| Copyright | (c) 2011 Patrick Bahr |
|---|---|
| License | BSD3 |
| Maintainer | Patrick Bahr <paba@diku.dk> |
| Stability | experimental |
| Portability | non-portable (GHC Extensions) |
| Safe Haskell | None |
| Language | Haskell98 |
Data.Comp.Multi.Algebra
Description
This module defines the notion of algebras and catamorphisms, and their generalizations to e.g. monadic versions and other (co)recursion schemes. All definitions are generalised versions of those in Data.Comp.Algebra.
Synopsis
- type Alg f e = f e :-> e
- free :: forall f h a b. HFunctor f => Alg f b -> (a :-> b) -> Cxt h f a :-> b
- cata :: forall f a. HFunctor f => Alg f a -> HFix f :-> a
- cata' :: HFunctor f => Alg f e -> Cxt h f e :-> e
- appCxt :: HFunctor f => Context f (Cxt h f a) :-> Cxt h f a
- type AlgM m f e = NatM m (f e) e
- freeM :: forall f m h a b. (HTraversable f, Monad m) => AlgM m f b -> NatM m a b -> NatM m (Cxt h f a) b
- cataM :: forall f m a. (HTraversable f, Monad m) => AlgM m f a -> NatM m (HFix f) a
- cataM' :: forall m h a f. (Monad m, HTraversable f) => AlgM m f a -> NatM m (Cxt h f a) a
- liftMAlg :: forall m f. (Monad m, HTraversable f) => Alg f I -> Alg f m
- type CxtFun f g = forall h. SigFun (Cxt h f) (Cxt h g)
- type SigFun f g = forall (a :: * -> *). f a :-> g a
- type Hom f g = SigFun f (Context g)
- appHom :: forall f g. (HFunctor f, HFunctor g) => Hom f g -> CxtFun f g
- appHom' :: forall f g. HFunctor g => Hom f g -> CxtFun f g
- compHom :: (HFunctor g, HFunctor h) => Hom g h -> Hom f g -> Hom f h
- appSigFun :: forall f g. HFunctor f => SigFun f g -> CxtFun f g
- appSigFun' :: forall f g. HFunctor g => SigFun f g -> CxtFun f g
- compSigFun :: SigFun g h -> SigFun f g -> SigFun f h
- hom :: HFunctor g => SigFun f g -> Hom f g
- compAlg :: HFunctor g => Alg g a -> Hom f g -> Alg f a
- type CxtFunM m f g = forall h. SigFunM m (Cxt h f) (Cxt h g)
- type SigFunM m f g = forall (a :: * -> *). NatM m (f a) (g a)
- type HomM m f g = SigFunM m f (Context g)
- sigFunM :: Monad m => SigFun f g -> SigFunM m f g
- hom' :: (HFunctor f, HFunctor g, Monad m) => SigFunM m f g -> HomM m f g
- appHomM :: forall f g m. (HTraversable f, HFunctor g, Monad m) => HomM m f g -> CxtFunM m f g
- appHomM' :: forall f g m. (HTraversable g, Monad m) => HomM m f g -> CxtFunM m f g
- homM :: (HFunctor g, Monad m) => SigFun f g -> HomM m f g
- appSigFunM :: forall f g m. (HTraversable f, Monad m) => SigFunM m f g -> CxtFunM m f g
- appSigFunM' :: forall f g m. (HTraversable g, Monad m) => SigFunM m f g -> CxtFunM m f g
- compHomM :: (HTraversable g, HFunctor h, Monad m) => HomM m g h -> HomM m f g -> HomM m f h
- compSigFunM :: Monad m => SigFunM m g h -> SigFunM m f g -> SigFunM m f h
- compAlgM :: (HTraversable g, Monad m) => AlgM m g a -> HomM m f g -> AlgM m f a
- compAlgM' :: (HTraversable g, Monad m) => AlgM m g a -> Hom f g -> AlgM m f a
- type Coalg f a = a :-> f a
- ana :: forall f a. HFunctor f => Coalg f a -> a :-> HFix f
- type CoalgM m f a = NatM m a (f a)
- anaM :: forall a m f. (HTraversable f, Monad m) => CoalgM m f a -> NatM m a (HFix f)
- type RAlg f a = f (HFix f :*: a) :-> a
- para :: forall f a. HFunctor f => RAlg f a -> HFix f :-> a
- type RAlgM m f a = NatM m (f (HFix f :*: a)) a
- paraM :: forall f m a. (HTraversable f, Monad m) => RAlgM m f a -> NatM m (HFix f) a
- type RCoalg f a = a :-> f (HFix f :+: a)
- apo :: forall f a. HFunctor f => RCoalg f a -> a :-> HFix f
- type RCoalgM m f a = NatM m a (f (HFix f :+: a))
- apoM :: forall f m a. (HTraversable f, Monad m) => RCoalgM m f a -> NatM m a (HFix f)
- type CVCoalg f a = a :-> f (Context f a)
- futu :: forall f a. HFunctor f => CVCoalg f a -> a :-> HFix f
- type CVCoalgM m f a = NatM m a (f (Context f a))
- futuM :: forall f a m. (HTraversable f, Monad m) => CVCoalgM m f a -> NatM m a (HFix f)
Algebras & Catamorphisms
type Alg f e = f e :-> e Source #
This type represents multisorted f-algebras with a family e
of carriers.
free :: forall f h a b. HFunctor f => Alg f b -> (a :-> b) -> Cxt h f a :-> b Source #
Construct a catamorphism for contexts over f with holes of type
b, from the given algebra.
cata :: forall f a. HFunctor f => Alg f a -> HFix f :-> a Source #
Construct a catamorphism from the given algebra.
cata' :: HFunctor f => Alg f e -> Cxt h f e :-> e Source #
A generalisation of cata from terms over f to contexts over
f, where the holes have the type of the algebra carrier.
appCxt :: HFunctor f => Context f (Cxt h f a) :-> Cxt h f a Source #
This function applies a whole context into another context.
Monadic Algebras & Catamorphisms
type AlgM m f e = NatM m (f e) e Source #
This type represents a monadic algebra. It is similar to Alg
but the return type is monadic.
freeM :: forall f m h a b. (HTraversable f, Monad m) => AlgM m f b -> NatM m a b -> NatM m (Cxt h f a) b Source #
Construct a monadic catamorphism for contexts over f with holes
of type b, from the given monadic algebra.
cataM :: forall f m a. (HTraversable f, Monad m) => AlgM m f a -> NatM m (HFix f) a Source #
This is a monadic version of cata.
liftMAlg :: forall m f. (Monad m, HTraversable f) => Alg f I -> Alg f m Source #
This function lifts a many-sorted algebra to a monadic domain.
Term Homomorphisms
type CxtFun f g = forall h. SigFun (Cxt h f) (Cxt h g) Source #
This type represents context function.
type SigFun f g = forall (a :: * -> *). f a :-> g a Source #
This type represents uniform signature function specification.
appHom :: forall f g. (HFunctor f, HFunctor g) => Hom f g -> CxtFun f g Source #
This function applies the given term homomorphism to a term/context.
appHom' :: forall f g. HFunctor g => Hom f g -> CxtFun f g Source #
This function applies the given term homomorphism to a
term/context. This is the top-down variant of appHom.
compHom :: (HFunctor g, HFunctor h) => Hom g h -> Hom f g -> Hom f h Source #
This function composes two term algebras.
appSigFun :: forall f g. HFunctor f => SigFun f g -> CxtFun f g Source #
This function applies a signature function to the given context.
appSigFun' :: forall f g. HFunctor g => SigFun f g -> CxtFun f g Source #
This function applies a signature function to the given
context. This is the top-down variant of appSigFun.
compSigFun :: SigFun g h -> SigFun f g -> SigFun f h Source #
This function composes two signature functions.
hom :: HFunctor g => SigFun f g -> Hom f g Source #
Lifts the given signature function to the canonical term homomorphism.
compAlg :: HFunctor g => Alg g a -> Hom f g -> Alg f a Source #
This function composes a term algebra with an algebra.
Monadic Term Homomorphisms
type CxtFunM m f g = forall h. SigFunM m (Cxt h f) (Cxt h g) Source #
This type represents monadic context function.
type SigFunM m f g = forall (a :: * -> *). NatM m (f a) (g a) Source #
This type represents monadic signature functions.
sigFunM :: Monad m => SigFun f g -> SigFunM m f g Source #
This function lifts the given signature function to a monadic signature function. Note that term algebras are instances of signature functions. Hence this function also applies to term algebras.
hom' :: (HFunctor f, HFunctor g, Monad m) => SigFunM m f g -> HomM m f g Source #
This function lifts the give monadic signature function to a monadic term algebra.
appHomM :: forall f g m. (HTraversable f, HFunctor g, Monad m) => HomM m f g -> CxtFunM m f g Source #
This function applies the given monadic term homomorphism to the given term/context.
appHomM' :: forall f g m. (HTraversable g, Monad m) => HomM m f g -> CxtFunM m f g Source #
This function applies the given monadic term homomorphism to the
given term/context. This is a top-down variant of appHomM.
homM :: (HFunctor g, Monad m) => SigFun f g -> HomM m f g Source #
This function lifts the given signature function to a monadic term algebra.
appSigFunM :: forall f g m. (HTraversable f, Monad m) => SigFunM m f g -> CxtFunM m f g Source #
This function applies the given monadic signature function to the given context.
appSigFunM' :: forall f g m. (HTraversable g, Monad m) => SigFunM m f g -> CxtFunM m f g Source #
This function applies the given monadic signature function to the
given context. This is a top-down variant of appSigFunM.
compHomM :: (HTraversable g, HFunctor h, Monad m) => HomM m g h -> HomM m f g -> HomM m f h Source #
This function composes two monadic term algebras.
compSigFunM :: Monad m => SigFunM m g h -> SigFunM m f g -> SigFunM m f h Source #
This function composes two monadic signature functions.
compAlgM :: (HTraversable g, Monad m) => AlgM m g a -> HomM m f g -> AlgM m f a Source #
This function composes a monadic term algebra with a monadic algebra
compAlgM' :: (HTraversable g, Monad m) => AlgM m g a -> Hom f g -> AlgM m f a Source #
This function composes a monadic term algebra with a monadic algebra.
Coalgebras & Anamorphisms
ana :: forall f a. HFunctor f => Coalg f a -> a :-> HFix f Source #
This function unfolds the given value to a term using the given
unravelling function. This is the unique homomorphism a -> Term f
from the given coalgebra of type a -> f a to the final coalgebra
Term f.
anaM :: forall a m f. (HTraversable f, Monad m) => CoalgM m f a -> NatM m a (HFix f) Source #
This function unfolds the given value to a term using the given
monadic unravelling function. This is the unique homomorphism a ->
Term f from the given coalgebra of type a -> f a to the final
coalgebra Term f.
R-Algebras & Paramorphisms
type RAlg f a = f (HFix f :*: a) :-> a Source #
This type represents r-algebras over functor f and with domain
a.
para :: forall f a. HFunctor f => RAlg f a -> HFix f :-> a Source #
This function constructs a paramorphism from the given r-algebra
type RAlgM m f a = NatM m (f (HFix f :*: a)) a Source #
This type represents monadic r-algebras over monad m and
functor f and with domain a.
paraM :: forall f m a. (HTraversable f, Monad m) => RAlgM m f a -> NatM m (HFix f) a Source #
This function constructs a monadic paramorphism from the given monadic r-algebra
R-Coalgebras & Apomorphisms
type RCoalg f a = a :-> f (HFix f :+: a) Source #
This type represents r-coalgebras over functor f and with
domain a.
apo :: forall f a. HFunctor f => RCoalg f a -> a :-> HFix f Source #
This function constructs an apomorphism from the given r-coalgebra.
type RCoalgM m f a = NatM m a (f (HFix f :+: a)) Source #
This type represents monadic r-coalgebras over monad m and
functor f with domain a.
apoM :: forall f m a. (HTraversable f, Monad m) => RCoalgM m f a -> NatM m a (HFix f) Source #
This function constructs a monadic apomorphism from the given monadic r-coalgebra.
CV-Coalgebras & Futumorphisms
type CVCoalg f a = a :-> f (Context f a) Source #
This type represents cv-coalgebras over functor f and with domain
a.
futu :: forall f a. HFunctor f => CVCoalg f a -> a :-> HFix f Source #
This function constructs the unique futumorphism from the given cv-coalgebra to the term algebra.