From 0ec8452b4d5838c68085114e5a62a400b8fb78f9 Mon Sep 17 00:00:00 2001 From: Zara Turtle Date: Wed, 8 Apr 2020 22:32:52 +0100 Subject: [PATCH] Variance, Applicative and Monads --- manuscript/book.org | 776 +++++++++++++---------- manuscript/complete.md | 469 ++++++++++++++ manuscript/images/cats-applicative.png | Bin 0 -> 659611 bytes manuscript/images/cats-apply.png | Bin 0 -> 647256 bytes manuscript/images/cats-commute.png | Bin 0 -> 710644 bytes manuscript/images/cats-core-cliques.png | Bin 386868 -> 355581 bytes manuscript/images/cats-core-loners.png | Bin 65384 -> 55511 bytes manuscript/images/cats-core-tree.png | Bin 784303 -> 715577 bytes manuscript/images/cats-mappable.png | Bin 609403 -> 581789 bytes manuscript/images/cats-variance.png | Bin 679111 -> 596886 bytes manuscript/images/scalaz-applicative.png | Bin 380135 -> 0 bytes manuscript/images/scalaz-apply.png | Bin 362759 -> 0 bytes 12 files changed, 918 insertions(+), 327 deletions(-) create mode 100644 manuscript/images/cats-applicative.png create mode 100644 manuscript/images/cats-apply.png create mode 100644 manuscript/images/cats-commute.png delete mode 100644 manuscript/images/scalaz-applicative.png delete mode 100644 manuscript/images/scalaz-apply.png diff --git a/manuscript/book.org b/manuscript/book.org index 306e0c8d..6eae6238 100644 --- a/manuscript/book.org +++ b/manuscript/book.org @@ -3185,6 +3185,7 @@ digraph G { ApplicativeError -> Applicative; MonadError; MonadError -> ApplicativeError; + MonadError -> Monad; Bimonad; Bimonad -> Monad; @@ -3627,16 +3628,16 @@ digraph G { Monad -> Applicative; Monad -> FlatMap; - UnorderedFoldable; + UnorderedFoldable [fontcolor=black,color=black]; Foldable [fontcolor=black,color=black]; - Foldable -> UnorderedFoldable; + Foldable -> UnorderedFoldable [color=black]; Reducible [fontcolor=black,color=black]; Reducible -> Foldable [color=black]; - UnorderedTraverse; + UnorderedTraverse [fontcolor=black,color=black]; Traverse [fontcolor=black,color=black]; Traverse -> Functor [color=black]; Traverse -> Foldable [color=black]; - Traverse -> UnorderedTraverse; + Traverse -> UnorderedTraverse [color=black]; NonEmptyTraverse [fontcolor=black,color=black]; NonEmptyTraverse -> Traverse [color=black]; NonEmptyTraverse -> Reducible [color=black]; @@ -3669,6 +3670,7 @@ digraph G { ApplicativeError -> Applicative; MonadError; MonadError -> ApplicativeError; + MonadError -> Monad; Bimonad; Bimonad -> Monad; @@ -4286,11 +4288,6 @@ trait TraverseFilter[F[_]] extends FunctorFilter[F] { } #+END_SRC -* Incomplete :final: - :PROPERTIES: - :EXPORT_FILE_NAME: incomplete.md - :END: - *** Variance We must return to =Functor= for a moment and discuss an ancestor that @@ -4304,11 +4301,11 @@ digraph G { edge [color=lightgray] Invariant [fontcolor=black,color=black]; - InvariantSemigroupal [fontcolor=black,color=black]; - InvariantSemigroupal -> Invariant [color=black]; - Semigroupal [fontcolor=black,color=black]; - InvariantSemigroupal -> Semigroupal [color=black]; - InvariantMonoidal [fontcolor=black,color=black]; + InvariantSemigroupal; + InvariantSemigroupal -> Invariant; + Semigroupal; + InvariantSemigroupal -> Semigroupal; + InvariantMonoidal; InvariantMonoidal -> InvariantSemigroupal; Functor [fontcolor=black,color=black]; @@ -4325,26 +4322,26 @@ digraph G { Monad -> Applicative; Monad -> FlatMap; - UnorderedFoldable; + UnorderedFoldable [fontcolor=lightslategray,color=lightslategray]; Foldable [fontcolor=lightslategray,color=lightslategray]; - Foldable -> UnorderedFoldable; + Foldable -> UnorderedFoldable [color=lightslategray]; Reducible [fontcolor=lightslategray,color=lightslategray]; Reducible -> Foldable [color=lightslategray]; - UnorderedTraverse; + UnorderedTraverse [fontcolor=lightslategray,color=lightslategray]; Traverse [fontcolor=lightslategray,color=lightslategray]; Traverse -> Functor [color=lightslategray]; Traverse -> Foldable [color=lightslategray]; - Traverse -> UnorderedTraverse; + Traverse -> UnorderedTraverse [color=lightslategray]; NonEmptyTraverse [fontcolor=lightslategray,color=lightslategray]; NonEmptyTraverse -> Traverse [color=lightslategray]; NonEmptyTraverse -> Reducible [color=lightslategray]; Contravariant [fontcolor=black,color=black]; Contravariant -> Invariant [color=black]; - ContravariantSemigroupal [fontcolor=black,color=black]; + ContravariantSemigroupal; ContravariantSemigroupal -> InvariantSemigroupal; ContravariantSemigroupal -> Contravariant; - ContravariantMonoidal [fontcolor=black,color=black]; + ContravariantMonoidal; ContravariantMonoidal -> InvariantMonoidal; ContravariantMonoidal -> ContravariantSemigroupal; @@ -4367,6 +4364,7 @@ digraph G { ApplicativeError -> Applicative; MonadError; MonadError -> ApplicativeError; + MonadError -> Monad; Bimonad; Bimonad -> Monad; @@ -4398,29 +4396,29 @@ digraph G { =Invariant= has a method =.imap= which says that given a function from =A= to =B=, and a function from =B= to =A=, then we can convert =F[A]= to =F[B]=. -=Functor= is a short name for what should be /covariant functor/. But -since =Functor= is so popular it gets the nickname. Likewise -=Contravariant= should really be /contravariant functor/. +=Functor= is a short name for what should be /covariant functor/. But since +=Functor= is so popular it gets the nickname. Likewise =Contravariant= should +really be /contravariant functor/, and =Invariant= an /invariant functor/. -=Functor= implements =xmap= with =map= and ignores the function from -=B= to =A=. =Contravariant=, on the other hand, implements =xmap= with -=contramap= and ignores the function from =A= to =B=: +=Functor= implements =.imap= with =.map= and ignores the function from +=B= to =A=. =Contravariant=, on the other hand, implements =.imap= with +=.contramap= and ignores the function from =A= to =B=: #+BEGIN_SRC scala -@typeclass trait InvariantFunctor[F[_]] { - def xmap[A, B](fa: F[A], f: A => B, g: B => A): F[B] +@typeclass trait Invariant[F[_]] { + def imap[A, B](fa: F[A], f: A => B, g: B => A): F[B] ... } -@typeclass trait Functor[F[_]] extends InvariantFunctor[F] { +@typeclass trait Functor[F[_]] extends Invariant[F] { def map[A, B](fa: F[A])(f: A => B): F[B] - def xmap[A, B](fa: F[A], f: A => B, g: B => A): F[B] = map(fa)(f) + def imap[A, B](fa: F[A], f: A => B, g: B => A): F[B] = map(fa)(f) ... } -@typeclass trait Contravariant[F[_]] extends InvariantFunctor[F] { +@typeclass trait Contravariant[F[_]] extends Invariant[F] { def contramap[A, B](fa: F[A])(f: B => A): F[B] - def xmap[A, B](fa: F[A], f: A => B, g: B => A): F[B] = contramap(fa)(g) + def imap[A, B](fa: F[A], f: A => B, g: B => A): F[B] = contramap(fa)(g) ... } #+END_SRC @@ -4433,11 +4431,11 @@ possible to map the contents of a structure =F[A]= into =F[B]=. Using =identity= we can see that =A= can be safely downcast (or upcast) into =B= depending on the variance of the functor. -=.map= may be understand by its contract "if you give me an =F= of =A= and a way +=.map= may be understood by its contract "if you give me an =F= of =A= and a way to turn an =A= into a =B=, then I can give you an =F= of =B=". Likewise, =.contramap= reads as "if you give me an =F= of =A= and a way to turn -a =B= into a =A=, then I can give you an =F= of =B=". +a =B= into an =A=, then I can give you an =F= of =B=". We will consider an example: in our application we introduce domain specific types =Alpha=, =Beta=, =Gamma=, etc, to ensure that we don't mix up numbers in a @@ -4470,105 +4468,151 @@ Methods on a typeclass can have their type parameters in /contravariant position/ (method parameters) or in /covariant position/ (return type). If a typeclass has a combination of covariant and contravariant positions, it might have an /invariant functor/. For example, =Semigroup= and =Monoid= have an -=InvariantFunctor=, but not a =Functor= or a =Contravariant=. - -*** Apply and Bind - -TODO - - +=Invariant=, but not a =Functor= or a =Contravariant=. +*** Semigroupal, Apply and FlatMap Consider this the warm-up act to =Applicative= and =Monad= -#+BEGIN_SRC dot :file images/scalaz-apply.png :exports results +#+BEGIN_SRC dot :file images/cats-apply.png :exports results digraph G { graph [dpi=300, rankdir=BT]; node [fontname=Hack]; node [fontcolor=lightgray,color=lightgray]; edge [color=lightgray] - Cobind; - Comonad; - Bind [fontcolor=black,color=black]; - Apply [fontcolor=black,color=black]; - Contravariant [fontcolor=lightslategray,color=lightslategray]; + Invariant [fontcolor=lightslategray,color=lightslategray]; + InvariantSemigroupal [fontcolor=black,color=black]; + InvariantSemigroupal -> Invariant [color=black]; + Semigroupal [fontcolor=black,color=black]; + InvariantSemigroupal -> Semigroupal [color=black]; + InvariantMonoidal [fontcolor=black,color=black]; + InvariantMonoidal -> InvariantSemigroupal [color=black]; + Functor [fontcolor=lightslategray,color=lightslategray]; - InvariantFunctor [fontcolor=lightslategray,color=lightslategray]; - Functor; - Divide; - Divisible; - Apply; - Align [fontcolor=lightslategray,color=lightslategray]; + Functor -> Invariant [color=lightslategray]; + Apply [fontcolor=black,color=black]; + Apply -> InvariantSemigroupal [color=black]; + Apply -> Functor [color=black]; Applicative; - Foldable1 [fontcolor=lightslategray,color=lightslategray]; + Applicative -> Apply; + Applicative -> InvariantMonoidal; + FlatMap [fontcolor=black,color=black]; + FlatMap -> Apply [color=black]; + Monad; + Monad -> Applicative; + Monad -> FlatMap; + + UnorderedFoldable [fontcolor=lightslategray,color=lightslategray]; Foldable [fontcolor=lightslategray,color=lightslategray]; + Foldable -> UnorderedFoldable [fontcolor=lightslategray,color=lightslategray]; + Reducible [fontcolor=lightslategray,color=lightslategray]; + Reducible -> Foldable [color=lightslategray]; + UnorderedTraverse [fontcolor=lightslategray,color=lightslategray]; Traverse [fontcolor=lightslategray,color=lightslategray]; - Traverse1 [fontcolor=lightslategray,color=lightslategray]; - Monad; - "Advanced Monads" [style=dashed]; - Plus; - PlusEmpty; - IsEmpty; - ApplicativePlus; - MonadPlus; + Traverse -> Functor [color=lightslategray]; + Traverse -> Foldable [color=lightslategray]; + Traverse -> UnorderedTraverse [fontcolor=lightslategray,color=lightslategray]; + NonEmptyTraverse [fontcolor=lightslategray,color=lightslategray]; + NonEmptyTraverse -> Traverse [color=lightslategray]; + NonEmptyTraverse -> Reducible [color=lightslategray]; - Cobind -> Functor; - Comonad -> Cobind; + Contravariant [fontcolor=lightslategray,color=lightslategray]; + Contravariant -> Invariant [color=lightslategray]; + ContravariantSemigroupal [fontcolor=black,color=black]; + ContravariantSemigroupal -> InvariantSemigroupal [color=black]; + ContravariantSemigroupal -> Contravariant [color=black]; + ContravariantMonoidal; + ContravariantMonoidal -> InvariantMonoidal; + ContravariantMonoidal -> ContravariantSemigroupal; - Bind -> Apply [color=black]; + CoflatMap; + CoflatMap -> Functor; + Comonad; + Comonad -> CoflatMap; - Contravariant -> InvariantFunctor [color=lightslategray]; - Functor -> InvariantFunctor [color=lightslategray]; + Distributive; + Distributive -> Functor; - Divide -> Contravariant; - Divisible -> Divide; - Apply -> Functor [color=black]; - Applicative -> Apply; + SemigroupK; + MonoidK; + MonoidK -> SemigroupK; + Alternative; + Alternative -> Applicative; + Alternative -> MonoidK; - Align -> Functor [color=lightslategray]; + ApplicativeError; + ApplicativeError -> Applicative; + MonadError; + MonadError -> ApplicativeError; + MonadError -> Monad; - Foldable1 -> Foldable [color=lightslategray]; - Traverse -> Functor [color=lightslategray]; - Traverse -> Foldable [color=lightslategray]; - Traverse1 -> Traverse [color=lightslategray]; - Traverse1 -> Foldable1 [color=lightslategray]; + Bimonad; + Bimonad -> Monad; + Bimonad -> Comonad; - Monad -> Applicative; - "Advanced Monads" -> Monad; - Monad -> Bind; + StackSafeMonad; + StackSafeMonad -> Monad; - PlusEmpty -> Plus; - IsEmpty -> PlusEmpty; + CommutativeApply; + CommutativeApply -> Apply; + CommutativeApplicative; + CommutativeApplicative -> Applicative; + CommutativeApplicative -> CommutativeApply; + CommutativeFlatMap; + CommutativeFlatMap -> FlatMap; + CommutativeFlatMap -> CommutativeApply; + CommutativeMonad; + CommutativeMonad -> Monad; + CommutativeMonad -> CommutativeFlatMap; + CommutativeMonad -> CommutativeApplicative; - ApplicativePlus -> Applicative; - ApplicativePlus -> PlusEmpty; - MonadPlus -> ApplicativePlus; - MonadPlus -> Monad; } #+END_SRC #+CAPTION: 100 #+RESULTS: -[[file:images/scalaz-apply.png]] +[[file:images/cats-apply.png]] + +**** Semigroupal + +=Semigroupal= looks at first sight to be similar to =Align= because it zips +together two values in the same context + +#+BEGIN_SRC scala +@typeclass trait Semigroupal[F[_]] { + def product[A, B](fa: F[A], fb: F[B]): F[(A, B)] +} +#+END_SRC + +however this is the Cartesian product, which means that every =A= is matched up +with every =B=. Compare how =Align.align= and =Semigroupal.product= differ in this +example + +#+BEGIN_SRC scala +scala> Align[List].align(List("a", "b", "c"), List(1,2,3,4)) +res: List[Ior[String,Int]] = List(Both(a,1), Both(b,2), Both(c,3), Right(4)) + +scala> Semigroupal[List].product(List("a", "b", "c"), List(1,2,3,4)) +res: List[(String, Int)] = List((a,1), (a,2), (a,3), (a,4), + (b,1), (b,2), (b,3), (b,4), + (c,1), (c,2), (c,3), (c,4)) +#+END_SRC **** Apply -=Apply= extends =Functor= by adding a method named =ap= which is -similar to =map= in that it applies a function to values. However, -with =ap=, the function is in the same context as the values. +=Apply= extends =Functor= and =Semigroupal= by adding a method named =.ap= which is +similar to =.map= in that it applies a function to values. However, +with =.ap=, the function is in the same context as the values. #+BEGIN_SRC scala -@typeclass trait Apply[F[_]] extends Functor[F] { - @op("<*>") def ap[A, B](fa: =>F[A])(f: =>F[A => B]): F[B] +@typeclass trait Apply[F[_]] extends Functor[F] with Semigroupal[F] { + @op("<*>") def ap[A, B](ff: F[A => B])(fa: F[A]): F[B] ... #+END_SRC -# https://mobile.twitter.com/mrkgrnao/status/962030468093767681 - #+BEGIN_ASIDE -~<*>~ is the Advanced TIE Fighter, as flown by Darth Vader. Appropriate since it -looks like an angry parent. Or a sad Pikachu. +~<*>~ is spoken as "the ap operator". #+END_ASIDE It is worth taking a moment to consider what that means for a simple data @@ -4576,102 +4620,66 @@ structure like =Option[A]=, having the following implementation of =.ap= #+BEGIN_SRC scala implicit def option[A]: Apply[Option[A]] = new Apply[Option[A]] { - override def ap[A, B](fa: =>Option[A])(f: =>Option[A => B]) = f match { - case Some(ff) => fa.map(ff) + override def ap[A, B](ff: Option[A => B], fa: Option[A])() = f match { + case Some(f) => fa.map(f) case None => None } ... } #+END_SRC -To implement ~.ap~, we must first extract the function ~ff: A => B~ from ~f: +To implement ~.ap~, we must first extract the function ~f: A => B~ from ~ff: Option[A => B]~, then we can map over ~fa~. The extraction of the function from the context is the important power that ~Apply~ brings, allowing multiple function to be combined inside the context. -Returning to =Apply=, we find =.applyX= boilerplate that allows us to combine +Returning to =Apply=, we find =.mapX= boilerplate that allows us to combine parallel functions and then map over their combined output: #+BEGIN_SRC scala -@typeclass trait Apply[F[_]] extends Functor[F] { +@typeclass trait Apply[F[_]] extends Functor[F] with Semigroupal[F] { ... - def apply2[A,B,C](fa: =>F[A], fb: =>F[B])(f: (A, B) => C): F[C] = ... - def apply3[A,B,C,D](fa: =>F[A],fb: =>F[B],fc: =>F[C])(f: (A,B,C) =>D): F[D] = ... + def map2[A,B,C](fa: F[A], fb: F[B])(f: (A, B) => Z): F[Z] = ... + def map3[A,B,C,D](fa: F[A], fb: F[B], fc: F[C])(f: (A,B,C) => Z): F[Z] = ... ... - def apply12[...] + def map21[...] #+END_SRC -Read =.apply2= as a contract promising: "if you give me an =F= of =A= and an =F= -of =B=, with a way of combining =A= and =B= into a =C=, then I can give you an -=F= of =C=". There are many uses for this contract and the two most important are: +Read =.map2= as a contract promising: "if you give me an =F= of =A= and an =F= +of =B=, with a way of combining =A= and =B= into a =Z=, then I can give you an +=F= of =Z=". There are many uses for this contract and the two most important are: -- constructing some typeclasses for a product type =C= from its constituents =A= +- constructing some typeclasses for a product type =Z= from its constituents =A= and =B= - performing /effects/ in parallel, like the drone and google algebras we created in Chapter 3, and then combining their results. -Indeed, =Apply= is so useful that it has special syntax: +Indeed, =Apply= is so useful that it has special syntax that is worth revisiting from Chapter 3: #+BEGIN_SRC scala -implicit class ApplyOps[F[_]: Apply, A](self: F[A]) { - def *>[B](fb: F[B]): F[B] = Apply[F].apply2(self,fb)((_,b) => b) - def <*[B](fb: F[B]): F[A] = Apply[F].apply2(self,fb)((a,_) => a) - def |@|[B](fb: F[B]): ApplicativeBuilder[F, A, B] = ... -} - -class ApplicativeBuilder[F[_]: Apply, A, B](a: F[A], b: F[B]) { - def tupled: F[(A, B)] = Apply[F].apply2(a, b)(Tuple2(_)) - def |@|[C](cc: F[C]): ApplicativeBuilder3[C] = ... - - sealed abstract class ApplicativeBuilder3[C](c: F[C]) { - ..ApplicativeBuilder4 - ... - ..ApplicativeBuilder12 -} +(d.getBacklog, d.getAgents, m.getManaged, m.getAlive, m.getTime).mapN { ... } #+END_SRC -which is exactly what we used in Chapter 3: - -#+BEGIN_SRC scala -(d.getBacklog |@| d.getAgents |@| m.getManaged |@| m.getAlive |@| m.getTime) -#+END_SRC +where the =.mapN= method will apply =.map5= here, because the compiler knows the +size of the tuple. We could also write -#+BEGIN_ASIDE -The =|@|= operator has many names. Some call it the /Cartesian Product Syntax/, -others call it the /Cinnamon Bun/, the /Admiral Ackbar/ or the /Macaulay -Culkin/. We prefer to call it /The Scream/ operator, after the Munch painting, -because it is also the sound our CPU makes when it is parallelising All The -Things. -#+END_ASIDE - -The syntax =<*= and =*>= (left bird and right bird) offer a convenient way to -ignore the output from one of two parallel effects. - -Unfortunately, although the =|@|= syntax is clear, there is a problem -in that a new =ApplicativeBuilder= object is allocated for each -additional effect. If the work is I/O-bound, the memory allocation -cost is insignificant. However, when performing CPU-bound work, use -the alternative /lifting with arity/ syntax, which does not produce -any intermediate objects: +or directly call =applyX= #+BEGIN_SRC scala -def ^[F[_]: Apply,A,B,C](fa: =>F[A],fb: =>F[B])(f: (A,B) =>C): F[C] = ... -def ^^[F[_]: Apply,A,B,C,D](fa: =>F[A],fb: =>F[B],fc: =>F[C])(f: (A,B,C) =>D): F[D] = ... -... -def ^^^^^^[F[_]: Apply, ...] +Apply[F].map5(d.getBacklog, d.getAgents, m.getManaged, m.getAlive, m.getTime) #+END_SRC -used like +=.productL= and =.productR= offer a way to ignore the output from one of two +effects: #+BEGIN_SRC scala -^^^^(d.getBacklog, d.getAgents, m.getManaged, m.getAlive, m.getTime) + @op("<*") def productL[A, B](fa: F[A])(fb: F[B]): F[A] = ... + @op("*>") def productR[A, B](fa: F[A])(fb: F[B]): F[B] = ... #+END_SRC -or directly call =applyX= - -#+BEGIN_SRC scala -Apply[F].apply5(d.getBacklog, d.getAgents, m.getManaged, m.getAlive, m.getTime) -#+END_SRC +#+BEGIN_ASIDE +=<*= and =*>= are spoken "product left" and "product right". +#+END_ASIDE Despite being more commonly used with effects, =Apply= works just as well with data structures. Consider rewriting @@ -4686,7 +4694,7 @@ data structures. Consider rewriting as #+BEGIN_SRC scala -(data.foo |@| data.bar)(_ + _.shows) +(data.foo, data.bar).mapN(_ + _.shows) #+END_SRC @@ -4694,221 +4702,225 @@ If we only want the combined output as a tuple, methods exist to do just that: #+BEGIN_SRC scala - @op("tuple") def tuple2[A,B](fa: =>F[A],fb: =>F[B]): F[(A,B)] = ... - def tuple3[A,B,C](fa: =>F[A],fb: =>F[B],fc: =>F[C]): F[(A,B,C)] = ... - ... - def tuple12[...] -#+END_SRC - -#+BEGIN_SRC scala -(data.foo tuple data.bar) : Option[(String, Int)] -#+END_SRC - -There are also the generalised versions of =ap= for more than two -parameters: - -#+BEGIN_SRC scala - def ap2[A,B,C](fa: =>F[A],fb: =>F[B])(f: F[(A,B) => C]): F[C] = ... - def ap3[A,B,C,D](fa: =>F[A],fb: =>F[B],fc: =>F[C])(f: F[(A,B,C) => D]): F[D] = ... + def tuple2[A,B](fa: F[A], fb: F[B]): F[(A,B)] = ... + def tuple3[A,B,C](fa: F[A], fb: F[B], fc: F[C]): F[(A,B,C)] = ... ... - def ap12[...] + def tuple21[...] #+END_SRC -along with =.lift= methods that take normal functions and lift them into the -=F[_]= context, the generalisation of =Functor.lift= - #+BEGIN_SRC scala - def lift2[A,B,C](f: (A,B) => C): (F[A],F[B]) => F[C] = ... - def lift3[A,B,C,D](f: (A,B,C) => D): (F[A],F[B],F[C]) => F[D] = ... - ... - def lift12[...] +Apply[Option].tuple2(data.foo, data.bar) : Option[(String, Int)] #+END_SRC -and =.apF=, a partially applied syntax for =ap= +**** FlatMap -#+BEGIN_SRC scala - def apF[A,B](f: =>F[A => B]): F[A] => F[B] = ... -#+END_SRC - -Finally =.forever= - -#+BEGIN_SRC scala - def forever[A, B](fa: F[A]): F[B] = ... -#+END_SRC - -repeating an effect without stopping. The instance of =Apply= must be -stack safe or we will get =StackOverflowError=. - -**** Bind - -=Bind= introduces =.bind=, synonymous with =.flatMap=, which allows functions +=FlatMap= introduces =.flatMap= which allows functions over the result of an effect to return a new effect, or for functions over the values of a data structure to return new data structures that are then joined. #+BEGIN_SRC scala -@typeclass trait Bind[F[_]] extends Apply[F] { +@typeclass trait FlatMap[F[_]] extends Apply[F] { - @op(">>=") def bind[A, B](fa: F[A])(f: A => F[B]): F[B] - def flatMap[A, B](fa: F[A])(f: A => F[B]): F[B] = bind(fa)(f) + @op(">>=") def flatMap[A, B](fa: F[A])(f: A => F[B]): F[B] - override def ap[A, B](fa: =>F[A])(f: =>F[A => B]): F[B] = - bind(f)(x => map(fa)(x)) - override def apply2[A, B, C](fa: =>F[A], fb: =>F[B])(f: (A, B) => C): F[C] = - bind(fa)(a => map(fb)(b => f(a, b))) + def flatten[A](ffa: F[F[A]]): F[A] = - def join[A](ffa: F[F[A]]): F[A] = bind(ffa)(identity) + override def map2[A, B, Z](fa: F[A], fb: F[B])(f: (A, B) => Z): F[Z] = + flatMap(fa)(a => map(fb)(b => f(a, b))) def mproduct[A, B](fa: F[A])(f: A => F[B]): F[(A, B)] = ... - def ifM[B](value: F[Boolean], t: =>F[B], f: =>F[B]): F[B] = ... - + def ifM[B](fa: F[Boolean])(ifTrue: => F[B], ifFalse: => F[B]): F[B] = ... } #+END_SRC -The =.join= may be familiar to users of =.flatten= in the stdlib, it takes a -nested context and squashes it into one. +=.flatten= takes a nested context and squashes it into one. -Derived combinators are introduced for =.ap= and =.apply2= that require -consistency with =.bind=. We will see later that this law has consequences for +Derived combinators are introduced for =.map2= that require consistency with +=.flatMap= ordering. We will see later that this law has consequences for parallelisation strategies. -=mproduct= is like =Functor.fproduct= and pairs the function's input -with its output, inside the =F=. +=.mproduct= is like =Functor.fproduct= and pairs the function's input with its +output, inside the =F=. -=ifM= is a way to construct a conditional data structure or effect: +=.ifM= is a way to construct a conditional data structure or effect: #+BEGIN_SRC scala scala> List(true, false, true).ifM(List(0), List(1, 1)) res: List[Int] = List(0, 1, 1, 0) #+END_SRC -=ifM= and =ap= are optimised to cache and reuse code branches, compare -to the longer form +If we want to ignore the result of the =.flatMap= effect, we can use =.flatTap=, +analagous to =.productL= #+BEGIN_SRC scala -scala> List(true, false, true).flatMap { b => if (b) List(0) else List(1, 1) } + def flatTap[A, B](fa: F[A])(f: A => F[B]): F[A] = ... #+END_SRC -which produces a fresh =List(0)= or =List(1, 1)= every time the branch -is invoked. +Finally =.foreverM= -#+BEGIN_ASIDE -These kinds of optimisations are possible in FP because all methods -are deterministic, also known as /referentially transparent/. +#+BEGIN_SRC scala + def foreverM[A, B](fa: F[A]): F[B] = ... +#+END_SRC + +repeating an effect without stopping. Instances of =FlatMap= are guaranteed to +be stack safe, in the sense that we will never get a =StackOverflowError= as a +result of calling =.foreverM=, because the tail recursive step must be +implemented -If a method returns a different value every time it is called, it is -/impure/ and breaks the reasoning and optimisations that we can -otherwise make. +#+BEGIN_SRC scala + def tailRecM[A, B](a: A)(f: A => F[Either[A, B]]): F[B] +#+END_SRC -If the =F= is an effect, perhaps one of our drone or Google algebras, -it does not mean that the output of the call to the algebra is cached. -Rather the reference to the operation is cached. The performance -optimisation of =ifM= is only noticeable for data structures, and more -pronounced with the difficulty of the work in each branch. +If our love of FP is not forever, we can exit the loop -We will explore the concept of determinism and value caching in more -detail in the next chapter. -#+END_ASIDE +#+BEGIN_SRC scala +def untilDefinedM[A](foa: F[Option[A]]): F[A] +#+END_SRC + +Only kidding, our love of FP *is* forever, we simply return =Some= love and +continue FPing! + +**** InvariantSemigroupal, InvariantMonoidal + +=InvariantSemigroupal= is a convenient typeclass that combines =Semigroupal= and +=Invariant= without adding any new methods, simply because it is so common to do +this. + +#+BEGIN_SRC scala +@typeclass trait InvariantSemigroupal[F[_]] extends Semigroupal[F] with Invariant[F] +#+END_SRC -=Bind= also has some special syntax +That leads to =InvariantMonoidal= which introduces =.point= as a way to wrap a +single value in a context. =.unit= is a convenience for =.point(())= (the =Unit= +type). #+BEGIN_SRC scala -implicit class BindOps[F[_]: Bind, A] (self: F[A]) { - def >>[B](b: =>F[B]): F[B] = Bind[F].bind(self)(_ => b) - def >>![B](f: A => F[B]): F[A] = Bind[F].bind(self)(a => f(a).map(_ => a)) +@typeclass trait InvariantMonoidal[F[_]] extends InvariantSemigroupal[F] { + def point[A](a: A): F[A] + def unit: F[Unit] } #+END_SRC -=>>= is when we wish to discard the input to =bind= and =>>!= is when -we want to run an effect but discard its output. +#+BEGIN_SRC scala +scala> InvariantMonoidal[Option].point(13) +res: Option[Int] = Some(13) +#+END_SRC *** Applicative and Monad -From a functionality point of view, =Applicative= is =Apply= with a -=pure= method, and =Monad= extends =Applicative= with =Bind=. +From an API point of view, =Applicative= is =Apply= with a =.pure= method, and +=Monad= extends =Applicative= with =FlatMap=. -#+BEGIN_SRC dot :file images/scalaz-applicative.png :exports results +#+BEGIN_SRC dot :file images/cats-applicative.png :exports results digraph G { graph [dpi=300, rankdir=BT]; node [fontname=Hack]; node [fontcolor=lightgray,color=lightgray]; edge [color=lightgray] - Cobind; - Comonad; - Bind [fontcolor=lightslategray,color=lightslategray]; - Apply [fontcolor=lightslategray,color=lightslategray]; - Contravariant [fontcolor=lightslategray,color=lightslategray]; + Invariant [fontcolor=lightslategray,color=lightslategray]; + InvariantSemigroupal [fontcolor=lightslategray,color=lightslategray]; + InvariantSemigroupal -> Invariant [color=lightslategray]; + Semigroupal [fontcolor=lightslategray,color=lightslategray]; + InvariantSemigroupal -> Semigroupal [color=lightslategray]; + InvariantMonoidal [fontcolor=lightslategray,color=lightslategray]; + InvariantMonoidal -> InvariantSemigroupal [color=lightslategray]; + Functor [fontcolor=lightslategray,color=lightslategray]; - InvariantFunctor [fontcolor=lightslategray,color=lightslategray]; - Functor; - Divide; - Divisible; - Apply; - Align [fontcolor=lightslategray,color=lightslategray]; + Functor -> Invariant [color=lightslategray]; + Apply [fontcolor=lightslategray,color=lightslategray]; + Apply -> InvariantSemigroupal [color=lightslategray]; + Apply -> Functor [color=lightslategray]; Applicative [fontcolor=black,color=black]; - Foldable1 [fontcolor=lightslategray,color=lightslategray]; - Foldable [fontcolor=lightslategray,color=lightslategray]; - Traverse [fontcolor=lightslategray,color=lightslategray]; - Traverse1 [fontcolor=lightslategray,color=lightslategray]; + Applicative -> Apply [color=black]; + Applicative -> InvariantMonoidal [color=black]; + FlatMap [fontcolor=lightslategray,color=lightslategray]; + FlatMap -> Apply [color=lightslategray]; Monad [fontcolor=black,color=black]; - "Advanced Monads" [style=dashed]; - Plus; - PlusEmpty; - IsEmpty; - ApplicativePlus; - MonadPlus; + Monad -> Applicative [color=black]; + Monad -> FlatMap [color=black]; - Bind -> Apply [color=lightslategray]; + UnorderedFoldable [fontcolor=lightslategray,color=lightslategray]; + Foldable [fontcolor=lightslategray,color=lightslategray]; + Foldable -> UnorderedFoldable [fontcolor=lightslategray,color=lightslategray]; + Reducible [fontcolor=lightslategray,color=lightslategray]; + Reducible -> Foldable [color=lightslategray]; + UnorderedTraverse [fontcolor=lightslategray,color=lightslategray]; + Traverse [fontcolor=lightslategray,color=lightslategray]; + Traverse -> Functor [color=lightslategray]; + Traverse -> Foldable [color=lightslategray]; + Traverse -> UnorderedTraverse [fontcolor=lightslategray,color=lightslategray]; + NonEmptyTraverse [fontcolor=lightslategray,color=lightslategray]; + NonEmptyTraverse -> Traverse [color=lightslategray]; + NonEmptyTraverse -> Reducible [color=lightslategray]; - Contravariant -> InvariantFunctor [color=lightslategray]; - Functor -> InvariantFunctor [color=lightslategray]; + Contravariant [fontcolor=lightslategray,color=lightslategray]; + Contravariant -> Invariant [color=lightslategray]; + ContravariantSemigroupal [fontcolor=lightslategray,color=lightslategray]; + ContravariantSemigroupal -> InvariantSemigroupal [color=lightslategray]; + ContravariantSemigroupal -> Contravariant [color=lightslategray]; + ContravariantMonoidal; + ContravariantMonoidal -> InvariantMonoidal; + ContravariantMonoidal -> ContravariantSemigroupal; - Divide -> Contravariant; - Divisible -> Divide; - Apply -> Functor [color=lightslategray]; - Applicative -> Apply [color=black]; + CoflatMap; + CoflatMap -> Functor; + Comonad; + Comonad -> CoflatMap; - Foldable1 -> Foldable [color=lightslategray]; - Traverse -> Functor [color=lightslategray]; - Traverse -> Foldable [color=lightslategray]; - Traverse1 -> Traverse [color=lightslategray]; - Traverse1 -> Foldable1 [color=lightslategray]; + Distributive; + Distributive -> Functor; - Monad -> Applicative [color=black]; - "Advanced Monads" -> Monad; - Monad -> Bind [color=black]; + SemigroupK; + MonoidK; + MonoidK -> SemigroupK; + Alternative; + Alternative -> Applicative; + Alternative -> MonoidK; - Cobind -> Functor; - Comonad -> Cobind; + ApplicativeError; + ApplicativeError -> Applicative; + MonadError; + MonadError -> ApplicativeError; + MonadError -> Monad; - PlusEmpty -> Plus; - IsEmpty -> PlusEmpty; + Bimonad; + Bimonad -> Monad; + Bimonad -> Comonad; - ApplicativePlus -> Applicative; - ApplicativePlus -> PlusEmpty; - MonadPlus -> ApplicativePlus; - MonadPlus -> Monad; + StackSafeMonad; + StackSafeMonad -> Monad; - Align -> Functor [color=lightslategray]; + CommutativeApply; + CommutativeApply -> Apply; + CommutativeApplicative; + CommutativeApplicative -> Applicative; + CommutativeApplicative -> CommutativeApply; + CommutativeFlatMap; + CommutativeFlatMap -> FlatMap; + CommutativeFlatMap -> CommutativeApply; + CommutativeMonad; + CommutativeMonad -> Monad; + CommutativeMonad -> CommutativeFlatMap; + CommutativeMonad -> CommutativeApplicative; } #+END_SRC #+CAPTION: 100 #+RESULTS: -[[file:images/scalaz-applicative.png]] +[[file:images/cats-applicative.png]] #+BEGIN_SRC scala -@typeclass trait Applicative[F[_]] extends Apply[F] { - def point[A](a: =>A): F[A] - def pure[A](a: =>A): F[A] = point(a) +@typeclass trait Applicative[F[_]] extends Apply[F] with InvariantMonoidal[F] { + def pure[A](a: A): F[A] } -@typeclass trait Monad[F[_]] extends Applicative[F] with Bind[F] +@typeclass trait Monad[F[_]] extends Applicative[F] with FlatMap[F] #+END_SRC In many ways, =Applicative= and =Monad= are the culmination of everything we've -seen in this chapter. =.pure= (or =.point= as it is more commonly known for data -structures) allows us to create effects or data structures from values. +seen in this chapter. =.pure= (aliased to =.point=) allows us to create effects +or data structures from values. Instances of =Applicative= must meet some laws, effectively asserting that all the methods are consistent: @@ -4924,15 +4936,14 @@ that all the methods are consistent: =Monad= adds additional laws: -- *Left Identity*: ~pure(a).bind(f) === f(a)~ -- *Right Identity*: ~a.bind(pure(_)) === a~ -- *Associativity*: ~fa.bind(f).bind(g) === fa.bind(a => f(a).bind(g))~ where +- *Left Identity*: ~pure(a).flatMap(f) === f(a)~ +- *Right Identity*: ~a.flatMap(pure(_)) === a~ +- *Associativity*: ~fa.flatMap(f).flatMap(g) === fa.flatMap(a => f(a).flatMap(g))~ where ~fa~ is an ~F[A]~, ~f~ is an ~A => F[B]~ and ~g~ is a ~B => F[C]~. -Associativity says that chained =bind= calls must agree with nested -=bind=. However, it does not mean that we can rearrange the order, -which would be /commutativity/. For example, recalling that =flatMap= -is an alias to =bind=, we cannot rearrange +Associativity says that chained =flatMap= calls must agree with nested +=flatMap=. However, it does not mean that we can rearrange the order, +which would be /commutativity/. For example we cannot rearrange #+BEGIN_SRC scala for { @@ -4950,11 +4961,11 @@ for { } yield true #+END_SRC -=start= and =stop= are *non*-/commutative/, because the intended +=.start= and =.stop= are *non*-/commutative/, because the intended effect of starting then stopping a node is different to stopping then starting it! -But =start= is commutative with itself, and =stop= is commutative with +But =.start= is commutative with itself, and =.stop= is commutative with itself, so we can rewrite #+BEGIN_SRC scala @@ -4982,7 +4993,7 @@ A practical consequence is that a =Monad= must be /commutative/ if its Chapter 3 when we ran these effects in parallel #+BEGIN_SRC scala -(d.getBacklog |@| d.getAgents |@| m.getManaged |@| m.getAlive |@| m.getTime) +(d.getBacklog, d.getAgents, m.getManaged, m.getAlive, m.getTime) #+END_SRC because we know that they are commutative among themselves. When it comes to @@ -4993,6 +5004,117 @@ implementation may choose to sequence the operations to be on the safe side. The subtleties of how we deal with (re)-ordering of effects, and what those effects are, deserves a dedicated chapter on Advanced Monads. +**** Commutativity + +Now that we have discussed commutativity in the context of =Monad= we can +understand the entire suite of =Commutative*= typeclasses in Cats: they add no +additional methods but add the constraint that the order of effects does not +matter. + +#+BEGIN_SRC dot :file images/cats-commute.png :exports results +digraph G { + graph [dpi=300, rankdir=BT]; + node [fontname=Hack]; + node [fontcolor=lightgray,color=lightgray]; + edge [color=lightgray] + + Invariant [fontcolor=lightslategray,color=lightslategray]; + InvariantSemigroupal [fontcolor=lightslategray,color=lightslategray]; + InvariantSemigroupal -> Invariant [color=lightslategray]; + Semigroupal [fontcolor=lightslategray,color=lightslategray]; + InvariantSemigroupal -> Semigroupal [color=lightslategray]; + InvariantMonoidal [fontcolor=lightslategray,color=lightslategray]; + InvariantMonoidal -> InvariantSemigroupal [color=lightslategray]; + + Functor [fontcolor=lightslategray,color=lightslategray]; + Functor -> Invariant [color=lightslategray]; + Apply [fontcolor=lightslategray,color=lightslategray]; + Apply -> InvariantSemigroupal [color=lightslategray]; + Apply -> Functor [color=lightslategray]; + Applicative [fontcolor=lightslategray,color=lightslategray]; + Applicative -> Apply [color=lightslategray]; + Applicative -> InvariantMonoidal [color=lightslategray]; + FlatMap [fontcolor=lightslategray,color=lightslategray]; + FlatMap -> Apply [color=lightslategray]; + Monad [fontcolor=lightslategray,color=lightslategray]; + Monad -> Applicative [color=lightslategray]; + Monad -> FlatMap [color=lightslategray]; + + UnorderedFoldable [fontcolor=lightslategray,color=lightslategray]; + Foldable [fontcolor=lightslategray,color=lightslategray]; + Foldable -> UnorderedFoldable [fontcolor=lightslategray,color=lightslategray]; + Reducible [fontcolor=lightslategray,color=lightslategray]; + Reducible -> Foldable [color=lightslategray]; + UnorderedTraverse [fontcolor=lightslategray,color=lightslategray]; + Traverse [fontcolor=lightslategray,color=lightslategray]; + Traverse -> Functor [color=lightslategray]; + Traverse -> Foldable [color=lightslategray]; + Traverse -> UnorderedTraverse [fontcolor=lightslategray,color=lightslategray]; + NonEmptyTraverse [fontcolor=lightslategray,color=lightslategray]; + NonEmptyTraverse -> Traverse [color=lightslategray]; + NonEmptyTraverse -> Reducible [color=lightslategray]; + + Contravariant [fontcolor=lightslategray,color=lightslategray]; + Contravariant -> Invariant [color=lightslategray]; + ContravariantSemigroupal [fontcolor=lightslategray,color=lightslategray]; + ContravariantSemigroupal -> InvariantSemigroupal [color=lightslategray]; + ContravariantSemigroupal -> Contravariant [color=lightslategray]; + ContravariantMonoidal; + ContravariantMonoidal -> InvariantMonoidal; + ContravariantMonoidal -> ContravariantSemigroupal; + + CoflatMap; + CoflatMap -> Functor; + Comonad; + Comonad -> CoflatMap; + + Distributive; + Distributive -> Functor; + + SemigroupK; + MonoidK; + MonoidK -> SemigroupK; + Alternative; + Alternative -> Applicative; + Alternative -> MonoidK; + + ApplicativeError; + ApplicativeError -> Applicative; + MonadError; + MonadError -> ApplicativeError; + MonadError -> Monad; + + Bimonad; + Bimonad -> Monad; + Bimonad -> Comonad; + + StackSafeMonad; + StackSafeMonad -> Monad; + + CommutativeApply [fontcolor=black,color=black]; + CommutativeApply -> Apply [color=black]; + CommutativeApplicative [fontcolor=black,color=black]; + CommutativeApplicative -> Applicative [color=black]; + CommutativeApplicative -> CommutativeApply [color=black]; + CommutativeFlatMap [fontcolor=black,color=black]; + CommutativeFlatMap -> FlatMap [color=black]; + CommutativeFlatMap -> CommutativeApply [color=black]; + CommutativeMonad [fontcolor=black,color=black]; + CommutativeMonad -> Monad [color=black]; + CommutativeMonad -> CommutativeFlatMap [color=black]; + CommutativeMonad -> CommutativeApplicative [color=black]; +} +#+END_SRC + +#+CAPTION: 100 +#+RESULTS: +[[file:images/cats-commute.png]] + +* Incomplete :final: + :PROPERTIES: + :EXPORT_FILE_NAME: incomplete.md + :END: + *** Divide and Conquer #+BEGIN_SRC dot :file images/scalaz-divide.png :exports results @@ -5008,7 +5130,7 @@ digraph G { Apply [fontcolor=lightslategray,color=lightslategray]; Contravariant [fontcolor=lightslategray,color=lightslategray]; Functor [fontcolor=lightslategray,color=lightslategray]; - InvariantFunctor [fontcolor=lightslategray,color=lightslategray]; + Invariant [fontcolor=lightslategray,color=lightslategray]; Functor; Divide [fontcolor=black,color=black]; Divisible [fontcolor=black,color=black]; @@ -5029,8 +5151,8 @@ digraph G { Bind -> Apply [color=lightslategray]; - Contravariant -> InvariantFunctor [color=lightslategray]; - Functor -> InvariantFunctor [color=lightslategray]; + Contravariant -> Invariant [color=lightslategray]; + Functor -> Invariant [color=lightslategray]; Divide -> Contravariant [color=black]; Divisible -> Divide [color=black]; @@ -5145,7 +5267,7 @@ digraph G { Apply [fontcolor=lightslategray,color=lightslategray]; Contravariant [fontcolor=lightslategray,color=lightslategray]; Functor [fontcolor=lightslategray,color=lightslategray]; - InvariantFunctor [fontcolor=lightslategray,color=lightslategray]; + Invariant [fontcolor=lightslategray,color=lightslategray]; Functor; Divide [fontcolor=lightslategray,color=lightslategray]; Divisible [fontcolor=lightslategray,color=lightslategray]; @@ -5166,8 +5288,8 @@ digraph G { Bind -> Apply [color=lightslategray]; - Contravariant -> InvariantFunctor [color=lightslategray]; - Functor -> InvariantFunctor [color=lightslategray]; + Contravariant -> Invariant [color=lightslategray]; + Functor -> Invariant [color=lightslategray]; Divide -> Contravariant [color=lightslategray]; Divisible -> Divide [color=lightslategray]; @@ -5491,7 +5613,7 @@ digraph G { Apply [fontcolor=lightslategray,color=lightslategray]; Contravariant [fontcolor=lightslategray,color=lightslategray]; Functor [fontcolor=lightslategray,color=lightslategray]; - InvariantFunctor [fontcolor=lightslategray,color=lightslategray]; + Invariant [fontcolor=lightslategray,color=lightslategray]; Functor [fontcolor=lightslategray,color=lightslategray]; Divide [fontcolor=lightslategray,color=lightslategray]; Divisible [fontcolor=lightslategray,color=lightslategray]; @@ -5512,8 +5634,8 @@ digraph G { Bind -> Apply [color=lightslategray]; - Contravariant -> InvariantFunctor [color=lightslategray]; - Functor -> InvariantFunctor [color=lightslategray]; + Contravariant -> Invariant [color=lightslategray]; + Functor -> Invariant [color=lightslategray]; Divide -> Contravariant [color=lightslategray]; Divisible -> Divide [color=lightslategray]; @@ -12373,7 +12495,7 @@ digraph G { node [fontcolor=lightgray,color=lightgray]; edge [color=lightgray] - InvariantFunctor; + Invariant; InvariantApplicative [fontcolor=black,color=black]; InvariantAlt [fontcolor=black,color=black]; @@ -12388,14 +12510,14 @@ digraph G { Monad; MonadError; - Contravariant -> InvariantFunctor; + Contravariant -> Invariant; Divisible -> Contravariant; Decidable -> Divisible [color=black]; - InvariantApplicative -> InvariantFunctor [color=black]; + InvariantApplicative -> Invariant [color=black]; - Functor -> InvariantFunctor; + Functor -> Invariant; Applicative -> Functor; Alt -> Applicative [color=black]; @@ -12422,11 +12544,11 @@ In Scalaz 7.3, =Applicative= and =Divisible= will inherit from =InvariantApplica Before we proceed, here is a quick recap of the core Scalaz typeclasses: #+BEGIN_SRC scala -@typeclass trait InvariantFunctor[F[_]] { +@typeclass trait Invariant[F[_]] { def xmap[A, B](fa: F[A], f: A => B, g: B => A): F[B] } -@typeclass trait Contravariant[F[_]] extends InvariantFunctor[F] { +@typeclass trait Contravariant[F[_]] extends Invariant[F] { def contramap[A, B](fa: F[A])(f: B => A): F[B] def xmap[A, B](fa: F[A], f: A => B, g: B => A): F[B] = contramap(fa)(g) } @@ -12438,7 +12560,7 @@ Before we proceed, here is a quick recap of the core Scalaz typeclasses: def divide22[...] = ... } -@typeclass trait Functor[F[_]] extends InvariantFunctor[F] { +@typeclass trait Functor[F[_]] extends Invariant[F] { def map[A, B](fa: F[A])(f: A => B): F[B] def xmap[A, B](fa: F[A], f: A => B, g: B => A): F[B] = map(fa)(f) } @@ -12517,11 +12639,11 @@ object Foo { #+END_SRC If a typeclass has parameters in both covariant and contravariant position, as -is the case with =Semigroup=, it may provide an =InvariantFunctor= +is the case with =Semigroup=, it may provide an =Invariant= #+BEGIN_SRC scala object Semigroup { - implicit val invariant = new InvariantFunctor[Semigroup] { + implicit val invariant = new Invariant[Semigroup] { def xmap[A, B](ma: Semigroup[A], f: A => B, g: B => A) = new Semigroup[B] { def append(x: B, y: =>B): B = f(ma.append(g(x), g(y))) } @@ -13062,7 +13184,7 @@ Returning to the =scalaz-deriving= typeclasses, the invariant parents of =Alt= and =Decidable= are: #+BEGIN_SRC scala -@typeclass trait InvariantApplicative[F[_]] extends InvariantFunctor[F] { +@typeclass trait InvariantApplicative[F[_]] extends Invariant[F] { def xproduct0[Z](f: =>Z): F[Z] def xproduct1[Z, A1](a1: =>F[A1])(f: A1 => Z, g: Z => A1): F[Z] = ... def xproduct2 ... @@ -13078,7 +13200,7 @@ and =Decidable= are: } #+END_SRC -supporting typeclasses with an =InvariantFunctor= like =Monoid= and =Semigroup=. +supporting typeclasses with an =Invariant= like =Monoid= and =Semigroup=. **** Arbitrary Arity and =@deriving= @@ -16249,7 +16371,7 @@ release. | Typeclass | Method | From | Given | To | |--------------------+-----------------+-----------------+------------------------+----------------| -| ~InvariantFunctor~ | ~xmap~ | ~F[A]~ | ~A => B, B => A~ | ~F[B]~ | +| ~Invariant~ | ~xmap~ | ~F[A]~ | ~A => B, B => A~ | ~F[B]~ | | ~Contravariant~ | ~contramap~ | ~F[A]~ | ~B => A~ | ~F[B]~ | | ~Functor~ | ~map~ | ~F[A]~ | ~A => B~ | ~F[B]~ | | ~Apply~ | ~ap~ / ~<*>~ | ~F[A]~ | ~F[A => B]~ | ~F[B]~ | diff --git a/manuscript/complete.md b/manuscript/complete.md index 1285d4fa..49bfb301 100644 --- a/manuscript/complete.md +++ b/manuscript/complete.md @@ -3703,3 +3703,472 @@ And similarly, `TraverseFilter` can filter the values while traversing or sequen ~~~~~~~~ +## Variance + +We must return to `Functor` for a moment and discuss an ancestor that +we previously ignored: + +{width=100%} +![](images/cats-variance.png) + +`Invariant` has a method `.imap` which says that given a function from `A` to +`B`, and a function from `B` to `A`, then we can convert `F[A]` to `F[B]`. + +`Functor` is a short name for what should be *covariant functor*. But since +`Functor` is so popular it gets the nickname. Likewise `Contravariant` should +really be *contravariant functor*, and `Invariant` an *invariant functor*. + +`Functor` implements `.imap` with `.map` and ignores the function from +`B` to `A`. `Contravariant`, on the other hand, implements `.imap` with +`.contramap` and ignores the function from `A` to `B`: + +{lang="text"} +~~~~~~~~ + @typeclass trait Invariant[F[_]] { + def imap[A, B](fa: F[A], f: A => B, g: B => A): F[B] + ... + } + + @typeclass trait Functor[F[_]] extends Invariant[F] { + def map[A, B](fa: F[A])(f: A => B): F[B] + def imap[A, B](fa: F[A], f: A => B, g: B => A): F[B] = map(fa)(f) + ... + } + + @typeclass trait Contravariant[F[_]] extends Invariant[F] { + def contramap[A, B](fa: F[A])(f: B => A): F[B] + def imap[A, B](fa: F[A], f: A => B, g: B => A): F[B] = contramap(fa)(g) + ... + } +~~~~~~~~ + +It is important to note that, although related at a theoretical level, +the words *covariant*, *contravariant* and *invariant* do not directly +refer to Scala type variance (i.e. `+` and `-` prefixes that may be +written in type signatures). *Invariance* here means that it is +possible to map the contents of a structure `F[A]` into `F[B]`. Using +`identity` we can see that `A` can be safely downcast (or upcast) into +`B` depending on the variance of the functor. + +`.map` may be understood by its contract "if you give me an `F` of `A` and a way +to turn an `A` into a `B`, then I can give you an `F` of `B`". + +Likewise, `.contramap` reads as "if you give me an `F` of `A` and a way to turn +a `B` into an `A`, then I can give you an `F` of `B`". + +We will consider an example: in our application we introduce domain specific +types `Alpha`, `Beta`, `Gamma`, etc, to ensure that we don't mix up numbers in a +financial calculation: + +{lang="text"} +~~~~~~~~ + final case class Alpha(value: Double) +~~~~~~~~ + +but now we're faced with the problem that we don't have any typeclasses for +these new types. If we use the values in JSON documents, we have to write +instances of `JsEncoder` and `JsDecoder`. + +However, `JsEncoder` has a `Contravariant` and `JsDecoder` has a `Functor`, so +we can derive instances. Filling in the contract: + +- "if you give me a `JsDecoder` for a `Double`, and a way to go from a `Double` + to an `Alpha`, then I can give you a `JsDecoder` for an `Alpha`". +- "if you give me a `JsEncoder` for a `Double`, and a way to go from an `Alpha` + to a `Double`, then I can give you a `JsEncoder` for an `Alpha`". + +{lang="text"} +~~~~~~~~ + object Alpha { + implicit val decoder: JsDecoder[Alpha] = JsDecoder[Double].map(Alpha(_)) + implicit val encoder: JsEncoder[Alpha] = JsEncoder[Double].contramap(_.value) + } +~~~~~~~~ + +Methods on a typeclass can have their type parameters in *contravariant +position* (method parameters) or in *covariant position* (return type). If a +typeclass has a combination of covariant and contravariant positions, it might +have an *invariant functor*. For example, `Semigroup` and `Monoid` have an +`Invariant`, but not a `Functor` or a `Contravariant`. + + +## Semigroupal, Apply and FlatMap + +Consider this the warm-up act to `Applicative` and `Monad` + +{width=100%} +![](images/cats-apply.png) + + +### Semigroupal + +`Semigroupal` looks at first sight to be similar to `Align` because it zips +together two values in the same context + +{lang="text"} +~~~~~~~~ + @typeclass trait Semigroupal[F[_]] { + def product[A, B](fa: F[A], fb: F[B]): F[(A, B)] + } +~~~~~~~~ + +however this is the Cartesian product, which means that every `A` is matched up +with every `B`. Compare how `Align.align` and `Semigroupal.product` differ in this +example + +{lang="text"} +~~~~~~~~ + scala> Align[List].align(List("a", "b", "c"), List(1,2,3,4)) + res: List[Ior[String,Int]] = List(Both(a,1), Both(b,2), Both(c,3), Right(4)) + + scala> Semigroupal[List].product(List("a", "b", "c"), List(1,2,3,4)) + res: List[(String, Int)] = List((a,1), (a,2), (a,3), (a,4), + (b,1), (b,2), (b,3), (b,4), + (c,1), (c,2), (c,3), (c,4)) +~~~~~~~~ + + +### Apply + +`Apply` extends `Functor` and `Semigroupal` by adding a method named `.ap` which is +similar to `.map` in that it applies a function to values. However, +with `.ap`, the function is in the same context as the values. + +{lang="text"} +~~~~~~~~ + @typeclass trait Apply[F[_]] extends Functor[F] with Semigroupal[F] { + @op("<*>") def ap[A, B](ff: F[A => B])(fa: F[A]): F[B] + ... +~~~~~~~~ + +A> `<*>` is spoken as "the ap operator". + +It is worth taking a moment to consider what that means for a simple data +structure like `Option[A]`, having the following implementation of `.ap` + +{lang="text"} +~~~~~~~~ + implicit def option[A]: Apply[Option[A]] = new Apply[Option[A]] { + override def ap[A, B](ff: Option[A => B], fa: Option[A])() = f match { + case Some(f) => fa.map(f) + case None => None + } + ... + } +~~~~~~~~ + +To implement `.ap`, we must first extract the function `f: A => B` from `ff: +Option[A => B]`, then we can map over `fa`. The extraction of the function from +the context is the important power that `Apply` brings, allowing multiple +function to be combined inside the context. + +Returning to `Apply`, we find `.mapX` boilerplate that allows us to combine +parallel functions and then map over their combined output: + +{lang="text"} +~~~~~~~~ + @typeclass trait Apply[F[_]] extends Functor[F] with Semigroupal[F] { + ... + def map2[A,B,C](fa: F[A], fb: F[B])(f: (A, B) => Z): F[Z] = ... + def map3[A,B,C,D](fa: F[A], fb: F[B], fc: F[C])(f: (A,B,C) => Z): F[Z] = ... + ... + def map21[...] +~~~~~~~~ + +Read `.map2` as a contract promising: "if you give me an `F` of `A` and an `F` +of `B`, with a way of combining `A` and `B` into a `Z`, then I can give you an +`F` of `Z`". There are many uses for this contract and the two most important are: + +- constructing some typeclasses for a product type `Z` from its constituents `A` + and `B` +- performing *effects* in parallel, like the drone and google algebras we + created in Chapter 3, and then combining their results. + +Indeed, `Apply` is so useful that it has special syntax that is worth revisiting from Chapter 3: + +{lang="text"} +~~~~~~~~ + (d.getBacklog, d.getAgents, m.getManaged, m.getAlive, m.getTime).mapN { ... } +~~~~~~~~ + +where the `.mapN` method will apply `.map5` here, because the compiler knows the +size of the tuple. We could also write + +or directly call `applyX` + +{lang="text"} +~~~~~~~~ + Apply[F].map5(d.getBacklog, d.getAgents, m.getManaged, m.getAlive, m.getTime) +~~~~~~~~ + +`.productL` and `.productR` offer a way to ignore the output from one of two +effects: + +{lang="text"} +~~~~~~~~ + @op("<*") def productL[A, B](fa: F[A])(fb: F[B]): F[A] = ... + @op("*>") def productR[A, B](fa: F[A])(fb: F[B]): F[B] = ... +~~~~~~~~ + +A> `<*` and `*>` are spoken "product left" and "product right". + +Despite being more commonly used with effects, `Apply` works just as well with +data structures. Consider rewriting + +{lang="text"} +~~~~~~~~ + for { + foo <- data.foo: Option[String] + bar <- data.bar: Option[Int] + } yield foo + bar.shows +~~~~~~~~ + +as + +{lang="text"} +~~~~~~~~ + (data.foo, data.bar).mapN(_ + _.shows) +~~~~~~~~ + +If we only want the combined output as a tuple, methods exist to do +just that: + +{lang="text"} +~~~~~~~~ + def tuple2[A,B](fa: F[A], fb: F[B]): F[(A,B)] = ... + def tuple3[A,B,C](fa: F[A], fb: F[B], fc: F[C]): F[(A,B,C)] = ... + ... + def tuple21[...] +~~~~~~~~ + +{lang="text"} +~~~~~~~~ + Apply[Option].tuple2(data.foo, data.bar) : Option[(String, Int)] +~~~~~~~~ + + +### FlatMap + +`FlatMap` introduces `.flatMap` which allows functions +over the result of an effect to return a new effect, or for functions over the +values of a data structure to return new data structures that are then joined. + +{lang="text"} +~~~~~~~~ + @typeclass trait FlatMap[F[_]] extends Apply[F] { + + @op(">>=") def flatMap[A, B](fa: F[A])(f: A => F[B]): F[B] + + def flatten[A](ffa: F[F[A]]): F[A] = + + override def map2[A, B, Z](fa: F[A], fb: F[B])(f: (A, B) => Z): F[Z] = + flatMap(fa)(a => map(fb)(b => f(a, b))) + + def mproduct[A, B](fa: F[A])(f: A => F[B]): F[(A, B)] = ... + def ifM[B](fa: F[Boolean])(ifTrue: => F[B], ifFalse: => F[B]): F[B] = ... + } +~~~~~~~~ + +`.flatten` takes a nested context and squashes it into one. + +Derived combinators are introduced for `.map2` that require consistency with +`.flatMap` ordering. We will see later that this law has consequences for +parallelisation strategies. + +`.mproduct` is like `Functor.fproduct` and pairs the function's input with its +output, inside the `F`. + +`.ifM` is a way to construct a conditional data structure or effect: + +{lang="text"} +~~~~~~~~ + scala> List(true, false, true).ifM(List(0), List(1, 1)) + res: List[Int] = List(0, 1, 1, 0) +~~~~~~~~ + +If we want to ignore the result of the `.flatMap` effect, we can use `.flatTap`, +analagous to `.productL` + +{lang="text"} +~~~~~~~~ + def flatTap[A, B](fa: F[A])(f: A => F[B]): F[A] = ... +~~~~~~~~ + +Finally `.foreverM` + +{lang="text"} +~~~~~~~~ + def foreverM[A, B](fa: F[A]): F[B] = ... +~~~~~~~~ + +repeating an effect without stopping. Instances of `FlatMap` are guaranteed to +be stack safe, in the sense that we will never get a `StackOverflowError` as a +result of calling `.foreverM`, because the tail recursive step must be +implemented + +{lang="text"} +~~~~~~~~ + def tailRecM[A, B](a: A)(f: A => F[Either[A, B]]): F[B] +~~~~~~~~ + +If our love of FP is not forever, we can exit the loop + +{lang="text"} +~~~~~~~~ + def untilDefinedM[A](foa: F[Option[A]]): F[A] +~~~~~~~~ + +Only kidding, our love of FP **is** forever, we simply return `Some` love and +continue FPing! + + +### InvariantSemigroupal, InvariantMonoidal + +`InvariantSemigroupal` is a convenient typeclass that combines `Semigroupal` and +`Invariant` without adding any new methods, simply because it is so common to do +this. + +{lang="text"} +~~~~~~~~ + @typeclass trait InvariantSemigroupal[F[_]] extends Semigroupal[F] with Invariant[F] +~~~~~~~~ + +That leads to `InvariantMonoidal` which introduces `.point` as a way to wrap a +single value in a context. `.unit` is a convenience for `.point(())` (the `Unit` +type). + +{lang="text"} +~~~~~~~~ + @typeclass trait InvariantMonoidal[F[_]] extends InvariantSemigroupal[F] { + def point[A](a: A): F[A] + def unit: F[Unit] + } +~~~~~~~~ + +{lang="text"} +~~~~~~~~ + scala> InvariantMonoidal[Option].point(13) + res: Option[Int] = Some(13) +~~~~~~~~ + + +## Applicative and Monad + +From an API point of view, `Applicative` is `Apply` with a `.pure` method, and +`Monad` extends `Applicative` with `FlatMap`. + +{width=100%} +![](images/cats-applicative.png) + +{lang="text"} +~~~~~~~~ + @typeclass trait Applicative[F[_]] extends Apply[F] with InvariantMonoidal[F] { + def pure[A](a: A): F[A] + } + + @typeclass trait Monad[F[_]] extends Applicative[F] with FlatMap[F] +~~~~~~~~ + +In many ways, `Applicative` and `Monad` are the culmination of everything we've +seen in this chapter. `.pure` (aliased to `.point`) allows us to create effects +or data structures from values. + +Instances of `Applicative` must meet some laws, effectively asserting +that all the methods are consistent: + +- **Identity**: `fa <*> pure(identity) === fa`, (where `fa` is an `F[A]`) i.e. + applying `pure(identity)` does nothing. +- **Homomorphism**: `pure(a) <*> pure(ab) === pure(ab(a))` (where `ab` is an `A => + B`), i.e. applying a `pure` function to a `pure` value is the same as applying + the function to the value and then using `pure` on the result. +- **Interchange**: `pure(a) <*> fab === fab <*> pure(f => f(a))`, (where `fab` is + an `F[A => B]`), i.e. `pure` is a left and right identity +- **Mappy**: `map(fa)(f) === fa <*> pure(f)` + +`Monad` adds additional laws: + +- **Left Identity**: `pure(a).flatMap(f) === f(a)` +- **Right Identity**: `a.flatMap(pure(_)) === a` +- **Associativity**: `fa.flatMap(f).flatMap(g) === fa.flatMap(a => f(a).flatMap(g))` where + `fa` is an `F[A]`, `f` is an `A => F[B]` and `g` is a `B => F[C]`. + +Associativity says that chained `flatMap` calls must agree with nested +`flatMap`. However, it does not mean that we can rearrange the order, +which would be *commutativity*. For example we cannot rearrange + +{lang="text"} +~~~~~~~~ + for { + _ <- machine.start(node1) + _ <- machine.stop(node1) + } yield true +~~~~~~~~ + +as + +{lang="text"} +~~~~~~~~ + for { + _ <- machine.stop(node1) + _ <- machine.start(node1) + } yield true +~~~~~~~~ + +`.start` and `.stop` are **non**-*commutative*, because the intended +effect of starting then stopping a node is different to stopping then +starting it! + +But `.start` is commutative with itself, and `.stop` is commutative with +itself, so we can rewrite + +{lang="text"} +~~~~~~~~ + for { + _ <- machine.start(node1) + _ <- machine.start(node2) + } yield true +~~~~~~~~ + +as + +{lang="text"} +~~~~~~~~ + for { + _ <- machine.start(node2) + _ <- machine.start(node1) + } yield true +~~~~~~~~ + +which are equivalent for our algebra, but not in general. We're making a lot of +assumptions about the Google Container API here, but this is a reasonable choice +to make. + +A practical consequence is that a `Monad` must be *commutative* if its +`applyX` methods can be allowed to run in parallel. We cheated in +Chapter 3 when we ran these effects in parallel + +{lang="text"} +~~~~~~~~ + (d.getBacklog, d.getAgents, m.getManaged, m.getAlive, m.getTime) +~~~~~~~~ + +because we know that they are commutative among themselves. When it comes to +interpreting our application, later in the book, we will have to provide +evidence that these effects are in fact commutative, or an asynchronous +implementation may choose to sequence the operations to be on the safe side. + +The subtleties of how we deal with (re)-ordering of effects, and what +those effects are, deserves a dedicated chapter on Advanced Monads. + + +### Commutativity + +Now that we have discussed commutativity in the context of `Monad` we can +understand the entire suite of `Commutative*` typeclasses in Cats: they add no +additional methods but add the constraint that the order of effects does not +matter. + +{width=100%} +![](images/cats-commute.png) + + diff --git a/manuscript/images/cats-applicative.png b/manuscript/images/cats-applicative.png new file mode 100644 index 0000000000000000000000000000000000000000..0e2e23a3a248d811ad9ba9729ccda2f79ec7b4fd GIT binary patch literal 659611 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