How can I give active characters definitions with expl3?

Question

It's very much glossed over in texdoc interface3…

\documentclass{article}
\usepackage{xparse}
\ExplSyntaxOn
\NewDocumentCommand \Digits { m m }{
  \group_begin:
    \char_set_catcode_active:N ;
    \char_set_catcode_active:N .
\def;{,\;}
\def.{\dots}

\langle #1 \rangle \sb #2

\group_end:
}
\ExplSyntaxOff
\begin{document}
Desired syntax:
  $\Digits{1,2,3;4,5,6;.;,7}{8}$
Desired output:
  $\langle 1,2,3, ; 4,5,6, ; \dots, ; 7 \rangle_8$
\end{document}

I will be doing fancy things with the comma list at a later point (using l3clist), but I'd like to get this syntax working first.

I imagined that this should work based on the TeX FAQ article, but I keep getting the following error:

ERROR: Missing control sequence inserted.
--- TeX said ---
<inserted text> 
                \inaccessible 
l.18 $\Digits{1,2,3;4,5,6;.;,7}{8}
                                  $
--- HELP ---
This is probably caused by a \newcommand, \renewcommand, \newlength,
or \newsavebox command whose first argument is not a command name.

---

Per cgnieder's comment, perhaps the following approach would work:

\NewDocumentCommand \Digits { s m m } {
  \group_begin:
    \seq_set_split:Nnn
      \l_digit_seq
      { #2 }
      { \IfBooleanTF { #1 } { } { , } }
\char_set_catcode_active:N ;
\char_set_catcode_active:N .

\def;{,\;}
\def.{\dots}

\real_digits:nn

}
\cs_new:Npn \real_digits:nn #1 #2 {
  \langle #1 \rangle \sb #2
  \group_end:
}
\ExplSyntaxOff

but alas, it does not.

Answer 1

It's no different from usual TeX programming. In this case we can exploit the possibility of making a character math active and the advanced management of active characters provided by expl3.

\documentclass{article}
\usepackage{xparse}
\ExplSyntaxOn

\cs_new:Npn \sean_active_define:NN #1 #2
 {
  % define the active version of character #1 to do #2
  \char_set_active_eq:nN { `#1 } #2
  % make #1 math active
  \char_set_mathcode:nn { `#1 } { "8000 }
}

\NewDocumentCommand \Digits { m m }
 {
  % limit the scope of the math active characters
  \group_begin:
  \sean_active_define:NN ; \__sean_active_semicolon:
  \sean_active_define:NN . \__sean_active_period:
  \langle #1 \rangle\sb{#2}
  \group_end:
 }

\cs_new_protected:Npn \__sean_active_semicolon: { ,\; }
\cs_new_protected:Npn \__sean_active_period: { \dots }
\ExplSyntaxOff

\begin{document}
\makebox[3cm][l]{Desired syntax:}
  \verb$\Digits{1,2,3;4,5,6;.;7}{8}$

\makebox[3cm][l]{Obtained output:}
  $\Digits{1,2,3;4,5,6;.;7}{8}$

\makebox[3cm][l]{Desired output:}
  $\langle 1,2,3, \; 4,5,6, \; \dots, \; 7 \rangle_8$
\end{document}

Note. The most recent version (September 2015) of the l3 kernel has improved the management of active characters.

---

OLD VERSION

It's no different from usual TeX programming; category codes are fixed at the moment the definition is stored, and in your definition ; and . have category code 12.

The procedure requires using \tl_to_lowercase:n in order to obtain the requested characters as active, while being able to use them in the replacement text.

Here's a possible implementation, with a generic activation function; the definition of \Digits shouldn't have parameters, in order to avoid absorbing the arguments after activating the characters.

\documentclass{article}
\usepackage{xparse}
\ExplSyntaxOn

\group_begin:
\char_set_catcode_active:N \^^@
\cs_new:Npn \sean_active_define:NN #1 #2
 {
  \group_begin: 
  \char_set_lccode:nn { `^^@ } { `#1 }
  \tl_to_lowercase:n
   {
    \group_end:
    \cs_set_eq:NN ^^@
   }
   #2 % the replacement text is better outside \tl_to_lowercase:n
   \char_set_catcode_active:N #1
}
\group_end:

\NewDocumentCommand \Digits { }
 {
  \group_begin:
  \sean_active_define:NN ; \__sean_active_semicolon:
  \sean_active_define:NN . \__sean_active_period:
  \sean_digits:nn
 }

\cs_new_protected:Npn \sean_digits:nn #1 #2
 {
  \langle #1 \rangle\sb{#2}
  \group_end:
 }

\cs_new_protected:Npn \__sean_active_semicolon: { ,\; }
\cs_new_protected:Npn \__sean_active_period: { \dots }
\ExplSyntaxOff

\begin{document}
\makebox[2.5cm][l]{Desired syntax:}
  $\Digits{1,2,3;4,5,6;.;7}{8}$

\makebox[2.5cm][l]{Desired output:}
  $\langle 1,2,3, \; 4,5,6, \; \dots, \; 7 \rangle_8$
\end{document}

You may avoid activating the characters by making them “math active”, which doesn't require absorbing the arguments in two stages.

\documentclass{article}
\usepackage{xparse}
\ExplSyntaxOn

\group_begin:
\char_set_catcode_active:N \^^@
\cs_new:Npn \sean_math_active_define:NN #1 #2
 {
  \group_begin: 
  \char_set_lccode:nn { `^^@ } { `#1 }
  \tl_to_lowercase:n
   {
    \group_end:
    \cs_set_eq:NN ^^@
   }
   #2 % the replacement text is better outside \tl_to_lowercase:n
   \char_set_mathcode:nn { `#1 } { "8000 }
}
\group_end:

\NewDocumentCommand \Digits { m m }
 {
  \group_begin:
  \sean_math_active_define:NN ; \__sean_active_semicolon:
  \sean_math_active_define:NN . \__sean_active_period:
  \langle #1 \rangle\sb{#2}
  \group_end:
 }

\cs_new_protected:Npn \__sean_active_semicolon: { ,\; }
\cs_new_protected:Npn \__sean_active_period: { \dots }

\ExplSyntaxOff

\begin{document}
\makebox[2.5cm][l]{Desired syntax:}
  $\Digits{1,2,3;4,5,6;.;7}{8}$

\makebox[2.5cm][l]{Desired output:}
  $\langle 1,2,3, \; 4,5,6, \; \dots, \; 7 \rangle_8$
\end{document}

NOTE: I changed slightly the implementation, making it a bit more powerful and easier to manage; now it's easy to define local active characters also with arguments.