User-Agent: Mozilla/5.0 (Macintosh; U; Intel Mac OS X 10.6; sv-SE; rv:1.9.2.22)
            Gecko/20110902 Thunderbird/3.1.14
MIME-Version: 1.0
Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Content-Transfer-Encoding: 8bit
Message-ID:  <4EA7DB88.7020906@residenset.net>
Date:         Wed, 26 Oct 2011 12:06:00 +0200
Reply-To: Mailing list for the LaTeX3 project
              <LATEX-L@listserv.uni-heidelberg.de>
Sender: Mailing list for the LaTeX3 project <LATEX-L@listserv.uni-heidelberg.de>
From: =?ISO-8859-1?Q?Lars_Hellstr=F6m?= <Lars.Hellstrom@RESIDENSET.NET>
Subject: S combinator and such
To: LATEX-L@listserv.uni-heidelberg.de
Precedence: list
Status: R

Continuing my reading up on Lambda calculus and related matters, I've now 
gotten to the Combinatory logic 
(http://en.wikipedia.org/wiki/Combinatory_logic), which also turns out to be 
much easier to understand when one transcribes the whole thing into LaTeX 
macros. :-) In particular, there are

\cs_new:Npn \combinator_I:n #1 { #1 }      % AKA \use_i:n
\cs_new:Npn \combinator_K:nn #1 #2 { #1 }  % AKA \use_i:nn
\cs_new:Npn \combinator_S:nnn #1 #2 #3 { #1{#3}{ #2{#3} } }
\cs_new:Npn \combinator_B:nnn #1 #2 #3 { #1{ #2{#3} } }
\cs_new:Npn \combinator_C:nnn #1 #2 #3 { #1{#3}{#2} }

My thought with this mail is mainly to ask whether there are any more of 
these (or other standard combinators) that are defined in LaTeX3 already. I 
seem to recall some discussion to the effect that the C combinator might be 
named \use_i_biii_bii:nnn, but I haven't seen any occurrence in the source 
of such _bii_ names (then again, maybe I'm not looking close enough). 
Continuing such a naming scheme, one would arrive at maybe

   \use_i_biibiii:nnn        for the B combinator
   \use_i_biii_biibiii:nnn   for the S combinator

which does not seem entirely practical...

David Carlisle mentioned in the Church booleans thread that
http://www.ctan.org/tex-archive/macros/plain/contrib/lambda provides what 
amounts to the B and C combinators under the (so far most comprehensible) 
names \Compose and \Twiddle respectively. It doesn't provide the S 
combinator up front, but rather

\cs_new:Npn \Lift #1 #2 #3 #4 { #1{#4} {#2} {#3} {#4} }

the primary intent for which is that #1{#4} should behave as a Church 
boolean, and thus select whether to apply #2 or #3 to the second #4. I 
suspect \Lift, \Twiddle, \Compose, and maybe some additional macro could be 
combined into an S combinator, but it is not immediately clear to me how.

(Reading that lambda.tex, I'm surprised that not more of it made it into 
fontinst -- both projects having the same original author -- but three years 
was a lot of time in TeX history, back then.)

Somewhat relatedly, it occurs to me that the process of converting 
lambda-terms to combinator formulae /might/ be the beginning of an approach 
to having "named command parameters" in (high-level) LaTeX without radical 
modifications of the TeX engine -- the idea being that the named parameters 
are (upon macro definition) removed from the replacement text as a matter of 
lambda elimination (conversion of a lambda term to an equivalent 
combinatorial term). Whether that would be practical is of course an 
entirely different matter. ;-)

Lars Hellström