
diff -r dc2f5eb33a9a -r 6d74d2a0a4b0 handouts/ho05.tex
--- a/handouts/ho05.tex	Wed Oct 23 21:39:11 2019 +0100
+++ b/handouts/ho05.tex	Wed Oct 23 22:23:10 2019 +0100
@@ -1,3 +1,4 @@
+% !TEX program = xelatex
 \documentclass{article}
 \usepackage{../style}
 \usepackage{../langs}
@@ -53,23 +54,23 @@
 languages are sets of strings). As indicated the set of regular
 languages are fully included inside the context-free languages,
 meaning every regular language is also context-free, but not vice
-versa. Below I will let you think for example what the context-free
+versa. Below I will let you think, for example, what the context-free
 grammar is for the language corresponding to the regular expression
 $(aaa)^*a$.
 
 Because of their convenience, context-free languages play an important
 role in `day-to-day' text processing and in programming
 languages. Context-free in this setting means that ``words'' have one
-meaning only ansd this meaning is independend from in which context
+meaning only and this meaning is independent from in which context
 the ``words'' appear. For example ambiguity issues like
 
 \begin{center}
-\tt Time flies like an arrow; fruit flies like a banana.
+\tt Time flies like an arrow; fruit flies like bananas.
 \end{center}  
 
 \noindent
 from natural languages were the meaning of \emph{flies} depend on the
-surounding \emph{context} are avoided as much as possible.
+surrounding \emph{context} are avoided as much as possible.
 
 Context-free languages are usually specified by grammars. For example
 a grammar for well-parenthesised expressions can be given as follows:
@@ -167,22 +168,22 @@
 
 \begin{center}
 \begin{tikzpicture}[level distance=8mm, black]
-  \node {$E$}
-    child {node {$E$} 
+  \node {\meta{E}}
+    child {node {\meta{E} } 
        child {node {$($}}
-       child {node {$E$}       
-         child {node {$E$} child {node {$N$} child {node {$1$}}}}
+       child {node {\meta{E} }       
+         child {node {\meta{E} } child {node {\meta{N} } child {node {$1$}}}}
          child {node {$+$}}
-         child {node {$E$} 
-            child {node {$N$} child {node {$2$}}}
-            child {node {$N$} child {node {$3$}}}
+         child {node {\meta{E} } 
+            child {node {\meta{N} } child {node {$2$}}}
+            child {node {\meta{N} } child {node {$3$}}}
             } 
         }
        child {node {$)$}}
      }
      child {node {$+$}}
-     child {node {$E$}
-        child {node {$N$} child {node {$4$}}}
+     child {node {\meta{E} }
+        child {node {\meta{N} } child {node {$4$}}}
      };
 \end{tikzpicture}
 \end{center}
@@ -192,18 +193,19 @@
 grammar. Before we come to the problem of constructing such
 parse-trees, we need to consider the following two properties
 of grammars. A grammar is \emph{left-recursive} if there is a
-derivation starting from a non-terminal, say $NT$ which leads
-to a string which again starts with $NT$. This means a
+derivation starting from a non-terminal, say \meta{NT} which leads
+to a string which again starts with \meta{NT}. This means a
 derivation of the form.
 
 \begin{center}
-$NT \rightarrow \ldots \rightarrow NT \cdot \ldots$
+$\meta{NT} \rightarrow \ldots \rightarrow \meta{NT} \cdot \ldots$
 \end{center}
 
 \noindent It can be easily seen that the grammar above for
 arithmetic expressions is left-recursive: for example the
-rules $E \rightarrow E\cdot + \cdot E$ and $N \rightarrow
-N\cdot N$ show that this grammar is left-recursive. But note
+rules $\meta{E} \rightarrow \meta{E}\cdot + \cdot \meta{E}$ and 
+$\meta{N} \rightarrow \meta{N}\cdot \meta{N}$ show that this 
+grammar is left-recursive. But note
 that left-recursiveness can involve more than one step in the
 derivation. The problem with left-recursive grammars is that
 some algorithms cannot cope with them: they fall into a loop.
@@ -214,12 +216,13 @@
 for numbers we might specify
 
 \begin{center}
-$N \;\;\rightarrow\;\; 0\;|\;\ldots\;|\;9\;|\;1\cdot N\;|\;2\cdot N\;|\;\ldots\;|\;9\cdot N$
+$\meta{N} \;\;\rightarrow\;\; 0\;|\;\ldots\;|\;9\;|\;
+1\cdot \meta{N}\;|\;2\cdot \meta{N}\;|\;\ldots\;|\;9\cdot \meta{N}$
 \end{center}
 
 \noindent Using this grammar we can still derive every number
 string, but we will never be able to derive a string of the
-form $N \to \ldots \to N \cdot \ldots$.
+form $\meta{N} \to \ldots \to \meta{N} \cdot \ldots$.
 
 The other property we have to watch out for is when a grammar
 is \emph{ambiguous}. A grammar is said to be ambiguous if
@@ -232,26 +235,26 @@
 \begin{center}
 \begin{tabular}{c@{\hspace{10mm}}c}
 \begin{tikzpicture}[level distance=8mm, black]
-  \node {$E$}
-    child {node {$E$} child {node {$N$} child {node {$1$}}}}
+  \node {\meta{E} }
+    child {node {\meta{E} } child {node {\meta{N} } child {node {$1$}}}}
     child {node {$+$}}
-    child {node {$E$}
-       child {node {$E$} child {node {$N$} child {node {$2$}}}}
+    child {node {\meta{E} }
+       child {node {\meta{E} } child {node {\meta{N} } child {node {$2$}}}}
        child {node {$+$}}
-       child {node {$E$} child {node {$N$} child {node {$3$}}}}
+       child {node {\meta{E} } child {node {\meta{N} } child {node {$3$}}}}
     }
     ;
 \end{tikzpicture} 
 &
 \begin{tikzpicture}[level distance=8mm, black]
-  \node {$E$}
-    child {node {$E$}
-       child {node {$E$} child {node {$N$} child {node {$1$}}}}
+  \node {\meta{E} }
+    child {node {\meta{E} }
+       child {node {\meta{E} } child {node {\meta{N} } child {node {$1$}}}}
        child {node {$+$}}
-       child {node {$E$} child {node {$N$} child {node {$2$}}}} 
+       child {node {\meta{E} } child {node {\meta{N} } child {node {$2$}}}} 
     }
     child {node {$+$}}
-    child {node {$E$} child {node {$N$} child {node {$3$}}}}
+    child {node {\meta{E} } child {node {\meta{N} } child {node {$3$}}}}
     ;
 \end{tikzpicture}
 \end{tabular} 
@@ -275,7 +278,7 @@
 a grammar and string. In what follows we explain \emph{parser
 combinators}, because they are easy to implement and closely
 resemble grammar rules. Imagine that a grammar describes the
-strings of natural numbers, such as the grammar $N$ shown
+strings of natural numbers, such as the grammar \meta{N}  shown
 above. For all such strings we want to generate the
 parse-trees or later on we actually want to extract the
 meaning of these strings, that is the concrete integers




