LAoP.Matrix.Type

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.

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)

Constructors

M (Matrix e (Normalize cols) (Normalize rows))

Instances

Num e => Category (Matrix e :: Type -> Type -> Type)

It isn't possible to implement the id function so it's implementation is undefined. However comp can be and this partial class implementation exists just to make the code more readable.

Please use identity instead.

Defined in LAoP.Matrix.Type

Methods

id :: Matrix e a a

(.) :: Matrix e b c -> Matrix e a b -> Matrix e a c

Eq e => Eq (Matrix e cols rows)

Defined in LAoP.Matrix.Type

Methods

(==) :: Matrix e cols rows -> Matrix e cols rows -> Bool

(/=) :: Matrix e cols rows -> Matrix e cols rows -> Bool

Num e => Num (Matrix e cols rows)

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)

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)

Defined in LAoP.Matrix.Type

Methods

showsPrec :: Int -> Matrix e cols rows -> ShowS

show :: Matrix e cols rows -> String

showList :: [Matrix e cols rows] -> ShowS

NFData e => NFData (Matrix e cols rows)

Defined in LAoP.Matrix.Type

Methods

rnf :: Matrix e cols rows -> ()

Constraint type aliases

type Countable a = KnownNat (Count a)

Constraint type synonyms to keep the type signatures less convoluted

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 (Normalize a) ~ Normalize (Normalize (Normalize a))

type TrivialP a b = Normalize (a, b) ~ Normalize (Normalize a, Normalize b)

Type aliases

type Zero = Void

Zero type alias

type One = ()

One type alias

Primitives

empty :: Matrix e Zero Zero

Empty matrix constructor

one :: e -> Matrix e One One

Unit matrix constructor

junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

Matrix Junc constructor

split :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

Matrix Split constructor

Auxiliary type families

type family FromNat (n :: Nat) :: Type where ...

Type family that computes of a given type dimension from a given natural

Thanks to Li-Yao Xia this type family is super fast.

Equations

FromNat 0 = Void

FromNat 1 = ()

FromNat n = FromNat' (Mod n 2 == 0) (FromNat (Div n 2))

type family Count (d :: Type) :: Nat where ...

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 ...

Type family that normalizes the representation of a given data structure

Equations

Normalize (Either a b) = Either (Normalize a) (Normalize b)

Normalize d = FromNat (Count d)

Matrix construction and conversion

class FromLists e cols rows

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 very much like an inductive definition but on types.

Minimal complete definition

fromLists

Instances

FromLists e () ()

Defined in LAoP.Matrix.Internal

Methods

fromLists :: [[e]] -> Matrix e () ()

FromLists e Void Void

Defined in LAoP.Matrix.Internal

Methods

fromLists :: [[e]] -> Matrix e Void Void

(FromLists e () a, FromLists e () b, Countable a) => FromLists e () (Either a b)

Defined in LAoP.Matrix.Internal

Methods

fromLists :: [[e]] -> Matrix e () (Either a b)

FromLists e () rows => FromLists e () (Either () rows)

Defined in LAoP.Matrix.Internal

Methods

fromLists :: [[e]] -> Matrix e () (Either () rows)

(FromLists e a (), FromLists e b (), Countable a) => FromLists e (Either a b) ()

Defined in LAoP.Matrix.Internal

Methods

fromLists :: [[e]] -> Matrix e (Either a b) ()

FromLists e cols () => FromLists e (Either () cols) ()

Defined in LAoP.Matrix.Internal

Methods

fromLists :: [[e]] -> Matrix e (Either () cols) ()

(FromLists e (Either a b) c, FromLists e (Either a b) d, Countable c) => FromLists e (Either a b) (Either c d)

Defined in LAoP.Matrix.Internal

Methods

fromLists :: [[e]] -> Matrix e (Either a b) (Either c d)

fromLists :: FromListsN e cols rows => [[e]] -> Matrix e cols rows

Build a matrix out of a list of list of elements. Throws a runtime error if the dimensions do not match.

toLists :: Matrix e cols rows -> [[e]]

Converts a matrix to a list of lists of elements.

toList :: Matrix e cols rows -> [e]

Converts a matrix to a list of elements.

matrixBuilder :: (FromListsN e cols rows, CountableDimensionsN cols rows) => ((Int, Int) -> e) -> Matrix e cols rows

Matrix builder function. Constructs a matrix provided with a construction function.

row :: FromListsN e cols () => [e] -> Matrix e cols One

Constructs a row vector matrix

col :: FromListsN e () rows => [e] -> Matrix e One rows

Constructs a column vector matrix

zeros :: (Num e, FromListsN e cols rows, CountableDimensionsN cols rows) => Matrix e cols rows

The zero matrix. A matrix wholly filled with zeros.

ones :: (Num e, FromListsN e cols rows, CountableDimensionsN cols rows) => Matrix e cols rows

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

The T (Top) row vector matrix.

point :: (Bounded a, Enum a, Eq a, Num e, Ord e, CountableN a, FromListsN e a One) => a -> Matrix e One a

Point constant relation

constant :: (Num e, FromListsN e cols rows, CountableDimensionsN cols rows) => e -> Matrix e cols rows

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

Functor instance equivalent function

bimapM :: (Liftable e a b, Liftable e c d, CountableDimensionsN a c, CountableDimensionsN b d, FromListsN e d c, FromListsN e b a) => (a -> b) -> (c -> d) -> Matrix e a c -> Matrix e b d

Bifunctor equivalent function

Applicative/Monoidal instance equivalent functions

unitM :: Num e => Matrix e () ()

Applicative instance equivalent unit function,

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)

Applicative instance equivalent unit function,

Selective equivalent instance function

selectM :: (Num e, FromListsN e b b, CountableN b) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b

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

Monad instance equivalent return function,

bindM :: Num e => Matrix e a b -> Matrix e b c -> Matrix e a c

Monad instance equivalent '(>>=)' function,

Misc

Get dimensions

columns :: CountableN cols => Matrix e cols rows -> Int

Obtain the number of columns.

NOTE: The KnownNat constaint is needed in order to obtain the dimensions in constant time.

TODO: A columns function that does not need the KnownNat constraint in exchange for performance.

rows :: CountableN rows => Matrix e cols rows -> Int

Obtain the number of rows.

NOTE: The KnownNat constaint is needed in order to obtain the dimensions in constant time.

TODO: A rows function that does not need the KnownNat constraint in exchange for performance.

Matrix Transposition

tr :: Matrix e cols rows -> Matrix e rows cols

Matrix transposition.

McCarthy's Conditional

cond :: (Trivial a, Trivial2 a, Trivial3 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

McCarthy's Conditional expresses probabilistic choice.

Matrix "abiding"

abideJS :: Matrix e cols rows -> Matrix e cols rows

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

Matrix "abiding" followin the 'Split'-'Junc' abide law.

Law:

Split (Junc a b) (Junc c d) == Junc (Split a c) (Split b d)

Zip Matrices

zipWithM :: (e -> f -> g) -> Matrix e a b -> Matrix f a b -> Matrix g a b

Zip two matrices with a given binary function

Biproduct approach

Split

(===) :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b) infixl 2

Matrix Split constructor

Projections

p1 :: (Num e, CountableDimensionsN n m, FromListsN e n m, FromListsN e m m) => Matrix e (Either m n) m

Biproduct first component projection

p2 :: (Num e, CountableDimensionsN n m, FromListsN e m n, FromListsN e n n) => Matrix e (Either m n) n

Biproduct second component projection

Junc

(|||) :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows infixl 3

Matrix Junc constructor

Injections

i1 :: (Num e, CountableDimensionsN n m, FromListsN e n m, FromListsN e m m) => Matrix e m (Either m n)

Biproduct first component injection

i2 :: (Num e, CountableDimensionsN n m, FromListsN e m n, FromListsN e n n) => Matrix e n (Either m n)

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) => Matrix e n k -> Matrix e m j -> Matrix e (Either n m) (Either k j) infixl 5

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

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

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

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)

Khatri Rao Matrix product also known as matrix pairing.

NOTE: That this is not a true categorical product, see for instance:

               | kp1 comp khatri a b == a 
khatri a b ==> |
               | kp2 comp khatri a b == b

Emphasis on the implication symbol.

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

Identity matrix

comp :: Num e => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows

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

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

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

Lifts relation functions to Boolean Matrix

Matrix printing

pretty :: (CountableDimensionsN cols rows, Show e) => Matrix e cols rows -> String

Matrix pretty printer

prettyPrint :: (CountableDimensionsN cols rows, Show e) => Matrix e cols rows -> IO ()

Matrix pretty printer