# LC to CL, Semantically

Compilers abandoned bracket abstraction long ago in favour of generating custom combinators for each particular program, known as supercombinators.

We’ll buck this trend, partly for notoriety, but also for novelty: a bracket abstraction algorithm by Oleg Kiselyov breathes new life into the old approach.

Rather than treating lambda terms as a syntax to be rewritten as combinators, Kiselyov defines the meaning of a lambda term using combinators. The formal semantics can be viewed as a bracket abstraction algorithm.

Much of our previous compiler remains the same. We have standard definitions:

or f g x y = f x (g x y);
and f g x y = @C f y (g x y);
lsteq = @Y \r xs ys a b -> xs (ys a (\u u -> b)) (\x xt -> ys b (\y yt -> x(y(@=)) (r xt yt a b) b));
append = @Y \r xs ys -> xs ys (\x xt -> @: x (r xt ys));
pair x y f = f x y;
just x f g = g x;
foldr = @Y \r c n l -> l n (\h t -> c h(r c n t));

Scott encoding of a data structure we use to hold lambda terms:

-- data LC = R String | V String | A LC LC | L String LC
lcr s   = \a b c d -> a s;
lcv v   = \a b c d -> b v;
lca x y = \a b c d -> c x y;
lcl x y = \a b c d -> d x y;

Parser combinators library:

pure x inp = just (pair x inp);
bind f m = m @K (\x -> x f);
ap x y = \inp -> bind (\a t -> bind (\b u -> pure (a b) u) (y t)) (x inp);
fmap f x = ap (pure f) x;
alt x y = \inp -> (x inp) (y inp) just;
liftaa f x y = ap (fmap f x) y;
many = @Y \r p -> alt (liftaa @: p (r p)) (pure @K);
some p = liftaa @: p (many p);
liftki = liftaa (@K @I);
liftk = liftaa @K;
sat f inp = inp @K (\h t -> f h (pure h t) @K);

Parser:

char c = sat (\x -> x(c(@=)));
com = liftki (char #-) (liftki (char #-) (liftki (many (sat (\c -> @C (c(#
(@=)))))) (char #
)));
sp = many (alt (sat (\c -> or (c(# (@=))) (c(#
(@=))))) com);
spc f = liftk f sp;
spch = @B spc char;
var = spc ( some (sat (\x -> and (#z(x(@L))) (x(#a(@L))) )));
anyone = fmap (@C @: @K) (spc (sat (@K @K)));
pre = alt (liftki (char #@) anyone) (liftaa @: (char ##) anyone);
lam r = liftki (spch #\) (liftaa (@C (foldr lcl)) (some var) (liftki (char #-) (liftki (spch #>) r)));
atom r = alt (alt (alt (liftki (spch #() (liftk r (spch #)))) (lam r)) (fmap lcr pre)) (fmap lcv var);
apps = @Y \rr r -> alt (liftaa @T (atom r) (fmap (\vs v x -> vs (lca x v)) (rr r))) (pure @I);
expr = @Y \r -> liftaa @T (atom r) (apps r);
def = liftaa pair var (liftaa (@C (foldr lcl)) (many var) (liftki (spch #=) expr));
program = liftki sp (some (liftk def (spch #;)));

Finally, something new: conversion to De Bruijn notation:

-- data DB = Ze | Su DB | Pass LC | La DB | App DB DB
ze   = \    a b c d e -> a;
su   = \x   a b c d e -> b x;
pass = \x   a b c d e -> c x;
la   = \x   a b c d e -> d x;
app  = \x y a b c d e -> e x y;

debruijn = @Y (\r n e -> e
(\s -> pass (lcr s))
(\v -> foldr (\h m -> lsteq h v ze (su m)) (pass (lcv v)) n)
(\x y -> app (r n x) (r n y))
(\s t -> la (r (@: s n) t))
);

And Kiselyov’s bracket abstraction algorithm from Section 4 of the paper:

closed = \t a b c -> a t;
need   = \x a b c -> b x;
weak   = \x a b c -> c x;

lclo = \r d y -> y
(\dd -> closed (lca d dd))
(\e -> need (r (closed (lca (lcr (@:#B@K)) d)) e))
(\e -> weak (r (closed d) e))
;

lnee = \r e y -> y
(\d -> need (r (closed (lca (lcr (@:#R@K)) d)) e))
(\ee -> need (r (r (closed (lcr (@:#S@K))) e) ee))
(\ee -> need (r (r (closed (lcr (@:#C@K))) e) ee))
;

lwea = \r e y -> y
(\d -> weak (r e (closed d)))
(\ee -> need (r (r (closed (lcr (@:#B@K))) e) ee))
(\ee -> weak (r e ee))
;

babsa = @Y (\r x y -> x
(\d -> lclo r d y)
(\e -> lnee r e y)
(\e -> lwea r e y)
);

babs = @Y (\r t -> t
(need (closed (lcr (@:#I@K))))
(@B weak r)
closed
(\t -> r t
(\d -> closed (lca (lcr (@:#K@K)) d))
@I
(babsa (closed (lcr (@:#K@K)))))
(\x y -> babsa (r x) (r y))
);
nolam x = babs (debruijn @K x) @I @? @?;

That leaves the code generator and the main function tying everything together:

rank v ds = foldr (\d t -> lsteq v (d @K) (\n -> @: #@ (@: n @K)) (@B t \n -> # (#!(@-))(n(@+)) )) (@K v) ds # ;
show = @Y \r ds t -> t @I (\v -> rank v ds) (\x y -> @:#(append (r ds x) (r ds y))) @?;
dump = @Y \r tab ds -> ds @K \h t -> append (show tab (nolam (h (@K @I)))) (@: #; (r tab t));
main s = program s (@:#?@K) (@B (\ds -> dump ds ds) (@T @K));`

The grammar is identical, but the generated code is far smaller. Without garbage collection, the previous compiler requires over 87 million 32-bit words on the heap to compile the above, while this compiler compiles itself using under 11 million 32-bit words.

If our computers were less powerful, it would be better to skip our previous compilers using classic bracket abstraction and go straight to the above, at the cost of converting many more LC terms to CL terms by hand.

Ben Lynn blynn@cs.stanford.edu 💡