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Mark 2023-08-07 08:53:27 -07:00
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Advanced/DFAs/parts/0 DFA.tex
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@ -41,7 +41,7 @@ $A$ will go through the states $a - b - c - b - b$ while processing this string.
\problem{} \problem{}
Which of the following strings are accepted by $A$? \\ Which of the following strings are accepted by $A$? \par
\begin{itemize} \begin{itemize}
\item \texttt{1} \item \texttt{1}
\item \texttt{1010} \item \texttt{1010}
@ -99,7 +99,7 @@ It starts in the state $s$ and has two accepting states $a_1$ and $b_1$.
\end{center} \end{center}
\problem{} \problem{}
Which of the following strings are accepted by $B$: Which of the following strings are accepted by $B$?
\begin{itemize} \begin{itemize}
\item \texttt{aa} \item \texttt{aa}
\item \texttt{abba} \item \texttt{abba}
@ -171,7 +171,7 @@ For example, $\{\texttt{0}, \texttt{1}\}^*$ is the set $\{\varepsilon, \texttt{0
Note that this set contains the empty string. Note that this set contains the empty string.
\definition{} \definition{}
A \textit{language} over an alphabet $Q$ is a subset of $Q^*$. \\ A \textit{language} over an alphabet $Q$ is a subset of $Q^*$. \par
For example, the language \say{strings of length 2} over $\{\texttt{0}, \texttt{1}\}$ is $\{\texttt{00}, \texttt{01}, \texttt{10}, \texttt{11}\}$ For example, the language \say{strings of length 2} over $\{\texttt{0}, \texttt{1}\}$ is $\{\texttt{00}, \texttt{01}, \texttt{10}, \texttt{11}\}$
\definition{} \definition{}
@ -432,7 +432,7 @@ Draw a DFA over an alphabet $\{\texttt{a}, \texttt{b}, \texttt{@}, \texttt{.}\}$
\pagebreak \pagebreak
\problem{} \problem{}
Draw a state diagram for a DFA over an alphabet of your choice that accepts exactly $f(n)$ strings of length $n$ if \\ Draw a state diagram for a DFA over an alphabet of your choice that accepts exactly $f(n)$ strings of length $n$ if \par
\begin{itemize} \begin{itemize}
\item $f(n) = n$ \item $f(n) = n$
\item $f(n) = n+1$ \item $f(n) = n+1$
@ -441,7 +441,7 @@ Draw a state diagram for a DFA over an alphabet of your choice that accepts exac
\item $f(n)$ is a Tribonacci number. \par \item $f(n)$ is a Tribonacci number. \par
Tribonacci numbers are defined by the sequence $f(0) = 0$, $f(1) = 1$, $f(2) = 1$, Tribonacci numbers are defined by the sequence $f(0) = 0$, $f(1) = 1$, $f(2) = 1$,
and $f(n) = f(n-1)+f(n-2)+f(n-3)$ for $n \ge 3$ \par and $f(n) = f(n-1)+f(n-2)+f(n-3)$ for $n \ge 3$ \par
\hint{Fibonacci numbers are given by the automaton prohibiting two \texttt{'a'}s in a row.} \hint{Fibonacci numbers are given by the automaton prohibiting two \texttt{`a'}s in a row.}
\end{itemize} \end{itemize}