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Cleaned up DFA handout

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2023-05-19 17:03:20 -07:00
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Advanced/DFAs/parts/0 DFA.tex
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@ -111,7 +111,7 @@ Which of the following strings are accepted by $B$:
\problem{}
\problem{}<SameStartAndEnd>
Describe the strings accepted by $B$.
\begin{solution}
@ -122,38 +122,7 @@ Describe the strings accepted by $B$.
\pagebreak
\definition{}
An \textit{alphabet} is a finite set of symbols. \par
\definition{}
A \textit{string} over an alphabet $Q$ is a finite sequence of symbols from $Q$. \par
We denote the empty string $\varepsilon$. \par
\vspace{2mm}
$Q^*$ is the set of all possible strings over $Q$. \par
For example, $\{\texttt{0}, \texttt{1}\}^*$ is the set $\{\varepsilon, \texttt{0}, \texttt{1}, \texttt{00}, \texttt{01}, \texttt{10}, \texttt{11}, \texttt{000},... \}$ \par
Note that this set contains the empty string.
\definition{}
A \textit{language} over an alphabet $Q$ is a subset of $Q^*$. \\
For example, the language \say{strings of length 2} over $\{\texttt{0}, \texttt{1}\}$ is $\{\texttt{00}, \texttt{01}, \texttt{10}, \texttt{11}\}$
\definition{}
We say a language $L$ is \textit{recognized} by a DFA if that DFA accepts a string $w$ if and only if $w \in L$.
%\begin{remark}
%A machine, such as DFA or Turing machine, may accept several strings, but it always recognizes only one language. If the machine %accepts no strings, it still recognizes one language — namely, the empty language $\emptyset$.
%\end{remark}
\vspace{8mm}
\problem{}
\problem{}<fibonacci>
How many strings of length $n$ are accepted by the automaton $C$?
\begin{center}
@ -186,6 +155,28 @@ How many strings of length $n$ are accepted by the automaton $C$?
%\end{remark}
\vfill
\definition{}
An \textit{alphabet} is a finite set of symbols. \par
\definition{}
A \textit{string} over an alphabet $Q$ is a finite sequence of symbols from $Q$. \par
We denote the empty string $\varepsilon$. \par
\vspace{2mm}
$Q^*$ is the set of all possible strings over $Q$. \par
For example, $\{\texttt{0}, \texttt{1}\}^*$ is the set $\{\varepsilon, \texttt{0}, \texttt{1}, \texttt{00}, \texttt{01}, \texttt{10}, \texttt{11}, \texttt{000},... \}$ \par
Note that this set contains the empty string.
\definition{}
A \textit{language} over an alphabet $Q$ is a subset of $Q^*$. \\
For example, the language \say{strings of length 2} over $\{\texttt{0}, \texttt{1}\}$ is $\{\texttt{00}, \texttt{01}, \texttt{10}, \texttt{11}\}$
\definition{}
We say a language $L$ is \textit{recognized} by a DFA if that DFA accepts a string $w$ if and only if $w \in L$.
\pagebreak
\problem{}