107   negation and back-references.

   107   negation and back-references.

   108 \end{abstract}

   108 \end{abstract}

   109 

   109 

   110 

   110 

   111 \section{Introduction}

   111 \section{Introduction}

   112 While we believe derivatives of regular expressions, written

   114 While we believe derivatives of regular expressions, written

   113 $r\backslash s$, are a beautiful concept (in terms of ease of

   115 $r\backslash s$, are a beautiful concept (in terms of ease of

   114 implementing them in functional programming languages and in terms of

   116 implementing them in functional programming languages and in terms of

   115 reasoning about them formally), they have one major drawback: every

   117 reasoning about them formally), they have one major drawback: every

   116 derivative step can make regular expressions grow drastically in

   118 derivative step can make regular expressions grow drastically in

   141 

   143 

   142 

   144 

   143 \section{Recapitulation of Concepts From the Last Report}

   145 \section{Recapitulation of Concepts From the Last Report}

   144 

   146 

   145 \subsection*{Regular Expressions and Derivatives}

   147 \subsection*{Regular Expressions and Derivatives}

   147 Suppose (basic) regular expressions are given by the following grammar:

   148 Suppose (basic) regular expressions are given by the following grammar:

   148 

   149 

   149 \[			r ::=   \ZERO \mid  \ONE

   150 \[			r ::=   \ZERO \mid  \ONE

   150 			 \mid  c  

   151 			 \mid  c  

   151 			 \mid  r_1 \cdot r_2

   152 			 \mid  r_1 \cdot r_2