--- a/cws/cw05.tex	Thu Dec 06 21:49:43 2018 +0000
+++ b/cws/cw05.tex	Thu Dec 06 22:51:46 2018 +0000
@@ -97,7 +97,7 @@
 set of instructions---just 8 instructions in total and all of which
 are single characters. Despite the minimalism, this language has been
 shown to be Turing complete\ldots{}if this doesn't ring any bell with
-you: it roughly means that every algorithm we know can, in principle,
+you: it roughly means that every(!) algorithm can, in principle,
 be implemented in brainf***. It just takes a lot of determination and
 quite a lot of memory resources. Some relatively sophisticated sample
 programs in brainf*** are given in the file \texttt{bf.scala}, including
@@ -136,9 +136,9 @@
 \subsubsection*{Tasks (file bf.scala)}
 
 \begin{itemize}
-\item[(1)]  Write a function that takes a file name as an argument
+\item[(1)]  Write a function that takes a filename (a string) as an argument
   and requests the corresponding file from disk. It returns the
-  content of the file as a String. If the file does not exists,
+  content of the file as a string. If the file does not exists,
   the function should return the empty string.\\
   \mbox{}\hfill[1 Mark]
   
@@ -150,21 +150,21 @@
   convention is that if we query the memory at a location that is
   \emph{not} defined in the \texttt{Map}, we return \texttt{0}. Write
   a function, \texttt{sread}, that takes a memory (a \texttt{Map}) and
-  a memory pointer (an \texttt{Int}) as argument, and `safely' reads the
+  a memory pointer (an \texttt{Int}) as arguments, and `safely' reads the
   corresponding memory location. If the \texttt{Map} is not defined at
   the memory pointer, \texttt{sread} returns \texttt{0}.
 
   Write another function \texttt{write}, which takes a memory, a
-  memory pointer and an integer value as argument and updates the
-  \texttt{Map} with the value at the given memory location. As usual
+  memory pointer and an integer value as arguments and updates the
+  \texttt{Map} with the value at the given memory location. As usual,
   the \texttt{Map} is not updated `in-place' but a new map is created
-  with the same data, except the value is stored at the given memory
+  with the same data, except the new value is stored at the given memory
   pointer.\hfill[1 Mark]
 
 \item[(3)] Write two functions, \texttt{jumpRight} and
-  \texttt{jumpLeft} that are needed to implement the loop constructs
-  of brainf***. They take a program (a \texttt{String}) and a program
-  counter (an \texttt{Int}) as argument and move right (respectively
+  \texttt{jumpLeft}, that are needed to implement the loop constructs
+  in brainf***. They take a program (a \texttt{String}) and a program
+  counter (an \texttt{Int}) as arguments and move right (respectively
   left) in the string in order to find the \textbf{matching}
   opening/closing bracket. For example, given the following program
   with the program counter indicated by an arrow:
@@ -180,7 +180,7 @@
   \texttt{--[..+>--]\barbelow{,}>,++}
   \end{center}
 
-  meaning it jumps to after the loop. Similarly, if you are in 8th position
+  meaning it jumps to after the loop. Similarly, if you are in 8th position,
   then \texttt{jumpLeft} is supposed to jump to just after the opening
   bracket (that is jumping to the beginning of the loop):
 
@@ -340,9 +340,9 @@
   For this write a function \texttt{jtable} that precomputes the ``jump
   table'' for a bf-program. This function takes a bf-program 
   as an argument and returns a \texttt{Map[Int, Int]}. The 
-  purpose of this Map is to record the information
-  that given on the pc-position $pc$ is a '\texttt{[}' or a '\texttt{]}',
-  then to which pc-position do we need to jump next?
+  purpose of this Map is to record the information, in cases
+  a pc-position points to a '\texttt{[}' or a '\texttt{]}',
+  to which pc-position do we need to jump next?
  
   For example for the program
     
@@ -374,20 +374,22 @@
   the function \texttt{jumpLeft} and \texttt{jumpRight} for
   calculating the \texttt{jtable}.\hfill{[1 Mark]}
 
-\item[(6)] Compilers try to eliminate any ``dead'' code that could slow
-  down programs and also perform what is often called
-  \emph{peephole optimisations}.\footnote{\url{https://en.wikipedia.org/wiki/Peephole_optimization}} While it is difficult for compilers to comprehend what
-  is intended with whole programs, they are very good at finding out
-  what small snippets of code do, and then try to generate
-  faster code for such snippets.
+\item[(6)] Compilers try to eliminate any ``dead'' code that could
+  slow down programs and also perform what is often called
+  \emph{peephole
+    optimisations}.\footnote{\url{https://en.wikipedia.org/wiki/Peephole_optimization}}
+  For the latter consider that it is difficult for compilers to
+  comprehend what is intended with whole programs, they are very good
+  at finding out what small snippets of code do, and then try to
+  generate faster code for such snippets.
 
   In our case, dead code is everything that is not a bf-command.
   Therefore write a function \texttt{optimise} which deletes such
-  dead code. Moreover this function should replace every substring
+  dead code from a bf-program. Moreover this function should replace every substring
   of the form \pcode{[-]} by a new command \texttt{0}. 
   The idea is that the loop \pcode{[-]} just resets the
   memory at the current location to 0. It is more efficient
-  to do this in a single step, rather than incrementally as in
+  to do this in a single step, rather than stepwise in a loop as in
   the original bf-programs.
 
   In the extended \texttt{compute3} and \texttt{run3} functions you should
@@ -396,18 +398,18 @@
   The easiest way to modify a string in this way is to use the regular
   expression \pcode{"""[^<>+-.,\\[\\]]"""}, which recognises everything that is 
   not a bf-command. Similarly, the
-  regular expression \pcode{"""\\[-\\]"""} finds all occurrences of \pcode{[-]} and 
-  by using the Scala method \pcode{.replaceAll} you can replace it with the 
-  string \pcode{"0"} standing for the new bf-command.\hfill{[1 Mark]}
+  regular expression \pcode{"""\\[-\\]"""} finds all occurrences of \pcode{[-]}.  By using the Scala method \pcode{.replaceAll} you can replace substrings
+  with new strings.\\
+  \mbox{}\hfill{[1 Mark]}
 
 \item[(7)] Finally, real compilers try to take advantage of CPUs which often
   provide complex operations in hardware that can combine many smaller
   instructions into a single faster instruction.
 
-  In our case we can optimise the several single increments of a
+  In our case we can optimise the several single increments performed at a
   memory cell, for example \pcode{++++}, by a single ``increment by
   4''. For this optimisation we just have to make sure these single
-  increments are all next to each other. Similarly optimisations should apply
+  increments are all next to each other. Similar optimisations should apply
   for the bf-commands \pcode{-}, \pcode{<} and
   \pcode{>}, which can all be replaced by extended versions that take
   the amount of the increment (decrement) into account. We will do
@@ -427,20 +429,22 @@
   \end{center}  
 
 
-  Similarly for \pcode{-}, \pcode{<} and \pcode{>}. If there are more
-  than 26 \pcode{+}'s in a row, then more than on ``two-character''
+  If there are more
+  than 26 \pcode{+}'s in a row, then more than one ``two-character''
   bf-commands need to be generated (the idea is that more than
   26 copies of a single bf-command in a row is a rare occurrence in
-  actual bf-programs). All other bf-commands should be unaffected by this
+  actual bf-programs). Similar replacements apply
+  for \pcode{-}, \pcode{<} and \pcode{>}, but
+  all other bf-commands should be unaffected by this
   change. 
 
   For this write a function \texttt{combine} which replaces sequences
   of repeated increment and decrement commands by appropriate
-  two-character commands.  In the functions \pcode{compute4} and
+  two-character commands. In the functions \pcode{compute4} and
   \pcode{run4}, the ``combine'' and the optimisation from (6) should
   be performed. Make sure that when a two-character bf-command is
   encountered you need to increase the \pcode{pc}-counter by two in
-  order to process the next command. For example
+  order to progress to the next command. For example
 
   \begin{center}
   \pcode{combine(optimise(load_bff("benchmark.bf")))}  
@@ -459,7 +463,7 @@
   \end{center}    
 
   As you can see, the compiler bets on saving so much time on the
-  \pcode{+B} and \pcode{+M} steps so that the optimisations is
+  \pcode{+B} and \pcode{+M} steps such that the optimisations is
   worthwhile overall (of course for the \pcode{>A}'s and so on, the compiler incurs a
   penalty). Luckily, after you have performed all
   optimisations in (5) - (7), you can expect that the