@@ -1,3 +1,4 @@
1+{-# LANGUAGE StrictData #-}
12{-# LANGUAGE TypeFamilyDependencies #-}
23{-# LANGUAGE ConstraintKinds #-}
34{-# LANGUAGE AllowAmbiguousTypes #-}
@@ -29,7 +30,7 @@
2930-- motivation behind the library, the underlying theory, and implementation details.
3031--
3132-- This module offers many of the combinators mentioned in the work of
32−-- Macedo (2012) and Oliveira (2012).
33+-- Macedo (2012) and Oliveira (2012).
3334--
3435-- This is an Internal module and it is no supposed to be imported.
3536--
@@ -42,17 +43,26 @@ module LAoP.Matrix.Internal
4243 --
4344 -- There exists two type families that make it easier to write
4445 -- matrix dimensions: 'FromNat' and 'Count'. This approach
45− -- leads to a very straightforward implementation
46− -- of LAoP combinators.
46+ -- leads to a very straightforward implementation
47+ -- of LAoP combinators.
4748
4849 -- * Type safe matrix representation
4950 Matrix (..),
5051
52+ -- * Constraint type aliases
53+ Countable,
54+ CountableDims,
55+ CountableN,
56+ CountableDimsN,
57+ FL,
58+ FLN,
59+ Liftable,
60+ Trivial,
61+
5162 -- * Primitives
52− empty,
5363 one,
54− junc,
55− split,
64+ join,
65+ fork,
5666
5767 -- * Auxiliary type families
5868 FromNat,
@@ -65,6 +75,7 @@ module LAoP.Matrix.Internal
6575 toLists,
6676 toList,
6777 matrixBuilder,
78+ matrixBuilder',
6879 row,
6980 col,
7081 zeros,
@@ -75,11 +86,17 @@ module LAoP.Matrix.Internal
7586 -- * Misc
7687 -- ** Get dimensions
7788 columns,
89+ columns',
7890 rows,
91+ rows',
7992
8093 -- ** Matrix Transposition
8194 tr,
8295
96+ -- ** Scalar multiplication/division of matrices
97+ (.|),
98+ (./),
99+
83100 -- ** Selective operator
84101 select,
85102 branch,
@@ -88,16 +105,19 @@ module LAoP.Matrix.Internal
88105 cond,
89106
90107 -- ** Matrix "abiding"
91− abideJS,
92− abideSJ,
108+ abideJF,
109+ abideFJ,
110+
111+ -- ** Zip matrices
112+ zipWithM,
93113
94114 -- * Biproduct approach
95− -- ** Split
115+ -- ** Fork
96116 (===),
97117 -- *** Projections
98118 p1,
99119 p2,
100− -- ** Junc
120+ -- ** Join
101121 (|||),
102122 -- *** Injections
103123 i1,
@@ -110,12 +130,13 @@ module LAoP.Matrix.Internal
110130
111131 -- | Note that given the restrictions imposed it is not possible to
112132 -- implement the standard type classes present in standard Haskell.
133+
113134 -- *** Matrix pairing projections
114− kp1,
115− kp2,
135+ fstM,
136+ sndM,
116137
117138 -- *** Matrix pairing
118− khatri,
139+ kr,
119140
120141 -- * Matrix composition and lifting
121142
@@ -123,10 +144,10 @@ module LAoP.Matrix.Internal
123144
124145 -- | Note that given the restrictions imposed it is not possible to
125146 -- implement the standard type classes present in standard Haskell.
126− identity,
147+ iden,
127148 comp,
128− fromF,
129149 fromF',
150+ fromF,
130151
131152 -- * Matrix printing
132153 pretty,
@@ -153,21 +174,22 @@ import LAoP.Utils.Internal
153174import Data.Bool
154175import Data.Kind
155176import Data.List
177+import Data.Maybe
156178import Data.Proxy
179+import Data.Typeable
157180import Data.Void
158181import GHC.TypeLits
159182import Data.Type.Equality
160183import GHC.Generics
161184import Control.DeepSeq
162−import Control.Category
163−import Prelude hiding ((.))
185+import Prelude hiding (id, (.))
186+import qualified Prelude (id, (.))
164187
165188-- | LAoP (Linear Algebra of Programming) Inductive Matrix definition.
166189data Matrix e cols rows where
167− Empty :: Matrix e Void Void
168190 One :: e -> Matrix e () ()
169− Junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
170− 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)
171193
172194deriving instance (Show e) => Show (Matrix e cols rows)
173195
@@ -194,120 +216,95 @@ type family Count (d :: Type) :: Nat where
194216--
195217-- Thanks to Li-Yao Xia this type family is super fast.
196218type family FromNat (n :: Nat) :: Type where
197− FromNat 0 = Void
198219 FromNat 1 = ()
199220 FromNat n = FromNat' (Mod n 2 == 0) (FromNat (Div n 2))
200221
201222type family FromNat' (b :: Bool) (m :: Type) :: Type where
202− FromNat' 'True m = Either m m
223+ FromNat' 'True m = Either m m
203224 FromNat' 'False m = Either () (Either m m)
204225
205226-- | Type family that normalizes the representation of a given data
206227-- structure
207228type family Normalize (d :: Type) :: Type where
208− Normalize d = FromNat (Count d)
229+ Normalize (Either a b) = Either (Normalize a) (Normalize b)
230+ Normalize d = FromNat (Count d)
209231
210232-- | Constraint type synonyms to keep the type signatures less convoluted
211233type Countable a = KnownNat (Count a)
212234type CountableN a = KnownNat (Count (Normalize a))
213−type CountableDimensions a b = (Countable a, Countable b)
214−type CountableDimensionsN a b = (CountableN a, CountableN b)
215−type FromListsN e a b = FromLists e (Normalize a) (Normalize b)
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)
216239type 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
217241
218−-- | It isn't possible to implement the 'id' function so it's
219−-- implementation is 'undefined'. However 'comp' can be and this partial
220−-- class implementation exists just to make the code more readable.
221−--
222−-- Please use 'identity' instead.
242+-- | It is possible to implement a constrained version of the category type
243+-- class.
223244instance (Num e) => Category (Matrix e) where
224− id = undefined
225− (.) = comp
245+ type Object (Matrix e) a = (FL a a, Countable a)
246+ id = iden
247+ (.) = comp
226248
227249instance NFData e => NFData (Matrix e cols rows) where
228− rnf Empty = ()
229− rnf (One e) = rnf e
230− rnf (Junc a b) = rnf a `seq` rnf b
231− 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
232253
233254instance Eq e => Eq (Matrix e cols rows) where
234− Empty == Empty = True
235− (One a) == (One b) = a == b
236− (Junc a b) == (Junc c d) = a == c && b == d
237− (Split a b) == (Split c d) = a == c && b == d
238− x@(Split _ _) == y@(Junc _ _) = x == abideJS y
239− 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
240260
241261instance 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
242267
243− Empty + Empty = Empty
244− (One a) + (One b) = One (a + b)
245− (Junc a b) + (Junc c d) = Junc (a + c) (b + d)
246− (Split a b) + (Split c d) = Split (a + c) (b + d)
247− x@(Split _ _) + y@(Junc _ _) = x + abideJS y
248− x@(Junc _ _) + y@(Split _ _) = abideJS x + y
249−
250− Empty - Empty = Empty
251− (One a) - (One b) = One (a - b)
252− (Junc a b) - (Junc c d) = Junc (a - c) (b - d)
253− (Split a b) - (Split c d) = Split (a - c) (b - d)
254− x@(Split _ _) - y@(Junc _ _) = x - abideJS y
255− x@(Junc _ _) - y@(Split _ _) = abideJS x - y
256−
257− Empty * Empty = Empty
258− (One a) * (One b) = One (a * b)
259− (Junc a b) * (Junc c d) = Junc (a * c) (b * d)
260− (Split a b) * (Split c d) = Split (a * c) (b * d)
261− x@(Split _ _) * y@(Junc _ _) = x * abideJS y
262− x@(Junc _ _) * y@(Split _ _) = abideJS x * y
263−
264− abs Empty = Empty
265− abs (One a) = One (abs a)
266− abs (Junc a b) = Junc (abs a) (abs b)
267− abs (Split a b) = Split (abs a) (abs b)
268−
269− signum Empty = Empty
270− signum (One a) = One (signum a)
271− signum (Junc a b) = Junc (signum a) (signum b)
272− signum (Split a b) = Split (signum a) (signum b)
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)
271+
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)
273275
274276instance Ord e => Ord (Matrix e cols rows) where
275− Empty <= Empty = True
276− (One a) <= (One b) = a <= b
277− (Junc a b) <= (Junc c d) = (a <= c) && (b <= d)
278− (Split a b) <= (Split c d) = (a <= c) && (b <= d)
279− x@(Split _ _) <= y@(Junc _ _) = x <= abideJS y
280− 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
281282
282283-- Primitives
283284
284−-- | Empty matrix constructor
285−empty :: Matrix e Void Void
286−empty = Empty
287−
288285-- | Unit matrix constructor
289286one :: e -> Matrix e () ()
290287one = One
291288
292−-- | Matrix 'Junc' constructor
293−junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
294−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
295292
296293infixl 3 |||
297294
298−-- | Matrix 'Junc' constructor
295+-- | Matrix 'Join' constructor
299296(|||) :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
300−(|||) = Junc
297+(|||) = Join
301298
302−-- | Matrix 'Split' constructor
303−split :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
304−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
305302
306303infixl 2 ===
307304
308−-- | Matrix 'Split' constructor
305+-- | Matrix 'Fork' constructor
309306(===) :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
310−(===) = Split
307+(===) = Fork
311308
312309-- Construction
313310
@@ -315,84 +312,101 @@ infixl 2 ===
315312--
316313-- Given that it is not possible to branch on types at the term level type
317314-- classes are needed very much like an inductive definition but on types.
318−class FromLists e cols rows where
315+class FromLists cols rows where
319316 -- | Build a matrix out of a list of list of elements. Throws a runtime
320317 -- error if the dimensions do not match.
321318 fromLists :: [[e]] -> Matrix e cols rows
322319
323−instance FromLists e Void Void where
324− fromLists [] = Empty
325− fromLists _ = error "Wrong dimensions"
326−
327−instance {-# OVERLAPPING #-} FromLists e () () where
320+instance FromLists () () where
328321 fromLists [[e]] = One e
329322 fromLists _ = error "Wrong dimensions"
330323
331−instance {-# OVERLAPPING #-} (FromLists e cols ()) => FromLists e (Either () cols) () where
332− 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])
333326 fromLists _ = error "Wrong dimensions"
334327
335−instance {-# OVERLAPPABLE #-} (FromLists e a (), FromLists e b (), Countable a) => FromLists e (Either a b) () where
336− fromLists [l] =
328+instance {-# OVERLAPPABLE #-}
329+ (FromLists a (), FromLists b (), Countable a) => FromLists (Either a b) () where
330+ fromLists [l] =
337331 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))
338− in Junc (fromLists [take rowsA l]) (fromLists [drop rowsA l])
332+ in Join (fromLists [take rowsA l]) (fromLists [drop rowsA l])
339333 fromLists _ = error "Wrong dimensions"
340334
341−instance {-# OVERLAPPING #-} (FromLists e () rows) => FromLists e () (Either () rows) where
342− 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)
343337 fromLists _ = error "Wrong dimensions"
344338
345−instance {-# OVERLAPPABLE #-} (FromLists e () a, FromLists e () b, Countable a) => FromLists e () (Either a b) where
346− fromLists l@([_] : _) =
339+instance {-# OVERLAPPABLE #-}
340+ (FromLists () a, FromLists () b, Countable a) => FromLists () (Either a b) where
341+ fromLists l@([_] : _) =
347342 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))
348− in Split (fromLists (take rowsA l)) (fromLists (drop rowsA l))
343+ in Fork (fromLists (take rowsA l)) (fromLists (drop rowsA l))
349344 fromLists _ = error "Wrong dimensions"
350345
351−instance {-# OVERLAPPABLE #-} (FromLists e (Either a b) c, FromLists e (Either a b) d, Countable 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
352348 fromLists l@(h : t) =
353349 let lh = length h
354350 rowsC = fromInteger (natVal (Proxy :: Proxy (Count c)))
355− condition = all (== lh) (map length t)
351+ condition = all ((== lh) . length) t
356352 in if lh > 0 && condition
357− then Split (fromLists (take rowsC l)) (fromLists (drop rowsC l))
353+ then Fork (fromLists (take rowsC l)) (fromLists (drop rowsC l))
358354 else error "Not all rows have the same length"
359355
360356-- | Matrix builder function. Constructs a matrix provided with
361−-- a construction function.
362−matrixBuilder ::
357+-- a construction function that operates with indices.
358+matrixBuilder' ::
363359 forall e cols rows.
364− ( FromLists e cols rows,
365− CountableDimensions cols rows
366− ) =>
367− ((Int, Int) -> e) ->
368− Matrix e cols rows
369−matrixBuilder f =
360+ ( FL cols rows,
361+ CountableDims cols rows
362+ ) => ((Int, Int) -> e) -> Matrix e cols rows
363+matrixBuilder' f =
370364 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))
371365 r = fromInteger $ natVal (Proxy :: Proxy (Count rows))
372366 positions = [(a, b) | a <- [0 .. (r - 1)], b <- [0 .. (c - 1)]]
373367 in fromLists . map (map f) . groupBy (\(x, _) (w, _) -> x == w) $ positions
374368
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+
375389-- | Constructs a column vector matrix
376−col :: (FromLists e () rows) => [e] -> Matrix e () rows
390+col :: (FL () rows) => [e] -> Matrix e () rows
377391col = fromLists . map (: [])
378392
379393-- | Constructs a row vector matrix
380−row :: (FromLists e cols ()) => [e] -> Matrix e cols ()
394+row :: (FL cols ()) => [e] -> Matrix e cols ()
381395row = fromLists . (: [])
382396
383397-- | Lifts functions to matrices with arbitrary dimensions.
384398--
385399-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
386400-- types @a@ or @b@ allows.
387−fromF ::
401+fromF' ::
388402 forall a b cols rows e.
389403 ( Liftable e a b,
390− CountableDimensions cols rows,
391− FromLists e rows cols
404+ CountableDims cols rows,
405+ FL rows cols
392406 ) =>
393407 (a -> b) ->
394408 Matrix e cols rows
395−fromF f =
409+fromF' f =
396410 let minA = minBound @a
397411 maxA = maxBound @a
398412 minB = minBound @b
@@ -402,8 +416,8 @@ fromF f =
402416 elementsA = take ccols [minA .. maxA]
403417 elementsB = take rrows [minB .. maxB]
404418 combinations = (,) <$> elementsA <*> elementsB
405− combAp = map snd . sort . map (\(a, b) -> if f a == b
406− 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)
407421 else ((fromEnum a, fromEnum b), 0)) $ combinations
408422 mList = buildList combAp rrows
409423 in tr $ fromLists mList
@@ -413,41 +427,23 @@ fromF f =
413427
414428-- | Lifts functions to matrices with dimensions matching @a@ and @b@
415429-- cardinality's.
416−fromF' ::
430+fromF ::
417431 forall a b e.
418432 ( Liftable e a b,
419− CountableDimensionsN a b,
420− FromListsN e b a
433+ CountableDimsN a b,
434+ FLN b a
421435 ) =>
422436 (a -> b) ->
423437 Matrix e (Normalize a) (Normalize b)
424−fromF' f =
425− let minA = minBound @a
426− maxA = maxBound @a
427− minB = minBound @b
428− maxB = maxBound @b
429− ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))
430− rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))
431− elementsA = take ccols [minA .. maxA]
432− elementsB = take rrows [minB .. maxB]
433− combinations = (,) <$> elementsA <*> elementsB
434− combAp = map snd . sort . map (\(a, b) -> if f a == b
435− then ((fromEnum a, fromEnum b), 1)
436− else ((fromEnum a, fromEnum b), 0)) $ combinations
437− mList = buildList combAp rrows
438− in tr $ fromLists mList
439− where
440− buildList [] _ = []
441− buildList l r = take r l : buildList (drop r l) r
438+fromF = fromF'
442439
443440-- Conversion
444441
445442-- | Converts a matrix to a list of lists of elements.
446443toLists :: Matrix e cols rows -> [[e]]
447−toLists Empty = []
448444toLists (One e) = [[e]]
449−toLists (Split l r) = toLists l ++ toLists r
450−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)
451447
452448-- | Converts a matrix to a list of elements.
453449toList :: Matrix e cols rows -> [e]
@@ -456,37 +452,38 @@ toList = concat . toLists
456452-- Zeros Matrix
457453
458454-- | The zero matrix. A matrix wholly filled with zeros.
459−zeros :: (Num e, FromLists e cols rows, CountableDimensions cols rows) => Matrix e cols rows
460−zeros = matrixBuilder (const 0)
455+zeros :: (Num e, FL cols rows, CountableDims cols rows) => Matrix e cols rows
456+zeros = matrixBuilder' (const 0)
461457
462458-- Ones Matrix
463459
464460-- | The ones matrix. A matrix wholly filled with ones.
465461--
466462-- Also known as T (Top) matrix.
467−ones :: (Num e, FromLists e cols rows, CountableDimensions cols rows) => Matrix e cols rows
468−ones = matrixBuilder (const 1)
463+ones :: (Num e, FL cols rows, CountableDims cols rows) => Matrix e cols rows
464+ones = matrixBuilder' (const 1)
469465
470466-- Const Matrix
471467
472468-- | The constant matrix constructor. A matrix wholly filled with a given
473469-- value.
474−constant :: (Num e, FromLists e cols rows, CountableDimensions cols rows) => e -> Matrix e cols rows
475−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)
476472
477473-- Bang Matrix
478474
479475-- | The T (Top) row vector matrix.
480−bang :: forall e cols. (Num e, Enum e, FromLists e cols (), Countable cols) => Matrix e cols ()
476+bang :: forall e cols. (Num e, Enum e, FL cols (), Countable cols) => Matrix e cols ()
481477bang =
482478 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))
483479 in fromLists [take c [1, 1 ..]]
484480
485−-- Identity Matrix
481+-- iden Matrix
486482
487−-- | Identity matrix.
488−identity :: (Num e, FromLists e cols cols, Countable cols) => Matrix e cols cols
489−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 #-}
490487
491488-- Matrix composition (MMM)
492489
@@ -495,60 +492,90 @@ identity = matrixBuilder (bool 0 1 . uncurry (==))
495492-- This definition takes advantage of divide-and-conquer and fusion laws
496493-- from LAoP.
497494comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
498−comp Empty Empty = Empty
499−comp (One a) (One b) = One (a * b)
500−comp (Junc a b) (Split c d) = comp a c + comp b d -- Divide-and-conquer law
501−comp (Split a b) c = Split (comp a c) (comp b c) -- Split fusion law
502−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)
503522
504523-- Projections
505524
506525-- | Biproduct first component projection
507−p1 :: forall e m n. (Num e, CountableDimensions n m, FromLists e n m, FromLists e m m) => Matrix e (Either m n) m
508−p1 =
509− let iden = identity :: Matrix e m m
510− zero = zeros :: Matrix e n m
511− 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
512528
513529-- | Biproduct second component projection
514−p2 :: forall e m n. (Num e, CountableDimensions n m, FromLists e m n, FromLists e n n) => Matrix e (Either m n) n
515−p2 =
516− let iden = identity :: Matrix e n n
517− zero = zeros :: Matrix e m n
518− 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
519532
520533-- Injections
521534
522535-- | Biproduct first component injection
523−i1 :: (Num e, CountableDimensions n 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)
524537i1 = tr p1
525538
526539-- | Biproduct second component injection
527−i2 :: (Num e, CountableDimensions n 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)
528541i2 = tr p2
529542
530543-- Dimensions
531544
532545-- | Obtain the number of rows.
533546--
534−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the
535−-- dimensions in constant time.
536−--
537−-- TODO: A 'rows' function that does not need the 'KnownNat' constraint in
538−-- exchange for performance.
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''.
539550rows :: forall e cols rows. (Countable rows) => Matrix e cols rows -> Int
540551rows _ = fromInteger $ natVal (Proxy :: Proxy (Count rows))
541552
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+
542562-- | Obtain the number of columns.
543−--
544−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the
545−-- dimensions in constant time.
546563--
547−-- TODO: A 'columns' function that does not need the 'KnownNat' constraint in
548−-- exchange for performance.
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''.
549567columns :: forall e cols rows. (Countable cols) => Matrix e cols rows -> Int
550568columns _ = fromInteger $ natVal (Proxy :: Proxy (Count cols))
551569
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+
552579-- Coproduct Bifunctor
553580
554581infixl 5 -|-
@@ -557,175 +584,172 @@ infixl 5 -|-
557584(-|-) ::
558585 forall e n k m j.
559586 ( Num e,
560− CountableDimensions j k,
561− FromLists e k k,
562− FromLists e j k,
563− FromLists e k j,
564− FromLists e j j
587+ CountableDims j k,
588+ FL k k,
589+ FL j k,
590+ FL k j,
591+ FL j j
565592 ) =>
566593 Matrix e n k ->
567594 Matrix e m j ->
568595 Matrix e (Either n m) (Either k j)
569−(-|-) a b = Junc (i1 . a) (i2 . b)
596+(-|-) a b = Join (i1 . a) (i2 . b)
570597
571598-- Khatri Rao Product and projections
572599
573600-- | Khatri Rao product first component projection matrix.
574−kp1 ::
601+fstM ::
575602 forall e m k .
576603 ( Num e,
577− CountableDimensions k m,
578− FromLists e (Normalize (m, k)) m,
604+ CountableDims k m,
605+ FL (Normalize (m, k)) m,
579606 CountableN (m, k)
580607 ) => Matrix e (Normalize (m, k)) m
581−kp1 = matrixBuilder f
608+fstM = matrixBuilder' f
582609 where
583610 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))
584611 f (x, y)
585612 | y >= (x * offset) && y <= (x * offset + offset - 1) = 1
586− | otherwise = 0
613+ | otherwise = 0
587614
588615-- | Khatri Rao product second component projection matrix.
589−kp2 ::
616+sndM ::
590617 forall e m k .
591618 ( Num e,
592− CountableDimensions k m,
593− FromLists e (Normalize (m, k)) k,
619+ CountableDims k m,
620+ FL (Normalize (m, k)) k,
594621 CountableN (m, k)
595622 ) => Matrix e (Normalize (m, k)) k
596−kp2 = matrixBuilder f
623+sndM = matrixBuilder' f
597624 where
598625 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))
599626 f (x, y)
600627 | x == y || mod (y - x) offset == 0 = 1
601− | otherwise = 0
628+ | otherwise = 0
602629
603630-- | Khatri Rao Matrix product also known as matrix pairing.
604631--
605632-- NOTE: That this is not a true categorical product, see for instance:
606−--
633+--
607634-- @
608−-- | kp1 . khatri a b == a
609−-- khatri a b ==> |
610−-- | kp2 . khatri a b == b
635+-- | fstM . kr a b == a
636+-- kr a b ==> |
637+-- | sndM . kr a b == b
611638-- @
612639--
613640-- __Emphasis__ on the implication symbol.
614−khatri ::
615− forall e cols a b.
641+kr ::
642+ forall e cols a b.
616643 ( Num e,
617− CountableDimensions a b,
644+ CountableDims a b,
618645 CountableN (a, b),
619− FromLists e (Normalize (a, b)) a,
620− FromLists e (Normalize (a, b)) b
646+ FL (Normalize (a, b)) a,
647+ FL (Normalize (a, b)) b
621648 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Normalize (a, b))
622−khatri a b =
623− let kp1' = kp1 @e @a @b
624− kp2' = kp2 @e @a @b
625− 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)
626653
627654-- Product Bifunctor (Kronecker)
628655
629656infixl 4 ><
630657
631658-- | Matrix product functor also known as kronecker product
632−(><) ::
633− forall e m p n q.
659+(><) ::
660+ forall e m p n q.
634661 ( Num e,
635− CountableDimensions m n,
636− CountableDimensions p q,
637− CountableDimensionsN (m, n) (p, q),
638− FromLists e (Normalize (m, n)) m,
639− FromLists e (Normalize (m, n)) n,
640− FromLists e (Normalize (p, q)) p,
641− FromLists e (Normalize (p, q)) q
642− )
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+ )
643670 => Matrix e m p -> Matrix e n q -> Matrix e (Normalize (m, n)) (Normalize (p, q))
644671(><) a b =
645− let kp1' = kp1 @e @m @n
646− kp2' = kp2 @e @m @n
647− 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')
648675
649−-- Matrix abide Junc Split
676+-- Matrix abide Join Fork
650677
651−-- | Matrix "abiding" followin the 'Junc'-'Split' abide law.
652−--
678+-- | Matrix "abiding" following the 'Join'-'Fork' exchange law.
679+--
653680-- Law:
654681--
655682-- @
656−-- '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)
657684-- @
658−abideJS :: Matrix e cols rows -> Matrix e cols rows
659−abideJS (Junc (Split a c) (Split b d)) = Split (Junc (abideJS a) (abideJS b)) (Junc (abideJS c) (abideJS d)) -- Junc-Split abide law
660−abideJS Empty = Empty
661−abideJS (One e) = One e
662−abideJS (Junc a b) = Junc (abideJS a) (abideJS b)
663−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)
664690
665−-- Matrix abide Split Junc
691+-- Matrix abide Fork Join
666692
667−-- | Matrix "abiding" followin the 'Split'-'Junc' abide law.
668−--
693+-- | Matrix "abiding" followin the 'Fork'-'Join' abide law.
694+--
669695-- @
670−-- '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)
671697-- @
672−abideSJ :: Matrix e cols rows -> Matrix e cols rows
673−abideSJ (Split (Junc a b) (Junc c d)) = Junc (Split (abideSJ a) (abideSJ c)) (Split (abideSJ b) (abideSJ d)) -- Split-Junc abide law
674−abideSJ Empty = Empty
675−abideSJ (One e) = One e
676−abideSJ (Junc a b) = Junc (abideSJ a) (abideSJ b)
677−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)
678703
679704-- Matrix transposition
680705
681706-- | Matrix transposition.
682707tr :: Matrix e cols rows -> Matrix e rows cols
683−tr Empty = Empty
684−tr (One e) = One e
685−tr (Junc a b) = Split (tr a) (tr b)
686−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)
687711
688712-- Selective 'select' operator
689713
690714-- | Selective functors 'select' operator equivalent inspired by the
691715-- ArrowMonad solution presented in the paper.
692−select :: (Num e, FromLists e b b, Countable b) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b
693−select (Split a b) y = y . a + b -- Divide-and-conquer law
694−select (Junc (Split a c) (Split b d)) y = junc (y . a + c) (y . b + d) -- Pattern matching + DnC law
695−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
696720
697−branch ::
721+branch ::
698722 ( Num e,
699− CountableDimensions a b,
700− CountableDimensions c (Either b c),
701− FromLists e c b,
702− FromLists e a b,
703− FromLists e a a,
704− FromLists e b b,
705− FromLists e c c,
706− FromLists e b a,
707− FromLists e b c,
708− FromLists e (Either b c) b,
709− FromLists e (Either b c) c
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
710734 )
711735 => Matrix e cols (Either a b) -> Matrix e a c -> Matrix e b c -> Matrix e cols c
712736branch x l r = f x `select` g l `select` r
713737 where
714− f :: (Num e, Countable a, CountableDimensions b c, FromLists e a b, FromLists e c b, FromLists e b b, FromLists e b a, FromLists e a a)
738+ f :: (Num e, Countable a, CountableDims b c, FL a b, FL c b, FL b b, FL b a, FL a a)
715739 => Matrix e cols (Either a b) -> Matrix e cols (Either a (Either b c))
716− f m = split (tr i1) (i1 . tr i2) . m
717− g :: (Num e, CountableDimensions b c, FromLists e b c, FromLists e c c) => Matrix e a c -> Matrix e 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)
718742 g m = i2 . m
719743
720744-- McCarthy's Conditional
721745
722746-- | McCarthy's Conditional expresses probabilistic choice.
723747cond ::
724− ( cols ~ Normalize cols,
748+ ( Trivial cols,
725749 Countable cols,
726− FromLists e () cols,
727− FromLists e cols (),
728− FromLists e cols cols,
750+ FL () cols,
751+ FL cols (),
752+ FL cols cols,
729753 Bounded a,
730754 Enum a,
731755 Num e,
@@ -733,14 +757,14 @@ cond ::
733757 )
734758 =>
735759 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows
736−cond p f g = junc f g . grd p
760+cond p f g = join f g . grd p
737761
738−grd ::
739− ( q ~ Normalize q,
762+grd ::
763+ ( Trivial q,
740764 Countable q,
741− FromLists e () q,
742− FromLists e q (),
743− FromLists e q q,
765+ FL () q,
766+ FL q (),
767+ FL q q,
744768 Bounded a,
745769 Enum a,
746770 Num e,
@@ -748,60 +772,86 @@ grd ::
748772 )
749773 =>
750774 (a -> Bool) -> Matrix e q (Either q q)
751−grd f = split (corr f) (corr (not . f))
775+grd f = fork (corr f) (corr (not . f))
752776
753−corr ::
754− forall e a q .
755− ( q ~ Normalize q,
777+corr ::
778+ forall e a q .
779+ ( Trivial q,
756780 Countable q,
757− FromLists e () q,
758− FromLists e q (),
759− FromLists e q q,
781+ FL () q,
782+ FL q (),
783+ FL q q,
760784 Liftable e a Bool
761− )
785+ )
762786 => (a -> Bool) -> Matrix e q q
763−corr p = let f = fromF p :: Matrix e q ()
764− 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)
765789
766790-- Pretty print
767791
768792prettyAux :: Show e => [[e]] -> [[e]] -> String
769793prettyAux [] _ = ""
770−prettyAux [[e]] m = "│ " ++ fill (show e) ++ " │\n"
794+prettyAux [[e]] m = "│ " ++ fill (show e) ++ " │\n"
771795 where
772796 v = fmap show m
773797 widest = maximum $ fmap length v
774798 fill str = replicate (widest - length str - 2) ' ' ++ str
775−prettyAux [h] m = "│ " ++ fill (unwords $ map show h) ++ " │\n"
799+prettyAux [h] m = "│ " ++ fill (unwords $ map show h) ++ " │\n"
776800 where
777− v = fmap show m
778− widest = maximum $ fmap length v
801+ v = fmap show m
802+ widest = maximum $ fmap length v
779803 fill str = replicate (widest - length str - 2) ' ' ++ str
780−prettyAux (h : t) l = "│ " ++ fill (unwords $ map show h) ++ " │\n" ++
804+prettyAux (h : t) l = "│ " ++ fill (unwords $ map show h) ++ " │\n" ++
781805 prettyAux t l
782806 where
783− v = fmap show l
784− widest = maximum $ fmap length v
807+ v = fmap show l
808+ widest = maximum $ fmap length v
785809 fill str = replicate (widest - length str - 2) ' ' ++ str
786810
787811-- | Matrix pretty printer
788−pretty :: (Countable cols, Show e) => Matrix e cols rows -> String
789−pretty m = "┌ " ++ unwords (replicate (columns m) blank) ++ " ┐\n" ++
790− prettyAux (toLists m) (toLists m) ++
791− "└ " ++ unwords (replicate (columns m) blank) ++ " ┘"
792− where
793− v = fmap show (toList m)
794− 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
795822 fill str = replicate (widest - length str) ' ' ++ str
796− 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)
797839
798840-- | Matrix pretty printer
799−prettyPrint :: (Countable cols, Show e) => Matrix e cols rows -> IO ()
841+prettyPrint :: (CountableDims cols rows, Show e) => Matrix e cols rows -> IO ()
800842prettyPrint = putStrLn . pretty
801843
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+
802852-- Relational operators functions
803853
804−type Boolean = Natural 0 1
854+type Boolean = Natural 0 1
805855type Relation a b = Matrix Boolean a b
806856
807857-- | Helper conversion function
@@ -817,54 +867,47 @@ fromBool False = nat 0
817867
818868-- | Relational negation
819869negateM :: Relation cols rows -> Relation cols rows
820−negateM Empty = Empty
821−negateM (One (Nat 0)) = One (Nat 1)
822−negateM (One (Nat 1)) = One (Nat 0)
823−negateM (Junc a b) = Junc (negateM a) (negateM b)
824−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)
825873
826874-- | Relational addition
827875orM :: Relation cols rows -> Relation cols rows -> Relation cols rows
828−orM Empty Empty = Empty
829−orM (One a) (One b) = One (fromBool (toBool a || toBool b))
830−orM (Junc a b) (Junc c d) = Junc (orM a c) (orM b d)
831−orM (Split a b) (Split c d) = Split (orM a c) (orM b d)
832−orM x@(Split _ _) y@(Junc _ _) = orM x (abideJS y)
833−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
834881
835882-- | Relational multiplication
836883andM :: Relation cols rows -> Relation cols rows -> Relation cols rows
837−andM Empty Empty = Empty
838−andM (One a) (One b) = One (fromBool (toBool a && toBool b))
839−andM (Junc a b) (Junc c d) = Junc (andM a c) (andM b d)
840−andM (Split a b) (Split c d) = Split (andM a c) (andM b d)
841−andM x@(Split _ _) y@(Junc _ _) = andM x (abideJS y)
842−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
843889
844890-- | Relational subtraction
845891subM :: Relation cols rows -> Relation cols rows -> Relation cols rows
846−subM Empty Empty = Empty
847−subM (One a) (One b) = if a - b < nat 0 then One (nat 0) else One (a - b)
848−subM (Junc a b) (Junc c d) = Junc (subM a c) (subM b d)
849−subM (Split a b) (Split c d) = Split (subM a c) (subM b d)
850−subM x@(Split _ _) y@(Junc _ _) = subM x (abideJS y)
851−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
852897
853898-- | Matrix relational composition.
854899compRel :: Relation cr rows -> Relation cols cr -> Relation cols rows
855−compRel Empty Empty = Empty
856−compRel (One a) (One b) = One (fromBool (toBool a && toBool b))
857−compRel (Junc a b) (Split c d) = orM (compRel a c) (compRel b d) -- Divide-and-conquer law
858−compRel (Split a b) c = Split (compRel a c) (compRel b c) -- Split fusion law
859−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
860904
861905-- | Matrix relational right division
862906divR :: Relation b c -> Relation b a -> Relation a c
863−divR Empty Empty = Empty
864907divR (One a) (One b) = One (fromBool (not (toBool b) || toBool a)) -- b implies a
865−divR (Junc a b) (Junc c d) = andM (divR a c) (divR b d)
866−divR (Split a b) c = Split (divR a c) (divR b c)
867−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)
868911
869912-- | Matrix relational left division
870913divL :: Relation c b -> Relation a b -> Relation a c
@@ -880,86 +923,52 @@ divS s r = divL r s `intersection` divR (tr r) (tr s)
880923--
881924-- NOTE: Be careful to not ask for a relation bigger than the cardinality of
882925-- types @a@ or @b@ allows.
883−fromFRel ::
926+fromFRel' ::
884927 forall a b cols rows.
885928 ( Liftable Boolean a b,
886− CountableDimensions cols rows,
887− FromLists Boolean rows cols
929+ CountableDims cols rows,
930+ FL rows cols
888931 ) =>
889932 (a -> b) ->
890933 Relation cols rows
891−fromFRel f =
892− let minA = minBound @a
893− maxA = maxBound @a
894− minB = minBound @b
895− maxB = maxBound @b
896− ccols = fromInteger $ natVal (Proxy :: Proxy (Count cols))
897− rrows = fromInteger $ natVal (Proxy :: Proxy (Count rows))
898− elementsA = take ccols [minA .. maxA]
899− elementsB = take rrows [minB .. maxB]
900− combinations = (,) <$> elementsA <*> elementsB
901− combAp = map snd . sort . map (\(a, b) -> if f a == b
902− then ((fromEnum a, fromEnum b), nat 1)
903− else ((fromEnum a, fromEnum b), nat 0)) $ combinations
904− mList = buildList combAp rrows
905− in tr $ fromLists mList
906− where
907− buildList [] _ = []
908− buildList l r = take r l : buildList (drop r l) r
934+fromFRel' = fromF'
909935
910936-- | Lifts functions to relations with dimensions matching @a@ and @b@
911937-- cardinality's.
912−fromFRel' ::
938+fromFRel ::
913939 forall a b.
914940 ( Liftable Boolean a b,
915− CountableDimensionsN a b,
916− FromLists Boolean (Normalize b) (Normalize a)
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− CountableDimensionsN a b,
948− FromListsN Boolean b a
949− )
956+ CountableDims a b,
957+ FLN b a
958+ )
950959 => (a -> b -> Bool) -> Relation (Normalize a) (Normalize b)
951960toRel f =
952961 let minA = minBound @a
953962 maxA = maxBound @a
954963 minB = minBound @b
955964 maxB = maxBound @b
956− ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))
957− rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))
965+ ccols = fromInteger $ natVal (Proxy :: Proxy (Count a))
966+ rrows = fromInteger $ natVal (Proxy :: Proxy (Count b))
958967 elementsA = take ccols [minA .. maxA]
959968 elementsB = take rrows [minB .. maxB]
960969 combinations = (,) <$> elementsA <*> elementsB
961− combAp = map snd . sort . map (\(a, b) -> if uncurry f (a, b)
962− 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)
963972 else ((fromEnum a, fromEnum b), nat 0)) $ combinations
964973 mList = buildList combAp rrows
965974 in tr $ fromLists mList