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\section*{Homework 4}

\begin{enumerate}
\item Assume you have an alphabet consisting of the letters $a$, $b$ and $c$ only.
(a) Find a regular expression that recognises the two strings $ab$ and $ac$. (b)
Find a regular expression that matches all strings \emph{except} these two strings.
Note, you can only use regular expressions of the form 
\begin{center}
$r ::= \varnothing \;|\; \epsilon \;|\; c  \;|\; r_1 + r_2  \;|\; r_1 \cdot r_2 \;|\; r^*$
\end{center}

\item Define the function $zeroable$ which takes a regular expression as argument
and returns a boolean.\footnote{In an earlier version there was an error.} The 
function should satisfy the following property:
\begin{center}
$zeroable(r)$ \;if and only if\; $L(r) = \varnothing$
\end{center}

\item Define the tokens and regular expressions for a language
consisting of numbers, left-parenthesis (, right-parenthesis ),
identifiers and the operations $+$, $-$ and $*$. Can the following strings 
in this language be lexed?

\begin{itemize}
\item \texttt{"}$(a + 3) * b$\texttt{"}
\item \texttt{"}$)()++ -33$\texttt{"}
\item \texttt{"}$(a / 3) * 3$\texttt{"}
\end{itemize}

\item (Optional) The tokenizer in \texttt{regexp3.scala} takes as
argument a string and a list of rules. The result is a list of tokens. Improve this tokenizer so 
that it filters out all comments and whitespace from the result.

\item (Optional) Modify the tokenizer in \texttt{regexp2.scala} so that it
implements the \texttt{findAll} function. This function takes a regular
expressions and a string, and returns all substrings in this string that 
match the regular expression.
\end{enumerate}



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