--- a/handouts/ho05.tex	Mon Oct 02 23:10:56 2023 +0100
+++ b/handouts/ho05.tex	Tue Oct 03 14:29:12 2023 +0100
@@ -84,7 +84,7 @@
 \noindent
 from natural languages were the meaning of \emph{flies} depends on the
 surrounding \emph{context} are avoided as much as possible. Here is
-an interesting video about C++ being not a context-free language
+an interesting video about C++ not being a context-free language
 
 \begin{center}
 \url{https://www.youtube.com/watch?v=OzK8pUu4UfM}
@@ -99,7 +99,7 @@
 \end{plstx}
 
 \noindent 
-or a grammar for recognising strings consisting of ones is
+or a grammar for recognising strings consisting of ones (at least one) is
 
 \begin{plstx}[margin=3cm]
 : \meta{O} ::= 1 \cdot  \meta{O} 
@@ -378,10 +378,11 @@
 \end{plstx}
 
 \noindent
-In this grammar all $\meta{E}$xpressions, $\meta{T}$erms and $\meta{F}$actors
-are in some way protected from being left-recusive. For example if you
-start $\meta{E}$ you can derive another one by going through $\meta{T}$, then 
-$\meta{F}$, but then $\meta{E}$ is protected by the open-parenthesis.
+In this grammar all $\meta{E}$xpressions, $\meta{T}$erms and
+$\meta{F}$actors are in some way protected from being
+left-recusive. For example if you start $\meta{E}$ you can derive
+another one by going through $\meta{T}$, then $\meta{F}$, but then
+$\meta{E}$ is protected by the open-parenthesis in the last rule.
 
 \subsection*{Removing $\epsilon$-Rules and CYK-Algorithm}
 
@@ -395,7 +396,7 @@
 \noindent
 The transformation made the original grammar non-left-recursive, but at
 the expense of introducing an $\epsilon$ in the second rule. Having an
-explicit $\epsilon$-rule is annoying to, not in terms of looping, but in
+explicit $\epsilon$-rule is annoying, not in terms of looping, but in
 terms of efficiency. The reason is that the $\epsilon$-rule always
 applies but since it recognises the empty string, it does not make any
 progress with recognising a string. Better are rules like $( \cdot
@@ -455,14 +456,15 @@
 \noindent
 I let you think about whether this grammar can still recognise all
 binary numbers and whether this grammar is non-left-recursive. The
-precise statement for the transformation of removing $\epsilon$-rules is
-that if the original grammar was able to recognise only non-empty
+precise statement for the transformation of removing $\epsilon$-rules
+is that if the original grammar was able to recognise only non-empty
 strings, then the transformed grammar will be equivalent (matching the
 same set of strings); if the original grammar was able to match the
 empty string, then the transformed grammar will be able to match the
-same strings, \emph{except} the empty string. So the  $\epsilon$-removal
-does not preserve equivalence of grammars, but the small defect with the
-empty string is not important for practical purposes.
+same strings, \emph{except} the empty string. So the
+$\epsilon$-removal does not preserve equivalence of grammars in
+general, but the small defect with the empty string is not important
+for practical purposes.
 
 So why are these transformations all useful? Well apart from making the 
 parser combinators work (remember they cannot deal with left-recursion and
@@ -563,10 +565,10 @@
 \noindent
 The last row contains the information about all words and their
 corresponding non-terminals. For example the field for \texttt{trains}
-contains the information $\meta{N}$ and $\meta{V}$ because it can be a
+contains the information $\meta{N}$ and $\meta{V}$ because \texttt{trains} can be a
 ``verb'' and a ``noun'' according to the grammar.  The row above,
 let's call the corresponding fields 5a to 5e, contains information
-about 2-word parts of the sentence, namely
+about \underline{2-word} parts of the sentence, namely
 
 \begin{center}
 \begin{tabular}{llll}