Lisp programs have their own particular style, often involving mutable state, macros, meta-programming and more, which is to be expected of such a flexible language. Haskell, on the other hand, seems to be a different world altogether, encapsulating effects with monads, a strong type system, and occasionally, use of Template Haskell.

So how difficult would it be to translate a Common Lisp program to Haskell, in a way that makes the translated code seem idiomatic? The answer is through a careful choice of what Haskell features to use, in this case, monad transformers, but also a lesser-known technique—the tagless final style.

Get the full code here.

The Common Lisp program: an assembler for FRACTRAN

A couple of years ago, malisper wrote a blog post on writing an assembler for the esoteric programming language FRACTRAN in Common Lisp. It’s quite a nice display of the power of Common Lisp, particularly the macro system. We’re going to display a Common Lisp code block followed by its translated Haskell variant, when possible.

(defparameter *cur-inst-prime* nil)
(defparameter *next-inst-prime* nil)
(defparameter *lisptran-labels* nil)
(defparameter *lisptran-vars* nil)
(defparameter *next-new-prime* nil)

Unsurprisingly, this global state is encapsulated with the state monad. ExceptT appears too because we want to be able to throw an error when a label is not found in the map. We’re also using the Math.NumberTheory.Primes package to generate the infinite stream of primes for us.

import qualified Data.Map as M
import qualified Math.NumberTheory.Primes as P
data CompState =
    { currInstPrime, nextInstPrime :: Integer
    , labels, vars :: M.Map String Integer
    , primes :: [P.Prime Integer]
    , gensymCount :: Integer
  deriving (Show)
newtype Comp a =
  Comp { runComp :: ExceptT String (State CompState) a }
  deriving ( Functor, Applicative, Monad, 
           , MonadState CompState, MonadError String)

new-prime generates a fresh prime and places it in *next-new-prime*. In our case, we have an infinite list of primes, so newPrime should just advance the list and return the old head.

(defun new-prime ()
  "Returns a new prime we haven't used yet."
  (prog1 *next-new-prime*
    (setf *next-new-prime*
          (loop for i from (+ *next-new-prime* 1)
                if (prime i)
                  return i))))
newPrime :: Comp Integer
newPrime = do
  l <- gets primes
  modify (\s -> s {primes = tail l})
  return (P.unPrime (head l))

advance is a sequence of assignments, so the translation is straightforward.

(defun advance ()
  (setf *cur-inst-prime* *next-inst-prime*
        *next-inst-prime* (new-prime)))
advance :: Comp ()
advance = do
  c <- gets nextInstPrime
  p <- newPrime
  modify (\s -> s {currInstPrime = c, nextInstPrime = p})

prime-for-label looks up a label and returns its value if found, and inserts it otherwise. prime-for-var is defined similarly.

(defun prime-for-label (label)
  (or (gethash label *lisptran-labels*)
      (setf (gethash label *lisptran-labels*)
primeForLabel :: String -> Comp Integer
primeForLabel label = do
  labels <- gets labels
  case M.lookup label labels of
    Just p -> return p
    Nothing -> do
      p <- newPrime
      modify (\s -> s {labels = M.insert label p labels})
      return p

An awkward step

Now we run into a little bit of an issue;

(defmacro deftran (name args &body body)
  "Define a Lisptran macro."
  (setf (gethash ',name *lisptran-macros*)
        (lambda ,args ,@body)))

We don’t have macros in Haskell! This is where our translation starts to diverge. In such a case, it is useful to read the rest of the Lisp code and see the larger structures at play, in this case, how the deftran macro is used, for instance, in the definitions of addi, subi and >=i.

(deftran addi (x y)
  (prog1 (list (/ (* *next* (expt (prime-for-var x) y))
(deftran subi (x y) ((addi x ,(- y))))
(deftran >=i (var val label)
  (prog1 (let ((restore (new-prime)))
           (list (/ restore
                    (expt (prime-for-var var) val)
                 (/ (* (prime-for-label label)
                       (expt (prime-for-var var) val))
                 (/ *next-inst-prime* *cur-inst-prime*)))

It would seem that we are stuck. We could generate lists of Rationals, but the use of advance forces us to use the State monad. Furthermore, in subi, it calls addi!

One approach would be to express the instructions as a data type;

type Var = String
type Label = String
data Instr = Addi Var Int
           | Subi Var Int
           | Jge Var Int Label

But we lose a critical feature of macros, that they can be used in other macros, such as subi calling addi, and when we add a new instruction, we have to go through the entire codebase to handle the extra case, this is the infamous expression problem. Fortunately, much work has been carried out in attempting to resolve this, with one promising approach being the tagless final approach. That is, can express we addi, subi and more using a typeclass, rather than a data declaration? The answer is a resounding yes.

A macro is a tagless final encoding!

class MonadState repr => FracComp repr where
  lit :: Integer -> repr [Rational]
  label :: String -> repr [Rational]
  addi :: String -> Integer -> repr [Rational]
  jge :: String -> Integer -> String -> repr [Rational]
  gensym :: repr String
  subi :: String -> Integer -> repr [Rational]
  subi x y = addi x (-y)

Now the definition of subi looks just like the Lisp one! What’s going on in this typeclass is that repr is a higher-kinded type, repr :: * -> *. The FracComp typeclass has a constraint, repr has to support being a State monad, because we will need a notion of sequencing label generation to assemble programs correctly.

This extends naturally to deftran definitions that have side effects, for instance, gensym in goto.

(deftran goto (label) `((>=i ,(gensym) 0 ,label)))
goto :: FracComp repr => String -> repr [Rational]
goto dest = do
  g <- gensym
  jge g 0 dest

That’s neat, but now we only have a typeclass, we need to actually instantiate it. Indeed, Comp can be made an instance of FracComp.

instance FracComp Comp where
  addi x 0 = primeForVar x $> []
  addi x y = do
    g <- (^ abs y) <$> primeForVar x
    f <-
      if y < 0
        then (%) <$> gets nextInstPrime <*> ((* g) <$> gets currInstPrime)
        else (%) <$> ((* g) <$> gets nextInstPrime) <*> gets currInstPrime
    return [f]
  gensym = newsym
newsym = do
  n <- gets gensymCount
  modify (\s -> s {gensymCount = n + 1})
  return ('t' : show n)

There’s a little bit of a hiccup when y is negative, because raising to a negative exponent raises an error. Otherwise, the code is remarkably close to Lisp.

Now we need to actually assemble a program. assemble initializes the state to the initial state.

(defun assemble (insts)
  "Compile the given Lisptran program into Fractran. 
   Returns two values. The first is the Fractran program 
   and the second is the alphabet of the program."
  (let* ((*cur-prime* 2)
         (*cur-inst-prime* (new-prime))
         (*next-inst-prime* (new-prime))
         (*lisptran-labels* (make-hash-table))
         (*lisptran-vars* (make-hash-table)))
    (values (assemble-helper insts)
            (alphabet *lisptran-vars*))))
initState =
  let (c:n:p) = P.primes
   in (CompState
         { currInstPrime = P.unPrime c
         , nextInstPrime = P.unPrime n
         , primes = p
         , labels = mempty
         , vars = mempty
         , gensymCount = 0
run a = a & runComp & runExceptT & (`evalState` initState)

Now, we want to run the assembler. Something like this;

λ> [addi "x" 3] :: FracComp repr => [repr [Rational]]
λ> assemble [addi "x" 3] :: FracComp f => f [Rational]

So, assemble should have the following type:

assemble :: FracComp repr => [repr [Rational]] -> repr [Rational]

We can calculate it as follows;

λ> :t [addi "x" 3]
it :: FracComp repr => [repr [Rational]]
λ> :t sequence [addi "x" 3]
it :: FracComp m => m [[Rational]]
λ> :t concat <$> sequence [addi "x" 3]
it :: FracComp f => f [Rational]

And for kicks, we can generalize concat to join, yielding our final result.

assemble :: (Traversable m, Monad m, Monad f) => m (f (m a)) -> f (m a)
assemble l = join <$> sequence l
λ> run (assemble [addi "x" 3])
Right [375 % 2]

The genius of the tagless final approach is that it lets us define new data variants, in this case, new modular pieces of FRACTRAN code.

Some examples;

(deftran while (test &rest body)
  (let ((gstart (gensym))
        (gend (gensym)))
    `((goto ,gend)
      (,@test ,gstart))))

(deftran zero (var)
  `((while (>=i ,var 1)
      (subi ,var 1))))

(deftran move (to from)
  (let ((gtemp (gevnsym)))
    `((zero ,to)
      (while (>=i ,from 1)
        (addi ,gtemp 1)
        (subi ,from 1))
      (while (>=i ,gtemp 1)
        (addi ,to 1)
        (addi ,from 1)
        (subi ,gtemp 1)))))
while test body = do
  gstart <- gensym
  gend <- gensym
    (concat [[goto gend,
              label gstart],
              [label gend, test gstart]])

zero var = while (jge var 1) [subi var 1]

move to from = do
  gtemp <- gensym
    [ zero to
    , while (jge from 1)
        [addi gtemp 1, subi from 1]
    , while (jge gtemp 1)
        [addi to 1, addi from 1, subi gtemp 1]

adds a b = do
  gtemp <- gensym
    [ while (jge b 1) [addi gtemp 1, subi b 1]
    , while (jge gtemp 1) [addi a 1, addi b 1, subi gtemp 1]

Because this is a deep embedding, we can write Haskell functions that generate FRACTRAN programs. For instance, a function that takes an integer n and returns a FRACTRAN program that computes the sum of the numbers from 1 to n inclusive.

sumTo :: FracComp repr => Integer -> [repr [Rational]]
sumTo n = [ addi "c" 0
          , addi "n" n
          , while (jge "n" 0)
              [adds "c" "n", subi "n" 1]]

Now let’s see the assembler in action!

λ> runAssembler (sumTo 10)

Right [847425747 % 2,13 % 3,19 % 13,11 % 3,11 % 29,31 % 11,41 % 31,
23 % 11,23 % 47,2279 % 23,59 % 301,59 % 41,67 % 413,329 % 67,61 % 59,
73 % 61,83 % 73,71 % 61,71 % 97,445 % 71,707 % 89,103 % 5353,
103 % 83,109 % 5459,5141 % 109,107 % 103,113 % 749,113 % 19,
131 % 113,29 % 131,127 % 113]

Going beyond: a pretty printer

We’re done. Let’s see what directions we can take our newly translated FRACTRAN assembler. Since we used the tagless final approach, we can do cool things such as interpreting the values under a different semantic domain. In other words, a fully assembled and final (pun intended) program FracComp f => f [Rational] has a concrete type that depends on the appropriate choice of f, which in turn depends on the call site! In particular, we can let f be the newtype S, defined as

newtype S a = S { unS :: StateT Int (Writer [Doc]) a }
          deriving (Functor, Applicative, Monad,
                    MonadWriter [Doc], MonadState Int)

And write the FracComp instance for S.

instance FracComp S where
  lit i = tell [text (show i)] $> []
  label l = tell ["label" <+> text l] $> []
  addi l x = tell ["addi"  <+> text l <+> text (show x)] $> []
  jge l x dest = tell ["jge" <+> text l <+> text (show x) <+> text dest] $> []
  gensym = gets (('g' :) . show) <* modify (+ 1)

pretty :: Traversable t => t (S a) -> Doc
pretty x = x
         & (unS <$>)
         & sequence
         & (`evalStateT` 0)
         & execWriter
         & vcat

pretty works by unwrapping the t (S a) to a stateful writer, then handling the state and writing.

-- Traversable t
pretty x = x                :: t (S a)
         & unS <$>          :: t (StateT Int (Writer [Doc]) a)
         & sequence         :: StateT Int (Writer [Doc]) (t a)
         & (`evalStateT` 0) :: Writer [Doc] (t a)
         & execWriter       :: [Doc]
         & vcat             :: Doc
λ> pretty (sumTo 10)
addi n 10
jge g2 0 g1
label g0
jge g6 0 g5
label g4
addi g3 1
addi n -1
label g5
jge n 1 g4
jge g9 0 g8
label g7
addi c 1
addi n 1
addi g3 -1
label g8
jge g3 1 g7
addi n -1
label g1
jge n 0 g0

But we have just defined the pretty printers for the basic opcodes, let’s also write specialized printers for the high-level constructs like while. Once again, tagless final helps us achieve this.

instance FracComp S where
  lit i = tell [text (show i)] $> []
  label l = tell ["label" <+> text l] $> []
  addi l x = tell [text l <+> "+=" <+> text (show x)] $> []
  jge l x dest = tell [text l <+> ">=" <+> (text (show x) <+> text dest)] $> []
  gensym = gets (('g' :) . show) <* modify (+ 1)
  jle l x dest = tell [text l <+> "<=" <+> (text (show x) <+> text dest)] $> []
  adds l x = tell [text l <+> "+=" <+> text x] $> []
  subi l x = tell [text l <+> "-=" <+> text (show x)] $> []
  goto l = tell ["goto" <+> text l] $> []
  while test body = do
    censor ((\x -> "while " <> x <> "{") <$>) (test "")
    censor (nest 2 <$>) (sequence body)
    tell ["}"]
    return []

As a result, we can now output FRACTRAN programs in a language resembling C.

λ> pretty (sumTo 10)
c += 0
n += 10
while n >= 0 {
  c += n
  n -= 1


Porting code can be challenging, as there are multiple facets to consider, for instance, what if the target language lacked a feature of the source language? Keeping it idiomatic across paradigms adds additional challenges. In this translation, some Lisp functions were omitted entirely, either because they were not needed or did not fit with the model (for instance, the assemble-helper function). Nevertheless, code translation is a (in my opinion) good way to deepen understanding and practice.

Get the full code here.