Copyright | (c) Armando Santos 2019-2020 |
---|---|
Maintainer | armandoifsantos@gmail.com |
Stability | experimental |
Safe Haskell | None |
Language | Haskell2010 |
LAoP.Matrix.Type
Contents
- Type safe matrix representation
- Constraint type aliases
- Type aliases
- Primitives
- Auxiliary type families
- Matrix construction and conversion
- Functor instance equivalent function
- Applicative/Monoidal instance equivalent functions
- Selective equivalent instance function
- Monad equivalent instance function
- Misc
- Biproduct approach
- Matrix composition and lifting
- Relation
- Matrix printing
Description
The LAoP discipline generalises relations and functions treating them as Boolean matrices and in turn consider these as arrows.
LAoP is a library for algebraic (inductive) construction and manipulation of matrices in Haskell. See my Msc Thesis for the motivation behind the library, the underlying theory, and implementation details.
This module offers a newtype wrapper around Matrix
that
uses arbitrary types instead of canonical data types for the matrices
dimensions.
NOTE: If the types in the dimensions are custom they must need to
implement a Generic
instance.
Synopsis
- newtype Matrix e (cols :: Type) (rows :: Type) = M (Matrix e (Normalize cols) (Normalize rows))
- type Countable a = KnownNat (Count a)
- type CountableDimensions a b = (Countable a, Countable b)
- type CountableN a = KnownNat (Count (Normalize a))
- type CountableDimensionsN a b = (CountableN a, CountableN b)
- type FromListsN e a b = FromLists e (Normalize a) (Normalize b)
- type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)
- type Trivial a = Normalize a ~ Normalize (Normalize a)
- type TrivialE a b = Normalize (Either a b) ~ Either (Normalize a) (Normalize b)
- type TrivialP a b = Normalize (a, b) ~ Normalize (Normalize a, Normalize b)
- type Zero = Void
- type One = ()
- empty :: Matrix e Zero Zero
- one :: e -> Matrix e One One
- junc :: TrivialE a b => Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
- split :: TrivialE a b => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
- type family FromNat (n :: Nat) :: Type where ...
- type family Count (d :: Type) :: Nat where ...
- type family Normalize (d :: Type) :: Type where ...
- class FromLists e cols rows
- fromLists :: FromListsN e cols rows => [[e]] -> Matrix e cols rows
- toLists :: Matrix e cols rows -> [[e]]
- toList :: Matrix e cols rows -> [e]
- matrixBuilder :: (FromListsN e cols rows, CountableDimensionsN cols rows) => ((Int, Int) -> e) -> Matrix e cols rows
- row :: FromListsN e cols () => [e] -> Matrix e cols One
- col :: FromListsN e () rows => [e] -> Matrix e One rows
- zeros :: (Num e, FromListsN e cols rows, CountableDimensionsN cols rows) => Matrix e cols rows
- ones :: (Num e, FromListsN e cols rows, CountableDimensionsN cols rows) => Matrix e cols rows
- bang :: forall e cols. (Num e, Enum e, FromListsN e cols (), CountableN cols) => Matrix e cols One
- constant :: (Num e, FromListsN e cols rows, CountableDimensionsN cols rows) => e -> Matrix e cols rows
- fmapM :: (Liftable e a b, CountableDimensionsN a b, FromListsN e b a) => (a -> b) -> Matrix e c a -> Matrix e c b
- unitM :: Num e => Matrix e () ()
- multM :: (CountableDimensionsN a b, CountableN (a, b), Num e, FromListsN e (a, b) a, FromListsN e (a, b) b, TrivialP a b) => Matrix e c a -> Matrix e c b -> Matrix e c (a, b)
- selectM :: (Num e, FromListsN e b b, CountableN b, TrivialE a b) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b
- returnM :: forall e a. (Num e, Enum e, Enum a, FromListsN e () a, Countable a) => a -> Matrix e One a
- bindM :: Num e => Matrix e a b -> Matrix e b c -> Matrix e a c
- columns :: CountableN cols => Matrix e cols rows -> Int
- rows :: CountableN rows => Matrix e cols rows -> Int
- tr :: Matrix e cols rows -> Matrix e rows cols
- cond :: (Trivial a, CountableN a, FromListsN e () a, FromListsN e a (), FromListsN e a a, Liftable e a Bool) => (a -> Bool) -> Matrix e a b -> Matrix e a b -> Matrix e a b
- abideJS :: Matrix e cols rows -> Matrix e cols rows
- abideSJ :: Matrix e cols rows -> Matrix e cols rows
- (===) :: TrivialE a b => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
- p1 :: (Num e, CountableDimensionsN n m, FromListsN e n m, FromListsN e m m, TrivialE m n) => Matrix e (Either m n) m
- p2 :: (Num e, CountableDimensionsN n m, FromListsN e m n, FromListsN e n n, TrivialE m n) => Matrix e (Either m n) n
- (|||) :: TrivialE a b => Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
- i1 :: (Num e, CountableDimensionsN n m, FromListsN e n m, FromListsN e m m, TrivialE m n) => Matrix e m (Either m n)
- i2 :: (Num e, CountableDimensionsN n m, FromListsN e m n, FromListsN e n n, TrivialE m n) => Matrix e n (Either m n)
- (-|-) :: (Num e, CountableDimensionsN j k, FromListsN e k k, FromListsN e j k, FromListsN e k j, FromListsN e j j, TrivialE n m, TrivialE k j) => Matrix e n k -> Matrix e m j -> Matrix e (Either n m) (Either k j)
- (><) :: forall e m p n q. (Num e, CountableDimensionsN m n, CountableDimensionsN p q, CountableDimensionsN (m, n) (p, q), FromListsN e (m, n) m, FromListsN e (m, n) n, FromListsN e (p, q) p, FromListsN e (p, q) q, TrivialP m n, TrivialP p q) => Matrix e m p -> Matrix e n q -> Matrix e (m, n) (p, q)
- kp1 :: forall e m k. (Num e, CountableDimensionsN m k, CountableN (m, k), FromListsN e (m, k) m, TrivialP m k) => Matrix e (m, k) m
- kp2 :: forall e m k. (Num e, CountableDimensionsN k m, CountableN (m, k), FromListsN e (m, k) k, TrivialP m k) => Matrix e (m, k) k
- khatri :: forall e cols a b. (Num e, CountableDimensionsN a b, CountableN (a, b), FromListsN e (a, b) a, FromListsN e (a, b) b, TrivialP a b) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a, b)
- identity :: (Num e, FromListsN e a a, CountableN a) => Matrix e a a
- comp :: Num e => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
- fromF :: (Liftable e a b, CountableDimensionsN cols rows, FromListsN e rows cols) => (a -> b) -> Matrix e cols rows
- fromF' :: (Liftable e a b, CountableDimensionsN a b, FromListsN e b a) => (a -> b) -> Matrix e a b
- toRel :: (Liftable (Natural 0 1) a b, CountableDimensionsN a b, FromListsN (Natural 0 1) b a) => (a -> b -> Bool) -> Matrix (Natural 0 1) a b
- pretty :: (CountableN cols, Show e) => Matrix e cols rows -> String
- prettyPrint :: (CountableN cols, Show e) => Matrix e cols rows -> IO ()
Documentation
LAoP (Linear Algebra of Programming) Inductive Matrix definition.
LAoP generalises relations and functions treating them as Boolean matrices and in turn consider these as arrows. This library offers many of the combinators mentioned in the work of Macedo (2012) and Oliveira (2012).
This definition is a wrapper around Type
but
dimensions are arbitrary data types. Type inference might not
be as desired.
Type safe matrix representation
newtype Matrix e (cols :: Type) (rows :: Type) Source #
Instances
Num e => Category (Matrix e :: Type -> Type -> Type) Source # | It isn't possible to implement the Please use |
Eq e => Eq (Matrix e cols rows) Source # | |
Num e => Num (Matrix e cols rows) Source # | |
Defined in LAoP.Matrix.Type Methods (+) :: Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows # (-) :: Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows # (*) :: Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows # negate :: Matrix e cols rows -> Matrix e cols rows # abs :: Matrix e cols rows -> Matrix e cols rows # signum :: Matrix e cols rows -> Matrix e cols rows # fromInteger :: Integer -> Matrix e cols rows # | |
Ord e => Ord (Matrix e cols rows) Source # | |
Defined in LAoP.Matrix.Type Methods compare :: Matrix e cols rows -> Matrix e cols rows -> Ordering # (<) :: Matrix e cols rows -> Matrix e cols rows -> Bool # (<=) :: Matrix e cols rows -> Matrix e cols rows -> Bool # (>) :: Matrix e cols rows -> Matrix e cols rows -> Bool # (>=) :: Matrix e cols rows -> Matrix e cols rows -> Bool # max :: Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows # min :: Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows # | |
Show e => Show (Matrix e cols rows) Source # | |
NFData e => NFData (Matrix e cols rows) Source # | |
Defined in LAoP.Matrix.Type |
Constraint type aliases
type Countable a = KnownNat (Count a) Source #
Constraint type synonyms to keep the type signatures less convoluted
type CountableDimensions a b = (Countable a, Countable b) Source #
type CountableDimensionsN a b = (CountableN a, CountableN b) Source #
Type aliases
Primitives
junc :: TrivialE a b => Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows Source #
Matrix Junc
constructor
split :: TrivialE a b => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b) Source #
Matrix Split
constructor
Auxiliary type families
type family FromNat (n :: Nat) :: Type where ... Source #
Type family that computes of a given type dimension from a given natural
Thanks to Li-Yao Xia this type family is super fast.
type family Count (d :: Type) :: Nat where ... Source #
Type family that computes the cardinality of a given type dimension.
It can also count the cardinality of custom types that implement the
Generic
instance.
Equations
Count (Natural n m) = (m - n) + 1 | |
Count (List a) = (^) 2 (Count a) | |
Count (Either a b) = (+) (Count a) (Count b) | |
Count (a, b) = * (Count a) (Count b) | |
Count (a -> b) = (^) (Count b) (Count a) | |
Count (M1 _ _ f p) = Count (f p) | |
Count (K1 _ _ _) = 1 | |
Count (V1 _) = 0 | |
Count (U1 _) = 1 | |
Count ((:*:) a b p) = Count (a p) * Count (b p) | |
Count ((:+:) a b p) = Count (a p) + Count (b p) | |
Count d = Count (Rep d R) |
type family Normalize (d :: Type) :: Type where ... Source #
Type family that normalizes the representation of a given data structure
Matrix construction and conversion
class FromLists e cols rows Source #
Type class for defining the fromList
conversion function.
Given that it is not possible to branch on types at the term level type classes are needed bery much like an inductive definition but on types.
Minimal complete definition
Instances
FromLists e () () Source # | |
Defined in LAoP.Matrix.Internal | |
FromLists e Void Void Source # | |
(FromLists e () a, FromLists e () b, Countable a) => FromLists e () (Either a b) Source # | |
FromLists e () rows => FromLists e () (Either () rows) Source # | |
(FromLists e a (), FromLists e b (), Countable a) => FromLists e (Either a b) () Source # | |
FromLists e cols () => FromLists e (Either () cols) () Source # | |
(FromLists e (Either a b) c, FromLists e (Either a b) d, Countable c) => FromLists e (Either a b) (Either c d) Source # | |
fromLists :: FromListsN e cols rows => [[e]] -> Matrix e cols rows Source #
Build a matrix out of a list of list of elements. Throws a runtime error if the dimensions do not match.
matrixBuilder :: (FromListsN e cols rows, CountableDimensionsN cols rows) => ((Int, Int) -> e) -> Matrix e cols rows Source #
Matrix builder function. Constructs a matrix provided with a construction function.
zeros :: (Num e, FromListsN e cols rows, CountableDimensionsN cols rows) => Matrix e cols rows Source #
The zero matrix. A matrix wholly filled with zeros.
ones :: (Num e, FromListsN e cols rows, CountableDimensionsN cols rows) => Matrix e cols rows Source #
The ones matrix. A matrix wholly filled with ones.
Also known as T (Top) matrix.
bang :: forall e cols. (Num e, Enum e, FromListsN e cols (), CountableN cols) => Matrix e cols One Source #
The T (Top) row vector matrix.
constant :: (Num e, FromListsN e cols rows, CountableDimensionsN cols rows) => e -> Matrix e cols rows Source #
The constant matrix constructor. A matrix wholly filled with a given value.
Functor instance equivalent function
fmapM :: (Liftable e a b, CountableDimensionsN a b, FromListsN e b a) => (a -> b) -> Matrix e c a -> Matrix e c b Source #
Functor instance equivalent function
Applicative/Monoidal instance equivalent functions
multM :: (CountableDimensionsN a b, CountableN (a, b), Num e, FromListsN e (a, b) a, FromListsN e (a, b) b, TrivialP a b) => Matrix e c a -> Matrix e c b -> Matrix e c (a, b) Source #
Applicative instance equivalent unit
function,
Selective equivalent instance function
selectM :: (Num e, FromListsN e b b, CountableN b, TrivialE a b) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b Source #
Selective functors select
operator equivalent inspired by the
ArrowMonad solution presented in the paper.
Monad equivalent instance function
returnM :: forall e a. (Num e, Enum e, Enum a, FromListsN e () a, Countable a) => a -> Matrix e One a Source #
Monad instance equivalent return
function,
bindM :: Num e => Matrix e a b -> Matrix e b c -> Matrix e a c Source #
Monad instance equivalent '(>>=)' function,
Misc
Get dimensions
Matrix Transposition
McCarthy's Conditional
cond :: (Trivial a, CountableN a, FromListsN e () a, FromListsN e a (), FromListsN e a a, Liftable e a Bool) => (a -> Bool) -> Matrix e a b -> Matrix e a b -> Matrix e a b Source #
McCarthy's Conditional expresses probabilistic choice.
Matrix "abiding"
abideJS :: Matrix e cols rows -> Matrix e cols rows Source #
Matrix "abiding" followin the 'Junc'-'Split' abide law.
Law:
Junc
(Split
a c) (Split
b d) ==Split
(Junc
a b) (Junc
c d)
abideSJ :: Matrix e cols rows -> Matrix e cols rows Source #
Matrix "abiding" followin the 'Split'-'Junc' abide law.
Law:
Split
(Junc
a b) (Junc
c d) ==Junc
(Split
a c) (Split
b d)
Biproduct approach
Split
(===) :: TrivialE a b => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b) infixl 2 Source #
Matrix Split
constructor
Projections
p1 :: (Num e, CountableDimensionsN n m, FromListsN e n m, FromListsN e m m, TrivialE m n) => Matrix e (Either m n) m Source #
Biproduct first component projection
p2 :: (Num e, CountableDimensionsN n m, FromListsN e m n, FromListsN e n n, TrivialE m n) => Matrix e (Either m n) n Source #
Biproduct second component projection
Junc
(|||) :: TrivialE a b => Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows infixl 3 Source #
Matrix Junc
constructor
Injections
i1 :: (Num e, CountableDimensionsN n m, FromListsN e n m, FromListsN e m m, TrivialE m n) => Matrix e m (Either m n) Source #
Biproduct first component injection
i2 :: (Num e, CountableDimensionsN n m, FromListsN e m n, FromListsN e n n, TrivialE m n) => Matrix e n (Either m n) Source #
Biproduct second component injection
Bifunctors
(-|-) :: (Num e, CountableDimensionsN j k, FromListsN e k k, FromListsN e j k, FromListsN e k j, FromListsN e j j, TrivialE n m, TrivialE k j) => Matrix e n k -> Matrix e m j -> Matrix e (Either n m) (Either k j) infixl 5 Source #
Matrix coproduct functor also known as matrix direct sum.
(><) :: forall e m p n q. (Num e, CountableDimensionsN m n, CountableDimensionsN p q, CountableDimensionsN (m, n) (p, q), FromListsN e (m, n) m, FromListsN e (m, n) n, FromListsN e (p, q) p, FromListsN e (p, q) q, TrivialP m n, TrivialP p q) => Matrix e m p -> Matrix e n q -> Matrix e (m, n) (p, q) infixl 4 Source #
Matrix product functor also known as kronecker product
Applicative matrix combinators
Note that given the restrictions imposed it is not possible to implement the standard type classes present in standard Haskell. *** Matrix pairing projections
kp1 :: forall e m k. (Num e, CountableDimensionsN m k, CountableN (m, k), FromListsN e (m, k) m, TrivialP m k) => Matrix e (m, k) m Source #
Khatri Rao product first component projection matrix.
kp2 :: forall e m k. (Num e, CountableDimensionsN k m, CountableN (m, k), FromListsN e (m, k) k, TrivialP m k) => Matrix e (m, k) k Source #
Khatri Rao product second component projection matrix.
Matrix pairing
khatri :: forall e cols a b. (Num e, CountableDimensionsN a b, CountableN (a, b), FromListsN e (a, b) a, FromListsN e (a, b) b, TrivialP a b) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a, b) Source #
Matrix composition and lifting
Arrow matrix combinators
Note that given the restrictions imposed it is not possible to implement the standard type classes present in standard Haskell.
identity :: (Num e, FromListsN e a a, CountableN a) => Matrix e a a Source #
Identity matrix
comp :: Num e => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows Source #
Matrix composition. Equivalent to matrix-matrix multiplication.
This definition takes advantage of divide-and-conquer and fusion laws from LAoP.
fromF :: (Liftable e a b, CountableDimensionsN cols rows, FromListsN e rows cols) => (a -> b) -> Matrix e cols rows Source #
Lifts functions to matrices with arbitrary dimensions.
NOTE: Be careful to not ask for a matrix bigger than the cardinality of
types a
or b
allows.
fromF' :: (Liftable e a b, CountableDimensionsN a b, FromListsN e b a) => (a -> b) -> Matrix e a b Source #
Lifts functions to matrices with dimensions matching a
and b
cardinality's.
Relation
toRel :: (Liftable (Natural 0 1) a b, CountableDimensionsN a b, FromListsN (Natural 0 1) b a) => (a -> b -> Bool) -> Matrix (Natural 0 1) a b Source #
Lifts relation functions to Boolean Matrix
Matrix printing
prettyPrint :: (CountableN cols, Show e) => Matrix e cols rows -> IO () Source #
Matrix pretty printer