# A Combinatory Compiler

The compiler below accepts a Turing-complete language and produces WebAssembly. The source should consist of lambda calculus definitions including a function main that outputs a Church-encoded integer.

intermediate form:

wasm:

## Parser

{-# LANGUAGE CPP #-}
#ifdef __HASTE__
import Haste.DOM
import Haste.Events
import Haste.Foreign
import Numeric
#else
#endif
import Data.Char
import qualified Data.IntMap as I
import Data.List
import Data.Maybe
import Text.Parsec

infixl 5 :@
data Expr = Expr :@ Expr | Var String | Lam String Expr deriving Eq

source :: Parsec String () [(String, Expr)]
source = catMaybes <$> many maybeLet where maybeLet = between ws newline$ optionMaybe $(,) <$> v <*> (str "=" >> term)
term = lam <|> app
lam = flip (foldr Lam) <$> between lam0 lam1 (many1 v) <*> term where lam0 = str "\\" <|> str "\955" lam1 = str "->" <|> str "." app = foldl1' (:@) <$> many1
((Var <$> v) <|> between (str "(") (str ")") term) v = many1 alphaNum <* ws str = (>> ws) . string ws = many (oneOf " \t") >> optional (try$ string "--" >> many (noneOf "\n"))


## Combinators

We recap part of our notes on combinatory logic. Define the combinators:

• $$S = \lambda x y z . x z (y z)$$

• $$K = \lambda x y . x$$

The classic bracket abstraction algorithm with the K-optimization is:

\begin{align} \lceil \lambda x . M \rceil &= K M \quad (x \notin M) \\ \lceil \lambda x . x \rceil &= S K K \\ \lceil \lambda x . M N \rceil &= S \lceil \lambda x . M \rceil \lceil \lambda x . N \rceil \end{align}

Any closed lambda term can be rewritten in terms of $$S$$ and $$K$$ by applying the above rules starting from the innermost lambda abstraction and working outwards:

lacks x t = case t of
Var s | s == x -> False
u :@ v -> lacks x u && lacks x v
_ -> True

babs0 env (Lam x e)
| lacks x t   = Var "k" :@ t
| otherwise   = case t of
Var y  -> Var "s" :@ Var "k" :@ Var "k"
m :@ n -> Var "s" :@ babs0 env (Lam x m) :@ babs0 env (Lam x n)
where t = babs0 env e
babs0 env (Var s)
| Just t <- lookup s env = babs0 env t
| otherwise              = Var s
babs0 env (m :@ n) = babs0 env m :@ babs0 env n


We also mentioned David Turner found more optimizations, enough to make bracket abstraction practical. However, he used more combinators than just $$S$$ and $$K$$. Luckily, John Tromp ported the rules to $$S$$ and $$K$$:

babs env (Lam x e) = go $babs env e where go t | Var "s" :@ Var "k" :@ _ <- t = Var "s" :@ Var "k" | lacks x t = Var "k" :@ t | Var y <- t, x == y = Var "s" :@ Var "k" :@ Var "k" | m :@ Var y <- t, x == y, lacks x m = m | Var y :@ m :@ Var z <- t, x == y, x == z = go$ Var "s" :@ Var "s" :@ Var "k" :@ Var x :@ m
| m :@ (n :@ l) <- t, isComb m, isComb n =
go $Var "s" :@ go m :@ n :@ l | (m :@ n) :@ l <- t, isComb m, isComb l = go$ Var "s" :@ m :@ go l :@ n
| (m :@ l) :@ (n :@ l') <- t, l == l', isComb m, isComb n =
go $Var "s" :@ m :@ n :@ l | m :@ n <- t = Var "s" :@ go m :@ go n babs env (Var s) | Just t <- lookup s env = babs env t | otherwise = Var s babs env (m :@ n) = babs env m :@ babs env n isComb t = case t of Var "s" -> True Var "k" -> True Var _ -> False u :@ v -> isComb u && isComb v  The above assumes we have no recursive let definitions and that s and k are reserved keywords. Enforcing this is left as an exercise. A few lines in the Either monad glues together our parser and our bracket abstraction routine: toSK s = do env <- parse source "" (s ++ "\n") case lookup "main" env of Nothing -> Left$ error "missing main"
Just t -> pure $babs env t :@ Var "u" :@ Var "z"  We’ve introduced two more combinators: u and z, which we think of as the successor function and zero respectively. Given a Church encoding M of an integer n, the expression Muz evaluates to u(u(...u(z)...)), where there are n occurrences of u. We make u increment a counter, and we make z return it, so when evaluated in normal order it returns n. ## Graph Reduction We encode the tree representing our program into an array, then write WebAssembly to manipulate this tree. In other words, we model computation as graph reduction. We view linear memory as an array of 32-bit integers. The values 0-3 represent leaf nodes z,u,k,s in that order, while any other value n represents an internal node with children represented by the 32-bit integers stored in linear memory at n and n + 4. We encode the tree so that address 4 holds the root of the tree. Since 0 represents a leaf node, the first 4 bytes of linear memory cannot be addressed, so their contents are initialized to zero and ignored. toArr n (Var "z") =  toArr n (Var "u") =  toArr n (Var "k") =  toArr n (Var "s") =  toArr n (x@(Var _) :@ y@(Var _)) = toArr n x ++ toArr n y toArr n (x@(Var _) :@ y) = toArr n x ++ [n + 2] ++ toArr (n + 2) y toArr n (x :@ y@(Var _)) = n + 2 : toArr n y ++ toArr (n + 2) x toArr n (x :@ y) = [n + 2, nl] ++ l ++ toArr nl y where l = toArr (n + 2) x nl = n + 2 + length l encodeTree :: Expr -> [Int] encodeTree e = concatMap f$ 0 : toArr 4 e where
f n | n < 4     = [n, 0, 0, 0]
| otherwise = toU32 $(n - 3) * 4 toU32 = take 4 . byteMe byteMe n | n < 256 = n : repeat 0 | otherwise = n mod 256 : byteMe (n div 256)  Our run function takes the current and a stack of addresses state of linear memory, and simulates what our assembly code will do. For the z combinator, we return 0. For the u combinator we return 1 plus the result of evaluating its argument. For the k combinator, we pop off the last two stack elements and push the evaluation of its first argument. For s we create two internal nodes representing xz and yz on the the heap hp, where x,y,z are the arguments of s. Then we lazily evaluate: we rewrite the immediate children of the parent of the z node to apply the first of the newly created nodes to the other. For internal nodes, we push the first child on the stack then recurse. We assume the input program is well-formed, that is, every k is given exactly 2 arguments, every s is given exactly 3 arguments, and so on. run m (p:sp) = case p of 0 -> 0 1 -> 1 + run m (arg 0 : sp) 2 -> run m$ arg 0 : drop 2 sp
3 -> run m' $hp:drop 2 sp where m' = insList m$
zip [hp..]    (concatMap toU32 [arg 0, arg 2, arg 1, arg 2]) ++
zip [sp!!2..] (concatMap toU32 [hp, hp + 8])
hp = I.size m
_ -> run m $get p:p:sp where arg k = get (sp!!k + 4) get n = sum$ zipWith (*) ((m I.!) <$> [n..n+3]) ((256^) <$> [0..3])
insList = foldr (\(k, a) m -> I.insert k a m)


## Machine Code

We convert the above to assembly. First, a few constants and helpers:

compile :: [Int] -> [Int]
compile heap = let
typeFunc = 0x60
typeI32  = 0x7f
br       = 0xc
getlocal = 0x20
setlocal = 0x21
teelocal = 0x22
i32store = 0x36
i32const = 0x41
i32sub   = 0x6b
i32mul   = 0x6c
i32shl   = 0x74
i32shr_s = 0x75
i32shr_u = 0x76
i64const = 0x42
i64store = 0x37
i64shl   = 0x86
i64extendui32 = 0xac
nPages = 8
leb128 n | n < 64   = [n]
| n < 128  = [128 + n, 0]
| otherwise = 128 + (n mod 128) : leb128 (n div 128)
varlen xs = leb128 $length xs lenc xs = varlen xs ++ xs encStr s = lenc$ ord <> s encSig ins outs = typeFunc : lenc ins ++ lenc outs sect t xs = t : lenc (varlen xs ++ concat xs)   in concat [ [0, 0x61, 0x73, 0x6d, 1, 0, 0, 0], -- Magic string, version. -- Type section. sect 1 [encSig [typeI32] [], encSig [] []], -- Import section. -- [0, 0] = external_kind Function, index 0. sect 2 [encStr "i" ++ encStr "f" ++ [0, 0]], -- Function section. --  = Type index. sect 3 [], -- Memory section. -- 0 = no-maximum sect 5 [[0, nPages]], -- Export section. -- [0, 1] = external_kind Function, index 1. sect 7 [encStr "e" ++ [0, 1]],  We compile the run function by hand. Initially, our tree is encoded at the bottom of the linear memory, and the stack pointer is at the top. We encounter features of WebAssembly may surprise those who accustomed to other instruction sets. Load and store instructions must be given alignment and offset arguments. There are no explicit labels or jumps. Instead, labels are implicitly defined by declaring well-nested block-end and loop-end blocks, and branch statements break out a given number of blocks.  -- Code section. -- Locals let sp = 0 -- stack pointer hp = 1 -- heap pointer ax = 2 -- accumulator in sect 10 [lenc [1, 3, typeI32,
-- SP = 65536 * nPages - 4
-- [SP] = 4
i32const] ++ leb128 (65536 * nPages - 4) ++ [teelocal, sp,
i32const, 4, i32store, 2, 0,
i32const] ++ varlen heap ++ [setlocal, hp,
3, 0x40,  -- loop
2, 0x40,  -- block 4
2, 0x40,  -- block 3
2, 0x40,  -- block 2
2, 0x40,  -- block 1
2, 0x40,  -- block 0
0xe,4,0,1,2,3,4, -- br_table
0xb,  -- end 0
-- Zero.
getlocal, ax, 0x10, 0,  -- call function 0
br, 5,  -- br function
0xb,  -- end 1
-- Successor.
getlocal, ax, i32const, 1, i32add, setlocal, ax,
-- SP = SP + 4
-- [SP] = [[SP] + 4]
getlocal, sp, i32const, 4, i32add, teelocal, sp,
br, 3,  -- br loop
0xb,  -- end 2
-- K combinator.
-- [SP + 8] = [[SP + 4] + 4]
getlocal, sp,
i32store, 2, 8,
-- SP = SP + 8
getlocal, sp, i32const, 8, i32add, setlocal, sp,
br, 2,  -- br loop
0xb,  -- end 3
-- S combinator.
-- [HP] = [[SP + 4] + 4]
-- [HP + 4] = [[SP + 12] + 4]
getlocal, hp,
i64const, 32, i64shl, i64add, i64store, 3, 0,
-- [HP + 8] = [[SP + 8] + 4]
-- [HP + 12] = [HP + 4]
getlocal, hp,
i64const, 32, i64shl, i64add, i64store, 3, 8,
-- SP = SP + 12
-- [[SP]] = HP
-- [[SP] + 4] = HP + 8
getlocal, sp, i32const, 12, i32add, teelocal, sp,
getlocal, hp, i64extendui32,
i64extendui32, i64const, 32, i64shl, i64add, i64store, 3, 0,
-- HP = HP + 16
getlocal, hp, i32const, 16, i32add, setlocal, hp,
br, 1,  -- br loop
0xb,  -- end 4
-- Application.
-- SP = SP - 4
-- [SP] = [[SP + 4]]
getlocal, sp, i32const, 4, i32sub,
teelocal, sp, getlocal, sp, i32load, 2, 4, i32load, 2, 0, i32store, 2, 0,
br, 0,
0xb,    -- end loop
0xb]],  -- end function


The data section initializes the linear memory so our encoded tree sits at the bottom.

  -- Data section.
sect 11 [[0, i32const, 0, 0xb] ++ lenc heap]]


To keep the code simple, we ignore garbage collection. Because we represent numbers in unary, and also because we only ask for a few pages of memory, our demo only works on relatively small programs.

## User Interface

For the demo, we add a couple of helpers to show the intermediate form and assembly opcodes.

showSK (Var s)  = s
showSK (x :@ y) = showSK x ++ showR y where
showR (Var s) = s
showR _       = "(" ++ showSK y ++ ")"

#ifdef __HASTE__
dump asm = unwords $xxShow <$> asm where
xxShow c = reverse $take 2$ reverse $'0' : showHex c "" main = withElems ["input", "output", "sk", "asm", "evalB"]$
\[iEl, oEl, skEl, aEl, evalB] -> do
let
setResult :: Int -> IO ()
setResult = setProp oEl "value" . show
export "setResult" setResult
evalB onEvent Click $const$ do
setProp oEl "value" ""
setProp skEl "value" ""
setProp aEl "value" ""
s <- getProp iEl "value"
case toSK s of
Left err -> setProp skEl "value" $"error: " ++ show err Right sk -> do let asm = compile$ encodeTree sk
setProp skEl "value" $showSK sk setProp aEl "value"$ dump asm
ffi "runWasmInts" asm :: IO ()
#else
main = interact $\s -> case toSK s of Left err -> "error: " ++ show err Right sk -> unlines [ showSK sk , show$ compile $encodeTree sk , show$ run (I.fromAscList $zip [0..]$ encodeTree sk) 
]
#endif


During development, a REPL for the intermediate language was helpful:

#ifndef __HASTE__
expr :: Parser Expr
expr = foldl1 (:@) <$> many1 ((Var . pure <$> letter) <|> between (char '(') (char ')') expr)

skRepl :: InputT IO ()
skRepl = do
ms <- getInputLine "> "
case ms of
Nothing -> outputStrLn ""
Just s  -> do
let Right e = parse expr "" s
outputStrLn $show$ encodeTree e
outputStrLn $show$ compile $encodeTree e outputStrLn$ show $run (I.fromAscList$ zip [0..] \$ encodeTree e) 
skRepl
#endif


Ben Lynn blynn@cs.stanford.edu 💡