Macros with # as the last parameter

Question

Knuth hid a special rule for delimited arguments in exercise 20.5 of the TeXbook.

If the very last character of the parameter text is #, so that this # is immediately followed by {, TeX will behave as if the { had been inserted at the right end of both the parameter text and the replacement text.

This means that a macro can be defined as,

\def\a#1#{#1}

calling it as \a 10 will give a runaway argument error whereas calling it as \a{12}, will compile with no trouble.

I struggled to find a practical application for such macros even after looking at TeX by Topic, TeXbook and LaTeX source.

Here is my take on it, create some commands to typeset and do some calculations for fractions, as for example those found in basic arithmetic texbooks. You type this,

\[\FRAC ADD{3}{8}+{1}{7}\]
\[\FRAC SUB{5}{8}-{1}{7}\]
\[\FRAC MUL{5}{8}x{13}{1201}\]

and you get this:

Here is the code,

\documentclass{article} 
\begin{document}
%% macro factory
\def\FRAC#1#{\csname #1\endcsname}
%% add
\def\ADD#1#2+#3#4{%
  \frac{#1}{#2}+\frac{#3}{#4}=
  \frac{\the\numexpr(#4*#1)+(#3*#2)}{\the\numexpr#2*#4}
}
%% subtract
\def\SUB#1#2-#3#4{%ok top
  \frac{#1}{#2}-\frac{#3}{#4}=
  \frac{\the\numexpr(#4*#1)-(#3*#2)}{\the\numexpr(#2*#4)}
}
%% multiply
\def\MUL#1#2x#3#4{%
  \frac{#1}{#2}\times\frac{#3}{#4}=
  \frac{\the\numexpr(#1*#3)}
  {\the\numexpr#2*#4}
}
%% testing
\[\FRAC ADD{3}{8}+{1}{7}\]
\[\FRAC SUB{5}{8}-{1}{7}\]
\[\FRAC MUL{5}{8}x{13}{1201}\]
\end{document}

Are there any practical applications for such macros? Are there any special precautions one should take? Why would Knuth include this facility in the first place?

Answer 1

This trick of catching until the first open brace can be used in many situations.

• As other posters said, it allows to catch an optional argument expandably if it is not the last. It is possible in principle to get this to be fully robust with respect to nesting, but not implemented yet.

• It can be used to parse the body of a definition provided by the user, to change it to fit your purposes while keeping a natural syntax. (More on parsing def below.)

• I've used this trick primarily to parse a token list expandably without losing any brace. For instance,

  • to expandably uppercase or lowercase a given string (see this answer)

  • to fully expand a token list expandably (almost as well as the luatex primitive \expanded (see this answer)

  • to expand tokens selectively, or in the reverse order

  • to write a primitive macro expander (i.e. take a file, and expand user-defined macros)

  so basically any situation where you need to be careful with braces, but cannot use \futurelet.

This trick only works if there is only one character with catcode 1 (begin group character). Also, we need to be able to put a sentinel at the end of the token list that we are manipulating: otherwise, in the absence of opening brace, we would get a runaway argument.

On parsing a definition: say that you want to give the user an easy way of defining a macro which possibly takes arguments, and always produces a boxed math result. Say that you also want the parameter text to be arbitrary. Either you let the user do everything, or you parse the definition using the trick you ask about.

%non-user-friendly
\def\foo_#1^#2#3#4{\fbox{$\sum_{#1}^{#2} \frac{#3}{#4}$}}

%more user friendly (perhaps)
\boxeddef\foo_#1^#2#3#4{\sum_{#1}^{#2} \frac{#3}{#4}}

To do that:

\documentclass{article}
\makeatletter
\def\boxeddef#1#2#{\boxeddef@aux{#1}{#2}}
\def\boxeddef@aux#1#2#3{\def#1#2{\fbox{$#3$}}}
\makeatother
\begin{document}
%\def\foo#1#2#3#4{\fbox{$\sum_{#1}^{#2} \frac{#3}{#4}$}}
\boxeddef\foo_#1^#2#3#4{\sum_{#1}^{#2} \frac{#3}{#4}}
\[
\foo_{a}^{b}{C}{D}
\]
\end{document}

Answer 2

The #{ trick can allow to make macros which behave somewhat like \hbox{...}, in the sense that they can have, for example, verbatim inside their argument (also, the argument is not read beforehand). There’s a nice example in the Tugboat article The TeX Hierarchy (Volume 15, 1994, p. 7-9):

Compared to the Wizard’s version, the Guru’s use of #{ makes sure that the character eaten by \let\next= will always be a brace, never something else.

Answer 3

an example from latex.ltx

\def\usepackage#1#{%
  \@latex@error
    {\noexpand \usepackage before \string\documentclass}%
    {\noexpand \usepackage may only appear in the document
      preamble, i.e.,\MessageBreak
      between \noexpand\documentclass and
      \string\begin{document}.}%
  \@gobble}

with the argument setting you can handle optional arguments, eg \usepackage[foo]{bar} without defining the different cases

Answer 4

Reading everything up the opening { as argument like \def\A#1#{...} would do, it is not very common, but useful when you want to read optional arguments of macro which normal argument must start with { anyway. This can be very useful in cases when the macro should be fully expandable and therefore \futurelet (used by \@ifnextchar) can't be used.

Funnily, I just started to use this TeX feature this week: In my up-coming package 'filemod' I define expandable and non-expandable macros to read and compare file modification dates. The expandable implementation of \filemodNewest takes an optional argument and list of file names {{filename1}{filename2}...{filename3}} and expands the the name of the newest file. The #{ syntax is used to read the potential optional argument (a number in this case):

\def\filemodNewest#1#{%
  \expandafter\expandafter
  \expandafter\@filemodNewest
  \csname
    @%
  \ifx\@nnil#1\@nnil
    first%
  \else
    second%
  \fi
    oftwo%
  \endcsname
    {[\filemodcmpdefault]}%
    {#1}%
}

It should be noted that this works only for simple optional arguments which do not include braces. This excludes e.g. complex PGF keys etc.

Also the etextools defines another way of expandable macros with optional arguments using eTeX \detokenize.

Answer 5

I just used this construction in my expandable sanitizer: Can one define an expandable command that removes control sequences from its argument?. The purpose there is so that I can parse some arbitrary (well-formed) input for the first group it contains, without actually entering the group. Without \futurelet or \catcode changes, it's impossible to grab a single brace token, but this way, I can expand up until a group, leaving the braces there, and then continue processing with the sure knowledge that the next thing on the menu is a group. It has exactly the same purpose as \futurelet for this one token.

Answer 6

I use this possibility in the next case. I can't use delimiters like () because the parenthesis are nested. With

\documentclass{minimal}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc} 

\def\macro#1#{%
  <#1>\par
  \macrobis
}
\def\macrobis#1{and <#1>}

\begin{document}
\macro(exp(1),ln(2)){A}
\end{document}

The first #1 catches (exp(1),ln(2))