fac n = if n == 0 then 1 else n * fac (n-1)
fac = (\(n) -> (if ((==) n 0) then 1 else ((*) n (fac ((-) n 1)))))
fac 0 = 1 fac (n+1) = (n+1) * fac n
fac 0 = 1 fac n = n * fac (n-1)
fac n = foldr (*) 1 [1..n]
fac n = foldl (*) 1 [1..n]
-- using foldr to simulate foldl fac n = foldr (\x g n -> g (x*n)) id [1..n] 1
facs = scanl (*) 1 [1..] fac n = facs !! n
fac = foldr (*) 1 . enumFromTo 1
fac n = result (for init next done) where init = (0,1) next (i,m) = (i+1, m * (i+1)) done (i,_) = i==n result (_,m) = m for i n d = until d n i
fac n = snd (until ((>n) . fst) (\(i,m) -> (i+1, i*m)) (1,1))
facAcc a 0 = a facAcc a n = facAcc (n*a) (n-1) fac = facAcc 1
facCps k 0 = k 1 facCps k n = facCps (k . (n *)) (n-1) fac = facCps id
y f = f (y f) fac = y (\f n -> if (n==0) then 1 else n * f (n-1))
s f g x = f x (g x) k x y = x b f g x = f (g x) c f g x = f x g y f = f (y f) cond p f g x = if p x then f x else g x fac = y (b (cond ((==) 0) (k 1)) (b (s (*)) (c b pred)))
arb = () -- "undefined" is also a good RHS, as is "arb" :) listenc n = replicate n arb listprj f = length . f . listenc listprod xs ys = [ i (x,y) | x<-xs, y<-ys ] where i _ = arb facl  = listenc 1 facl n@(_:pred) = listprod n (facl pred) fac = listprj facl
-- a dynamically-typed term language data Term = Occ Var | Use Prim | Lit Integer | App Term Term | Abs Var Term | Rec Var Term type Var = String type Prim = String -- a domain of values, including functions data Value = Num Integer | Bool Bool | Fun (Value -> Value) instance Show Value where show (Num n) = show n show (Bool b) = show b show (Fun _) = "
" prjFun (Fun f) = f prjFun _ = error "bad function value" prjNum (Num n) = n prjNum _ = error "bad numeric value" prjBool (Bool b) = b prjBool _ = error "bad boolean value" binOp inj f = Fun (\i -> (Fun (\j -> inj (f (prjNum i) (prjNum j))))) -- environments mapping variables to values type Env = [(Var, Value)] getval x env = case lookup x env of Just v -> v Nothing -> error ("no value for " ++ x) -- an environment-based evaluation function eval env (Occ x) = getval x env eval env (Use c) = getval c prims eval env (Lit k) = Num k eval env (App m n) = prjFun (eval env m) (eval env n) eval env (Abs x m) = Fun (\v -> eval ((x,v) : env) m) eval env (Rec x m) = f where f = eval ((x,f) : env) m -- a (fixed) "environment" of language primitives times = binOp Num (*) minus = binOp Num (-) equal = binOp Bool (==) cond = Fun (\b -> Fun (\x -> Fun (\y -> if (prjBool b) then x else y))) prims = [ ("*", times), ("-", minus), ("==", equal), ("if", cond) ] -- a term representing factorial and a "wrapper" for evaluation facTerm = Rec "f" (Abs "n" (App (App (App (Use "if") (App (App (Use "==") (Occ "n")) (Lit 0))) (Lit 1)) (App (App (Use "*") (Occ "n")) (App (Occ "f") (App (App (Use "-") (Occ "n")) (Lit 1)))))) fac n = prjNum (eval  (App facTerm (Lit n)))
-- static Peano constructors and numerals data Zero data Succ n type One = Succ Zero type Two = Succ One type Three = Succ Two type Four = Succ Three -- dynamic representatives for static Peanos zero = undefined :: Zero one = undefined :: One two = undefined :: Two three = undefined :: Three four = undefined :: Four -- addition, a la Prolog class Add a b c | a b -> c where add :: a -> b -> c instance Add Zero b b instance Add a b c => Add (Succ a) b (Succ c) -- multiplication, a la Prolog class Mul a b c | a b -> c where mul :: a -> b -> c instance Mul Zero b Zero instance (Mul a b c, Add b c d) => Mul (Succ a) b d -- factorial, a la Prolog class Fac a b | a -> b where fac :: a -> b instance Fac Zero One instance (Fac n k, Mul (Succ n) k m) => Fac (Succ n) m -- try, for "instance" (sorry): -- -- :t fac four
-- the natural numbers, a la Peano data Nat = Zero | Succ Nat -- iteration and some applications iter z s Zero = z iter z s (Succ n) = s (iter z s n) plus n = iter n Succ mult n = iter Zero (plus n) -- primitive recursion primrec z s Zero = z primrec z s (Succ n) = s n (primrec z s n) -- two versions of factorial fac = snd . iter (one, one) (\(a,b) -> (Succ a, mult a b)) fac' = primrec one (mult . Succ) -- for convenience and testing (try e.g. "fac five") int = iter 0 (1+) instance Show Nat where show = show . int (zero : one : two : three : four : five : _) = iterate Succ Zero
-- (curried, list) fold and an application fold c n  = n fold c n (x:xs) = c x (fold c n xs) prod = fold (*) 1 -- (curried, boolean-based, list) unfold and an application unfold p f g x = if p x then  else f x : unfold p f g (g x) downfrom = unfold (==0) id pred -- hylomorphisms, as-is or "unfolded" (ouch! sorry ...) refold c n p f g = fold c n . unfold p f g refold' c n p f g x = if p x then n else c (f x) (refold' c n p f g (g x)) -- several versions of factorial, all (extensionally) equivalent fac = prod . downfrom fac' = refold (*) 1 (==0) id pred fac'' = refold' (*) 1 (==0) id pred
-- (product-based, list) catamorphisms and an application cata (n,c)  = n cata (n,c) (x:xs) = c (x, cata (n,c) xs) mult = uncurry (*) prod = cata (1, mult) -- (co-product-based, list) anamorphisms and an application ana f = either (const ) (cons . pair (id, ana f)) . f cons = uncurry (:) downfrom = ana uncount uncount 0 = Left () uncount n = Right (n, n-1) -- two variations on list hylomorphisms hylo f g = cata g . ana f hylo' f (n,c) = either (const n) (c . pair (id, hylo' f (c,n))) . f pair (f,g) (x,y) = (f x, g y) -- several versions of factorial, all (extensionally) equivalent fac = prod . downfrom fac' = hylo uncount (1, mult) fac'' = hylo' uncount (1, mult)
-- explicit type recursion based on functors newtype Mu f = Mu (f (Mu f)) deriving Show in x = Mu x out (Mu x) = x -- cata- and ana-morphisms, now for *arbitrary* (regular) base functors cata phi = phi . fmap (cata phi) . out ana psi = in . fmap (ana psi) . psi -- base functor and data type for natural numbers, -- using a curried elimination operator data N b = Zero | Succ b deriving Show instance Functor N where fmap f = nelim Zero (Succ . f) nelim z s Zero = z nelim z s (Succ n) = s n type Nat = Mu N -- conversion to internal numbers, conveniences and applications int = cata (nelim 0 (1+)) instance Show Nat where show = show . int zero = in Zero suck = in . Succ -- pardon my "French" (Prelude conflict) plus n = cata (nelim n suck ) mult n = cata (nelim zero (plus n)) -- base functor and data type for lists data L a b = Nil | Cons a b deriving Show instance Functor (L a) where fmap f = lelim Nil (\a b -> Cons a (f b)) lelim n c Nil = n lelim n c (Cons a b) = c a b type List a = Mu (L a) -- conversion to internal lists, conveniences and applications list = cata (lelim  (:)) instance Show a => Show (List a) where show = show . list prod = cata (lelim (suck zero) mult) upto = ana (nelim Nil (diag (Cons . suck)) . out) diag f x = f x x fac = prod . upto
-- explicit type recursion with functors and catamorphisms newtype Mu f = In (f (Mu f)) unIn (In x) = x cata phi = phi . fmap (cata phi) . unIn -- base functor and data type for natural numbers, -- using locally-defined "eliminators" data N c = Z | S c instance Functor N where fmap g Z = Z fmap g (S x) = S (g x) type Nat = Mu N zero = In Z suck n = In (S n) add m = cata phi where phi Z = m phi (S f) = suck f mult m = cata phi where phi Z = zero phi (S f) = add m f -- explicit products and their functorial action data Prod e c = Pair c e outl (Pair x y) = x outr (Pair x y) = y fork f g x = Pair (f x) (g x) instance Functor (Prod e) where fmap g = fork (g . outl) outr -- comonads, the categorical "opposite" of monads class Functor n => Comonad n where extr :: n a -> a dupl :: n a -> n (n a) instance Comonad (Prod e) where extr = outl dupl = fork id outr -- generalized catamorphisms, zygomorphisms and paramorphisms gcata :: (Functor f, Comonad n) => (forall a. f (n a) -> n (f a)) -> (f (n c) -> c) -> Mu f -> c gcata dist phi = extr . cata (fmap phi . dist . fmap dupl) zygo chi = gcata (fork (fmap outl) (chi . fmap outr)) para :: Functor f => (f (Prod (Mu f) c) -> c) -> Mu f -> c para = zygo In -- factorial, the *hard* way! fac = para phi where phi Z = suck zero phi (S (Pair f n)) = mult f (suck n) -- for convenience and testing int = cata phi where phi Z = 0 phi (S f) = 1 + f instance Show (Mu N) where show = show . int
fac n = product [1..n]
Some time later, I came across Iavor’s "jokes" page, including a funny bit called “The Evolution of a Programmer” in which the traditional imperative "Hello, world" program is developed through several variations, from simple beginnings to a ridiculously complex extreme. A moment’s thought turned up the factorial function as the best functional counterpart of "Hello, world". Suddenly the Muse struck and I knew I must write out these examples, culminating (well, almost) in the heavily generalized categorical version of factorial provided by Uustalu, Vene and Pardo.
I suppose this is what you’d have to call “small-audience” humour.
PS: I’ve put all the code into a better-formatted text file for those who might like to experiment with the different variations (you could also just cut and paste a section from your browser).
PPS: As noted above, Iavor is not the original author of “The Evolution of a Programmer.” A quick web search suggests that there are thousands of copies floating around and it appears (unattributed) in humor newsgroups as far back as 1995. But I suspect some version of it goes back much further than that. Of course, if anyone does know who wrote the original, please let me know so that I may credit them here.
The first version (straight recursion with conditionals) is probably familiar to programmers of all stripes; fans of LISP and Scheme will find the sophomore version especially readable, except for the funny spelling of “lambda” and the absence of “define” (or “defun”). The use of patterns may seem only a slight shift in perspective, but in addition to mirroring mathematical notation, patterns encourage the view of data types as initial algebras (or as inductively defined).
The use of more “structural” recursion combinators (such as foldr and foldl) is square in the spirit of functional programming: these higher-order functions abstract away from the common details of different instances of recursive definitions, recovering the specifics through function arguments. The “points-free” style (defining functions without explicit reference to their formal parameters) can be compelling, but it can also be over-done; here the intent is to foreshadow similar usage in some of the later, more stridently algebraic variations.
The accumulating-parameter version illustrates a traditional technique for speeding up functional code. It is the second fastest implementation here, at least as measured in terms of number of reductions reported by Hugs, with the iterative versions coming in third. Although the latter run somewhat against the spirit of functional programming, they do give the flavor of the functional simulation of state as used in denotational semantics or, for that matter, in monads. (Monads are woefully un-represented here; I would be grateful if someone could contribute a few (progressive) examples in the spirit of the development above.) The continuation-passing version recalls a denotational account of control (the references are to Steele’s RABBIT compiler for Scheme and the SML/NJ compiler).
The fixed-point version demonstrates that we can isolate recursion in a general Y combinator. The combinatory version provides an extreme take on the points-free style inspired by Combinatory Logic, isolating dependence on variable names to the definitions of a few combinators. Of course we could go further, defining the Naturals and Booleans in combinatory terms, but note that the predecessor function will be a bit hard to accomodate (this is one good justification for algebraic types). Also note that we cannot define the Y combinator in terms of the others without running into typing problems (due essentially to issues of self-application). Interestingly, this is the fastest of all of the implementations, perhaps reflecting the underlying graph reduction mechanisms used in the implementation.
The list-encoded version exploits the simple observation that we can count in unary by using lists of arbitrary elements, so that the length of a list encodes a natural number. In some sense this idea foreshadows later versions based on recursive type definitions for Peano’s naturals, since lists of units are isomorphic to naturals. The only interesting thing here is that multiplication (numeric product) is seen to arise naturally out of combination (Cartesian product) by way of cardinality. Typing issues make it hard to express this correspondence as directly as we’d like: the following definition of listprod would break the definition of the facl function due to an occurs-check/infinite type:
Of course we could also simplify as follows, but only at the expense of obscuring the relationship between the two kinds of products:
listprod xs ys = [ (x,y) | x<-xs, y<-ys ]
listprod xs ys = [ arb | x<-xs, y<-ys ]
The interpretive version implements a small object language rich enough to express factorial, and then implements an interpreter for it based on a simple environment model. Exercises along these lines run all through the latter half of the Friedman, Wand and Haynes text (), albeit expressed there in Scheme. We used to get flack from students at Oberlin when we made them implement twelve interpreters in a single week-long lab, successively exposing more of the implementation by moving the real work from the meta-language to the interpreter. This implementation leaves a whole lot on the shoulders of the meta-language, corresponding to about Tuesday or Wednesday in their week. Industrious readers are invited to implement a compiler for a Squiggol-like language of polytypic folds and unfolds, targeting (and simulating) a suitable categorical abstract machine (see ), and then to implement factorial in that setting (but don't blame me if it makes you late for lunch ...).
The statically-computed version uses type classes and functional dependencies to facilitate computation at compile time (the latter are recent extensions to the Haskell 98 standard by Mark Jones, and are available in Hugs and GHC). The same kinds of techniques can also be used to encode behaviors more often associated with dependent types and polytypic programming, and are thus a topic of much recent interest in the Haskell community. The code shown here is based on an account by Thomas Hallgren (see ), extended to include factorial. Prolog fans will find the definitions particularly easy to read, if a bit backwards.
The first of the “graduate” versions gets more serious about recursion, defining natural numbers as a recursive algebraic datatype and highlighting the difference between iteration and primitive recursion. The “origamist” and “cartesian” variations take a small step backwards in this regard, as they return to the use of internal integer and list types. They serve, however, to introduce anamorphic and hylomorphic notions in a more familiar context.
The “Ph.D” example employs the categorical style of BMF/Squiggol in a serious way (we could actually go a bit further, by using co-products more directly, and thus eliminate some of the overt dependence on the “internal sums” of the data type definition mechanism).
By the time we arrive at the “pičce de résistance”, the comonadic version of Uustalu, Vene and Pardo, we have covered most of the underlying ideas and can (hopefully) concentrate better on their specific contributions. The final version, using the Prelude-defined product function and ellipsis notation, is how I think the function is most clearly expressed, presuming some knowledge of the language and Prelude definitions. (This definition also dates back at least to David Turner’s KRC* language .)
It is comforting to know that the Prelude ultimately uses a recursion combinator (foldl', the strict version of foldl) to define product. I guess we can all hope to see the day when the Prelude will define gcatamorphic, zygomorphic and paramorphic combinators for us, so that factorial can be defined both conveniently and with greater dignity :) .