## LATEX-L@LISTSERV.UNI-HEIDELBERG.DE

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Last week, I was reading up a bit on lambda calculus (a subject with which I
suspect there are list members far more familiar than I am). One thing that
caught my eye was the definitions of True and False, using the so-called
Church booleans

True = \lambda xy . x
False = \lambda xy . y

These may look strange, but turn out to be two objects that are very
familiar to us: \use_i:nn and \use_ii:nn respectively (or \@firstoftwo and
\@secondoftwo, for those who still think in 2e terms (like me)). Indeed,
putting my mind in LaTeX mode made the lambda calculus concepts much easier
to digest.

But with such thinking comes also the converse association: might LaTeX
programming benefit from borrowing concepts from lambda calculus? In
particular: might the Church booleans be useful as general-purpose booleans
in LaTeX3?!?

(A rather early thought then was that Frank's generation probably knows all
But I don't recall having seen it discussed anywhere.)

One thing that is really nice about the Church booleans is how they support
the logical operations. One can define

\cs_new:Npn \and:nnTF #1 #2 { #1 {#2} {\use_ii:nn} }
\cs_new:Npn  \or:nnTF #1 #2 { #1 {\use_i:nn} {#2} }
\cs_new:Npn \not:nTF  #1 { #1 {\use_ii:nn} {\use_i:nn} }

and go

\cs_set_eq:NN \foo_a_bool \use_i:nn
\cs_set_eq:NN \foo_b_bool \use_ii:nn
\or:nnTF {
\and:nnTF { \foo_a_bool } { \foo_b_bool }
} {
\not:nTF { \foo_b_bool }
} {True} {False}

to get the correct result True. These \and:, \or:, and \not: can even
operate on general \...:TF constructions, and they work correctly regardless
of whether the \...:TF is expandable or needs to be executed! (Like e.g.
\regex_match:nnTF.) This is already much nicer than what one can easily get
from the primitive \if... \fi conditionals. Using this for LaTeX3 booleans
would also have the interesting property that "predicates" and TF forms
become the exact same thing!

I can imagine that a downside of using this would be that it suggests a
coding style that could be considered "too exotic". It sometimes seems that
l3 tries to make things look very much like they would in a traditional
imperative programming language, even though what goes on under the hood can
be quite different. A concept like Church booleans rather furthers
functional idioms of programming, where a lot of the "data" you pass around
is also "functions" (or partially applied functions), that can be applied
with little or no syntactic framing in the source code.

Now I wonder: Would it be useful? (I think it would.) Would there still be
time to change? Does it seem worth it?

Lars Hellström