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% explain what is a context-free grammar and the language it generates 

%

\section*{Homework 5}

\HEADER

\begin{enumerate}

\item Consider the basic regular expressions

\begin{center}

$r ::= \ZERO \;|\; \ONE \;|\; c  \;|\; r_1 + r_2  \;|\; r_1 \cdot r_2 \;|\; r^*$

\end{center}

      and suppose you want to show a property $P(r)$ for all

      regular expressions $r$ by structural induction. Write

      down which cases do you need to analyse. State clearly

      the induction hypotheses if applicable in a case.

\item Define a regular expression, written $ALL$, that can

      match every string. This definition should be in terms

      of the following extended regular expressions:

\begin{center}

$r ::= \ZERO \;|\; 

       \ONE \;|\;  

       c  \;|\; 

       r_1 + r_2 \;|\; 

       r_1 \cdot r_2 \;|\; 

       r^* \;|\;

       \sim r$

\end{center}

%\item Assume the delimiters for comments are \texttt{$\slash$*}

%and \texttt{*$\slash$}. Give a regular expression that can

%recognise comments of the form

%

%\begin{center}

%\texttt{$\slash$*~\ldots{}~*$\slash$} 

%\end{center}

%

%where the three dots stand for arbitrary characters, but not

%comment delimiters.

\item Define the following regular expressions 

\begin{center}

\begin{tabular}{ll}

$r^+$ & (one or more matches)\\

$r^?$   & (zero or one match)\\

$r^{\{n\}}$ & (exactly $n$ matches)\\

$r^{\{m, n\}}$ & (at least $m$ and maximal $n$ matches, with the\\

&  \phantom{(}assumption $m \le n$)\\

\end{tabular}

\end{center}

in terms of the usual basic regular expressions

\begin{center}

$r ::= \ZERO \;|\; \ONE \;|\; c  \;|\; r_1 + r_2  \;|\; r_1 \cdot r_2 \;|\; r^*$

\end{center}

\item Give the regular expressions for lexing a language

      consisting of identifiers, left-parenthesis \texttt{(},

      right-parenthesis \texttt{)}, numbers that can be either

      positive or negative, and the operations \texttt{+},

      \texttt{-} and \texttt{*}. 

      Decide whether the following strings can be lexed in

      this language?

\begin{enumerate}

\item \texttt{"(a3+3)*b"}

\item \texttt{")()++-33"}

\item \texttt{"(b42/3)*3"}

\end{enumerate}

In case they can, give the corresponding token sequences. (Hint: 

Observe the maximal munch rule and the priorities of your regular

expressions that make the process of lexing unambiguous.)

\item Suppose the following context-free grammar $G$

\begin{plstx}[margin=1cm]

  : \meta{S\/} ::= \meta{A\/}\cdot\meta{S\/}\cdot\meta{B\/} \;\mid\;

                   \meta{B\/}\cdot\meta{S\/}\cdot\meta{A\/} \;\mid\; \epsilon\\

  : \meta{A\/} ::= a \mid \epsilon\\

  : \meta{B\/} ::= b\\

\end{plstx}

where the starting symbol is $\meta{S}$.

Which of the following strings are in the language of $G$?

\begin{itemize}

\item[$\bullet$] $a$

\item[$\bullet$] $b$

\item[$\bullet$] $ab$

\item[$\bullet$] $ba$

\item[$\bullet$] $bb$

\item[$\bullet$] $baa$

\end{itemize}

\item Suppose the following context-free grammar 

  \begin{plstx}[margin=1cm]

  : \meta{S\/} ::= a\cdot \meta{S\/}\cdot a\;\mid\;

                   b\cdot \meta{S\/}\cdot b\;\mid\; \epsilon\\

  \end{plstx}

Describe which language is generated by this grammar.

\item Remember we have specified identifiers with regular expressions as

  strings that start with a letter followed by letters, digits and

  underscores. This can also be specified by a grammar rule or rules.

  What would the rule(s) look like for identifiers? 

  \solution{

  \begin{plstx}[margin=1cm]

  : \meta{Id\/} ::= \meta{Let\/}\cdot \meta{R}\\

  : \meta{Let\/} ::= a \;\mid\; \dots \;\mid\; z\\

  : \meta{Dig\/} ::= 0 \;\mid\; \dots \;\mid\; 9\\

  : \meta{R\/} ::= \meta{Let\/} \cdot \meta{R\/} \;\mid\;

                  \meta{Dig\/} \cdot \meta{R\/} \;\mid\;

                  $\_$ \cdot \meta{R\/} \;\mid\; \epsilon\\

  \end{plstx}  

}

\item If we specify keywords, identifiers (see above) and programs

  by grammar rules, are there any problems you need to be careful

  about when using a parser for identifying tokens?

  \solution{Parsers do not have the POSIX rules (e.g.~longest munch

    rule) built in. I am not aware that any parser does this out of

    the box and you would need to build in such constraints into the

    grammar rules or parsing mechanism.}

\item {\bf(Optional)} Recall the definitions for $Der$ and $der$

      from the lectures. Prove by induction on $r$ the

      property that 

\[

L(der\,c\,r) = Der\,c\,(L(r))

\]

      holds.

\item \POSTSCRIPT        

\end{enumerate}

\end{document}

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