@@ -1,4 +1,6 @@
1+{-# LANGUAGE StrictData #-}
12{-# LANGUAGE TypeFamilyDependencies #-}
3+{-# LANGUAGE ConstraintKinds #-}
24{-# LANGUAGE AllowAmbiguousTypes #-}
35{-# LANGUAGE DataKinds #-}
46{-# LANGUAGE FlexibleContexts #-}
@@ -28,7 +30,7 @@
2830-- motivation behind the library, the underlying theory, and implementation details.
2931--
3032-- This module offers many of the combinators mentioned in the work of
31−-- Macedo (2012) and Oliveira (2012).
33+-- Macedo (2012) and Oliveira (2012).
3234--
3335-- This is an Internal module and it is no supposed to be imported.
3436--
@@ -41,17 +43,26 @@ module LAoP.Matrix.Internal
4143 --
4244 -- There exists two type families that make it easier to write
4345 -- matrix dimensions: 'FromNat' and 'Count'. This approach
44− -- leads to a very straightforward implementation
45− -- of LAoP combinators.
46+ -- leads to a very straightforward implementation
47+ -- of LAoP combinators.
4648
4749 -- * Type safe matrix representation
4850 Matrix (..),
4951
52+ -- * Constraint type aliases
53+ Countable,
54+ CountableDims,
55+ CountableN,
56+ CountableDimsN,
57+ FL,
58+ FLN,
59+ Liftable,
60+ Trivial,
61+
5062 -- * Primitives
51− empty,
5263 one,
53− junc,
54− split,
64+ join,
65+ fork,
5566
5667 -- * Auxiliary type families
5768 FromNat,
@@ -64,6 +75,7 @@ module LAoP.Matrix.Internal
6475 toLists,
6576 toList,
6677 matrixBuilder,
78+ matrixBuilder',
6779 row,
6880 col,
6981 zeros,
@@ -74,28 +86,38 @@ module LAoP.Matrix.Internal
7486 -- * Misc
7587 -- ** Get dimensions
7688 columns,
89+ columns',
7790 rows,
91+ rows',
7892
7993 -- ** Matrix Transposition
8094 tr,
8195
96+ -- ** Scalar multiplication/division of matrices
97+ (.|),
98+ (./),
99+
82100 -- ** Selective operator
83− select,
101+ select,
102+ branch,
84103
85104 -- ** McCarthy's Conditional
86105 cond,
87106
88107 -- ** Matrix "abiding"
89− abideJS,
90− abideSJ,
108+ abideJF,
109+ abideFJ,
110+
111+ -- ** Zip matrices
112+ zipWithM,
91113
92114 -- * Biproduct approach
93− -- ** Split
115+ -- ** Fork
94116 (===),
95117 -- *** Projections
96118 p1,
97119 p2,
98− -- ** Junc
120+ -- ** Join
99121 (|||),
100122 -- *** Injections
101123 i1,
@@ -108,12 +130,13 @@ module LAoP.Matrix.Internal
108130
109131 -- | Note that given the restrictions imposed it is not possible to
110132 -- implement the standard type classes present in standard Haskell.
133+
111134 -- *** Matrix pairing projections
112− kp1,
113− kp2,
135+ fstM,
136+ sndM,
114137
115138 -- *** Matrix pairing
116− khatri,
139+ kr,
117140
118141 -- * Matrix composition and lifting
119142
@@ -121,10 +144,10 @@ module LAoP.Matrix.Internal
121144
122145 -- | Note that given the restrictions imposed it is not possible to
123146 -- implement the standard type classes present in standard Haskell.
124− identity,
147+ iden,
125148 comp,
126− fromF,
127149 fromF',
150+ fromF,
128151
129152 -- * Matrix printing
130153 pretty,
@@ -151,21 +174,22 @@ import LAoP.Utils.Internal
151174import Data.Bool
152175import Data.Kind
153176import Data.List
177+import Data.Maybe
154178import Data.Proxy
179+import Data.Typeable
155180import Data.Void
156181import GHC.TypeLits
157182import Data.Type.Equality
158183import GHC.Generics
159184import Control.DeepSeq
160−import Control.Category
161−import Prelude hiding ((.))
185+import Prelude hiding (id, (.))
186+import qualified Prelude (id, (.))
162187
163188-- | LAoP (Linear Algebra of Programming) Inductive Matrix definition.
164189data Matrix e cols rows where
165− Empty :: Matrix e Void Void
166190 One :: e -> Matrix e () ()
167− Junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
168− Split :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
191+ Join :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
192+ Fork :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
169193
170194deriving instance (Show e) => Show (Matrix e cols rows)
171195
@@ -175,7 +199,7 @@ deriving instance (Show e) => Show (Matrix e cols rows)
175199-- 'Generic' instance.
176200type family Count (d :: Type) :: Nat where
177201 Count (Natural n m) = (m - n) + 1
178− Count (Powerset a) = (^) (Count a) (Count a)
202+ Count (List a) = (^) 2 (Count a)
179203 Count (Either a b) = (+) (Count a) (Count b)
180204 Count (a, b) = (*) (Count a) (Count b)
181205 Count (a -> b) = (^) (Count b) (Count a)
@@ -192,205 +216,197 @@ type family Count (d :: Type) :: Nat where
192216--
193217-- Thanks to Li-Yao Xia this type family is super fast.
194218type family FromNat (n :: Nat) :: Type where
195− FromNat 0 = Void
196219 FromNat 1 = ()
197220 FromNat n = FromNat' (Mod n 2 == 0) (FromNat (Div n 2))
198221
199222type family FromNat' (b :: Bool) (m :: Type) :: Type where
200− FromNat' 'True m = Either m m
223+ FromNat' 'True m = Either m m
201224 FromNat' 'False m = Either () (Either m m)
202225
203226-- | Type family that normalizes the representation of a given data
204227-- structure
205228type family Normalize (d :: Type) :: Type where
206− Normalize d = FromNat (Count d)
207−
208−-- | It isn't possible to implement the 'id' function so it's
209−-- implementation is 'undefined'. However 'comp' can be and this partial
210−-- class implementation exists just to make the code more readable.
211−--
212−-- Please use 'identity' instead.
229+ Normalize (Either a b) = Either (Normalize a) (Normalize b)
230+ Normalize d = FromNat (Count d)
231+
232+-- | Constraint type synonyms to keep the type signatures less convoluted
233+type Countable a = KnownNat (Count a)
234+type CountableN a = KnownNat (Count (Normalize a))
235+type CountableDims a b = (Countable a, Countable b)
236+type CountableDimsN a b = (CountableN a, CountableN b)
237+type FL a b = FromLists a b
238+type FLN a b = FromLists (Normalize a) (Normalize b)
239+type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)
240+type Trivial a = FromNat (Count a) ~ a
241+
242+-- | It is possible to implement a constrained version of the category type
243+-- class.
213244instance (Num e) => Category (Matrix e) where
214− id = undefined
215− (.) = comp
245+ type Object (Matrix e) a = (FL a a, Countable a)
246+ id = iden
247+ (.) = comp
216248
217249instance NFData e => NFData (Matrix e cols rows) where
218− rnf Empty = ()
219− rnf (One e) = rnf e
220− rnf (Junc a b) = rnf a `seq` rnf b
221− rnf (Split a b) = rnf a `seq` rnf b
250+ rnf (One e) = rnf e
251+ rnf (Join a b) = rnf a `seq` rnf b
252+ rnf (Fork a b) = rnf a `seq` rnf b
222253
223254instance Eq e => Eq (Matrix e cols rows) where
224− Empty == Empty = True
225− (One a) == (One b) = a == b
226− (Junc a b) == (Junc c d) = a == c && b == d
227− (Split a b) == (Split c d) = a == c && b == d
228− x@(Split _ _) == y@(Junc _ _) = x == abideJS y
229− x@(Junc _ _) == y@(Split _ _) = abideJS x == y
255+ (One a) == (One b) = a == b
256+ (Join a b) == (Join c d) = a == c && b == d
257+ (Fork a b) == (Fork c d) = a == c && b == d
258+ x@(Fork _ _) == y@(Join _ _) = x == abideJF y
259+ x@(Join _ _) == y@(Fork _ _) = abideJF x == y
230260
231261instance Num e => Num (Matrix e cols rows) where
262+ a + b = zipWithM (+) a b
263+
264+ a - b = zipWithM (-) a b
265+
266+ a * b = zipWithM (*) a b
267+
268+ abs (One a) = One (abs a)
269+ abs (Join a b) = Join (abs a) (abs b)
270+ abs (Fork a b) = Fork (abs a) (abs b)
232271
233− Empty + Empty = Empty
234− (One a) + (One b) = One (a + b)
235− (Junc a b) + (Junc c d) = Junc (a + c) (b + d)
236− (Split a b) + (Split c d) = Split (a + c) (b + d)
237− x@(Split _ _) + y@(Junc _ _) = x + abideJS y
238− x@(Junc _ _) + y@(Split _ _) = abideJS x + y
239−
240− Empty - Empty = Empty
241− (One a) - (One b) = One (a - b)
242− (Junc a b) - (Junc c d) = Junc (a - c) (b - d)
243− (Split a b) - (Split c d) = Split (a - c) (b - d)
244− x@(Split _ _) - y@(Junc _ _) = x - abideJS y
245− x@(Junc _ _) - y@(Split _ _) = abideJS x - y
246−
247− Empty * Empty = Empty
248− (One a) * (One b) = One (a * b)
249− (Junc a b) * (Junc c d) = Junc (a * c) (b * d)
250− (Split a b) * (Split c d) = Split (a * c) (b * d)
251− x@(Split _ _) * y@(Junc _ _) = x * abideJS y
252− x@(Junc _ _) * y@(Split _ _) = abideJS x * y
253−
254− abs Empty = Empty
255− abs (One a) = One (abs a)
256− abs (Junc a b) = Junc (abs a) (abs b)
257− abs (Split a b) = Split (abs a) (abs b)
258−
259− signum Empty = Empty
260− signum (One a) = One (signum a)
261− signum (Junc a b) = Junc (signum a) (signum b)
262− signum (Split a b) = Split (signum a) (signum b)
272+ signum (One a) = One (signum a)
273+ signum (Join a b) = Join (signum a) (signum b)
274+ signum (Fork a b) = Fork (signum a) (signum b)
263275
264276instance Ord e => Ord (Matrix e cols rows) where
265− Empty <= Empty = True
266− (One a) <= (One b) = a <= b
267− (Junc a b) <= (Junc c d) = (a <= c) && (b <= d)
268− (Split a b) <= (Split c d) = (a <= c) && (b <= d)
269− x@(Split _ _) <= y@(Junc _ _) = x <= abideJS y
270− x@(Junc _ _) <= y@(Split _ _) = abideJS x <= y
277+ (One a) <= (One b) = a <= b
278+ (Join a b) <= (Join c d) = (a <= c) && (b <= d)
279+ (Fork a b) <= (Fork c d) = (a <= c) && (b <= d)
280+ x@(Fork _ _) <= y@(Join _ _) = x <= abideJF y
281+ x@(Join _ _) <= y@(Fork _ _) = abideJF x <= y
271282
272283-- Primitives
273284
274−-- | Empty matrix constructor
275−empty :: Matrix e Void Void
276−empty = Empty
277−
278285-- | Unit matrix constructor
279286one :: e -> Matrix e () ()
280287one = One
281288
282−-- | Matrix 'Junc' constructor
283−junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
284−junc = Junc
289+-- | Matrix 'Join' constructor
290+join :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
291+join = Join
285292
286293infixl 3 |||
287294
288−-- | Matrix 'Junc' constructor
295+-- | Matrix 'Join' constructor
289296(|||) :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
290−(|||) = Junc
297+(|||) = Join
291298
292−-- | Matrix 'Split' constructor
293−split :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
294−split = Split
299+-- | Matrix 'Fork' constructor
300+fork :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
301+fork = Fork
295302
296303infixl 2 ===
297304
298−-- | Matrix 'Split' constructor
305+-- | Matrix 'Fork' constructor
299306(===) :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
300−(===) = Split
307+(===) = Fork
301308
302309-- Construction
303310
304311-- | Type class for defining the 'fromList' conversion function.
305312--
306313-- Given that it is not possible to branch on types at the term level type
307−-- classes are needed bery much like an inductive definition but on types.
308−class FromLists e cols rows where
314+-- classes are needed very much like an inductive definition but on types.
315+class FromLists cols rows where
309316 -- | Build a matrix out of a list of list of elements. Throws a runtime
310317 -- error if the dimensions do not match.
311318 fromLists :: [[e]] -> Matrix e cols rows
312319
313−instance FromLists e Void Void where
314− fromLists [] = Empty
315− fromLists _ = error "Wrong dimensions"
316−
317−instance {-# OVERLAPPING #-} FromLists e () () where
320+instance FromLists () () where
318321 fromLists [[e]] = One e
319322 fromLists _ = error "Wrong dimensions"
320323
321−instance {-# OVERLAPPING #-} (FromLists e cols ()) => FromLists e (Either () cols) () where
322− fromLists [h : t] = Junc (One h) (fromLists [t])
324+instance (FromLists cols ()) => FromLists (Either () cols) () where
325+ fromLists [h : t] = Join (One h) (fromLists [t])
323326 fromLists _ = error "Wrong dimensions"
324327
325−instance {-# OVERLAPPABLE #-} (FromLists e a (), FromLists e b (), KnownNat (Count a)) => FromLists e (Either a b) () where
326− fromLists [l] =
328+instance {-# OVERLAPPABLE #-}
329+ (FromLists a (), FromLists b (), Countable a) => FromLists (Either a b) () where
330+ fromLists [l] =
327331 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))
328− in Junc (fromLists [take rowsA l]) (fromLists [drop rowsA l])
332+ in Join (fromLists [take rowsA l]) (fromLists [drop rowsA l])
329333 fromLists _ = error "Wrong dimensions"
330334
331−instance {-# OVERLAPPING #-} (FromLists e () rows) => FromLists e () (Either () rows) where
332− fromLists ([h] : t) = Split (One h) (fromLists t)
335+instance (FromLists () rows) => FromLists () (Either () rows) where
336+ fromLists ([h] : t) = Fork (One h) (fromLists t)
333337 fromLists _ = error "Wrong dimensions"
334338
335−instance {-# OVERLAPPABLE #-} (FromLists e () a, FromLists e () b, KnownNat (Count a)) => FromLists e () (Either a b) where
336− fromLists l@([_] : _) =
339+instance {-# OVERLAPPABLE #-}
340+ (FromLists () a, FromLists () b, Countable a) => FromLists () (Either a b) where
341+ fromLists l@([_] : _) =
337342 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))
338− in Split (fromLists (take rowsA l)) (fromLists (drop rowsA l))
343+ in Fork (fromLists (take rowsA l)) (fromLists (drop rowsA l))
339344 fromLists _ = error "Wrong dimensions"
340345
341−instance {-# OVERLAPPABLE #-} (FromLists e (Either a b) c, FromLists e (Either a b) d, KnownNat (Count c)) => FromLists e (Either a b) (Either c d) where
346+instance (FromLists (Either a b) c, FromLists (Either a b) d, Countable c)
347+ => FromLists (Either a b) (Either c d) where
342348 fromLists l@(h : t) =
343349 let lh = length h
344350 rowsC = fromInteger (natVal (Proxy :: Proxy (Count c)))
345− condition = all (== lh) (map length t)
351+ condition = all ((== lh) . length) t
346352 in if lh > 0 && condition
347− then Split (fromLists (take rowsC l)) (fromLists (drop rowsC l))
353+ then Fork (fromLists (take rowsC l)) (fromLists (drop rowsC l))
348354 else error "Not all rows have the same length"
349355
350356-- | Matrix builder function. Constructs a matrix provided with
351−-- a construction function.
352−matrixBuilder ::
357+-- a construction function that operates with indices.
358+matrixBuilder' ::
353359 forall e cols rows.
354− ( FromLists e cols rows,
355− KnownNat (Count cols),
356− KnownNat (Count rows)
357− ) =>
358− ((Int, Int) -> e) ->
359− Matrix e cols rows
360−matrixBuilder f =
360+ ( FL cols rows,
361+ CountableDims cols rows
362+ ) => ((Int, Int) -> e) -> Matrix e cols rows
363+matrixBuilder' f =
361364 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))
362365 r = fromInteger $ natVal (Proxy :: Proxy (Count rows))
363366 positions = [(a, b) | a <- [0 .. (r - 1)], b <- [0 .. (c - 1)]]
364367 in fromLists . map (map f) . groupBy (\(x, _) (w, _) -> x == w) $ positions
365368
369+-- | Matrix builder function. Constructs a matrix provided with
370+-- a construction function that operates with arbitrary types.
371+matrixBuilder ::
372+ forall e a b.
373+ ( FLN a b,
374+ Enum a,
375+ Enum b,
376+ Bounded a,
377+ Bounded b,
378+ Eq a,
379+ Countable b
380+ ) => ((a, b) -> e) -> Matrix e (Normalize a) (Normalize b)
381+matrixBuilder f =
382+ let r = fromInteger $ natVal (Proxy :: Proxy (Count b))
383+ positions = [(a, b) | a <- [minBound .. maxBound], b <- [minBound .. maxBound]]
384+ in fromLists . map (map f) . transpose . buildList r $ positions
385+ where
386+ buildList _ [] = []
387+ buildList r l = take r l : buildList r (drop r l)
388+
366389-- | Constructs a column vector matrix
367−col :: (FromLists e () rows) => [e] -> Matrix e () rows
390+col :: (FL () rows) => [e] -> Matrix e () rows
368391col = fromLists . map (: [])
369392
370393-- | Constructs a row vector matrix
371−row :: (FromLists e cols ()) => [e] -> Matrix e cols ()
394+row :: (FL cols ()) => [e] -> Matrix e cols ()
372395row = fromLists . (: [])
373396
374397-- | Lifts functions to matrices with arbitrary dimensions.
375398--
376399-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
377400-- types @a@ or @b@ allows.
378−fromF ::
401+fromF' ::
379402 forall a b cols rows e.
380− ( Bounded a,
381− Bounded b,
382− Enum a,
383− Enum b,
384− Eq b,
385− Num e,
386− Ord e,
387− KnownNat (Count cols),
388− KnownNat (Count rows),
389− FromLists e rows cols
403+ ( Liftable e a b,
404+ CountableDims cols rows,
405+ FL rows cols
390406 ) =>
391407 (a -> b) ->
392408 Matrix e cols rows
393−fromF f =
409+fromF' f =
394410 let minA = minBound @a
395411 maxA = maxBound @a
396412 minB = minBound @b
@@ -400,8 +416,8 @@ fromF f =
400416 elementsA = take ccols [minA .. maxA]
401417 elementsB = take rrows [minB .. maxB]
402418 combinations = (,) <$> elementsA <*> elementsB
403− combAp = map snd . sort . map (\(a, b) -> if f a == b
404− then ((fromEnum a, fromEnum b), 1)
419+ combAp = map snd . sort . map (\(a, b) -> if f a == b
420+ then ((fromEnum a, fromEnum b), 1)
405421 else ((fromEnum a, fromEnum b), 0)) $ combinations
406422 mList = buildList combAp rrows
407423 in tr $ fromLists mList
@@ -411,48 +427,23 @@ fromF f =
411427
412428-- | Lifts functions to matrices with dimensions matching @a@ and @b@
413429-- cardinality's.
414−fromF' ::
430+fromF ::
415431 forall a b e.
416− ( Bounded a,
417− Bounded b,
418− Enum a,
419− Enum b,
420− Eq b,
421− Num e,
422− Ord e,
423− KnownNat (Count (Normalize a)),
424− KnownNat (Count (Normalize b)),
425− FromLists e (Normalize b) (Normalize a)
432+ ( Liftable e a b,
433+ CountableDimsN a b,
434+ FLN b a
426435 ) =>
427436 (a -> b) ->
428437 Matrix e (Normalize a) (Normalize b)
429−fromF' f =
430− let minA = minBound @a
431− maxA = maxBound @a
432− minB = minBound @b
433− maxB = maxBound @b
434− ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))
435− rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))
436− elementsA = take ccols [minA .. maxA]
437− elementsB = take rrows [minB .. maxB]
438− combinations = (,) <$> elementsA <*> elementsB
439− combAp = map snd . sort . map (\(a, b) -> if f a == b
440− then ((fromEnum a, fromEnum b), 1)
441− else ((fromEnum a, fromEnum b), 0)) $ combinations
442− mList = buildList combAp rrows
443− in tr $ fromLists mList
444− where
445− buildList [] _ = []
446− buildList l r = take r l : buildList (drop r l) r
438+fromF = fromF'
447439
448440-- Conversion
449441
450442-- | Converts a matrix to a list of lists of elements.
451443toLists :: Matrix e cols rows -> [[e]]
452−toLists Empty = []
453444toLists (One e) = [[e]]
454−toLists (Split l r) = toLists l ++ toLists r
455−toLists (Junc l r) = zipWith (++) (toLists l) (toLists r)
445+toLists (Fork l r) = toLists l ++ toLists r
446+toLists (Join l r) = zipWith (++) (toLists l) (toLists r)
456447
457448-- | Converts a matrix to a list of elements.
458449toList :: Matrix e cols rows -> [e]
@@ -461,37 +452,38 @@ toList = concat . toLists
461452-- Zeros Matrix
462453
463454-- | The zero matrix. A matrix wholly filled with zeros.
464−zeros :: (Num e, FromLists e cols rows, KnownNat (Count cols), KnownNat (Count rows)) => Matrix e cols rows
465−zeros = matrixBuilder (const 0)
455+zeros :: (Num e, FL cols rows, CountableDims cols rows) => Matrix e cols rows
456+zeros = matrixBuilder' (const 0)
466457
467458-- Ones Matrix
468459
469460-- | The ones matrix. A matrix wholly filled with ones.
470461--
471462-- Also known as T (Top) matrix.
472−ones :: (Num e, FromLists e cols rows, KnownNat (Count cols), KnownNat (Count rows)) => Matrix e cols rows
473−ones = matrixBuilder (const 1)
463+ones :: (Num e, FL cols rows, CountableDims cols rows) => Matrix e cols rows
464+ones = matrixBuilder' (const 1)
474465
475466-- Const Matrix
476467
477468-- | The constant matrix constructor. A matrix wholly filled with a given
478469-- value.
479−constant :: (Num e, FromLists e cols rows, KnownNat (Count cols), KnownNat (Count rows)) => e -> Matrix e cols rows
480−constant e = matrixBuilder (const e)
470+constant :: (Num e, FL cols rows, CountableDims cols rows) => e -> Matrix e cols rows
471+constant e = matrixBuilder' (const e)
481472
482473-- Bang Matrix
483474
484475-- | The T (Top) row vector matrix.
485−bang :: forall e cols. (Num e, Enum e, FromLists e cols (), KnownNat (Count cols)) => Matrix e cols ()
476+bang :: forall e cols. (Num e, Enum e, FL cols (), Countable cols) => Matrix e cols ()
486477bang =
487478 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))
488479 in fromLists [take c [1, 1 ..]]
489480
490−-- Identity Matrix
481+-- iden Matrix
491482
492−-- | Identity matrix.
493−identity :: (Num e, FromLists e cols cols, KnownNat (Count cols)) => Matrix e cols cols
494−identity = matrixBuilder (bool 0 1 . uncurry (==))
483+-- | iden matrix.
484+iden :: forall e cols . (Num e, FL cols cols, Countable cols) => Matrix e cols cols
485+iden = matrixBuilder' (bool 0 1 . uncurry (==))
486+{-# NOINLINE iden #-}
495487
496488-- Matrix composition (MMM)
497489
@@ -500,60 +492,90 @@ identity = matrixBuilder (bool 0 1 . uncurry (==))
500492-- This definition takes advantage of divide-and-conquer and fusion laws
501493-- from LAoP.
502494comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
503−comp Empty Empty = Empty
504−comp (One a) (One b) = One (a * b)
505−comp (Junc a b) (Split c d) = comp a c + comp b d -- Divide-and-conquer law
506−comp (Split a b) c = Split (comp a c) (comp b c) -- Split fusion law
507−comp c (Junc a b) = Junc (comp c a) (comp c b) -- Junc fusion law
495+comp (One a) (One b) = One (a * b)
496+comp (Join a b) (Fork c d) = comp a c + comp b d -- Divide-and-conquer law
497+comp (Fork a b) c = Fork (comp a c) (comp b c) -- Fork fusion law
498+comp c (Join a b) = Join (comp c a) (comp c b) -- Join fusion law
499+{-# NOINLINE comp #-}
500+{-# RULES
501+ "comp/iden1" forall m. comp m iden = m ;
502+ "comp/iden2" forall m. comp iden m = m
503+#-}
504+
505+-- Scalar multiplication of matrices
506+
507+infixl 7 .|
508+-- | Scalar multiplication of matrices.
509+(.|) :: Num e => e -> Matrix e cols rows -> Matrix e cols rows
510+(.|) e (One a) = One (e * a)
511+(.|) e (Join a b) = Join (e .| a) (e .| b)
512+(.|) e (Fork a b) = Fork (e .| a) (e .| b)
513+
514+-- Scalar division of matrices
515+
516+infixl 7 ./
517+-- | Scalar multiplication of matrices.
518+(./) :: Fractional e => Matrix e cols rows -> e -> Matrix e cols rows
519+(./) (One a) e = One (a / e)
520+(./) (Join a b) e = Join (a ./ e) (b ./ e)
521+(./) (Fork a b) e = Fork (a ./ e) (b ./ e)
508522
509523-- Projections
510524
511525-- | Biproduct first component projection
512−p1 :: forall e m n. (Num e, KnownNat (Count n), KnownNat (Count m), FromLists e n m, FromLists e m m) => Matrix e (Either m n) m
513−p1 =
514− let iden = identity :: Matrix e m m
515− zero = zeros :: Matrix e n m
516− in junc iden zero
526+p1 :: (Num e, CountableDims n m, FL n m, FL m m) => Matrix e (Either m n) m
527+p1 = join id zeros
517528
518529-- | Biproduct second component projection
519−p2 :: forall e m n. (Num e, KnownNat (Count n), KnownNat (Count m), FromLists e m n, FromLists e n n) => Matrix e (Either m n) n
520−p2 =
521− let iden = identity :: Matrix e n n
522− zero = zeros :: Matrix e m n
523− in junc zero iden
530+p2 :: (Num e, CountableDims n m, FL m n, FL n n) => Matrix e (Either m n) n
531+p2 = join zeros id
524532
525533-- Injections
526534
527535-- | Biproduct first component injection
528−i1 :: (Num e, KnownNat (Count n), KnownNat (Count m), FromLists e n m, FromLists e m m) => Matrix e m (Either m n)
536+i1 :: (Num e, CountableDims n m, FL n m, FL m m) => Matrix e m (Either m n)
529537i1 = tr p1
530538
531539-- | Biproduct second component injection
532−i2 :: (Num e, KnownNat (Count n), KnownNat (Count m), FromLists e m n, FromLists e n n) => Matrix e n (Either m n)
540+i2 :: (Num e, CountableDims n m, FL m n, FL n n) => Matrix e n (Either m n)
533541i2 = tr p2
534542
535543-- Dimensions
536544
537545-- | Obtain the number of rows.
538546--
539−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the
540−-- dimensions in constant time.
541−--
542−-- TODO: A 'rows' function that does not need the 'KnownNat' constraint in
543−-- exchange for performance.
544−rows :: forall e cols rows. (KnownNat (Count rows)) => Matrix e cols rows -> Int
547+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the
548+-- dimensions in constant time. For a version that doesn't require the
549+-- constraint see 'rows''.
550+rows :: forall e cols rows. (Countable rows) => Matrix e cols rows -> Int
545551rows _ = fromInteger $ natVal (Proxy :: Proxy (Count rows))
546552
553+-- | Obtain the number of rows in an inefficient manner, but without any
554+-- constraints.
555+--
556+-- For a more efficient version see 'rows'.
557+rows' :: Matrix e cols rows -> Int
558+rows' (One _) = 1
559+rows' (Join lhs _) = rows' lhs
560+rows' (Fork top bottom) = rows' top + rows' bottom
561+
547562-- | Obtain the number of columns.
548−--
549−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the
550−-- dimensions in constant time.
551563--
552−-- TODO: A 'columns' function that does not need the 'KnownNat' constraint in
553−-- exchange for performance.
554−columns :: forall e cols rows. (KnownNat (Count cols)) => Matrix e cols rows -> Int
564+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the
565+-- dimensions in constant time. For a version that doesn't require the
566+-- constraint see 'columns''.
567+columns :: forall e cols rows. (Countable cols) => Matrix e cols rows -> Int
555568columns _ = fromInteger $ natVal (Proxy :: Proxy (Count cols))
556569
570+-- | Obtain the number of columns in an inefficient manner, but without any
571+-- constraints.
572+--
573+-- For a more efficient version see 'columns'.
574+columns' :: Matrix e cols rows -> Int
575+columns' (One _) = 1
576+columns' (Join lhs rhs) = columns' lhs + columns' rhs
577+columns' (Fork top _) = columns' top
578+
557579-- Coproduct Bifunctor
558580
559581infixl 5 -|-
@@ -562,157 +584,172 @@ infixl 5 -|-
562584(-|-) ::
563585 forall e n k m j.
564586 ( Num e,
565− KnownNat (Count j),
566− KnownNat (Count k),
567− FromLists e k k,
568− FromLists e j k,
569− FromLists e k j,
570− FromLists e j j
587+ CountableDims j k,
588+ FL k k,
589+ FL j k,
590+ FL k j,
591+ FL j j
571592 ) =>
572593 Matrix e n k ->
573594 Matrix e m j ->
574595 Matrix e (Either n m) (Either k j)
575−(-|-) a b = Junc (i1 . a) (i2 . b)
596+(-|-) a b = Join (i1 . a) (i2 . b)
576597
577598-- Khatri Rao Product and projections
578599
579600-- | Khatri Rao product first component projection matrix.
580−kp1 ::
601+fstM ::
581602 forall e m k .
582603 ( Num e,
583− KnownNat (Count k),
584− FromLists e (FromNat (Count m * Count k)) m,
585− KnownNat (Count m),
586− KnownNat (Count (Normalize (m, k)))
604+ CountableDims k m,
605+ FL (Normalize (m, k)) m,
606+ CountableN (m, k)
587607 ) => Matrix e (Normalize (m, k)) m
588−kp1 = matrixBuilder f
608+fstM = matrixBuilder' f
589609 where
590610 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))
591611 f (x, y)
592612 | y >= (x * offset) && y <= (x * offset + offset - 1) = 1
593− | otherwise = 0
613+ | otherwise = 0
594614
595615-- | Khatri Rao product second component projection matrix.
596−kp2 ::
616+sndM ::
597617 forall e m k .
598618 ( Num e,
599− KnownNat (Count k),
600− FromLists e (FromNat (Count m * Count k)) k,
601− KnownNat (Count m),
602− KnownNat (Count (Normalize (m, k)))
619+ CountableDims k m,
620+ FL (Normalize (m, k)) k,
621+ CountableN (m, k)
603622 ) => Matrix e (Normalize (m, k)) k
604−kp2 = matrixBuilder f
623+sndM = matrixBuilder' f
605624 where
606625 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))
607626 f (x, y)
608627 | x == y || mod (y - x) offset == 0 = 1
609− | otherwise = 0
628+ | otherwise = 0
610629
611630-- | Khatri Rao Matrix product also known as matrix pairing.
612631--
613632-- NOTE: That this is not a true categorical product, see for instance:
614−--
633+--
615634-- @
616−-- | kp1 . khatri a b == a
617−-- khatri a b ==> |
618−-- | kp2 . khatri a b == b
635+-- | fstM . kr a b == a
636+-- kr a b ==> |
637+-- | sndM . kr a b == b
619638-- @
620639--
621640-- __Emphasis__ on the implication symbol.
622−khatri ::
623− forall e cols a b.
641+kr ::
642+ forall e cols a b.
624643 ( Num e,
625− KnownNat (Count a),
626− KnownNat (Count b),
627− KnownNat (Count (Normalize (a, b))),
628− FromLists e (Normalize (a, b)) a,
629− FromLists e (Normalize (a, b)) b
644+ CountableDims a b,
645+ CountableN (a, b),
646+ FL (Normalize (a, b)) a,
647+ FL (Normalize (a, b)) b
630648 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Normalize (a, b))
631−khatri a b =
632− let kp1' = kp1 @e @a @b
633− kp2' = kp2 @e @a @b
634− in (tr kp1') . a * (tr kp2') . b
649+kr a b =
650+ let fstM' = fstM @e @a @b
651+ sndM' = sndM @e @a @b
652+ in (tr fstM' . a) * (tr sndM' . b)
635653
636654-- Product Bifunctor (Kronecker)
637655
638656infixl 4 ><
639657
640658-- | Matrix product functor also known as kronecker product
641−(><) ::
642− forall e m p n q.
659+(><) ::
660+ forall e m p n q.
643661 ( Num e,
644− KnownNat (Count m),
645− KnownNat (Count n),
646− KnownNat (Count p),
647− KnownNat (Count q),
648− KnownNat (Count (Normalize (m, n))),
649− FromLists e (Normalize (m, n)) m,
650− FromLists e (Normalize (m, n)) n,
651− KnownNat (Count (Normalize (p, q))),
652− FromLists e (Normalize (p, q)) p,
653− FromLists e (Normalize (p, q)) q
654− )
662+ CountableDims m n,
663+ CountableDims p q,
664+ CountableDimsN (m, n) (p, q),
665+ FL (Normalize (m, n)) m,
666+ FL (Normalize (m, n)) n,
667+ FL (Normalize (p, q)) p,
668+ FL (Normalize (p, q)) q
669+ )
655670 => Matrix e m p -> Matrix e n q -> Matrix e (Normalize (m, n)) (Normalize (p, q))
656671(><) a b =
657− let kp1' = kp1 @e @m @n
658− kp2' = kp2 @e @m @n
659− in khatri (a . kp1') (b . kp2')
672+ let fstM' = fstM @e @m @n
673+ sndM' = sndM @e @m @n
674+ in kr (a . fstM') (b . sndM')
660675
661−-- Matrix abide Junc Split
676+-- Matrix abide Join Fork
662677
663−-- | Matrix "abiding" followin the 'Junc'-'Split' abide law.
664−--
678+-- | Matrix "abiding" following the 'Join'-'Fork' exchange law.
679+--
665680-- Law:
666681--
667682-- @
668−-- 'Junc' ('Split' a c) ('Split' b d) == 'Split' ('Junc' a b) ('Junc' c d)
683+-- 'Join' ('Fork' a c) ('Fork' b d) == 'Fork' ('Join' a b) ('Join' c d)
669684-- @
670−abideJS :: Matrix e cols rows -> Matrix e cols rows
671−abideJS (Junc (Split a c) (Split b d)) = Split (Junc (abideJS a) (abideJS b)) (Junc (abideJS c) (abideJS d)) -- Junc-Split abide law
672−abideJS Empty = Empty
673−abideJS (One e) = One e
674−abideJS (Junc a b) = Junc (abideJS a) (abideJS b)
675−abideJS (Split a b) = Split (abideJS a) (abideJS b)
685+abideJF :: Matrix e cols rows -> Matrix e cols rows
686+abideJF (Join (Fork a c) (Fork b d)) = Fork (Join (abideJF a) (abideJF b)) (Join (abideJF c) (abideJF d)) -- Join-Fork abide law
687+abideJF (One e) = One e
688+abideJF (Join a b) = Join (abideJF a) (abideJF b)
689+abideJF (Fork a b) = Fork (abideJF a) (abideJF b)
676690
677−-- Matrix abide Split Junc
691+-- Matrix abide Fork Join
678692
679−-- | Matrix "abiding" followin the 'Split'-'Junc' abide law.
680−--
693+-- | Matrix "abiding" followin the 'Fork'-'Join' abide law.
694+--
681695-- @
682−-- 'Split' ('Junc' a b) ('Junc' c d) == 'Junc' ('Split' a c) ('Split' b d)
696+-- 'Fork' ('Join' a b) ('Join' c d) == 'Join' ('Fork' a c) ('Fork' b d)
683697-- @
684−abideSJ :: Matrix e cols rows -> Matrix e cols rows
685−abideSJ (Split (Junc a b) (Junc c d)) = Junc (Split (abideSJ a) (abideSJ c)) (Split (abideSJ b) (abideSJ d)) -- Split-Junc abide law
686−abideSJ Empty = Empty
687−abideSJ (One e) = One e
688−abideSJ (Junc a b) = Junc (abideSJ a) (abideSJ b)
689−abideSJ (Split a b) = Split (abideSJ a) (abideSJ b)
698+abideFJ :: Matrix e cols rows -> Matrix e cols rows
699+abideFJ (Fork (Join a b) (Join c d)) = Join (Fork (abideFJ a) (abideFJ c)) (Fork (abideFJ b) (abideFJ d)) -- Fork-Join abide law
700+abideFJ (One e) = One e
701+abideFJ (Join a b) = Join (abideFJ a) (abideFJ b)
702+abideFJ (Fork a b) = Fork (abideFJ a) (abideFJ b)
690703
691704-- Matrix transposition
692705
693706-- | Matrix transposition.
694707tr :: Matrix e cols rows -> Matrix e rows cols
695−tr Empty = Empty
696−tr (One e) = One e
697−tr (Junc a b) = Split (tr a) (tr b)
698−tr (Split a b) = Junc (tr a) (tr b)
708+tr (One e) = One e
709+tr (Join a b) = Fork (tr a) (tr b)
710+tr (Fork a b) = Join (tr a) (tr b)
699711
700712-- Selective 'select' operator
701713
702714-- | Selective functors 'select' operator equivalent inspired by the
703715-- ArrowMonad solution presented in the paper.
704−select :: (Num e, FromLists e b b, KnownNat (Count b)) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b
705−select m y = junc y identity . m
716+select :: (Num e, FL b b, Countable b) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b
717+select (Fork a b) y = y . a + b -- Divide-and-conquer law
718+select (Join (Fork a c) (Fork b d)) y = join (y . a + c) (y . b + d) -- Pattern matching + DnC law
719+select m y = join y id . m
720+
721+branch ::
722+ ( Num e,
723+ CountableDims a b,
724+ CountableDims c (Either b c),
725+ FL c b,
726+ FL a b,
727+ FL a a,
728+ FL b b,
729+ FL c c,
730+ FL b a,
731+ FL b c,
732+ FL (Either b c) b,
733+ FL (Either b c) c
734+ )
735+ => Matrix e cols (Either a b) -> Matrix e a c -> Matrix e b c -> Matrix e cols c
736+branch x l r = f x `select` g l `select` r
737+ where
738+ f :: (Num e, Countable a, CountableDims b c, FL a b, FL c b, FL b b, FL b a, FL a a)
739+ => Matrix e cols (Either a b) -> Matrix e cols (Either a (Either b c))
740+ f m = fork (tr i1) (i1 . tr i2) . m
741+ g :: (Num e, CountableDims b c, FL b c, FL c c) => Matrix e a c -> Matrix e a (Either b c)
742+ g m = i2 . m
706743
707744-- McCarthy's Conditional
708745
709746-- | McCarthy's Conditional expresses probabilistic choice.
710747cond ::
711− ( cols ~ FromNat (Count cols),
712− KnownNat (Count cols),
713− FromLists e () cols,
714− FromLists e cols (),
715− FromLists e cols cols,
748+ ( Trivial cols,
749+ Countable cols,
750+ FL () cols,
751+ FL cols (),
752+ FL cols cols,
716753 Bounded a,
717754 Enum a,
718755 Num e,
@@ -720,14 +757,14 @@ cond ::
720757 )
721758 =>
722759 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows
723−cond p f g = junc f g . grd p
724−
725−grd ::
726− ( q ~ FromNat (Count q),
727− KnownNat (Count q),
728− FromLists e () q,
729− FromLists e q (),
730− FromLists e q q,
760+cond p f g = join f g . grd p
761+
762+grd ::
763+ ( Trivial q,
764+ Countable q,
765+ FL () q,
766+ FL q (),
767+ FL q q,
731768 Bounded a,
732769 Enum a,
733770 Num e,
@@ -735,63 +772,86 @@ grd ::
735772 )
736773 =>
737774 (a -> Bool) -> Matrix e q (Either q q)
738−grd f = split (corr f) (corr (not . f))
739−
740−corr ::
741− forall e a q .
742− ( q ~ FromNat (Count q),
743− KnownNat (Count q),
744− FromLists e () q,
745− FromLists e q (),
746− FromLists e q q,
747− Bounded a,
748− Enum a,
749− Num e,
750− Ord e
751− )
775+grd f = fork (corr f) (corr (not . f))
776+
777+corr ::
778+ forall e a q .
779+ ( Trivial q,
780+ Countable q,
781+ FL () q,
782+ FL q (),
783+ FL q q,
784+ Liftable e a Bool
785+ )
752786 => (a -> Bool) -> Matrix e q q
753−corr p = let f = fromF p :: Matrix e q ()
754− in khatri f (identity :: Matrix e q q)
787+corr p = let f = fromF' p :: Matrix e q ()
788+ in kr f (id :: Matrix e q q)
755789
756790-- Pretty print
757791
758792prettyAux :: Show e => [[e]] -> [[e]] -> String
759793prettyAux [] _ = ""
760−prettyAux [[e]] m = "│ " ++ fill (show e) ++ " │\n"
794+prettyAux [[e]] m = "│ " ++ fill (show e) ++ " │\n"
761795 where
762796 v = fmap show m
763797 widest = maximum $ fmap length v
764798 fill str = replicate (widest - length str - 2) ' ' ++ str
765−prettyAux [h] m = "│ " ++ fill (unwords $ map show h) ++ " │\n"
799+prettyAux [h] m = "│ " ++ fill (unwords $ map show h) ++ " │\n"
766800 where
767− v = fmap show m
768− widest = maximum $ fmap length v
801+ v = fmap show m
802+ widest = maximum $ fmap length v
769803 fill str = replicate (widest - length str - 2) ' ' ++ str
770−prettyAux (h : t) l = "│ " ++ fill (unwords $ map show h) ++ " │\n" ++
804+prettyAux (h : t) l = "│ " ++ fill (unwords $ map show h) ++ " │\n" ++
771805 prettyAux t l
772806 where
773− v = fmap show l
774− widest = maximum $ fmap length v
807+ v = fmap show l
808+ widest = maximum $ fmap length v
775809 fill str = replicate (widest - length str - 2) ' ' ++ str
776810
777811-- | Matrix pretty printer
778−pretty :: (KnownNat (Count cols), Show e) => Matrix e cols rows -> String
779−pretty m = "┌ " ++ unwords (replicate (columns m) blank) ++ " ┐\n" ++
780− prettyAux (toLists m) (toLists m) ++
781− "└ " ++ unwords (replicate (columns m) blank) ++ " ┘"
782− where
783− v = fmap show (toList m)
784− widest = maximum $ fmap length v
812+pretty :: (CountableDims cols rows, Show e) => Matrix e cols rows -> String
813+pretty m = concat
814+ [ "┌ ", unwords (replicate (columns m) blank), " ┐\n"
815+ , unlines
816+ [ "│ " ++ unwords (fmap (\j -> fill $ show $ getElem i j m) [1..columns m]) ++ " │" | i <- [1..rows m] ]
817+ , "└ ", unwords (replicate (columns m) blank), " ┘"
818+ ]
819+ where
820+ strings = map show (toList m)
821+ widest = maximum $ map length strings
785822 fill str = replicate (widest - length str) ' ' ++ str
786− blank = fill ""
823+ blank = fill ""
824+ safeGet i j m
825+ | i > rows m || j > columns m || i < 1 || j < 1 = Nothing
826+ | otherwise = Just $ unsafeGet i j m (toList m)
827+ unsafeGet i j m l = l !! encode (columns m) (i,j)
828+ encode m (i,j) = (i-1)*m + j - 1
829+ getElem i j m =
830+ fromMaybe
831+ (error $
832+ "getElem: Trying to get the "
833+ ++ show (i, j)
834+ ++ " element from a "
835+ ++ show (rows m) ++ "x" ++ show (columns m)
836+ ++ " matrix."
837+ )
838+ (safeGet i j m)
787839
788840-- | Matrix pretty printer
789−prettyPrint :: (KnownNat (Count cols), Show e) => Matrix e cols rows -> IO ()
841+prettyPrint :: (CountableDims cols rows, Show e) => Matrix e cols rows -> IO ()
790842prettyPrint = putStrLn . pretty
791843
844+-- | Zip two matrices with a given binary function
845+zipWithM :: (e -> f -> g) -> Matrix e cols rows -> Matrix f cols rows -> Matrix g cols rows
846+zipWithM f (One a) (One b) = One (f a b)
847+zipWithM f (Join a b) (Join c d) = Join (zipWithM f a c) (zipWithM f b d)
848+zipWithM f (Fork a b) (Fork c d) = Fork (zipWithM f a c) (zipWithM f b d)
849+zipWithM f x@(Fork _ _) y@(Join _ _) = zipWithM f x (abideJF y)
850+zipWithM f x@(Join _ _) y@(Fork _ _) = zipWithM f (abideJF x) y
851+
792852-- Relational operators functions
793853
794−type Boolean = Natural 0 1
854+type Boolean = Natural 0 1
795855type Relation a b = Matrix Boolean a b
796856
797857-- | Helper conversion function
@@ -807,54 +867,47 @@ fromBool False = nat 0
807867
808868-- | Relational negation
809869negateM :: Relation cols rows -> Relation cols rows
810−negateM Empty = Empty
811−negateM (One (Nat 0)) = One (Nat 1)
812−negateM (One (Nat 1)) = One (Nat 0)
813−negateM (Junc a b) = Junc (negateM a) (negateM b)
814−negateM (Split a b) = Split (negateM a) (negateM b)
870+negateM (One (Nat p)) = One (Nat (negate p))
871+negateM (Join a b) = Join (negateM a) (negateM b)
872+negateM (Fork a b) = Fork (negateM a) (negateM b)
815873
816874-- | Relational addition
817875orM :: Relation cols rows -> Relation cols rows -> Relation cols rows
818−orM Empty Empty = Empty
819−orM (One a) (One b) = One (fromBool (toBool a || toBool b))
820−orM (Junc a b) (Junc c d) = Junc (orM a c) (orM b d)
821−orM (Split a b) (Split c d) = Split (orM a c) (orM b d)
822−orM x@(Split _ _) y@(Junc _ _) = orM x (abideJS y)
823−orM x@(Junc _ _) y@(Split _ _) = orM (abideJS x) y
876+orM (One a) (One b) = One (fromBool (toBool a || toBool b))
877+orM (Join a b) (Join c d) = Join (orM a c) (orM b d)
878+orM (Fork a b) (Fork c d) = Fork (orM a c) (orM b d)
879+orM x@(Fork _ _) y@(Join _ _) = orM x (abideJF y)
880+orM x@(Join _ _) y@(Fork _ _) = orM (abideJF x) y
824881
825882-- | Relational multiplication
826883andM :: Relation cols rows -> Relation cols rows -> Relation cols rows
827−andM Empty Empty = Empty
828−andM (One a) (One b) = One (fromBool (toBool a && toBool b))
829−andM (Junc a b) (Junc c d) = Junc (andM a c) (andM b d)
830−andM (Split a b) (Split c d) = Split (andM a c) (andM b d)
831−andM x@(Split _ _) y@(Junc _ _) = andM x (abideJS y)
832−andM x@(Junc _ _) y@(Split _ _) = andM (abideJS x) y
884+andM (One a) (One b) = One (fromBool (toBool a && toBool b))
885+andM (Join a b) (Join c d) = Join (andM a c) (andM b d)
886+andM (Fork a b) (Fork c d) = Fork (andM a c) (andM b d)
887+andM x@(Fork _ _) y@(Join _ _) = andM x (abideJF y)
888+andM x@(Join _ _) y@(Fork _ _) = andM (abideJF x) y
833889
834890-- | Relational subtraction
835891subM :: Relation cols rows -> Relation cols rows -> Relation cols rows
836−subM Empty Empty = Empty
837−subM (One a) (One b) = if a - b < nat 0 then One (nat 0) else One (a - b)
838−subM (Junc a b) (Junc c d) = Junc (subM a c) (subM b d)
839−subM (Split a b) (Split c d) = Split (subM a c) (subM b d)
840−subM x@(Split _ _) y@(Junc _ _) = subM x (abideJS y)
841−subM x@(Junc _ _) y@(Split _ _) = subM (abideJS x) y
892+subM (One a) (One b) = if a - b < nat 0 then One (nat 0) else One (a - b)
893+subM (Join a b) (Join c d) = Join (subM a c) (subM b d)
894+subM (Fork a b) (Fork c d) = Fork (subM a c) (subM b d)
895+subM x@(Fork _ _) y@(Join _ _) = subM x (abideJF y)
896+subM x@(Join _ _) y@(Fork _ _) = subM (abideJF x) y
842897
843898-- | Matrix relational composition.
844899compRel :: Relation cr rows -> Relation cols cr -> Relation cols rows
845−compRel Empty Empty = Empty
846−compRel (One a) (One b) = One (fromBool (toBool a && toBool b))
847−compRel (Junc a b) (Split c d) = orM (compRel a c) (compRel b d) -- Divide-and-conquer law
848−compRel (Split a b) c = Split (compRel a c) (compRel b c) -- Split fusion law
849−compRel c (Junc a b) = Junc (compRel c a) (compRel c b) -- Junc fusion law
900+compRel (One a) (One b) = One (fromBool (toBool a && toBool b))
901+compRel (Join a b) (Fork c d) = orM (compRel a c) (compRel b d) -- Divide-and-conquer law
902+compRel (Fork a b) c = Fork (compRel a c) (compRel b c) -- Fork fusion law
903+compRel c (Join a b) = Join (compRel c a) (compRel c b) -- Join fusion law
850904
851905-- | Matrix relational right division
852906divR :: Relation b c -> Relation b a -> Relation a c
853−divR Empty Empty = Empty
854907divR (One a) (One b) = One (fromBool (not (toBool b) || toBool a)) -- b implies a
855−divR (Junc a b) (Junc c d) = andM (divR a c) (divR b d)
856−divR (Split a b) c = Split (divR a c) (divR b c)
857−divR c (Split a b) = Junc (divR c a) (divR c b)
908+divR (Join a b) (Join c d) = andM (divR a c) (divR b d)
909+divR (Fork a b) c = Fork (divR a c) (divR b c)
910+divR c (Fork a b) = Join (divR c a) (divR c b)
858911
859912-- | Matrix relational left division
860913divL :: Relation c b -> Relation a b -> Relation a c
@@ -870,97 +923,52 @@ divS s r = divL r s `intersection` divR (tr r) (tr s)
870923--
871924-- NOTE: Be careful to not ask for a relation bigger than the cardinality of
872925-- types @a@ or @b@ allows.
873−fromFRel ::
926+fromFRel' ::
874927 forall a b cols rows.
875− ( Bounded a,
876− Bounded b,
877− Enum a,
878− Enum b,
879− Eq b,
880− KnownNat (Count cols),
881− KnownNat (Count rows),
882− FromLists (Natural 0 1) rows cols
928+ ( Liftable Boolean a b,
929+ CountableDims cols rows,
930+ FL rows cols
883931 ) =>
884932 (a -> b) ->
885933 Relation cols rows
886−fromFRel f =
887− let minA = minBound @a
888− maxA = maxBound @a
889− minB = minBound @b
890− maxB = maxBound @b
891− ccols = fromInteger $ natVal (Proxy :: Proxy (Count cols))
892− rrows = fromInteger $ natVal (Proxy :: Proxy (Count rows))
893− elementsA = take ccols [minA .. maxA]
894− elementsB = take rrows [minB .. maxB]
895− combinations = (,) <$> elementsA <*> elementsB
896− combAp = map snd . sort . map (\(a, b) -> if f a == b
897− then ((fromEnum a, fromEnum b), nat 1)
898− else ((fromEnum a, fromEnum b), nat 0)) $ combinations
899− mList = buildList combAp rrows
900− in tr $ fromLists mList
901− where
902− buildList [] _ = []
903− buildList l r = take r l : buildList (drop r l) r
934+fromFRel' = fromF'
904935
905936-- | Lifts functions to relations with dimensions matching @a@ and @b@
906937-- cardinality's.
907−fromFRel' ::
938+fromFRel ::
908939 forall a b.
909− ( Bounded a,
910− Bounded b,
911− Enum a,
912− Enum b,
913− Eq b,
914− KnownNat (Count (Normalize a)),
915− KnownNat (Count (Normalize b)),
916− FromLists (Natural 0 1) (Normalize b) (Normalize a)
940+ ( Liftable Boolean a b,
941+ CountableDimsN a b,
942+ FLN b a
917943 ) =>
918944 (a -> b) ->
919945 Relation (Normalize a) (Normalize b)
920−fromFRel' f =
921− let minA = minBound @a
922− maxA = maxBound @a
923− minB = minBound @b
924− maxB = maxBound @b
925− ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))
926− rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))
927− elementsA = take ccols [minA .. maxA]
928− elementsB = take rrows [minB .. maxB]
929− combinations = (,) <$> elementsA <*> elementsB
930− combAp = map snd . sort . map (\(a, b) -> if f a == b
931− then ((fromEnum a, fromEnum b), nat 1)
932− else ((fromEnum a, fromEnum b), nat 0)) $ combinations
933− mList = buildList combAp rrows
934− in tr $ fromLists mList
935− where
936− buildList [] _ = []
937− buildList l r = take r l : buildList (drop r l) r
946+fromFRel = fromFRel'
938947
939948-- | Lifts a relation function to a Boolean Matrix
940−toRel ::
949+toRel ::
941950 forall a b.
942951 ( Bounded a,
943952 Bounded b,
944953 Enum a,
945954 Enum b,
946955 Eq b,
947− KnownNat (Count (Normalize a)),
948− KnownNat (Count (Normalize b)),
949− FromLists (Natural 0 1) (Normalize b) (Normalize a)
950− )
956+ CountableDims a b,
957+ FLN b a
958+ )
951959 => (a -> b -> Bool) -> Relation (Normalize a) (Normalize b)
952960toRel f =
953961 let minA = minBound @a
954962 maxA = maxBound @a
955963 minB = minBound @b
956964 maxB = maxBound @b
957− ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))
958− rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))
965+ ccols = fromInteger $ natVal (Proxy :: Proxy (Count a))
966+ rrows = fromInteger $ natVal (Proxy :: Proxy (Count b))
959967 elementsA = take ccols [minA .. maxA]
960968 elementsB = take rrows [minB .. maxB]
961969 combinations = (,) <$> elementsA <*> elementsB
962− combAp = map snd . sort . map (\(a, b) -> if uncurry f (a, b)
963− then ((fromEnum a, fromEnum b), nat 1)
970+ combAp = map snd . sort . map (\(a, b) -> if uncurry f (a, b)
971+ then ((fromEnum a, fromEnum b), nat 1)
964972 else ((fromEnum a, fromEnum b), nat 0)) $ combinations
965973 mList = buildList combAp rrows
966974 in tr $ fromLists mList