MemoTrie-0.4.8: src/Data/MemoTrie.hs
{-# LANGUAGE GADTs, TypeFamilies, TypeOperators, ScopedTypeVariables, CPP #-}
{-# OPTIONS_GHC -Wall -fenable-rewrite-rules #-}
-- ScopedTypeVariables works around a 6.10 bug. The forall keyword is
-- supposed to be recognized in a RULES pragma.
----------------------------------------------------------------------
-- |
-- Module : Data.MemoTrie
-- Copyright : (c) Conal Elliott 2008
-- License : BSD3
--
-- Maintainer : conal@conal.net
-- Stability : experimental
--
-- Trie-based memoizer
-- Adapted from sjanssen's paste: \"a lazy trie\" <http://hpaste.org/3839>.
----------------------------------------------------------------------
module Data.MemoTrie
( HasTrie(..), domain, idTrie, (@.@)
-- , trie2, trie3, untrie2, untrie3
, memo, memo2, memo3, mup
, inTrie, inTrie2, inTrie3
-- , untrieBits
) where
import Data.Bits
import Data.Word
import Data.Int
import Control.Applicative
import Control.Arrow (first,(&&&))
import Data.Monoid
import Data.Function (on)
-- import Prelude hiding (id,(.))
-- import Control.Category
-- import Control.Arrow
infixr 0 :->:
-- | Mapping from all elements of @a@ to the results of some function
class HasTrie a where
-- | Representation of trie with domain type @a@
data (:->:) a :: * -> *
-- | Create the trie for the entire domain of a function
trie :: (a -> b) -> (a :->: b)
-- | Convert a trie to a function, i.e., access a field of the trie
untrie :: (a :->: b) -> (a -> b)
-- | List the trie elements. Order of keys (@:: a@) is always the same.
enumerate :: (a :->: b) -> [(a,b)]
-- | Domain elements of a trie
domain :: HasTrie a => [a]
domain = map fst (enumerate (trie (const oops)))
where
oops = error "Data.MemoTrie.domain: range element evaluated."
-- Hm: domain :: [Bool] doesn't produce any output.
instance (HasTrie a, Eq b) => Eq (a :->: b) where
(==) = (==) `on` (map snd . enumerate)
instance (HasTrie a, Show a, Show b) => Show (a :->: b) where
show t = "Trie: " ++ show (enumerate t)
{-
trie2 :: (HasTrie a, HasTrie b) =>
(a -> b -> c) -> (a :->: b :->: c)
-- trie2 h = trie $ \ a -> trie $ \ b -> h a b
-- trie2 h = trie $ \ a -> trie (h a)
trie2 h = trie (trie . h)
-- trie2 h = trie (fmap trie h)
-- trie2 = (fmap.fmap) trie trie
trie3 :: (HasTrie a, HasTrie b, HasTrie c) =>
(a -> b -> c -> d) -> (a :->: b :->: c :->: d)
trie3 h = trie (trie2 . h)
untrie2 :: (HasTrie a, HasTrie b) =>
(a :->: b :->: c)-> (a -> b -> c)
untrie2 tt = untrie . untrie tt
untrie3 :: (HasTrie a, HasTrie b, HasTrie c) =>
(a :->: b :->: c :->: d)-> (a -> b -> c -> d)
untrie3 tt = untrie2 . untrie tt
-}
{-# RULES
"trie/untrie" forall t. trie (untrie t) = t
#-}
-- Don't include the dual rule:
-- "untrie/trie" forall f. untrie (trie f) = f
-- which would defeat memoization.
--
-- TODO: experiment with rule application. Maybe re-enable "untrie/trie"
-- but fiddle with phases, so it won't defeat 'memo'.
-- | Trie-based function memoizer
memo :: HasTrie t => (t -> a) -> (t -> a)
memo = untrie . trie
-- | Memoize a binary function, on its first argument and then on its
-- second. Take care to exploit any partial evaluation.
memo2 :: (HasTrie s,HasTrie t) => (s -> t -> a) -> (s -> t -> a)
-- | Memoize a ternary function on successive arguments. Take care to
-- exploit any partial evaluation.
memo3 :: (HasTrie r,HasTrie s,HasTrie t) => (r -> s -> t -> a) -> (r -> s -> t -> a)
-- | Lift a memoizer to work with one more argument.
mup :: HasTrie t => (b -> c) -> (t -> b) -> (t -> c)
mup mem f = memo (mem . f)
memo2 = mup memo
memo3 = mup memo2
-- | Apply a unary function inside of a trie
inTrie :: (HasTrie a, HasTrie c) =>
((a -> b) -> (c -> d))
-> ((a :->: b) -> (c :->: d))
inTrie = untrie ~> trie
-- | Apply a binary function inside of a trie
inTrie2 :: (HasTrie a, HasTrie c, HasTrie e) =>
((a -> b) -> (c -> d) -> (e -> f))
-> ((a :->: b) -> (c :->: d) -> (e :->: f))
inTrie2 = untrie ~> inTrie
-- | Apply a ternary function inside of a trie
inTrie3 :: (HasTrie a, HasTrie c, HasTrie e, HasTrie g) =>
((a -> b) -> (c -> d) -> (e -> f) -> (g -> h))
-> ((a :->: b) -> (c :->: d) -> (e :->: f) -> (g :->: h))
inTrie3 = untrie ~> inTrie2
---- Instances
instance HasTrie () where
data () :->: a = UnitTrie a
trie f = UnitTrie (f ())
untrie (UnitTrie a) = \ () -> a
enumerate (UnitTrie a) = [((),a)]
-- Proofs of inverse properties:
{-
untrie (trie f)
== { trie def }
untrie (UnitTrie (f ()))
== { untrie def }
\ () -> (f ())
== { const-unit }
f
trie (untrie (UnitTrie a))
== { untrie def }
trie (\ () -> a)
== { trie def }
UnitTrie ((\ () -> a) ())
== { beta-reduction }
UnitTrie a
Oops -- the last step of the first direction is bogus when f is non-strict.
Can be fixed by using @const a@ in place of @\ () -> a@, but I can't do
the same for other types, like integers or sums.
All of these proofs have this same bug, unless we restrict ourselves to
memoizing hyper-strict functions.
-}
instance HasTrie Bool where
data Bool :->: x = BoolTrie x x
trie f = BoolTrie (f False) (f True)
untrie (BoolTrie f t) = if' f t
enumerate (BoolTrie f t) = [(False,f),(True,t)]
-- | Conditional with boolean last.
-- Spec: @if' (f False) (f True) == f@
if' :: x -> x -> Bool -> x
if' t _ False = t
if' _ e True = e
{-
untrie (trie f)
== { trie def }
untrie (BoolTrie (f False) (f True))
== { untrie def }
if' (f False) (f True)
== { if' spec }
f
trie (untrie (BoolTrie f t))
== { untrie def }
trie (if' f t)
== { trie def }
BoolTrie (if' f t False) (if' f t True)
== { if' spec }
BoolTrie f t
-}
instance (HasTrie a, HasTrie b) => HasTrie (Either a b) where
data (Either a b) :->: x = EitherTrie (a :->: x) (b :->: x)
trie f = EitherTrie (trie (f . Left)) (trie (f . Right))
untrie (EitherTrie s t) = either (untrie s) (untrie t)
enumerate (EitherTrie s t) = enum' Left s `weave` enum' Right t
enum' :: (HasTrie a) => (a -> a') -> (a :->: b) -> [(a', b)]
enum' f = (fmap.first) f . enumerate
weave :: [a] -> [a] -> [a]
[] `weave` as = as
as `weave` [] = as
(a:as) `weave` bs = a : (bs `weave` as)
{-
untrie (trie f)
== { trie def }
untrie (EitherTrie (trie (f . Left)) (trie (f . Right)))
== { untrie def }
either (untrie (trie (f . Left))) (untrie (trie (f . Right)))
== { untrie . trie }
either (f . Left) (f . Right)
== { either }
f
trie (untrie (EitherTrie s t))
== { untrie def }
trie (either (untrie s) (untrie t))
== { trie def }
EitherTrie (trie (either (untrie s) (untrie t) . Left))
(trie (either (untrie s) (untrie t) . Right))
== { either }
EitherTrie (trie (untrie s)) (trie (untrie t))
== { trie . untrie }
EitherTrie s t
-}
instance (HasTrie a, HasTrie b) => HasTrie (a,b) where
data (a,b) :->: x = PairTrie (a :->: (b :->: x))
trie f = PairTrie (trie (trie . curry f))
untrie (PairTrie t) = uncurry (untrie . untrie t)
enumerate (PairTrie tt) =
[ ((a,b),x) | (a,t) <- enumerate tt , (b,x) <- enumerate t ]
{-
untrie (trie f)
== { trie def }
untrie (PairTrie (trie (trie . curry f)))
== { untrie def }
uncurry (untrie . untrie (trie (trie . curry f)))
== { untrie . trie }
uncurry (untrie . trie . curry f)
== { untrie . untrie }
uncurry (curry f)
== { uncurry . curry }
f
trie (untrie (PairTrie t))
== { untrie def }
trie (uncurry (untrie . untrie t))
== { trie def }
PairTrie (trie (trie . curry (uncurry (untrie . untrie t))))
== { curry . uncurry }
PairTrie (trie (trie . untrie . untrie t))
== { trie . untrie }
PairTrie (trie (untrie t))
== { trie . untrie }
PairTrie t
-}
instance (HasTrie a, HasTrie b, HasTrie c) => HasTrie (a,b,c) where
data (a,b,c) :->: x = TripleTrie (((a,b),c) :->: x)
trie f = TripleTrie (trie (f . trip))
untrie (TripleTrie t) = untrie t . detrip
enumerate (TripleTrie t) = enum' trip t
trip :: ((a,b),c) -> (a,b,c)
trip ((a,b),c) = (a,b,c)
detrip :: (a,b,c) -> ((a,b),c)
detrip (a,b,c) = ((a,b),c)
instance HasTrie x => HasTrie [x] where
data [x] :->: a = ListTrie (Either () (x,[x]) :->: a)
trie f = ListTrie (trie (f . list))
untrie (ListTrie t) = untrie t . delist
enumerate (ListTrie t) = enum' list t
list :: Either () (x,[x]) -> [x]
list = either (const []) (uncurry (:))
delist :: [x] -> Either () (x,[x])
delist [] = Left ()
delist (x:xs) = Right (x,xs)
#define WordInstance(Type,TrieType)\
instance HasTrie Type where \
data Type :->: a = TrieType ([Bool] :->: a);\
trie f = TrieType (trie (f . unbits));\
untrie (TrieType t) = untrie t . bits;\
enumerate (TrieType t) = enum' unbits t
WordInstance(Word,WordTrie)
WordInstance(Word8,Word8Trie)
WordInstance(Word16,Word16Trie)
WordInstance(Word32,Word32Trie)
WordInstance(Word64,Word64Trie)
-- instance HasTrie Word where
-- data Word :->: a = WordTrie ([Bool] :->: a)
-- trie f = WordTrie (trie (f . unbits))
-- untrie (WordTrie t) = untrie t . bits
-- enumerate (WordTrie t) = enum' unbits t
-- | Extract bits in little-endian order
bits :: Bits t => t -> [Bool]
bits 0 = []
bits x = testBit x 0 : bits (shiftR x 1)
-- | Convert boolean to 0 (False) or 1 (True)
unbit :: Num t => Bool -> t
unbit False = 0
unbit True = 1
-- | Bit list to value
unbits :: Bits t => [Bool] -> t
unbits [] = 0
unbits (x:xs) = unbit x .|. shiftL (unbits xs) 1
instance HasTrie Char where
data Char :->: a = CharTrie (Int :->: a)
untrie (CharTrie t) n = untrie t (fromEnum n)
trie f = CharTrie (trie (f . toEnum))
enumerate (CharTrie t) = enum' toEnum t
-- Although Int is a Bits instance, we can't use bits directly for
-- memoizing, because the "bits" function gives an infinite result, since
-- shiftR (-1) 1 == -1. Instead, convert between Int and Word, and use
-- a Word trie. Any Integral type can be handled similarly.
#define IntInstance(IntType,WordType,TrieType) \
instance HasTrie IntType where \
data IntType :->: a = TrieType (WordType :->: a); \
untrie (TrieType t) n = untrie t (fromIntegral n); \
trie f = TrieType (trie (f . fromIntegral)); \
enumerate (TrieType t) = enum' fromIntegral t
IntInstance(Int,Word,IntTrie)
IntInstance(Int8,Word8,Int8Trie)
IntInstance(Int16,Word16,Int16Trie)
IntInstance(Int32,Word32,Int32Trie)
IntInstance(Int64,Word64,Int64Trie)
-- For unbounded integers, we don't have a corresponding Word type, so
-- extract the sign bit.
instance HasTrie Integer where
data Integer :->: a = IntegerTrie ((Bool,[Bool]) :->: a)
trie f = IntegerTrie (trie (f . unbitsZ))
untrie (IntegerTrie t) = untrie t . bitsZ
enumerate (IntegerTrie t) = enum' unbitsZ t
unbitsZ :: (Bits n) => (Bool,[Bool]) -> n
unbitsZ (positive,bs) = sig (unbits bs)
where
sig | positive = id
| otherwise = negate
bitsZ :: (Ord n, Bits n) => n -> (Bool,[Bool])
bitsZ = (>= 0) &&& (bits . abs)
-- bitsZ n = (sign n, bits (abs n))
-- TODO: make these definitions more systematic.
---- Instances
{-
The \"semantic function\" 'untrie' is a morphism over 'Monoid', 'Functor',
'Applicative', 'Monad', 'Category', and 'Arrow', i.e.,
untrie mempty == mempty
untrie (s `mappend` t) == untrie s `mappend` untrie t
untrie (fmap f t) == fmap f (untrie t)
untrie (pure a) == pure a
untrie (tf <*> tx) == untrie tf <*> untrie tx
untrie (return a) == return a
untrie (u >>= k) == untrie u >>= untrie . k
untrie id == id
untrie (s . t) == untrie s . untrie t
untrie (arr f) == arr f
untrie (first t) == first (untrie t)
These morphism properties imply that all of the expected laws hold,
assuming that we interpret equality semantically (or observationally).
For instance,
untrie (mempty `mappend` a)
== untrie mempty `mappend` untrie a
== mempty `mappend` untrie a
== untrie a
untrie (fmap f (fmap g a))
== fmap f (untrie (fmap g a))
== fmap f (fmap g (untrie a))
== fmap (f.g) (untrie a)
== untrie (fmap (f.g) a)
The implementation instances then follow from applying 'trie' to both
sides of each of these morphism laws.
-}
{-
instance (HasTrie a, Monoid b) => Monoid (a :->: b) where
mempty = trie mempty
s `mappend` t = trie (untrie s `mappend` untrie t)
instance HasTrie a => Functor ((:->:) a) where
fmap f t = trie (fmap f (untrie t))
instance HasTrie a => Applicative ((:->:) a) where
pure b = trie (pure b)
tf <*> tx = trie (untrie tf <*> untrie tx)
instance HasTrie a => Monad ((:->:) a) where
return a = trie (return a)
u >>= k = trie (untrie u >>= untrie . k)
-- instance Category (:->:) where
-- id = trie id
-- s . t = trie (untrie s . untrie t)
-- instance Arrow (:->:) where
-- arr f = trie (arr f)
-- first t = trie (first (untrie t))
-}
-- Simplify, using inTrie, inTrie2
instance (HasTrie a, Monoid b) => Monoid (a :->: b) where
mempty = trie mempty
mappend = inTrie2 mappend
instance HasTrie a => Functor ((:->:) a) where
fmap f = inTrie (fmap f)
instance HasTrie a => Applicative ((:->:) a) where
pure b = trie (pure b)
(<*>) = inTrie2 (<*>)
instance HasTrie a => Monad ((:->:) a) where
return a = trie (return a)
u >>= k = trie (untrie u >>= untrie . k)
-- | Identity trie
idTrie :: HasTrie a => a :->: a
idTrie = trie id
infixr 9 @.@
-- | Trie composition
(@.@) :: (HasTrie a, HasTrie b) =>
(b :->: c) -> (a :->: b) -> (a :->: c)
(@.@) = inTrie2 (.)
-- instance Category (:->:) where
-- id = idTrie
-- (.) = (.:)
-- instance Arrow (:->:) where
-- arr f = trie (arr f)
-- first = inTrie first
{-
Correctness of these instances follows by applying 'untrie' to each side
of each definition and using the property @'untrie' . 'trie' == 'id'@.
The `Category` and `Arrow` instances don't quite work, however, because of
necessary but disallowed `HasTrie` constraints on the domain type.
-}
---- To go elsewhere
-- Matt Hellige's notation for @argument f . result g@.
-- <http://matt.immute.net/content/pointless-fun>
(~>) :: (a' -> a) -> (b -> b') -> ((a -> b) -> (a' -> b'))
g ~> f = (f .) . (. g)
{-
-- Examples
f1,f1' :: Int -> Int
f1 n = n + n
f1' = memo f1
-}