Minor edits

Advanced/DFAs/parts/0 DFA.tex

@@ -121,9 +121,33 @@ Describe the strings accepted by $B$.
 \vfill
 \pagebreak
 
+Before we continue, let's properly define all the words we've been using in the problems above.
+
+\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'll denote the empty string $\varepsilon$. \par
+
+
+\definition{}
+$Q^*$ 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 $Q^*$. \par
+For example, the language \say{strings of length 2} over $\{\texttt{0}, \texttt{1}\}$ is $\{\texttt{00}, \texttt{01}, \texttt{10}, \texttt{11}\}$
+
+\definition{}
+The language \textit{recognized} by a DFA is the set of strings that the DFA accepts.
+
+\vspace{5mm}
 
 \problem{}<fibonacci>
-How many strings of length $n$ are accepted by the automaton $C$?
+How many strings of length $n$ are accepted by the automaton below? \par
+\hint{Induction.}
 
 \begin{center}
 \begin{tikzpicture}
@@ -150,33 +174,9 @@ How many strings of length $n$ are accepted by the automaton $C$?
 	Computing initial conditions, we see that $A_n$ is an $n+2$-th Fibonacci number.
 \end{solution}
 
-%\begin{remark}
-%Note that all the states in our DFAs $A$, $B$ and $C$ from figures 1, 2, 3 have outgoing symbols for each letter of the alphabet. %Do the same for your DFAs.
-%\end{remark}
-
 \vfill
 
-\definition{}
+\vfill
-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 $Q^*$. \par
-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{}
@@ -193,7 +193,6 @@ Draw DFAs that recognize the following languages. In all parts, the alphabet is
 \begin{solution}
 	$\{w~ | ~w~ \text{begins with a \texttt{1} and ends with a \texttt{0}}\}$
 
-
 	\begin{center}
 	\begin{tikzpicture}
 		\begin{scope}[layer = nodes]
@@ -428,6 +427,13 @@ Draw a DFA over an alphabet $\{\texttt{a}, \texttt{b}, \texttt{@}, \texttt{.}\}$
 
 \end{solution}
 
+\vfill
+
+%\problem{}
+%Construct a DFA that accepts a binary integer iff it is divisible by two.
+%\vfill
+
+
 \vfill
 \pagebreak
 
@@ -441,7 +447,7 @@ Draw a state diagram for a DFA over an alphabet of your choice that accepts exac
 	\item $f(n)$ is a Tribonacci number. \par
 	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
-	\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 letters \say{\texttt{a}} in a row.}
 \end{itemize}
 
 
@@ -485,7 +491,7 @@ Draw a state diagram for a DFA over an alphabet of your choice that accepts exac
 
 	\textbf{Part 5:} Tribonacci numbers \par
 	Using the hint, we get the following automaton. \par
-	It rejects all strings with three \texttt{'a'}s in a row.
+	It rejects all strings with three \texttt{a}s in a row.
 
 	\begin{center}
 	\begin{tikzpicture}
@@ -510,7 +516,7 @@ Draw a state diagram for a DFA over an alphabet of your choice that accepts exac
 	\end{tikzpicture}
 	\end{center}
 
-	This automaton rejects all strings with three \texttt{'a'}s in a row. If we count accepted strings, we get the Tribonacci numbers with an offset: $f(0) = 1$, $f(1) = 2$, $f(2)=4$, ... \par
+	This automaton rejects all strings with three \texttt{a}s in a row. If we count accepted strings, we get the Tribonacci numbers with an offset: $f(0) = 1$, $f(1) = 2$, $f(2)=4$, ... \par
 
 	\pagebreak
 
@@ -626,5 +632,18 @@ Draw a state diagram for a DFA over an alphabet of your choice that accepts exac
 
 \end{solution}
 
+\vfill
+\pagebreak
+
+\problem{}<divsix>
+Construct a DFA that accepts an binary integer if and only if it is divisible by six. \par
+Strings are read from most to least significant digit.
+(that is, 18 will be read as 1,0,0,1,0)
+
+\vfill
+
+\problem{}
+Construct a DFA that satisfies \ref{divsix}, but reads digits in the opposite order.
+
 \vfill
 \pagebreak