{-# LANGUAGE MultiParamTypeClasses, GADTs, FlexibleContexts, FlexibleInstances, TypeFamilies #-} ----------------------------------------------------------------------------- -- | -- Module : GArrowSkeleton -- Copyright : none -- License : public domain -- -- Maintainer : Adam Megacz -- Stability : experimental -- -- | Sometimes it is convenient to be able to get your hands on the -- explicit boxes-and-wires representation of a GArrow-polymorphic -- term. GArrowSkeleton lets you do that. -- -- HOWEVER: technically this instance violates the laws (and RULEs) -- for Control.Category; the compiler might choose to optimize (f >>> -- id) into f, and this optimization would produce a change in -- behavior below -- you'd get (GAS_comp f GAS_id) instead of f. In -- practice this means that the user must be prepared for the skeleton -- TikZ diagram to be a nondeterministically-chosen boxes-and-wires -- diagram which ks *equivalent to* the term, rather than structurally -- exactly equal to it. -- module GArrowSkeleton (GArrowSkeleton(..), optimize) where import Prelude hiding ( id, (.), lookup ) import Control.Category import GHC.HetMet.GArrow import Unify import Control.Monad.State data GArrowSkeleton m :: * -> * -> * where GAS_const :: Int -> GArrowSkeleton m () Int GAS_id :: GArrowSkeleton m x x GAS_comp :: GArrowSkeleton m x y -> GArrowSkeleton m y z -> GArrowSkeleton m x z GAS_first :: GArrowSkeleton m x y -> GArrowSkeleton m (x,z) (y,z) GAS_second :: GArrowSkeleton m x y -> GArrowSkeleton m (z,x) (z,y) GAS_cancell :: GArrowSkeleton m ((),x) x GAS_cancelr :: GArrowSkeleton m (x,()) x GAS_uncancell :: GArrowSkeleton m x ((),x) GAS_uncancelr :: GArrowSkeleton m x (x,()) GAS_assoc :: GArrowSkeleton m ((x,y),z) (x,(y,z)) GAS_unassoc :: GArrowSkeleton m (x,(y,z)) ((x,y),z) GAS_drop :: GArrowSkeleton m x () GAS_copy :: GArrowSkeleton m x (x,x) GAS_swap :: GArrowSkeleton m (x,y) (y,x) GAS_merge :: GArrowSkeleton m (x,y) z GAS_loopl :: GArrowSkeleton m (z,x) (z,y) -> GArrowSkeleton m x y GAS_loopr :: GArrowSkeleton m (x,z) (y,z) -> GArrowSkeleton m x y GAS_misc :: m x y -> GArrowSkeleton m x y instance Category (GArrowSkeleton m) where id = GAS_id g . f = GAS_comp f g instance GArrow (GArrowSkeleton m) (,) () where ga_first = GAS_first ga_second = GAS_second ga_cancell = GAS_cancell ga_cancelr = GAS_cancelr ga_uncancell = GAS_uncancell ga_uncancelr = GAS_uncancelr ga_assoc = GAS_assoc ga_unassoc = GAS_unassoc instance GArrowDrop (GArrowSkeleton m) (,) () where ga_drop = GAS_drop instance GArrowCopy (GArrowSkeleton m) (,) () where ga_copy = GAS_copy instance GArrowSwap (GArrowSkeleton m) (,) () where ga_swap = GAS_swap instance GArrowLoop (GArrowSkeleton m) (,) () where ga_loopl = GAS_loopl ga_loopr = GAS_loopr type instance GArrowTensor (GArrowSkeleton m) = (,) type instance GArrowUnit (GArrowSkeleton m) = () type instance GArrowExponent (GArrowSkeleton m) = (->) instance GArrowSTKCL (GArrowSkeleton m) -- -- | Simple structural equality on skeletons. NOTE: two skeletons -- with the same shape but different types will nonetheless be "equal"; -- there's no way around this since types are gone at runtime. -- instance Eq ((GArrowSkeleton m) a b) where x == y = x === y where (===) :: (GArrowSkeleton m) a b -> (GArrowSkeleton m) c d -> Bool (GAS_id ) === (GAS_id ) = True (GAS_comp g f) === (GAS_comp g' f') = f===f' && g===g' (GAS_first f) === (GAS_first f') = f===f' (GAS_second f) === (GAS_second f') = f===f' GAS_cancell === GAS_cancell = True GAS_cancelr === GAS_cancelr = True GAS_uncancell === GAS_uncancell = True GAS_uncancelr === GAS_uncancelr = True GAS_drop === GAS_drop = True (GAS_const i) === (GAS_const i') = i==i' GAS_copy === GAS_copy = True GAS_merge === GAS_merge = True GAS_swap === GAS_swap = True GAS_assoc === GAS_assoc = True GAS_unassoc === GAS_unassoc = True (GAS_loopl f) === (GAS_loopl f') = f === f' (GAS_loopr f) === (GAS_loopr f') = f === f' _ === _ = False -- FIXME: GAS_misc's are never equal!!! -- -- | Performs some very simple-minded optimizations on a -- boxes-and-wires diagram. Preserves equivalence up to the GArrow -- laws, but no guarantees about which optimizations actually happen. -- optimize :: (GArrowSkeleton m) a b -> (GArrowSkeleton m) a b optimize x = let x' = optimize' x in if x == x' then x' else optimize x' where optimize' :: (GArrowSkeleton m) a b -> (GArrowSkeleton m) a b -- Some optimizations fail due to misparenthesization; we default to -- left-associativity and hope for the best optimize' (GAS_comp f (GAS_comp g h)) = GAS_comp (GAS_comp f g) h optimize' (GAS_comp (GAS_comp f (GAS_comp g h)) k) = GAS_comp (GAS_comp (GAS_comp f g) h) k optimize' (GAS_comp (GAS_comp GAS_unassoc (GAS_second g)) GAS_assoc) = GAS_second (GAS_second g) optimize' (GAS_comp (GAS_comp (GAS_comp f GAS_unassoc) (GAS_second g)) GAS_assoc) = GAS_comp f (GAS_second (GAS_second g)) optimize' (GAS_comp (GAS_comp f g) h) = case optimize_pair g h of Nothing -> GAS_comp (optimize' (GAS_comp f g)) h' Just ret' -> GAS_comp f' ret' where f' = optimize' f g' = optimize' g h' = optimize' h optimize' (GAS_comp f g ) = case optimize_pair f g of Nothing -> GAS_comp f' g' Just ret' -> ret' where f' = optimize' f g' = optimize' g optimize' (GAS_first GAS_id ) = GAS_id optimize' (GAS_second GAS_id ) = GAS_id optimize' (GAS_first f ) = GAS_first $ optimize' f optimize' (GAS_second f ) = GAS_second $ optimize' f optimize' (GAS_loopl GAS_id) = GAS_id optimize' (GAS_loopr GAS_id) = GAS_id optimize' (GAS_loopl f ) = GAS_loopl $ optimize' f optimize' (GAS_loopr f ) = GAS_loopr $ optimize' f optimize' x = x optimize_pair :: (GArrowSkeleton m) a b -> (GArrowSkeleton m) b c -> Maybe ((GArrowSkeleton m) a c) optimize_pair f GAS_drop = Just $ GAS_drop optimize_pair GAS_id f = Just $ f optimize_pair f GAS_id = Just $ f optimize_pair GAS_uncancell GAS_cancell = Just $ GAS_id optimize_pair GAS_uncancelr GAS_cancelr = Just $ GAS_id optimize_pair GAS_cancell GAS_uncancell = Just $ GAS_id optimize_pair GAS_cancelr GAS_uncancelr = Just $ GAS_id optimize_pair GAS_uncancelr GAS_cancell = Just $ GAS_id optimize_pair GAS_uncancell GAS_cancelr = Just $ GAS_id -- first priority: eliminate GAS_first optimize_pair (GAS_first f) GAS_cancelr = Just $ GAS_comp GAS_cancelr f optimize_pair (GAS_second f) GAS_cancell = Just $ GAS_comp GAS_cancell f optimize_pair GAS_uncancelr (GAS_first f) = Just $ GAS_comp f GAS_uncancelr optimize_pair GAS_uncancell (GAS_second f) = Just $ GAS_comp f GAS_uncancell -- second priority: push GAS_swap leftward optimize_pair (GAS_second f) GAS_swap = Just $ GAS_comp GAS_swap (GAS_first f) optimize_pair (GAS_first f) GAS_swap = Just $ GAS_comp GAS_swap (GAS_second f) optimize_pair GAS_swap GAS_swap = Just $ GAS_id optimize_pair GAS_swap GAS_cancell = Just $ GAS_cancelr optimize_pair GAS_swap GAS_cancelr = Just $ GAS_cancell optimize_pair (GAS_first f) (GAS_first g) = Just $ GAS_first (GAS_comp f g) optimize_pair (GAS_second f) (GAS_second g) = Just $ GAS_second (GAS_comp f g) optimize_pair GAS_assoc GAS_cancell = Just $ GAS_first GAS_cancell optimize_pair GAS_unassoc GAS_cancelr = Just $ GAS_second GAS_cancelr optimize_pair GAS_assoc (GAS_second GAS_cancell) = Just $ GAS_first GAS_cancelr optimize_pair GAS_unassoc (GAS_first GAS_cancell) = Just $ GAS_cancell -- FIXME: valid only for central morphisms --optimize_pair (GAS_second f) (GAS_first g) = Just $ GAS_comp (GAS_first g) (GAS_second f) optimize_pair (GAS_first g) (GAS_second f) = Just $ GAS_comp (GAS_second f) (GAS_first g) optimize_pair _ _ = Nothing