* b)^4 \} \]
\exc{}
\begin{subexcs}
\subexc{}
\begin{align*}
s_0 &\xrightarrow{a, 0} s_0 \\
s_0 &\xrightarrow{a, 0} s_0 \\
s_0 &\xrightarrow{b, 1} s_3 \\
s_3 &\xrightarrow{b, 0} s_3 \\
s_3 &\xrightarrow{c, 1} s_0 \\
s_0 &\xrightarrow{c, 1} s_2
\end{align*}
The output would be $001011$
\subexc{}
\includeDiagram[scale=1.2, width=13cm]{diagrams/ex6_b.tex}
\end{subexcs}
\exc{}
\begin{subexcs}
\subexc{}
Suppose we have $a \in A, b \in B$
\begin{align*}
AB^* &= \{a, ab, ab^2, ab^3, \ldots \} \\
&= \{a\} \cup \{ab, ab^2, ab^3, \ldots \} \\
&= A \cup \{ab, ab^2, ab^3, \ldots \} \\
&\Rightarrow A \subseteq AB^*
\end{align*}
\qed
\subexc{}
Since $A \subseteq B$, we can rewrite $B$ as $A \cup \overline{A}$ where $\overline{A} = \{b \mid b \in B, b \notin A \}$
\begin{align*}
B^* &= (A \cup \overline{A})^* \\
&= A^* \cap \overline{A}^* \cap B_1, \qquad B_1 = \{(B^*\ a\ B^*\ a_1\ B^*) \vee (B^*\ a_1\ B^*\ a\ B^*) \mid a \in A, a_1 \in \overline{A}\} \\
&\Rightarrow A^* \subseteq B^*
\end{align*}
\qed
\end{subexcs}
\end{excs}
\end{document}