Help to make a faster token replacing macro

Question

I have a macro called \logic which helps me write logical formulas faster by simplifying some common logical symbols. So when I write $\logic{p -> q}$ I obtain $p \to q$.

Here's the code:

\documentclass{article}
\makeatletter
%
% Command to format a formula with a succinct and natural syntax
%
\usepackage{xparse}
\usepackage{xstring}% for \StrSubstitute
\usepackage{pgffor}% for \foreach
% the strange \iffalse{\fi thing is a 'brace hack', to make this work inside
% tabular-like envs. See https://tex.stackexchange.com/questions/452442
\NewDocumentCommand\logic{+m}{\iffalse{\fi
  \def\gt@formula{#1}%
  \saveexpandmode
  \saveexploremode
  \expandarg
  \exploregroups
  \foreach \gt@keyword/\gt@operator in \gt@logic@operators {%
    \StrSubstitute{\gt@formula}{\gt@keyword}{\gt@operator}[@temp]%
    \global\let\gt@formula@temp
  }%
  \restoreexploremode
  \restoreexpandmode
  \ensuremath{\gt@formula}%
\iffalse}\fi}
\def\gt@logic@operators{}
\NewDocumentCommand\RegisterLogicOperators{m}{
  \def\gt@logic@operators{#1}
}
\makeatother
\RegisterLogicOperators{
  !/\neg,
  &&/\land,
  &/\land,
  &/\land,
  ||/\lor,
  |/\lor,
  <->/\longleftrigtharrow,
  ->/\to,
  ~>/\leadsto,
  <-/\impliedby,
  <</\llangle,
  >>/\rrangle,
  <=/\le,
  >=/\ge,
  ==/\equiv,
  [[/\llbracket,
  ]]/\rrbracket,
}
\begin{document}
This is a logic formula: $\logic{((p -> q) & p) -> q}$.
\end{document}

However, the macro is very slow. You don't notice when you use it once, but in a paper with heavy usage the compilation time goes up very quickly. I suppose the bottleneck here is the \StrSubstitute macro from the xstring package, which seems a bit overkill for what I have to do, but I do not know how to obtain the same result faster. Also note that the real-world usage has a much longer list of replacements in the call to \RegisterLogicOperators.

Is there a way to improve the performance of this macro?

Answer 1

You can try this solution in LaTeX3, which converts the entire string in one go. I tried to convert 1000 expressions at once, and it takes less than 5 seconds.

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{amsmath, amssymb}
\usepackage{expl3}
\ExplSyntaxOn
\prop_new:N \g_symbol_conv_prop
\prop_gset_from_keyval:Nn \g_symbol_conv_prop {
  !=\neg,
  &&=\land,
  &=\land,
  &=\land,
  ||=\lor,
  |=\lor,
  <->=\longleftrightarrow,
  ->=\to,
  ~>=\leadsto,
  <-=\impliedby,
  <<=\llangle,
  >>=\rrangle,
  {<=}=\le,
  {>=}=\ge,
  {==}=\equiv,
  [[=\llbracket,
  ]]=\rrbracket
}
% convert keys to string
\prop_new:N \g_str_symbol_conv_prop
\prop_map_inline:Nn \g_symbol_conv_prop {
  \prop_put:Nxx \g_str_symbol_conv_prop {\tl_to_str:n {#1}} {\exp_not:n {#2}}
}
\cs_set:Npn \mult_level_lookup_name:n #1 {
  __g_symbol_lookup_\int_to_alph:n{#1}_seq
}
% find longest symbol
\int_new:N \g_longest_symbol_len_int
\int_set:Nn \g_longest_symbol_len_int {0}
\prop_map_inline:Nn \g_str_symbol_conv_prop {
  \int_set:Nn \l_tmpa_int {\str_count:n {#1}}
  \int_set:Nn \g_longest_symbol_len_int {
    \int_max:nn {\g_longest_symbol_len_int} {\l_tmpa_int}
  }
}
% create multi-level look up tables
\int_step_inline:nn {\g_longest_symbol_len_int} {
  \seq_new:c {\mult_level_lookup_name:n {#1}}
}
% fill multi-level lookup tables
\prop_map_inline:Nn \g_str_symbol_conv_prop {
  \str_set:Nn \l_tmpa_str {#1}
  \int_step_inline:nn {\str_count:N \l_tmpa_str} {
    \seq_push:cx {\mult_level_lookup_name:n {##1}} {
      \str_item:Nn \l_tmpa_str {##1}
    }
  }
}
\tl_new:N \l_logic_result_tl
\int_new:N \l_current_level_int
\str_new:N \l_current_logic_str
\str_new:N \l_current_symbol_str
\msg_new:nnn {logic} {invalidsymbol} {"#1" is not a valid symbol}
\cs_generate_variant:Nn \tl_rescan:nn {nV}
\cs_set:Npn \logic_conv:n #1 {
  \str_set:Nn \l_current_logic_str {#1}
  \int_set:Nn \l_current_level_int {1}
  \tl_clear:N \l_logic_result_tl
  \str_clear:N \l_current_symbol_str
\bool_do_until:nn {\str_if_empty_p:N \l_current_logic_str} {
    % pop first item
    \str_set:Nx \l_tmpa_str {\str_head:N \l_current_logic_str}
    \str_set:Nx \l_current_logic_str {\str_tail:N \l_current_logic_str}
% check if this item can be found in the look up table
\seq_if_in:cVTF {\mult_level_lookup_name:n {\l_current_level_int}} \l_tmpa_str {
  % if it can be found, advance level and store it into \l_current_symbol_str
  \int_incr:N \l_current_level_int
  \str_put_right:NV \l_current_symbol_str \l_tmpa_str
}
{
  % if it cannot be found, it could be the end of a symbol
  % or just a normal string
  \str_if_empty:NF \l_current_symbol_str {
    % if \l_current_symbol_str is not empty, it is the end of a symbol
    % we need to insert the correct symbol and reset variables
    \prop_get:NVNTF \g_str_symbol_conv_prop \l_current_symbol_str \l_tmpa_tl {
      \int_set:Nn \l_current_level_int {1}
      \str_clear:N \l_current_symbol_str
      \tl_put_right:NV \l_logic_result_tl \l_tmpa_tl
    } {
      \tl_put_right:NV \l_logic_result_tl \l_current_symbol_str
      \int_set:Nn \l_current_level_int {1}
      \str_clear:N \l_current_symbol_str
    }
  }
  \tl_put_right:NV \l_logic_result_tl \l_tmpa_str
}

}
  % output last item
  \prop_get:NVNTF \g_str_symbol_conv_prop \l_current_symbol_str \l_tmpa_tl {
    \tl_put_right:NV \l_logic_result_tl \l_tmpa_tl
  } { \tl_put_right:NV \l_logic_result_tl \l_current_symbol_str }
%\tl_show:N \l_logic_result_tl
  \tl_rescan:nV {} \l_logic_result_tl
}
\newcommand{\logic}[1]{
  \logic_conv:n {#1}
}
\ExplSyntaxOff
\begin{document}
$\logic{((p -> q) & p | !r) <= >= == -> <- c | d \phi || \sigma}$
\end{document}