--- a/hws/hw06.tex	Sat Nov 04 18:28:09 2023 +0000
+++ b/hws/hw06.tex	Sat Nov 11 10:08:33 2023 +0000
@@ -1,6 +1,7 @@
 \documentclass{article}
 \usepackage{../style}
 \usepackage{../graphics}
+\usepackage{../grammar}
 
 \begin{document}
 
@@ -28,20 +29,38 @@
 
 \item Suppose the grammar
 
-\begin{center}
-\begin{tabular}{lcl}
-$E$ & $\rightarrow$ &  $F \;|\; F \cdot * \cdot F \;|\; F \cdot \backslash \cdot F$\\
-$F$ & $\rightarrow$ & $T \;|\; T \cdot \texttt{+} \cdot T \;|\; T \cdot \texttt{-} \cdot T$\\
-$T$ & $\rightarrow$ & $num \;|\; \texttt{(} \cdot E \cdot \texttt{)}$\\
-\end{tabular}
-\end{center}
+\begin{plstx}[margin=1cm]    
+:\meta{E} ::= \meta{F} \;|\; \meta{F} \cdot \texttt{*} \cdot \meta{F} \;|\; \meta{F} \cdot \backslash \cdot \meta{F}\\
+:\meta{F} ::= \meta{T} \;|\; \meta{T} \cdot \texttt{+} \cdot \meta{T} \;|\; \meta{T} \cdot \texttt{-} \cdot \meta{T}\\
+:\meta{T} ::= $num$ \;|\; \texttt{(} \cdot E \cdot \texttt{)}\\
+\end{plstx}
 
-where $E$, $F$ and $T$ are non-terminals, $E$ is the starting symbol of the grammar, and $num$ stands for
-a number token. Give a parse tree for the string \texttt{(3+3)+(2*3)}.
+where \meta{E}, \meta{F} and \meta{T} are non-terminals, \meta{E} is
+the starting symbol of the grammar, and $num$ stands for a number
+token. Give a parse tree for the string \texttt{(3+3)+(2*3)}. Note that
+\meta{F} and \meta{T} are ``exchanged'' in this grammar in comparison with
+the usual grammar for arithmetic expressions. What does this mean in terms
+of precedences of the operators?
+
+\solution{
+  For the second part "1+2*3" will be parsed as (1+2)*3, meaning + and - bind
+  tighter than * and $\backslash$
+}
 
 \item Define what it means for a grammar to be ambiguous. Give an example of
 an ambiguous grammar.
 
+\solution{
+  Already the grammar
+  \begin{plstx}[margin=1cm]
+    : \meta{E} ::= \meta{E}\cdot +\cdot\meta{E} | num\\
+  \end{plstx}
+
+  is ambiguous because a string like "1+2+3" can be parsed
+  as (1+2)+3 or 1+(2+3).
+  }
+
+
 \item Suppose boolean expressions are built up from
 
 \begin{center}
@@ -61,17 +80,34 @@
 
 \item Parsing combinators consist of atomic parsers, alternative
   parsers, sequence parsers and semantic actions.  What is the purpose
-  of (1) atomic parsers and of (2) semantic actions?
+  of (1) atomic parsers and of (2) map-parsers (also called semantic actions)?
 
+  \solution{
+    Atomic parsers look at a concrete prefix of the input, like num-tokens or identifiers.
+    Map-parsers apply a function to the output of a parser. In this way you can transform
+    the output from, for example, a string to an integer.
+  }
+
+  
 \item Parser combinators can directly be given a string as
       input, without the need of a lexer. What are the
       advantages to first lex a string and then feed a
       sequence of tokens as input to the parser?
 
-% Reason 1 you can filter out whitespaces and comments, which makes the grammar rules simpler. If you have to make sure that a whitespace comes after a variable say, then your parser rule for variables  gets more complicated. Same with comments which do not contribute anything to the parser tree.
-% Reason 2) The lexer can already classify tokens, for example as numbers, keywords or identifiers. This again makes the grammar rules more deterministic and as a result faster to parse.
-% The point is that parser combinators can be used to process strings, but in case of compilers where whitespaces and comments need to be filtered out, the lexing phase is actually quite useful.
+      \solution{ Reason 1 you can filter out whitespaces and comments,
+        which makes the grammar rules simpler. If you have to make
+        sure that a whitespace comes after a variable say, then your
+        parser rule for variables gets more complicated. Same with
+        comments which do not contribute anything to the parse tree.
+        
+        Reason 2) The lexer can already classify tokens, for example
+        as numbers, keywords or identifiers. This again makes the grammar
+        rules more deterministic and as a result faster to parse.
 
+        The point is that parser combinators can be used to process
+        strings, but in case of compilers where whitespaces and
+        comments need to be filtered out, the lexing phase is actually
+        quite useful.  }
       
 \item The injection function for sequence regular expressions is defined
       by three clauses:
@@ -84,8 +120,20 @@
 \end{tabular}
 \end{center}
 
-Explain why there are three cases in the injection function for sequence
-regular expressions. 
+     Explain why there are three cases in the injection function for sequence
+     regular expressions.
+
+     \solution{
+       This is because the derivative of sequences can be of the form
+
+       \begin{itemize}
+       \item $(der\,c\,r_1)\cdot r_2$
+       \item $(der\,c\,r_1)\cdot r_2 \;+\; der\,c\,r_2$  
+       \end{itemize}
+
+       In the first case the value needs to be of the form $Seq$, in the second case $\Left$ or $Right$.
+       Therefore 3 cases.
+       }
       
 \item \POSTSCRIPT        
 \end{enumerate}