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