Cases of different tokens having same meaning and same \string-representation that can occur in the stage of expansion

expansionstringstex-core

Please take this question for a moot point:

I'd like to know about edge cases where different tokens which TeX might encounter during expansion without delivering error-messages have the same meaning and the same \string-representation.

By "have the same meaning" I mean that

\ifx⟨token 1⟩⟨token 2⟩\expandafter\firstoftwo\else\expandafter\secondoftwo\fi

yields \firstoftwo.

By "same \string-representation" I mean that \string⟨token 1⟩ yields the same set of tokens as \string⟨token 2⟩.

Edge cases I came up with so far:

frozen-\relax and \relax-primitive.
the nameless control-sequence (producible via \csname\endcsname or via an escape-character (backslash) at the end of a line of .tex-input while \endlinechar has a negative value) and the control-sequence whose name is csname⟨escapechar⟩endcsname (producible via \csname csname\string\endcsname\endcsname) while those control-sequences have the same meaning.
active-character-token let equal to a non-active pendant.
one-letter-control-sequence let equal to an explicit character token where the character-code corresponds to the character which forms the name of the control-sequence while \escapechar has a negative value.

Are there more edge cases?

E.g., what about things like \inaccessible or tokens that TeX might insert while processing an alignment? You can define \inaccessible. Can you let it equal to TeX's \inaccessible?

Best Answer

Another such case are frozen font control sequences obtained by applying \the to a font command and the original font command itself. They fulfill all your criteria:

\documentclass{article}
\begin{document}
\makeatletter
% Let's assume that we loaded a font at some point:
\font\cmr cmr10
% Then we can get a second token for accessing the font using \the
\edef\tokens{\the\cmr\cmr}
% Compare their \string representations:
\edef\helpI{\expandafter\expandafter\expandafter\string\expandafter\@firstoftwo\tokens}
\edef\helpII{\expandafter\expandafter\expandafter\string\expandafter\@secondoftwo\tokens}
\ifx\helpI\helpII
  They have the same \texttt{\string\string} representation.
\else
  They have different \texttt{\string\string} representation.
\fi

\expandafter\ifx\tokens
  They have the same meaning.
\else
  They have different meaning.
\fi

% Now in case some people question that they are actually different, let's change the meaning of one of them and compare again.
\let\cmr\relax
\expandafter\ifx\tokens
  They are identical.
\else
  They are different tokens.
\fi
\makeatother
\end{document}

Related Solutions

Reliably obtain a control sequence token which is undefined

The following should work even without e-TeX. It builds the control sequence names inside a group, checks against \relax, if they are equal closes the group, rechecks against \relax and only if that latter check is false uses the resulting control sequence.

Control sequence names are build using a numeric name (and if every numeric names would be taken precedes with a dot, putting more and more dots in front of the name until an undefined cs is found).

\begingroup
\catcode`\@=11
\long\def\@firstofone#1{#1}
\@firstofone{\endgroup
\long\def\@secondofthree#1#2#3{#2}
\long\def\@secondoftwo#1#2{#2}
\newcount\undefinedto@count
\newcount\undefinedto@maxcount
\undefinedto@maxcount=2147483647
\long\def\undefinedto#1%
  {%
    \undefinedto@{}%
    {#1}%
  }
\def\undefinedto@
  {%
    \begingroup
    \global\undefinedto@count=-\undefinedto@maxcount
    \undefinedto@loop
  }
\def\undefinedto@loop#1%
  {%
    \expandafter\undefinedto@loop@
      \csname #1\the\undefinedto@count\endcsname
      {#1}%
  }
\long\def\undefinedto@loop@#1#2%
  {%
    \ifx#1\relax
      \expandafter\@secondofthree
    \fi
    \@secondoftwo
      {\undefinedto@test#1{#2}}%
      {\undefinedto@loop@next{#2}}%
  }
\def\undefinedto@loop@next#1%
  {%
    \ifnum\undefinedto@count=\undefinedto@maxcount
      \expandafter\@secondofthree
    \fi
    \@secondoftwo
      {%
        \global\undefinedto@count=-\undefinedto@maxcount
        \undefinedto@loop{#1.}%
      }%
      {%
        \global\advance\undefinedto@count1
        \undefinedto@loop{#1}%
      }
  }
\long\def\undefinedto@test#1#2%
  {%
    \endgroup
    \ifx#1\relax
      \expandafter\@secondofthree
    \fi
    \@secondoftwo
      {%
        \begingroup
        \undefinedto@loop@next{#2}%
      }%
      {\undefinedto@found#1}%
  }
\long\def\undefinedto@found#1#2{#2#1}
}

\undefinedto\show

How does \if work

The \if behavior can be clearly described by its equivalence classes. There is one equivalence class with all control sequences which are not equal (by \let) to a character token. Moreover, each character code has its own equivalence class with all tokens with the same character code (i.e. the category code is irrelevant) and with all control sequences which are equal to these tokens (by \let).

Try this:

\let\z=Z
\catcode`Z=13
\def Z{}
\if \noexpand Z\string Z YES\else NO\fi
\if \noexpand Z\z YES\else NO\fi

both cases gives YES. First case: there are two tokens Z, one with catcode 13 and second with catcode 12. Second case: token Z with catcode 13 and a control sequence equal to a token Z with catcode 11.

There is a special rule of \let behavior: if \let\cs=A is done, where A is an active character, then \cs isn't equal to this active character but to its current meaning. For example, if A is a macro then \cs is this macro after \let\cs=A. Such a \cs is element of the first equivalence class given by \if (i.e. \if\noexpand\cs\hbox gives true). On the other hand, if the meaning of the active A is a non-active character (by previous \let A=c) and \let\cs=A is done, then \cs belongs to the equivalence class where c is. Moreover, the active A itself is not equal to \string A in this case, but to its meaning c.

Try this:

\catcode`A=13
\def A{macro}
\let\cs=A
\if\noexpand\cs\hbox True\else False\fi     % True
\if\noexpand A\hbox True\else False\fi      % False
\if\noexpand A\string A True\else False\fi  % True

\let A=c
\let\cs=A
\if\noexpand\cs\hbox True\else False\fi % False
\if\noexpand\cs c True\else False\fi    % True

\if\noexpand A\string A True\else False \fi % False
\if\noexpand Ac True\else False\fi          % True

Best Answer

Related Solutions

Reliably obtain a control sequence token which is undefined

How does \if work

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