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Type level Roman numerals and skew binary random access lists
by Ivan Tomac 10 Aug '11

10 Aug '11

I've recently been playing with Sebastian Fischer's explicit sharing monad library (found at http://sebfisch.github.com/explicit-sharing) and after looking at the code noticed the implementation memoizes values into an IntMap of untyped values that are coerced into the correct type on retrieval, which is a sensible and practical way to implement this. Just for fun I thought I'd have a go at implementing a (very impractical) version without type casting but, disappointingly, I didn't quite get there. I ran out of steam by the time I got around to playing with parameterized monads by which stage the type signatures started looking a bit scary. That and I needed to get back to my projects. This has been sitting on my hard drive for a couple of weeks now and I've finally decided to clean it up and post it, maybe someone else will find it amusing. The post is about HList inspired type level implementation of Okasaki's skew binary random access lists indexed by Roman numerals, using ideas from Ralf Hinze's paper "Functional Pearl: Typed Quote/Antiquote - Or: Compile-time Parsing" and some fun tricks from the MLton wiki: http://www.cs.ox.ac.uk/ralf.hinze/publications/Quote.pdf http://mlton.org/Fold http://mlton.org/Fold01N http://mlton.org/VariableArityPolymorphism With that out of the way, we start of by declaring all the different extensions we'll need. > {-# LANGUAGE FlexibleInstances #-} > {-# LANGUAGE FunctionalDependencies #-} > {-# LANGUAGE MultiParamTypeClasses #-} > {-# LANGUAGE ScopedTypeVariables #-} > {-# LANGUAGE TypeFamilies #-} > {-# LANGUAGE TypeSynonymInstances #-} These two are ugly. We need undecidable instances for bunch of things and overlapping instances for the type equality trick borrowed from the HList paper: http://homepages.cwi.nl/~ralf/HList/paper.pdf > {-# LANGUAGE OverlappingInstances #-} > {-# LANGUAGE UndecidableInstances #-} For GHC 7 we also need to increase the context stack. > {-# OPTIONS_GHC -fcontext-stack=100 #-} Define Boolean values at the type level. > data T = T > data F = F And now Peano numbers. We don't actually use the constructors for anything but if we omitted them we'd have to use another extension, EmptyDataDecls. > data Z = Z > data S n = S n Here we create some types for heterogeneous lists and trees. Note that Pair has an extra type parameter. We provide two different versions of the cons operation. One requires that extra parameter but does not require undecidable instances. Unfortunately we still need undecidable instances later on for indexing. > data Pair i x y = Pair x y deriving Show > data Nil = Nil deriving Show > data Tree l x r = Node l x r deriving Show > data Leaf x = Leaf x deriving Show > > type Pair' x y = Pair () x y > > nil :: Nil > nil = Nil Equality, less than, addition, subtraction and multiplication by 2 type functions for Peano numbers. > type family Equals m n > > type instance Equals Z Z = T > type instance Equals Z (S n) = F > type instance Equals (S n) Z = F > type instance Equals (S m) (S n) = Equals m n > > type family LessThan m n > > type instance LessThan Z Z = F > type instance LessThan Z (S n) = T > type instance LessThan (S n) Z = F > type instance LessThan (S m) (S n) = LessThan m n > > type family Add m n > > type instance Add Z n = n > type instance Add n Z = n > type instance Add (S m) (S n) = S (S (Add m n)) > > type family Sub m n > > type instance Sub n Z = n > type instance Sub (S m) (S n) = Sub m n > > type family Mul2 n > > type instance Mul2 Z = Z > type instance Mul2 (S n) = S (S (Mul2 n)) And finally tree depth and weight/size. > type family Depth t > > type instance Depth (Leaf x ) = S Z > type instance Depth (Tree l x r) = S (Depth l) > > type family Weight t > > type instance Weight (Leaf x ) = S Z > type instance Weight (Tree l x r) = S (Mul2 (Weight l)) For reference, here is normal implementation of random access lists newtype List a = List { unList :: [(Int, Tree a)] } deriving Show data Tree a = Leaf a | Node (Tree a) a (Tree a) deriving Show cons :: a -> List a -> List a cons x (List wts) = List $ case wts of (v, l) : (w, r) : wts' | v == w -> (1 + w + w, Node l x r) : wts' _ -> (1 , Leaf x ) : wts List here is a list of pairs of balanced binary trees and their weights. The cons function has exactly two cases. The second case is trivial, we are mostly interested in the first case. If the weights of the two trees at the front of the list are the same, we combine them into a new tree with the newly inserted element at the top. Now we are ready to implement cons. This first version is simpler but requires undecidable instances. > class Cons' x y z | x y -> z where cons' :: x -> y -> z Here we cover the cases of inserting an element into an empty list, and inserting an element into a list with a single element. These cover two out of four different possibilities when inserting a new element. > instance Cons' x Nil (Pair' (Leaf x) Nil) where cons' = Pair . Leaf > > instance Cons' x (Pair' y Nil) (Pair' (Leaf x) (Pair' y Nil)) where > cons' = Pair . Leaf The remaining two cases occur when we have at least two elements in the list. This is where we need to compare weights of trees. First observation is that we don't need weight of a tree, depth is sufficient. Now we have a problem. We'd like to be able to write something like instance (Depth l ~ u, Depth r ~ v, Equals u v ~ T) => Cons' x (Pair' l (Pair' r y)) (Pair' (Tree l x r) y) where ... instance (Depth l ~ u, Depth r ~ v, Equals u v ~ F) => Cons' x y (Pair' (Leaf x) y) where ... but we get a functional dependency conflict in the second instance as the type variable y might stand for Pair' l (Pair' r y) and the functional dependency already says the result should be Pair' (Tree l x r) y, not Pair' (Leaf x) (Pair' l (Pair' r y)) We need to merge these two instances into one and then branch on the result of the comparison of tree depths. To do this, we introduce another type class to do the branching. > class ConsIf c x y z | c x y -> z where consIf :: c -> x -> y -> z And merge the instances. > instance ( Depth l ~ u > , Depth r ~ v > , Equals u v ~ c > , ConsIf c x (Pair' l (Pair' r y)) z ) => > > Cons' x (Pair' l (Pair' r y)) z where > > cons' = consIf (undefined :: c) > > instance ConsIf F x y (Pair' (Leaf x) y) where consIf _ = Pair . Leaf > > instance (Depth l ~ u, Depth r ~ u) => > > ConsIf T x (Pair' l (Pair' r y)) (Pair' (Tree l x r) y) where > > consIf _ x (Pair l (Pair r y)) = Pair (Node l x r) y We can now test our cons function. > t0' = cons' 'a' > $ cons' (33 :: Int) > $ cons' (4.4 :: Float) > $ cons' (88 :: Integer) > $ cons' ([1 .. 8] :: [Int]) > $ cons' "moose" > $ cons' ((6, 3) :: (Int, Int)) > $ nil And ghci tells us the type is *Main> :t t0' t0' :: Pair' (Tree (Tree (Leaf Float) Int (Leaf Integer)) Char (Tree (Leaf [Char]) [Int] (Leaf (Int, Int)))) Nil Now we get to the second version that doesn't need the undecidable instances extension. In this one we go a step back and attempt to keep track of depth information in the type of the list. When we again end up at the point where we need to compare if two trees at the front of the list are of the same depth, we run into a problem. instance Cons x (Pair i l (Pair i r y)) (Pair (S i) (Tree l x r) y) where ... instance Cons x (Pair i y (Pair j z w)) (Pair (S Z) x (Pair i y (Pair j z w))) where ... This clearly won't work as we again get a functional dependency conflict. The j type variable could be the same as i in which case we have two different rules for what the result of consing x with (Pair i l (Pair i r y)) should be. As we don't want to use undecidable instances, we are stuck. Unless we can find a way to disambiguate the two instances. Instead of keeping track of depth information, we could try to track how many elements we need to complete a new tree. That way instead of counting up, we are counting down towards zero and it becomes easy to differentiate the two instances as only the one where the depths of the two trees are equal will contain a Z. Keeping track of how many elements are missing turns out to be a problem, at least as long as we don't want to use undecidable instances. Fortunately we can replace the number of elements missing with the number of levels the tree at the front needs before its depth matches the tree that follows. > class Cons x y z | x y -> z where cons :: x -> y -> z Inserting an element into an empty list involves adding the element to the front and setting its the depth counter to 1. > instance Cons x Nil (Pair (S Z) (Leaf x) Nil) where cons = Pair . Leaf If we are inserting an element into a list that already has a single tree, we again add the new element to the front, set its counter to 1 and we decrement the counter of the tree that now follows the newly inserted element. > instance Cons x (Pair (S i) y Nil) (Pair (S Z) (Leaf x) (Pair i y Nil)) where > cons x (Pair y z) = Pair (Leaf x) (Pair y z) Here we have the case where the list already contains at least two trees but their depths are not equal. Like with the previous instance we insert the new element to the front and decrement the counter of the tree that now follows it. > instance Cons x (Pair (S i) y (Pair (S j) z w)) > (Pair (S Z) (Leaf x) (Pair i y (Pair (S j) z w))) where > > cons x (Pair y z) = Pair (Leaf x) (Pair y z) That means we are left with one more case to handle. The case when the depths of the two trees are the same, or rather when the counter reaches 0. As we merge the two trees and create a new tree, we must also decrement the counter of the tree that follows. What this means is that we really need two instances. One where there are only two trees of the same depth in the list, and one where those two trees are followed by at least one more tree. > instance (Depth l ~ i, Depth r ~ i) => > Cons x (Pair i l (Pair Z r Nil)) (Pair (S i) (Tree l x r) Nil) where > > cons x (Pair l (Pair r y)) = Pair (Node l x r) y As we merge the trees into a new tree, we reset and increment the depth counter. Note that we are using the first tree's counter. Reason for that is that the counter must represent the tree's depth as the tree at the front was either just inserted and so its counter is 1, or it was created when two smaller trees merged and its counter was set to its depth. Finally we must also decrement the counter of the tree that follows. > instance (Depth l ~ i, Depth r ~ i) => > Cons x (Pair i l (Pair Z r (Pair (S j) y z))) > (Pair (S i) (Tree l x r) (Pair j y z)) where > > cons x (Pair l (Pair r (Pair y z))) = Pair (Node l x r) (Pair y z) Now we can test the new cons function. > t0 = cons 'a' > $ cons (33 :: Int) > $ cons (4.4 :: Float) > $ cons (88 :: Integer) > $ cons ([1 .. 8] :: [Int]) > $ cons "moose" > $ cons ((6, 3) :: (Int, Int)) > $ nil The type is slightly more verbose. *Main> :t t0 t0 :: Pair (S (S (S Z))) (Tree (Tree (Leaf Float) Int (Leaf Integer)) Char (Tree (Leaf [Char]) [Int] (Leaf (Int, Int)))) Nil Here we introduce our wrapper for Roman numerals though we won't get to play with those just yet. > newtype R n = R { unR :: Int } deriving Show We want to get indexing and updates out of the way first. The reference implementation for the indexing function is a bit more complex than the cons function. Each of the cases we need to handle has been marked with a comment so we can see which instance matches which part of the code. index :: List a -> Int -> Maybe a index (List ((w, t) : wts)) i | i < w = index' w i t -- 1 | otherwise = index (List wts) (i - w) -- 2 where index' 1 0 (Leaf x ) = Just x -- 3 index' _ _ (Leaf _ ) = Nothing index' w i (Node l x r) | i == 0 = Just x -- 4 | i <= w' = index' w' (i - 1 ) l -- 5 | otherwise = index' w' (i - 1 - w') r -- 6 where w' = w `div` 2 index _ _ = Nothing We don't need to do anything when the index is out of range - the type system takes care of that for us. > class Index i x y | i y -> x where index :: y -> R i -> x > class IndexIf c i x y | c i y -> x where indexIf :: c -> y -> R i -> x First we handle the easy cases. > instance Index Z x (Leaf x ) where index (Leaf x ) _ = x -- 3 > instance Index Z x (Tree l x r) where index (Node _ x _) _ = x -- 4 Cases 5 and 6 require the use of a second type class so we can branch on the result of the comparison i <= w, though we rewrite it to w < i' + 1 to make use of the LessThan type family and to differentiate instance 6 from instance 4. Instead of halving the weight of the tree we are currently working on, we get the weight of one of its children. > instance ( Weight l ~ w > , LessThan w (S i) ~ c > , IndexIf c i x (Tree l y r) ) => Index (S i) x (Tree l y r) where > > index t _ = indexIf (undefined :: c) t (undefined :: R i) -- 5, 6 > > instance Index i x l => IndexIf F i x (Tree l y r) where > indexIf _ (Node l _ _) = index l -- 5 > > instance ( Weight r ~ w > , Sub i w ~ j > , Index j x r ) => IndexIf T i x (Tree l y r) where > > indexIf _ (Node _ _ r) _ = index r (undefined :: R j) -- 6 Cases 1 and 2 are handled in a similar way. > instance ( Weight t ~ w > , LessThan i w ~ c > , IndexIf c i x (Pair j t y) ) => Index i x (Pair j t y) where > > index = indexIf (undefined :: c) -- 1, 2 > > instance Index i x t => IndexIf T i x (Pair j t y) where > indexIf _ (Pair t _) = index t -- 1 > > instance ( Weight t ~ w > , Sub i w ~ j > , Index j x y ) => IndexIf F i x (Pair k t y) where > > indexIf _ (Pair _ y) _ = index y (undefined :: R j) -- 2 Update is similar. update :: Int -> a -> List a -> List a update i x (List ((w, t) : wts)) = List $ if i < w then (w, update' w i t) : wts -- 1 else (w, t) : unList (update (i - w) x (List wts)) -- 2 where update' 1 0 (Leaf _ ) = Leaf x -- 3 update' _ _ (Leaf _ ) = error "update" update' w i (Node l x' r) | i == 0 = Node l x r -- 4 | i <= w' = Node l' x' r -- 5 | otherwise = Node l x' r' -- 6 where w' = w `div` 2 l' = update' w' (i - 1 ) l r' = update' w' (i - 1 - w') r update _ _ _ = error "update" > class Index i x z => Update i x y z | i x y -> z where > update :: y -> R i -> x -> z > > class UpdateIf c i x y z | c i x y -> z where > updateIf :: c -> y -> R i -> x -> z > > instance Update Z x (Leaf y) (Leaf x) where > update _ _ x = Leaf x -- 3 > > instance Update Z x (Tree l y r) (Tree l x r) where > update (Node l _ r) _ x = Node l x r -- 4 > > instance ( Weight l ~ w > , LessThan w (S i) ~ c > , Index (S i) x t > , UpdateIf c i x (Tree l y r) t) => > > Update (S i) x (Tree l y r) t where > > update t _ = updateIf (undefined :: c) t (undefined :: R i) -- 5, 6 > > instance Update i x l l' => UpdateIf F i x (Tree l y r) (Tree l' y r) where > updateIf _ (Node l y r) i x = Node (update l i x) y r -- 5 > > instance ( Weight r ~ w > , Sub i w ~ j > , Update j x r r' ) => > > UpdateIf T i x (Tree l y r) (Tree l y r') where > > updateIf _ (Node l y r) _ x = -- 6 > Node l y (update r (undefined :: R j) x) > > instance ( Weight t ~ w > , LessThan i w ~ c > , Index i x p > , UpdateIf c i x (Pair j t y) p ) => Update i x (Pair j t y) p where > > update = updateIf (undefined :: c) -- 1, 2 > > instance Update i x t t' => UpdateIf T i x (Pair j t y) (Pair j t' y) where > updateIf _ (Pair t y) i x = Pair (update t i x) y -- 1 > > instance ( Weight t ~ w > , Sub i w ~ j > , Update j x y z ) => UpdateIf F i x (Pair k t y) (Pair k t z) where > > updateIf _ (Pair t y) _ x = -- 2 > Pair t (update y (undefined :: R j) x) Maybe we could combine Update and Index type classes into a single class? But we won't attempt that here. Now that we have indexing and updates, we need values we can use to index the list. Typing Z, S Z, S (S Z) and so on is too tedious and not very readable. We could use templates to predefine a whole bunch of type level numbers. That's kind of ugly though. And we can do it without using any extensions, in plain Haskell 98. We start off by making I a type synonym for S. Then we define types II to X, XX to C in steps of 10, CC to M in steps of 100 and finally we define MM and MMM. I don't know how Roman numerals are supposed to work above 3999 so that's the largest number we will be able to encode. Probably for the best as the type system doesn't seem too happy about having to deal with types of this sort. > type I n = S n > > type II n = I (I n) > type III n = I (II n) > type IV n = I (III n) > type V n = I (IV n) > type VI n = I (V n) > type VII n = I (VI n) > type VIII n = I (VII n) > type IX n = I (VIII n) > type X n = I (IX n) > > type XX n = X (X n) > type XXX n = X (XX n) > type XL n = X (XXX n) > type L n = X (XL n) > type LX n = X (L n) > type LXX n = X (LX n) > type LXXX n = X (LXX n) > type XC n = X (LXXX n) > type C n = X (XC n) > > type CC n = C (C n) > type CCC n = C (CC n) > type CD n = C (CCC n) > type D n = C (CD n) > type DC n = C (D n) > type DCC n = C (DC n) > type DCCC n = C (DCC n) > type CM n = C (DCCC n) > type M n = C (CM n) > > type MM n = M (M n) > type MMM n = M (MM n) Now for the fun part. We'd like to be able to construct only valid Roman numerals and that the type signatures match the values. We start of by defining the return and lift functions as described in Ralf Hinze's paper. We don't go all the way to creating a monad as these two functions are the only thing we need. > ret a f = f a > lift f = ret . f Now before we continue, we need a way to ensure that only valid numbers are constructed. We can group numbers in the range I to X into one group, let's call it ones, X to C into group tens, C to M into group hundreds and M, MM and MMM are in group thousands. A valid number can have at most 1 digit from each group. Digits also have to be ordered from largest to smallest, left to right. To do that, we could try and reuse the LessThen type family from before and add constraints so that a digit from the tens group can only be appended to a number less than ten. And a digit from the hundreds group can only be applied to a number less than a hundred. This approach works but it makes the GHC's type system choke and evaluating types of expressions takes minutes. We could also create a class for each of the groups and a set of combinators that can only be applied to a group of a lower rank. This works better. But we can go a step further than that - we don't actually need any type classes. All we need is one more phantom type for the rank of the number. So we introduce a second type that resembles R, only has one extra type variable. > newtype R' n s = R' { unR' :: Int } deriving Show Here we make use of ret and lift to create polyvariadic functions for constructing Roman numerals. rn starts the construction and nr finalizes it. > rn :: ((R' n s -> R n) -> r) -> r > rn f = ret (R . unR') f > > nr :: (R' Z (IV Z) -> r) -> r > nr f = f (R' 0) Trying them in ghci we get *Main> rn nr R {unR = 0} *Main> :t rn nr rn nr :: R Z We have a zero. Notice in the type signature for nr, rank of zero is 4. Now we need to generate the digits. We can use a single function to generate all of them. > mkR n = lift (\f -> f . R' . (+ n) . unR') And we encode the constraints in the type. Digits from the ones group can only be applied to a number of rank 4. There is only one number of rank 4 and that is zero. Applying a digit from the ones group to a zero gives us a number of rank 3. Digits from the tens group can be applied to numbers of ranks 3 or 4 and return a number of rank 2. And so on. > type RF n m s t k r = (R' n s -> k) -> ((R' m t -> k) -> r) -> r > > i :: RF (I n) n (III Z) (IV s) k r > ii :: RF (II n) n (III Z) (IV s) k r > iii :: RF (III n) n (III Z) (IV s) k r > iv :: RF (IV n) n (III Z) (IV s) k r > v :: RF (V n) n (III Z) (IV s) k r > vi :: RF (VI n) n (III Z) (IV s) k r > vii :: RF (VII n) n (III Z) (IV s) k r > viii :: RF (VIII n) n (III Z) (IV s) k r > ix :: RF (IX n) n (III Z) (IV s) k r > > x :: RF (X n) n (II Z) (III s) k r > xx :: RF (XX n) n (II Z) (III s) k r > xxx :: RF (XXX n) n (II Z) (III s) k r > xl :: RF (XL n) n (II Z) (III s) k r > l :: RF (L n) n (II Z) (III s) k r > lx :: RF (LX n) n (II Z) (III s) k r > lxx :: RF (LXX n) n (II Z) (III s) k r > lxxx :: RF (LXXX n) n (II Z) (III s) k r > xc :: RF (XC n) n (II Z) (III s) k r > > c :: RF (C n) n (I Z) (II s) k r > cc :: RF (CC n) n (I Z) (II s) k r > ccc :: RF (CCC n) n (I Z) (II s) k r > cd :: RF (CD n) n (I Z) (II s) k r > d :: RF (D n) n (I Z) (II s) k r > dc :: RF (DC n) n (I Z) (II s) k r > dcc :: RF (DCC n) n (I Z) (II s) k r > dccc :: RF (DCCC n) n (I Z) (II s) k r > cm :: RF (CM n) n (I Z) (II s) k r > > m :: RF (M n) n Z (I s) k r > mm :: RF (MM n) n Z (I s) k r > mmm :: RF (MMM n) n Z (I s) k r > > i = mkR 1 > ii = mkR 2 > iii = mkR 3 > iv = mkR 4 > v = mkR 5 > vi = mkR 6 > vii = mkR 7 > viii = mkR 8 > ix = mkR 9 > > x = mkR 10 > xx = mkR 20 > xxx = mkR 30 > xl = mkR 40 > l = mkR 50 > lx = mkR 60 > lxx = mkR 70 > lxxx = mkR 80 > xc = mkR 90 > > c = mkR 100 > cc = mkR 200 > ccc = mkR 300 > cd = mkR 400 > d = mkR 500 > dc = mkR 600 > dcc = mkR 700 > dccc = mkR 800 > cm = mkR 900 > > m = mkR 1000 > mm = mkR 2000 > mmm = mkR 3000 We can now try to construct a number such as 2011. > r2011 = rn mm x i nr *Main> r2011 R {unR = 2011} *Main> :t r2011 r2011 :: R (MM (X (I Z))) The type reflects the value we entered. Attempting to construct an invalid number gives us an error *Main> rn v vi nr <interactive>:1:0: Couldn't match expected type `Z' against inferred type `S s' Probable cause: `rn' is applied to too many arguments In the expression: rn v vi nr In the definition of `it': it = rn v vi nr I'll finish off this post with a generic map function. Here we use the type equality trick from the HList paper. > class GMap t p q | t p -> q where gmap :: t -> p -> q > class GMapIf c t p q | c t p -> q where gmapIf :: c -> t -> p -> q > > instance GMap (a -> b) Nil Nil where gmap _ = id > > instance ( TypeEq a x c > , GMapIf c (a -> b) (Leaf x) (Leaf y) ) => > > GMap (a -> b) (Leaf x) (Leaf y) where gmap = gmapIf (undefined :: c) > > instance ( TypeEq a x c > , GMapIf c (a -> b) (Tree l x r) (Tree l' y r') ) => > > GMap (a -> b) (Tree l x r) (Tree l' y r') where > > gmap = gmapIf (undefined :: c) > > instance ( GMap (a -> b) x y > , GMap (a -> b) p q ) => > > GMap (a -> b) (Pair i x p) (Pair i y q) where > > gmap f (Pair x p) = Pair (gmap f x) (gmap f p) > > instance GMapIf F (a -> b) (Leaf x) (Leaf x) where gmapIf _ _ = id > > instance GMapIf T (a -> b) (Leaf a) (Leaf b) where > gmapIf _ f (Leaf a) = Leaf (f a) > > instance ( GMap (a -> b) l l' > , GMap (a -> b) r r' ) => > > GMapIf F (a -> b) (Tree l x r) (Tree l' x r') where > > gmapIf _ f (Node l x r) = Node (gmap f l) x (gmap f r) > > instance ( GMap (a -> b) l l' > , GMap (a -> b) r r' ) => > > GMapIf T (a -> b) (Tree l a r) (Tree l' b r') where > > gmapIf _ f (Node l a r) = Node (gmap f l) (f a) (gmap f r) > > type family Cast a > type family Cast' t a > type family Cast'' t a > > type instance Cast a = Cast' () a > type instance Cast' t a = Cast'' t a > type instance Cast'' () a = a > > class TypeEq x y c | x y -> c > > instance Cast c ~ F => TypeEq x y c > > instance TypeEq Char Char T > instance TypeEq Int Int T > instance TypeEq Float Float T > instance TypeEq a a T => TypeEq [a] [a] T And a few tests to see if it all works. > t1 = gmap (length :: String -> Int) t0 > t2 = gmap (length :: [Int] -> Int) t1 > t3 = gmap ((+ 1000) :: Int -> Int) t2 > > t4 = cons (t0 `index` rn iv nr) t3 > > t5 = update t4 (rn iv nr) "asdf" > > t1' = gmap (length :: String -> Int) t0' > t2' = gmap (length :: [Int] -> Int) t1' > t3' = gmap ((+ 1000) :: Int -> Int) t2' > > t4' = cons' (t0' `index` rn iv nr) t3' > > t5' = update t4' (rn iv nr) "asdf" As a bonus the implementation of type-level Roman numerals in Standard ML. SML/NJ has a fit trying to load the code. It seems to cope with it if constructors for values above 100 are commented out. And it gives really nice type signatures. MLton has no problems compiling the code, but fully expands all type signatures. (*****************************************************************************) signature ROMAN = sig type z type 'n i type 'n ii = 'n i i type 'n iii = 'n i ii type 'n iv = 'n i iii type 'n v = 'n i iv type 'n vi = 'n i v type 'n vii = 'n i vi type 'n viii = 'n i vii type 'n ix = 'n i viii type 'n x = 'n i ix type 'n xx = 'n x x type 'n xxx = 'n x xx type 'n xl = 'n x xxx type 'n l = 'n x xl type 'n lx = 'n x l type 'n lxx = 'n x lx type 'n lxxx = 'n x lxx type 'n xc = 'n x lxxx type 'n c = 'n x xc type 'n cc = 'n c c type 'n ccc = 'n c cc type 'n cd = 'n c ccc type 'n d = 'n c cd type 'n dc = 'n c d type 'n dcc = 'n c dc type 'n dccc = 'n c dcc type 'n cm = 'n c dccc type 'n m = 'n c cm type 'n mm = 'n m m type 'n mmm = 'n m mm type 'n r type ('n, 's) r' val toInt : 'n r -> int val <| : ((('n, 's) r' -> 'n r) -> 'r) -> 'r val |> : ((z, z iv) r' -> 'r) -> 'r type ('n, 'm, 's, 't, 'k, 'r) rf = (('n, 's) r' -> 'k) -> ((('m, 't) r' -> 'k) -> 'r) -> 'r val i : ('n i , 'n, z iii, 's iv , 'k, 'r) rf val ii : ('n ii , 'n, z iii, 's iv , 'k, 'r) rf val iii : ('n iii , 'n, z iii, 's iv , 'k, 'r) rf val iv : ('n iv , 'n, z iii, 's iv , 'k, 'r) rf val v : ('n v , 'n, z iii, 's iv , 'k, 'r) rf val vi : ('n vi , 'n, z iii, 's iv , 'k, 'r) rf val vii : ('n vii , 'n, z iii, 's iv , 'k, 'r) rf val viii : ('n viii, 'n, z iii, 's iv , 'k, 'r) rf val ix : ('n ix , 'n, z iii, 's iv , 'k, 'r) rf val x : ('n x , 'n, z ii , 's iii, 'k, 'r) rf val xx : ('n xx , 'n, z ii , 's iii, 'k, 'r) rf val xxx : ('n xxx , 'n, z ii , 's iii, 'k, 'r) rf val xl : ('n xl , 'n, z ii , 's iii, 'k, 'r) rf val l : ('n l , 'n, z ii , 's iii, 'k, 'r) rf val lx : ('n lx , 'n, z ii , 's iii, 'k, 'r) rf val lxx : ('n lxx , 'n, z ii , 's iii, 'k, 'r) rf val lxxx : ('n lxxx, 'n, z ii , 's iii, 'k, 'r) rf val xc : ('n xc , 'n, z ii , 's iii, 'k, 'r) rf val c : ('n c , 'n, z i , 's ii , 'k, 'r) rf val cc : ('n cc , 'n, z i , 's ii , 'k, 'r) rf val ccc : ('n ccc , 'n, z i , 's ii , 'k, 'r) rf val cd : ('n cd , 'n, z i , 's ii , 'k, 'r) rf val d : ('n d , 'n, z i , 's ii , 'k, 'r) rf val dc : ('n dc , 'n, z i , 's ii , 'k, 'r) rf val dcc : ('n dcc , 'n, z i , 's ii , 'k, 'r) rf val dccc : ('n dccc, 'n, z i , 's ii , 'k, 'r) rf val cm : ('n cm , 'n, z i , 's ii , 'k, 'r) rf val m : ('n m , 'n, z , 's i , 'k, 'r) rf val mm : ('n mm , 'n, z , 's i , 'k, 'r) rf val mmm : ('n mmm , 'n, z , 's i , 'k, 'r) rf end structure Roman :> ROMAN = struct datatype z = Z datatype 'n i = I of 'n type 'n ii = 'n i i type 'n iii = 'n i ii type 'n iv = 'n i iii type 'n v = 'n i iv type 'n vi = 'n i v type 'n vii = 'n i vi type 'n viii = 'n i vii type 'n ix = 'n i viii type 'n x = 'n i ix type 'n xx = 'n x x type 'n xxx = 'n x xx type 'n xl = 'n x xxx type 'n l = 'n x xl type 'n lx = 'n x l type 'n lxx = 'n x lx type 'n lxxx = 'n x lxx type 'n xc = 'n x lxxx type 'n c = 'n x xc type 'n cc = 'n c c type 'n ccc = 'n c cc type 'n cd = 'n c ccc type 'n d = 'n c cd type 'n dc = 'n c d type 'n dcc = 'n c dc type 'n dccc = 'n c dcc type 'n cm = 'n c dccc type 'n m = 'n c cm type 'n mm = 'n m m type 'n mmm = 'n m mm type 'n r = int datatype ('n, 's) r' = R of int type ('n, 'm, 's, 't, 'k, 'r) rf = (('n, 's) r' -> 'k) -> ((('m, 't) r' -> 'k) -> 'r) -> 'r fun toInt n = n local fun unR (R n) = n fun ret a f = f a fun lift f = ret o f fun mkR n k = lift (fn g => g o R o (fn x => x + n) o unR) k in fun <| f = ret unR f fun |> f = f (R 0) fun i k = mkR 1 k fun ii k = mkR 2 k fun iii k = mkR 3 k fun iv k = mkR 4 k fun v k = mkR 5 k fun vi k = mkR 6 k fun vii k = mkR 7 k fun viii k = mkR 8 k fun ix k = mkR 9 k fun x k = mkR 10 k fun xx k = mkR 20 k fun xxx k = mkR 30 k fun xl k = mkR 40 k fun l k = mkR 50 k fun lx k = mkR 60 k fun lxx k = mkR 70 k fun lxxx k = mkR 80 k fun xc k = mkR 90 k fun c k = mkR 100 k fun cc k = mkR 200 k fun ccc k = mkR 300 k fun cd k = mkR 400 k fun d k = mkR 500 k fun dc k = mkR 600 k fun dcc k = mkR 700 k fun dccc k = mkR 800 k fun cm k = mkR 900 k fun m k = mkR 1000 k fun mm k = mkR 2000 k fun mmm k = mkR 3000 k end end

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Continuation Workshop 2011: Call for participation
by oleg＠okmij.org 10 Aug '11

10 Aug '11

ACM SIGPLAN Continuation Workshop 2011 http://logic.cs.tsukuba.ac.jp/cw2011/ co-located with ICFP 2011, Tokyo, Japan Saturday, September 24, 2011 Call for Participation Early Registration deadline is August 15! Continuations have been discovered many times, which highlights their many applications in programming language semantics and program analysis, linguistics, logic, parallel processing, compilation and web programming. Recently, there has been a surge of interest specifically in delimited continuations: new implementations (in Scala, Ruby, OCaml, Haskell), new applications (to probabilistic programming, event-driven distributed processing), substructural and constructive logics, natural language semantics. The goal of the Continuation Workshop is to make continuations more accessible and useful -- to practitioners and to researchers in various areas of computer science and outside computer science. We wish to promote communication among the implementors and users in many fields. We would like to publicize the applications of continuations in academic (logic, linguistics) and practical fields and various programming languages (OCaml, Haskell, Scala, Ruby, Scheme, etc.). Invited talks ------------- Mats Rooth, Cornell University http://conf.ling.cornell.edu/mr249/ >From Logic to Effects and Back Noam Zeilberger, Universite' Paris 7 http://www.pps.jussieu.fr/~noam/ Tutorials --------- In the evening before the workshop, there will be a tutorial session ``Introduction to Programming with Shift and Reset'' Tutorial date and time: Friday, September 23, 2011, 19:00-21:00 Tutorial place: IIJ (next to NII, the place of the ICFP conference) Tutorial speakers: Kenichi Asai and Oleg Kiselyov Presentations ------------- Non-Deterministic Search Library Kenichi ASAI, Chihiro KANEKO `Focus movement' by delimited continuations Daisuke BEKKI Swarm: transparent scalability through portable continuations James DOUGLAS Correctness of Functions with Shift and Reset Noriko HIROTA, Kenichi ASAI Yield, the control operator: applications and a conjecture Roshan P. JAMES, Amr SABRY Demonstration of Continuation based C on GCC Shinji KONO Modular rollback through free monads Conor McBRIDE, Olin SHIVERS, Aaron TURON Using delimited continuations for distributed computing with the CIEL engine Derek G. MURRAY, Malte SCHWARZKOPF, Christopher SMOWTON, Steven SMITH, Anil MADHAVAPEDDY, Steven HAND The limit of the CPS hierarchy Josef SVENNINGSSON Visualizing continuations Naoki TAKASHIMA, Yukiyoshi KAMEYAMA

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Error in the asynchronous exception operational semantics
by Edward Z. Yang 10 Aug '11

10 Aug '11

Hello all, I was recently reading "Asynchronous Exceptions as an Effect" by Harrison, Allwein, Gill and Procter, and noticed at the end that they found an error in the operational semantics described in "Asynchronous Exceptions in Haskell" by the Simons and Andrew Moran. Does anyone know what this error was, and whether or not it was corrected in the December 12, 2006 version of the paper floating around on the net? Thanks, Edward

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Re: [Haskell-cafe] Problems building lambdabot on osx
by John Lato 10 Aug '11

10 Aug '11

> From: Brandon Allbery <allbery.b(a)gmail.com> > > On Mon, Aug 8, 2011 at 21:38, Adam Turoff <adam.turoff(a)gmail.com> wrote: > >> First, there's the issue with linking against libiconv, which is solved >> this >> way: >> >> cabal install --extra-lib-dirs=/usr/lib >> >> That leaves a whole mess of link errors against libHSreadline: >> > > Yes, because now it's finding the system readline, which isn't actually > readline (Apple ships a "readline" which is actually BSD "editline", so you > get missing symbols for things editline doesn't support such as completion). > I think you'll need to temporarily deactivate MacPorts' iconv instead of > using --extra-lib-dirs. > > This can't really be fixed by either GHC or MacPorts; there doesn't seem to > be a good solution as yet, although the MacPorts folks may end up > implementing a hacky solution because Apple has introduced *another* > MacPorts-breaking library conflict in Lion (libnotify). Yep, this is Apple's problem and everyone else just has to put up with it. If you want to mix macports and a non-macports ghc, I think the best solution is to self-compile ghc against the macports libiconv. This is easy, just configure ghc with ./configure --with-iconv-includes=/opt/local/include --with-iconv-libraries=/opt/local/lib I haven't had a problem since I started building ghc this way (10.5 and 10.6). John L.

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Analyzing slow performance of a Haskell program
by Chris Yuen 10 Aug '11

10 Aug '11

Hello Cafe, I was trying to solve ITA Software's "Word Nubmers" puzzle ( http://www.itasoftware.com/careers/puzzle_archive.html) using a brute force approach. Here's a my version of a quick recap of the problem: > A wordNumber is defined as > > wordNumber 1 = "one" > wordNumber 2 = "onetwo" > wordNumber 3 = "onethree" > wordNumber 15 = "onetwothreefourfivesixseveneightnineteneleventwelvethirteenfourteenfifteen" > ... > > Problem: Find the 51-billion-th letter of (wordNumber Infinity); assume that letter is found at 'wordNumber x', also find 'sum [1..x]' >From an imperative perspective, a naive algorithm would be to have 2 counters, keep counting the length of each wordNumber and "break" to return the result. The imperative brute-force approach is implemented in C# here: http://ideone.com/JjCb3. It takes about 1.5 minutes to find the answer on my computer. Then I implemented a brute-force Haskell version: http://ideone.com/ngfFq. It cannot finish the calculation in 5 minutes on my machine. (Irony: it's has more lines than the C# version) Here is the `-p` profile of the Haskell program: http://hpaste.org/49934 The Question: Are there obvious mistakes I am making? (Note: I am fully aware that brute-forcing it is not the correct solution to this problem. I am mainly interested in making the Haskell version perform comparatively to the C# version. Right now it is at least 5x slower so obviously I am missing something obvious) (Note 2: It does not seem to be space leaking. The program runs with constant memory (about 2MB) on my computer) (Note 3: I am compiling with `ghc -O2 WordNumber.hs) Chris

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Snap Riak Extension v0.1
by Edward Tate 10 Aug '11

10 Aug '11

Hi, I released this library a while ago but didn't tell anyone, thought I would let people know. Would be cool to get feedback, suggestions etc. https://github.com/et4te/snap-extension-riak/ PS: There are no recent commits since I'm going to be overhauling the architecture once Snap 6 comes out with 'Snaplets'.

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Problems building lambdabot on osx
by Adam Turoff 10 Aug '11

10 Aug '11

I've tried off and on the last couple of days to build Lambdabot on Mac OS X (before and after the upgrade from 10.6 to 10.7) and I keep running into linker errors with the 64-bit Haskell Platform (ghc 7.0.3). First, there's the issue with linking against libiconv, which is solved this way: cabal install --extra-lib-dirs=/usr/lib That leaves a whole mess of link errors against libHSreadline: Linking dist/build/lambdabot/lambdabot ... ld: warning: could not create compact unwind for _ffi_call_unix64: does not use RBP or RSP based frame Undefined symbols for architecture x86_64: "_rl_insert_completions", referenced from: _readlinezm1zi0zi1zi0_SystemziConsoleziReadline_zdwa5_info in libHSreadline-1.0.1.0.a(Readline.o) "_rl_possible_completions", referenced from: _readlinezm1zi0zi1zi0_SystemziConsoleziReadline_zdwa7_info in libHSreadlin (full error report at http://hpaste.org/50036 ) I did manage to install readline through macports: $ port installed readline The following ports are currently installed: readline @6.2.000_0 (active) $ file /opt/local/lib/libreadline.dylib /opt/local/lib/libreadline.dylib: Mach-O 64-bit dynamically linked shared library x86_64 and I did manage to get readline installed properly by specifying the include/lib dirs from macports: cabal install readline --configure-option=--with-readline-libraries=/opt/local/lib --configure-option=--with-readline-includes=/opt/local/include and readline does work in ghci: $ ghci GHCi, version 7.0.3: http://www.haskell.org/ghc/ :? for help Prelude> :m + System.Console.Readline Prelude System.Console.Readline> readline "testing> " testing> exit exit Just "exit" Prelude System.Console.Readline> The only issue, which has me stumped, is that lambdabot can't seem to find the readline dependency. Adding the --with-readline options when trying to build lambdabot has no effect. Neither does adding --extra-lib-dirs=/opt/local/lib to cabal install. Anyone have a clue where to start to work around this? Would switching to ghc HEAD help here? Thanks, -- Adam

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Second CFP: PADL'12 - Practical Aspects of Declarative Languages 2012
by Claudio Russo 09 Aug '11

09 Aug '11

[Apologies if you receive multiple copies.] Call for Papers =============== 14th International Symposium on Practical Aspects of Declarative Languages (PADL 2012) http://research.microsoft.com/~crusso/padl12 Philadelphia, Pennsylvania, USA, January 23-24, 2012 Co-located with ACM POPL'12 Abstract/Paper submission deadline: September 10th/17th, 2011 Conference Description ====================== Declarative languages build on sound theoretical bases to provide attractive frameworks for application development. These languages have been successfully applied to many different real-world situations, ranging from data base management to active networks to software engineering to decision support systems. New developments in theory and implementation have opened up new application areas. At the same time, applications of declarative languages to novel problems raise numerous interesting research issues. Well-known questions include designing for scalability, language extensions for application deployment, and programming environments. Thus, applications drive the progress in the theory and implementation of declarative systems, and benefit from this progress as well. PADL is a forum for researchers and practitioners to present original work emphasizing novel applications and implementation techniques for all forms of declarative concepts, including, functional, logic, constraints, etc. Topics of interest include, but are not limited to: * Innovative applications of declarative languages * Declarative domain-specific languages and applications * Practical applications of theoretical results * New language developments and their impact on applications * Declarative languages and Software Engineering * Evaluation of implementation techniques on practical applications * Practical experiences and industrial applications * Novel uses of declarative languages in the classroom * Practical extensions such as constraint-based, probabilistic, and reactive languages. PADL'12 welcomes new ideas and approaches pertaining to applications and implementation of declarative languages. In this occasion PADL is co-located, as traditionally, with ACM POPL, which will be held immediately following PADL, January 25-27. The symposium will be held in Philadelphia, Pennsylvania, USA. Important Dates and Submission Guidelines ========================================= Abstract Submission: September 10, 2011 Paper Submission: September 17, 2011 Notification: October 22, 2011 Camera-ready: November 5, 2011 Symposium: January 23-24, 2012 Authors should submit an electronic copy of the full paper in PDF using the Springer LNCS format. The submission will be done through EasyChair conference system. If electronic submission is impossible, please contact the program chairs for information on how to submit hard copies. All submissions must be original work written in English. Submissions must be unpublished and not submitted for publication elsewhere. Work that already appeared in unpublished or informally published workshops proceedings may be submitted. PADL'12 will accept both technical and application papers: * Technical papers must describe original, previously unpublished research results. Technical papers must not exceed 15 pages in Springer LNCS format. * Application papers are a mechanism to present important practical applications of declarative languages that occur in industry or in areas of research other than Computer Science. Application papers will be published in the Springer-Verlag conference proceedings, and will be presented in a separate session. Application papers are expected to describe complex and/or real-world applications that rely on an innovative use of declarative languages. Application descriptions, engineering solutions and real-world experiences (both positive and negative) are solicited. The limit for application papers is 6 pages in Springer LNCS format. Program Committee ================= Marcello Balduccini, Intelligent Systems Department, Kodak Research Labs Edwin Brady, University of St Andrews, Scotland Henning Christiansen, Roskilde University, Denmark Agostino Dovier, University of Udine, Italy Matthew Flatt, University of Utah, USA Gopal Gupta, University of Texas at Dallas, USA John Hughes, Chalmers University of Technology, Sweden; Quviq AB Gabriele Keller, University of New South Wales, Australia Lunjin Lu, Oakland University, USA Marc Pouzet, École normale supérieure, France Ricardo Rocha, University of Porto, Portugal Andreas Rossberg, Google Germany GmbH, Germany Claudio Russo, Microsoft Research Cambridge, UK (co-chair) Kostis Sagonas, Uppsala Univeristy, Sweden Satnam Singh, Microsoft Research Cambridge, UK Zoltan Somogyi, University of Melbourne, Australia Eijiro Sumii, Tohoku University, Japan Terrance Swift, Universidade Nova de Lisboa, Portugal; Johns Hopkins University, USA Andrew Tolmach, Portland State University, USA Jan Wielemaker, University of Amsterdam, The Netherlands Roland Yap, National University of Singapore, Republic of Singapore Kwangkeun Yi, Seoul National University, Korea Neng-Fa Zhou, Brooklyn College, City University of New York, USA (co-chair) Contacts ======== For additional information about papers and submissions, please contact the Program Chairs: Claudio Russo Microsoft Research Cambridge,UK Email: crusso <AT> microsoft <DOT> com Neng-Fa Zhou Brooklyn College, The City University of New York, USA Email: zhou <AT> sci <DOT> brooklyn <DOT> cuny <DOT> edu With the Cooperation of ======================= The Association for Logic Programming (ALP) ACM SIGPLAN Microsoft Research, Cambridge ===================================

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PhD Position available at Strathclyde
by Conor McBride 09 Aug '11

09 Aug '11

[My colleague, Patricia Johann, advertises the following...] PhD Position in Category Theory and Functional Programming Department of Computer and Information Sciences University of Strathclyde, Scotland Applications are invited for one PhD position within the Mathematically Structured Programming group at the University of Strathclyde. The group comprises Prof. Neil Ghani, Dr. Patricia Johann, Dr. Conor McBride, Dr. Peter Hancock, Dr. Robert Atkey, and six PhD students. The PhD project centres around applications of categorical methods to functional programming languages. The project is under the direction of Patricia Johann. The successful applicant will have either a first-class degree or an MSc in Mathematics or Computing Science or a related subject with a strong Mathematics or Computing Science component. Ideally, they will also have a strong, documented interest in doing research. Strong mathematical background and problem-solving skills are essential; good programming skills are a plus. Prior knowledge of category theory and/ or functional programming is an advantage, but is not required. The PhD position is for 3 years; it starts in January 2012. The position is a fully-funded post for a UK or EU student, and includes both coverage of fees and an EPSRC-level stipend for each of the three years. More information about the department is available at http://www.strath.ac.uk/cis The University of Strathclyde (http://www.strath.ac.uk) is located in the heart of Glasgow, which Lonely Planet Travel Guides hail as "one of Britain's largest, liveliest and most interesting cities" (seehttp://www.lonelyplanet.com/worldguide/scotland/glasgow/) . Southern Scotland provides a particularly stimulating environment for researchers in theoretical computer science, with active groups in this area at Heriot-Watt University, the University of Edinburgh, the University of Glasgow, the University of St. Andrews, and the University of Strathclyde. Requests for further information and other informal enquiries can be sent to: Patricia Johann patricia at cis.strath.ac.uk Those interested in the position are asked to send e-mail to the address given above in the next short while.

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Question Formlets, file upload
by "Dr. Heinrich Hördegen" 09 Aug '11

09 Aug '11

Dear all, I am using the library Text.XHtml.Strict.Formlets to build a validating form. I use the function "file" to get a file upload widget. The encoding is set to multipart. I want to check, if the user entered something or left the widget empty. However, when entering nothing, instead of an empty string, I always get a string like "/tmp/12101" which corresponds to an empty file. It would be nice, if I could check for the empty string to see, weather the user entered something or not into the upload widget. I appreciate any hints to achive this and wish you a good day, Heinrich -- -- hoerdegen(a)funktional.info www.funktional.info --

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