Applied Functional Programming with Scala - Notes

1. Mastering Functions

A function is a mapping from one set, called a domain, to another set, called the codomain. A function associates every element in the domain with exactly one element in the codomain. In Scala, both domain and codomain are types.

val square : Int => Int = x => x * x

square(2) // 4

Higher-Order Functions

A higher-order function is a function that accepts or returns a function.

trait List[A] {
  def filter(f: A => Boolean): List[A]
}

Example: List[A].filter is a higher-order function that accepts the function A => Boolean and returns the value List[A].

Combinators

Function combinators are higher-order functions that accept and return functions.

type Conf[A] = ConfigReader => A

def string(name: String): Conf[String] = _.readString(name)

def both(left: Conf[A], right: Conf[B]): Conf[(A, B)] = c => (left(c), right(c))

Example: both is a combinator that takes two functions and returns a function.

Polymorphic Functions

A polymorphic function is one that is universally quantified over one or more type parameters. Scala has no support for polymorphic functions, but they can be emulated via polymorphic methods on traits. A trait modeling a polymorphic function usually has a single method called apply, so it can be applied with ordinary function application syntax.

case object identity {
  def apply[A](value: A): A = value
}
identity(3)   // 3
identity("3") // "3"

Example: This emulates a polymorphic function called id, which accepts one type parameter A, and a value of type A, and returns that same value.

2. Mastering Types

Type Theory 101

A type is a compile-time description of a set of values. Int is the set of all integers between -2147483648 and 2147483647. Values have types, which is to say, they may be a member of the set of values they represent.

2 : Int

Example: 2 is a member of the set of all Int. Equivalently, 2 has type Int.

Algebraic Data Types

An algebraic data type is a type formed by composing product and sum types.

Product Type

Product types are defined by a Cartesian cross product on 2 or more types.

type Point2D = (Int, Int)

Example: A two-dimensional point is a product of a number and a number; each value has both an x-coordinate and a y-coordinate.

Case Classes

In Scala, case classes are the idiomatic representation of product types. The terms of a case class are identified by name.

case class Person(name: String, age: Int)

Example: A person has both a string (name) and an integer (age).

Sum Types

Sum types are defined by a disjoint union on 2 or more types.

type RequestResult = Either[Error, HttpResponse]

Example: An request result is a sum of Error and HttpResponse; each value is either an error or an HTTP response (but not both).

Sealed Traits

In Scala, sealed traits are the idiomatic representation of sum types (pre-Dotty). The terms of a sum type are identified by constructor / deconstructor (and, incidentally, by subtype).

sealed trait AddressType
case object Home     extends AddressType
case object Business extends AddressType

Example: An AddressType is either a Home or a Business, but not both.

Subtyping

Type A is a subtype of B if $A \subseteq B$. In Scala, $A$ must extend $B$. The compiler allows subtypes to be used wherever a supertype is required.

sealed trait Shape {
  def width: Int
  def height: Int
}
case class Rectangle(corner�: Point2D, width: Int, height: Int) extends Shape

Example: A Rectangle is a subtype of Shape, because every Rectangle is a shape (but not every Shape is a Rectangle).

Supertyping

Type A is a supertype of B if $B \subseteq A$. In Scala, $B$ must extend $A$. The compiler allows supertypes to be used wherever a subtype is provided.

In the previous example, Shape is a supertype of Rectangle, and a variable of type Shape may be assigned to a value of type Rectangle.

Universals

A universally-quantified type defines a category (or kind) of types that are all parameterized by some arbitrary type. In Scala, type constructors (such as some traits) and methods may be universally quantified, although Scala methods do not have a type (they appear in types such as traits).

Type Constructors

A type constructor is a universally quantified type, which can be used to construct types.

sealed trait List[A]
case class Nil[A]() extends List[A]
case class Cons[A](head: A, tail: List[A]) extends List[A]

Example: List is type constructor, which defines a family of List-like types, including List[Boolean] (the type of lists of booleans). List is said to be universally quantified over its type variable A.

Methods

Polymorphic methods are also referred to as universally-quantified functions, because their domain is universally quantified over all types.

case object identity {
  def apply[A](value: A): A = value
}
identity(3)   // 3
identity("3") // "3"

Higher-Kinded Types

Type-Level Functions

Type constructors can be thought of as type-level functions, which accept types and return types. This interpretation is useful when reasoning about higher-kinded types.

Example: List is a type-level function that accepts one type A (the type of its elements), and returns another type List[A]. If you pass Boolean to List, you get back List[Boolean], the type of lists of boolean values.

Kinds

Kinds can be thought of as the type of types.

* — The kind of types (the set of all types).
* => * — The kind of type-level functions that accept 1 type (the set of all type-level functions that accept 1 type). The type constructor List has kind * => *, represented as _[_] in Scala.
[*, *] => * — The kind of type-level functions that accept 2 types (the set of all type-level functions that accept 2 types). The type constructor Either has kind [*, *] => *, represented as _[_, _] in Scala.

Note: Compare with the types of functions: A => B, (A, B) => C, (A, B, C) => D.

Higher-Order Kinds

Just like functions can be "higher-order", type constructors can be higher-order, too. Scala uses underscores to encode higher-order type constructors. The declaration trait CollectionModule[Collection[_]] specifies that CollectionModule's type constructor requires a type constructor of kind * -> * (such as List).

(* => *) => * — The kind of type constructors that accept a type constructor of kind * => *. For example, trait Functor[F[_]] { ... } and trait Monad[F[_]] { ... }.

Note: All higher-order kinds that return a type (*) are valid kinds.

Existentials

An existentially quantified type defines a type that depends on some definite but unknowable type. Existential types are useful for hiding type information that is not globally relevant.

trait ListMap[A] {
  type B
  val list : List[B]
  val mapf : B => A
  
  def run : List[A] = list.map(mapf)
}

Example: The type ListMap[A]#B is some definite type, but there is no way to know what that type is — it could be anything.

Skolemization

Every existential type can be encoded as a universal type. This process is called skolemization.

case class ListMap[B, A](list: List[B], mapf: B => A)

trait ListMapInspector[A, Z] {
  def apply[B](value: ListMap[B, A]): Z
}

case class AnyListMap[A] {
  def apply[Z](value: ListMapInspector[A, Z]): Z
}

Example: Instead of using ListMap directly, we use AnyListMap, which allows us to inspect a ListMap but only if we can handle any type parameter for B.

Type Lambdas

Functions may be partially applied with the underscore operator; e.g. zip(a, _). A type lambda is a way to partially apply a higher-kinded type, which yields another type constructor with fewer type parameters.

Type lambdas are to type constructors as lambdas are to functions. Type constructors and functions are declarations, while lambdas are expressions (either value expressions, or type expressions).

({type λ[α] = Either[String, α]})#λ

Example: This is the Either type, partially applied with a String as the first type parameter.

Note: In many (but not all) cases, you can use type aliases instead of type lambdas (e.g. type EitherString[A] = Either[String, A]).

Kind Projector

Kind Projector is a common compiler plugin for Scala that provides easier syntax to create type lambdas. For example, the type lambda ({type λ[α] = Either[String, α]})#λ can be represented with the syntax Either[String, ?]. Other syntax can be used to create more complex type lambdas.

https://github.com/non/kind-projector

3. Mastering Type Classes

A type class is a bundle of types and operations defined on them. Most type classes have laws that implementations are required to satisfy.

trait ShowRead[A] {
  def show(v: A): String
  def read(v: String): Either[String, A]
}
object ShowRead {
  def apply[A](implicit v: ShowRead[A]): ShowRead[A] = v
}

Example: The ShowRead[A] type class defines a way of "showing" a type A by rendering it to a string, and reading it by parsing it from a string (or producing an error message).

Right Identity
```
read(show(v)) == Right(v)
```

Left Partial Identity

read(v).map(show(_)).fold(_ => true, _ == v)

Instances

A type class instance, or simply instance, is an implementation of a type class for a given set of types. Such instances are usually made implicit so the compiler can thread them through functions that require them.

implicit val ShowReadString: ShowRead[String] = new ShowRead[String] {
  def show(v: String): String = v
  def read(v: String): Either[String, String] = Right(v)
}

ShowRead[String].show("foo") // foo

Syntax

Convenient syntax, sometimes called extension methods, can be added to types to make it easier to use type classes.

implicit class ShowOps[A: ShowRead](self: A) {
  def show: String = ShowRead[A].show(self)
}
implicit class ReadOps(self: String) {
  def read[A: ShowRead]: Either[String, A] = ShowRead[A].read(self)
}

true.show.read[Boolean] // Right(true)

4. Mastering Functional Patterns

Option, Either, Validation

These types are commonly used to describe optionality and partiality.

sealed trait Maybe[A]
final case class Just [A](value: A) extends Maybe[A]
final case class Empty[A]()         extends Maybe[A]

sealed trait \/[A, B]
final case class -\/ [A, B](value: A) extends \/[A, B]
final case class  \/-[B, B](value: B) extends \/[A, B]

type Either[A, B] = A \/ B

sealed trait Validation[A, B]
final case class Failure[A, B](value: A) extends Validation[A, B]
final case class Success[A, B](value: B) extends Validation[A, B]

Semigroup, Monoid

Semigroups allows combining two things of the same type into another thing of the same type. For example, addition forms a semigroup over integers. Monoids add the additional property of having an "zero" element, which you can append to a value without changing the value.

trait Semigroup[A] {
  def append(a1: A, a2: A): A
}
trait Monoid[A] extends Semigroup[A] {
  def zero: A
}

Associativity

append(a1, append(a2, a3)) == append(append(a1, a2), a3)

Identity
```
append(a, zero) == a
```

Functors

A functor F[_] is a type constructor of kind * -> *. In the most general case, an F[A] represents a recipe that may halt, run forever, or produce 0 or more A's.

trait Functor[F[_]] {
  def map[A, B](fa: F[A])(ab: A => B): F[B]
}

Note: Technically, this is a covariant endofunctor, and there are many other types of functors, including invariant and contravariant.

Identity
```
map(fa)(identity) == fa
```

Composition

map(map(fa)(ab))(bc) == map(fa)(ab.andThen(bc))

Example: List is a functor, and List[Int] is a trivial description of a computation producing some number of Int's.

Natural Transformations

A polymorphic function that maps from one functor F[_] to another functor G[_] is called a natural transformation, and is typically denoted using F ~> G. These functions are extremely important in higher-order functional programming.

trait NaturalTransformation[-F[_], +G[_]] {
  def apply[A](fa: F[A]): G[A]
}
type ~> [-F[_], +G[_]] = NaturalTransformation[F, G]

Note: If you have F ~> G, and G ~> H, you can compose them to get F ~> H. In addition, for any F, there is an identity F ~> F.

Functor Composition

Two functors can be composed together in a variety of ways to yield another functor.

Natural

The natural composition of two functors occurs when one is nested inside another.

case class Composite[F[_], G[_], A](run: F[G[A]])

Product

The product of two functors is a functor.

case class Product[F[_], G[_], A](run: (F[A], G[A]))

Coproduct

The sum (or coproduct) of two functors is a functor.

case class Coproduct[F[_], G[_], A](run: Either[F[A], G[A]])

Lifting

Often, for some value X, F[X] is referred to as the "lifted" version of X, because it is the same value, but placed "inside" of some functor F. For example, you can lift the function x => x * x inside List by writing List(x => x * x).

Apply

Some functors implement Apply, which provides a way of applying a lifted function F[A => B] to some lifted value F[A] to yield F[B].

trait Apply[F[_]] extends Functor[F] {
  def ap[A, B](fa: F[A])(fab: F[A => B]): F[B]
}

Associative Composition

ap(ap(fa)(fab))(fbc) ==
   ap(fa)(ap(fab)(map(fbc)(_.compose(_).curry))

Applicative

Some functors that implement Apply also implement Applicative, which provides the ability to lift any value into the functor.

trait Applicative[F[_]] extends Apply[F] {
  def point[A](a: A): F[A]
}

Identity
```
ap(fa)(point(_)) == fa
```
Homomorphism
```
ap(point(a))(point(ab)) == point(ab(a))
```

Interchange

ap(point(a))(fab) == ap(fab)(point(_.apply(a)))

Derived Map
```
map(fa)(ab) == ap(fa)(point(ab))
```

Bind

Some functors that implement Apply also implement Bind, which adds the ability to extend a recipe F[A] with a second recipe that depends on the result of A (A => F[B]), and collapse the result into a single recipe F[B].

trait Bind[F[_]] extends Apply[F] {
  def bind[A, B](fa: F[A])(afb: A => F[B]): F[B]
}

Associative Bind

bind(bind(fa)(afb))(bfc) == bind(fa)((a) => bind(afb(a))(bfc))

Derived Ap
```
ap(fa)(fab) == bind(fab)(map(fa)(_))
```

Monad

Some functors that implement Applicative and Bind are Monads.

trait Monad[F[_]] extends Applicative[F] with Bind[F]

Right Identity
```
bind(fa)(point(_)) == fa
```
Left Identity
```
bind(point(a))(afb) == afb(a)
```

Other Types of Functors

Invariant

trait InvariantFunctor[F[_]] {
  def xmap[A, B](ma: F[A], f: A => B, g: B => A): F[B]
}

Contravariant

trait Contravariant[F[_]] extends InvariantFunctor[F] {
  def contramap[A, B](r: F[A])(f: B => A): F[B]
}

Bifunctor

trait Bifunctor[F[_, _]] {
  def bimap[A, B, C, D](fab: F[A, B])(f: A => C, g: B => D): F[C, D]
}

Profunctor

trait Profunctor[P[_, _]] {
  def dimap[A, B, C, D](fab: P[A, B])(f: C => A)(g: B => D): P[C, D]
}

Functional Collections

Structures F[_] that are Foldable are "containers", whose elements can be inspected.

Foldable

trait Foldable[F[_]] {
  def foldMap[A,B](fa: F[A])(f: A => B)(implicit F: Monoid[B]): B
  def foldRight[A, B](fa: F[A], z: => B)(f: (A, => B) => B): B
}

Consistent Left Fold

foldMap(fa)(Vector(_)) == foldLeft(fa, Vector.empty[A])(_ :+ _)

Consistent Right Fold

foldMap(fa)(Vector(_)) == foldRight(fa, Vector.empty[A])(_ +: _)

Traversable

Containers F[_] that define Traverse can be traversed in a way that rebuilds the original structure. The typical usage is sequencing, which is a way of type-flipping while preserving structure, transforming F[G[A]] into G[F[A]] for some F[_] which has a Traverse.

trait Traverse[F[_]] extends Functor[F] with Foldable[F] {
  final def sequence[G[_]:Applicative,A](fga: F[G[A]]): G[F[A]] = ...
}

Notes: Laws include identity, sequential fusion, purity, naturality, and parallel fusion.

5. Mastering Data

Data & Codata

Data is a finite store of information, and may be non-recursive or recursive. Recursive data is sometimes called inductive data.

Codata is a description of a process for producing information, and may be non-corecursive or corecursive. Corecursive data is sometimes called coinductive data.

Finite data structures can be modeled with data or codata. Infinite data structures are modeled with codata.

Church Encoding

The lambda calculus, the basis of functional programming, is sufficient to represent all possible data structures in an encoding known as Church encoding. Church encoding can sometimes be used to speed up operations, and other times to give you a better understanding of the fundamental properties of some data structure. Church encoding must be faked in Scala because Scala does not have true polymorphic functions (only polymorphic methods).

Natural Numbers

trait Natural { self =>
  def fold[Z](zero: => Z, succ: Z => Z): Z
  
  def succ: Natural = new Natural { 
    def fold[Z](zero: => Z, succ: Z => Z): Z = succ(self.fold(zero, succ))
  }
  def + (that: Natural): Natural = new Natural {
    def fold[Z](zero: => Z, succ: Z => Z): Z = that.fold[Natural](self, _.succ).fold[Z](zero, succ)
  }
  def * (that: Natural): Natural = new Natural {
    def fold[Z](zero: => Z, succ: Z => Z): Z = 
      that.fold[Natural](Natural.zero, _ + self).fold[Z](zero, succ)
  }
  def isZero: Boolean = fold[Boolean](true, _ => false)
  def toInt: Int = fold[Int](0, _ + 1)
  override def toString = toInt.toString
}
object Natural {
  val zero = new Natural { def fold[Z](zero: => Z, succ: Z => Z): Z = zero }
  def of(v: Int): Natural = if (v == 0) zero else of(v - 1).succ
}

Optics

Optics provide highly-composable, purely-functional machinery for focusing in on a part of a substructure A inside a superstructure S and performing local transformations to yield a new substructure B inside a new superstructure T. Different types of optics compose (for example, two lenses can be composed to yield another lens, but a lens and a prism yield an optional).

Lens

Lenses provide a way to focus on a single term inside a product.

abstract class PLens[S, T, A, B] extends Serializable { self =>
  def get(s: S): A
  def set(b: B): S => T
  def modifyF[F[_]: Functor](f: A => F[B])(s: S): F[T]
  def modify(f: A => B): S => T
}
type Lens[S, A] = PLens[S, S, A, A]

Prisms

Prisms provide a way to focus on a single term inside a sum.

abstract class PPrism[S, T, A, B] extends Serializable { self =>
  def getOrModify(s: S): T \/ A
  def reverseGet(b: B): T
  def getOption(s: S): Option[A]
}
type Prism[S, A] = PPrism[S, S, A, A]

Traversal

Traversals provide a way to focus on zero or more elements.

abstract class PTraversal[S, T, A, B] extends Serializable { self =>
  def modifyF[F[_]: Applicative](f: A => F[B])(s: S): F[T]
}
type Traversal[S, A] = PTraversal[S, S, A, A]

Fold

Folds provide a way to fold over zero or more values embedded in a larger structure.

abstract class Fold[S, A] extends Serializable { self =>
  def foldMap[M: Monoid](f: A => M)(s: S): M
}

Getter

Getters provide a way to get a part of a larger structure.

abstract class Getter[S, A] extends Serializable { self =>
  def get(s: S): A
}

Setter

Setters provide a way to modify and set a part of a larger structure.

abstract class PSetter[S, T, A, B] extends Serializable { self =>
  def modify(f: A => B): S => T
  def set(b: B): S => T
}
type Setter[S, A] = PSetter[S, S, A, A]

Iso

Isos provide an isomorphism between an A in S and a B in T.

abstract class PIso[S, T, A, B] extends Serializable { self =>
  def get(s: S): A
  def reverseGet(b: B): T
  def reverse: PIso[B, A, T, S]
}
type Iso[S, A] = PIso[S, S, A, A]

Optional

Optionals provide either a way to get an A in S, or to get a T, and a way to set a B in S to get a T.

abstract class POptional[S, T, A, B] extends Serializable { self =>
  def getOrModify(s: S): T \/ A
  def set(b: B): S => T
  def getOption(s: S): Option[A]
  def modifyF[F[_]: Applicative](f: A => F[B])(s: S): F[T]
  def modify(f: A => B): S => T
}
type Optional[S, A] = POptional[S, S, A, A]

Optic Composition Table

Almost all optics compose with the other optics. The composition of one type and itself results in the same type of optic.

	Getter	Iso	Lens	Optional	Prism	Setter	Traversal
Getter	Getter	Getter	Getter	Fold	Fold	-	Fold
Iso	Getter	Iso	Lens	Optional	Prism	Setter	Traversal
Lens	Getter	Lens	Lens	Optional	Optional	Setter	Traversal
Optional	Fold	Optional	Optional	Optional	Optional	Setter	Traversal
Prism	Fold	Prism	Optional	Optional	Prism	Setter	Traversal
Setter	-	Setter	Setter	Setter	Setter	Setter	Setter
Traversal	Fold	Traversal	Traversal	Traversal	Traversal	Setter	Traversal

Recursion Schemes

Recursion schemes are the ways that recursive and corecursive structures may be traversed, for purposes of altering them or inspecting them. For example, you can "fold" over a list to yield a new value. Using fixed-point data types, you can leverage generic recursion schemes that can work on any type of recursive or corecursive data structure.

Fixed-Point Types

Fixed-point types factor out the recursion from recursive data structures. They operate on functors F[_] and extend them with recursion.

Fix — Inductive or coinductive recursive structures. The simplest fixed-point type.
```
final case class Fix[F[_]](unFix: F[Fix[F]])
```

Mu — Inductive (finite) recursive structures. For example, a JSON data structure.

type Algebra[F[_], A] = F[A] => A

final case class Mu[F[_]](unMu: Algebra[F, ?] ~> Id)

Nu — Coinductive (possibly infinite) recursive structures. For example, a stream of values.

sealed abstract class Nu[F[_]] {
  type A
  val a: A
  val unNu: A => F[A]
}

Cofree[F[_], A] — A tree-like recursive structure whose nodes are annotated by values of type A.
```
final case class Cofree[F[_], A](head: A, tail: F[Cofree[F, A]])
```

Free[F[_], A] — A tree-like recursive structure whose leaves may be values of type A.

sealed trait Free[F[_], A]
final case class Pure[F[_], A](value: A) extends Free[F, A]
final case class Wrap[F[_], A)(unwrap: F[Free[F, A]]) extends Free[F, A]

Type Classes

Recursive and corecursive type classes allow you to abstract over the fixed-point type used to encode recursion or corecursion.

Recursive — Mu and Nu are recursive.

trait Recursive[T[_[_]]] {
   def project[F[_]: Functor](t: T[F]): F[T[F]]
 }

Corecursive — Mu and Nu are corecursive.

trait Corecursive[T[_[_]]] {
  def embed[F[_]: Functor](t: F[T[F]]): T[F]
}

Algebras

Algebras are single-step folds from a structure F[_], while coalgebras are single-step unfolds to a structure F[_]. Both folds and unfolds may utilize some monadic effect M[_], which may be the Identity monad.

Algebra

type Algebra[F[_], A] = F[A] => A
type AlgebraM[M[_], F[_], A] = F[A] => M[A]

Generalized Algebra

type GAlgebra[W[_], F[_], A] = F[W[A]] => A
type GAlgebraM[W[_], M[_], F[_], A] = F[W[A]] => M[A]

Elgot Algebra

type ElgotAlgebra[W[_], F[_], A] = W[F[A]] => A
type ElgotAlgebraM[W[_], M[_], F[_], A] = W[F[A]] => M[A]

Coalgebra

type Coalgebra[F[_], A] = A => F[A]
type CoalgebraM[M[_], F[_], A] = A => M[F[A]]

Generalized Coalgebra

type GCoalgebra[N[_], F[_], A] = A => F[N[A]]
type GCoalgebraM[N[_], M[_], F[_], A] = A => M[F[N[A]]]

Elgot Coalgebra

type ElgotCoalgebraM[E[_], M[_], F[_], A] = A => M[E[F[A]]]
type ElgotCoalgebra[E[_], F[_], A] = A => E[F[A]]

Folds

Folds tear down a Recursive structure. These are some common schemes.

Catamorphism

    def cata[F[_]: Functor, A](t: T[F])(f: Algebra[F, A]): A
    ```

Prepromorphism

def prepro[F[_]: Functor, A](t: T[F])
  (e: F ~> F, f: Algebra[F, A])(implicit T: Corecursive[T]): A

Paramorphism

def para[F[_]: Functor, A](t: T[F])(f: GAlgebra[(T[F], ?), F, A]): A

Zygomorphism

def zygo[F[_]: Functor, A, B](t: T[F])(f: Algebra[F, B], g: GAlgebra[(B, ?), F, A]): A

Histomorphism

def histo[F[_]: Functor, A](t: T[F])(f: GAlgebra[Cofree[F, ?], F, A]): A

Unfolds

Unfolds build up a Corecursive structure. These are some common schemes.

Anamorphism

def ana[F[_]: Functor, A](a: A)(f: Coalgebra[F, A]): T[F]

Postpromorphism

def postpro[F[_]: Functor, A](a: A)
  (e: F ~> F, g: Coalgebra[F, A])(implicit T: Recursive[T]): T[F]

Apomorphism

def apo[F[_]: Functor, A](a: A)(f: GCoalgebra[T[F] \/ ?, F, A]): T[F]

Futumorphism

def futu[F[_]: Functor, A](a: A)(f: GCoalgebra[Free[F, ?], F, A]): T[F]

Refolds

Refolds tear down and build up a structure.

Hylomorphism

def hylo[F[_]: Functor, A, B](a: A)(f: Algebra[F, B], g: Coalgebra[F, A])): B

Dynamorphism

def dyna[F[_]: Functor, A, B](a: A)(φ: F[Cofree[F, B]] => B, ψ: Coalgebra[F, A]): B

Codynamorphism

def codyna[F[_]: Functor, A, B](a: A)(φ: Algebra[F, B], ψ: GCoalgebra[Free[F, ?], F, A]): B

Chronomorphism

def chrono[F[_]: Functor, A, B](a: A)
  (g: F[Cofree[F, B]] => B, f: A => F[Free[F, A]]): B

Elgot Algebra

def elgot[F[_]: Functor, A, B](a: A)(φ: Algebra[F, B], ψ: CoalgebraM[B \/ ?, F, A]): B

coElgot Algebra

def coelgot[F[_]: Functor, A, B](a: A)(φ: ElgotAlgebra[(A, ?), F, B], ψ: Coalgebra[F, A]): B

Transforms

Transforms are available on any type which acts as a functor transformer.

Trans Catamorphism

def transCataT[F[_]: Functor](t: T[F])(f: T[F] => T[F]): T[F]

Trans Anamorphism

def transAnaT[F[_]: Functor](t: T[F])(f: T[F] => T[F]): T[F]

Trans Apomorphism

def transApoT[F[_]: Functor](t: T[F])(f: T[F] => T[F] \/ T[F]): T[F]

Top-Down Catamorphism

def topDownCata[F[_]: Functor, A](t: T[F], a: A)(f: (A, T[F]) => (A, T[F])): T[F]

6. Mastering Effects

IO

An IO abstraction converts effectful computations into pure values. This lets you write purely functional programs that are useful. The values have to be eventually executed, but only in the main function of the application.

class IO[A](val unsafePerformIO: () => A) {
  def map[B](ab: A => B): IO[B] =
    new IO(() => ab(unsafePerformIO()))
  def flatMap[B](afb: A => IO[B]): IO[B] =
    new IO(() => afb(unsafePerformIO()).unsafePerformIO())
  def attempt: IO[Either[Throwable, A]] = new IO(() => {
    try Right(unsafePerformIO())
    catch {
      case t : Throwable => Left(t)
    }
  })
}
object IO {
  def apply[A](a: => A): IO[A] = new IO(() => a)

  def fail[A](t: Throwable): IO[A] = new IO(() => throw t)
}

State

A State monad allows retrieving and updating some arbitrary state S. Conceptually, the monad is a function from the old state, to both the new state and a value produced by the function.

case class State[S, A](runState: S => (S, A)) {
  def evalState(s: S): A = runState(s)._2
  def execState(s: S): S = runState(s)._1
  def map[B](ab: A => B): State[S, B] = 
    State(s => runState(s) match {
      case (s, a) => (s, ab(a))
    })
  def flatMap[B](afb: A => State[S, B]): State[S, B] =
    State(s => runState(s) match {
      case (s, a) => afb(a).runState(s)
    })
}
object State {
  def point[S, A](a: A): State[S, A] = State(s => (s, a))
  def get[S]: State[S, S] = State(s => (s, s))
  def put[S](s: S): State[S, Unit] = State(_ => (s, ()))
  def modify[S](ss: S => S): State[S, Unit] = 
    for { s <- get[S]; _ <- put(ss(s)) } yield ()
}

Reader

A Reader monad allows reading from some environment S.

case class Reader[S, A](runReader: S => A) {
  def map[B](ab: A => B): Reader[S, B] = Reader(s => ab(runReader(s)))
  def flatMap[B](afb: A => Reader[S, B]): Reader[S, B] =
    Reader(s => afb(runReader(s)).runReader(s))
}
object Reader {
  def point[S, A](a: => A): Reader[S, A] = Reader(_ => a)
  def read[S]: Reader[S, S] = Reader(s => s)
}

Combining Effects

In general, monadic effects do not combine. However, using type classes to abstract over effects enables composition of the requirements for an effect.

Classes

trait MonadIO[F[_]] {
  def captureEffect[A](a: A): F[A]
  
  def tryIO[A](fa: F[A]): F[Either[Throwable, A]]
}
trait MonadReader[F[_], S] {
  def ask: F[S]
  def local[A](f: S => S)(fa: F[A]): F[A]
}
trait MonadState[F[_], S] extends Monad[F] {
  def init: F[S]
  def get: F[S]
  def put(s: S): F[Unit]
}

def myFunction[M[_]: MonadIO: MonadReader[?, S]] = ...

Transformers

Monad transformers are abstractions that provide an effect (such as getting and setting state) that is layered on top of another, more powerful effect.

trait MonadTrans[F[_[_], _]] {
  def liftM[G[_] : Monad, A](a: G[A]): F[G, A]

  /** The [[scalaz.Monad]] implied by this transformer. */
  implicit def apply[G[_] : Monad]: Monad[F[G, ?]]
}

Identity

liftM(G.point(a)) == Monad[F[G, ?]].point(a)

Composition

liftM(Monad[G].bind(ga)(f)) == 
  Monad[F[G, ?]].bind(liftM(ga))(a => liftM(f(a)))

Common Transformers

StateT[F[_], S, A] — Adds stateful operations atop F[_].
StreamT[F[_], A] — Adds non-determinism atop F[_].
ReaderT[F[_], R, A] — Adds reading "config" atop F[_].
WriterT[F[_], W, A] — Adds adds "logging" of monoidal W atop F[_].
EitherT[F[_], E, A] — Adds "errors" atop F[_].
OptionT[F[_], A] — Adds optionality atop F[_].

7. Mastering Functional Architecture

Functional programs are comprised from smaller pieces. Small effects compose together to form larger effects. Small pieces of state compose together to form larger state. Small functions compose together to form "larger" functions.

Code that does not interact with external systems is easy to decompose. Code that interacts with external systems — so-called effects — can be decomposed in a way that depends on the application's overall architecture for effects.

Onion Architecture

The onion architecture for FP involves layering semantic domains of the application. On the inside, the application speaks at the level of the domain model; at the outside, the application speaks the language of external effects.

Interpreters translate from an inner layer to an outer layer.

The onion architecture can be implemented with monad classes and transformers, or with free monads and similar structures of reified computation.

Reified Computation

The purest and most powerful technique in representing effects involves reifying the structure and semantics of effectful computation. Standard abstractions such as the free monad and free applicative are useful.

Free Monad

A free monad provides a way to record and playback a sequential tree of operations described by some F[_]. Free monads provide a powerful way to describe all manner of effects using ordinary data structures that can be introspected, reflected, played back, and altered at runtime.

sealed trait Free[F[_], A] { self =>
  final def map[B](ab: A => B): Free[F, B] = Free.flatMap(self, ab andThen (Free.point[F, B](_)))
  final def flatMap[B](afb: A => Free[F, B]): Free[F, B] = Free.flatMap(self, afb)

  final def interpret[G[_]: Monad](fg: F ~> G): G[A] = self match {
    case Free.Point(a0) => a0().point[G]
    case Free.Effect(fa) => fg(fa)
    case fm : Free.FlatMap[F, A] =>
    	val ga0 = fm.fa.interpret[G](fg)
    	ga0.flatMap(a0 => fm.afb(a0).interpret[G](fg))
  }
}
object Free {
  def point[F[_], A](a: => A): Free[F, A] = Point[F, A](() => a)
  def liftF[F[_], A](fa: F[A]): Free[F, A] = Effect[F, A](fa)
  private final case class Point[F[_], A](a0: () => A) extends Free[F, A]
  private final case class Effect[F[_], A](fa: F[A]) extends Free[F, A]
  private sealed trait FlatMap[F[_], B] extends Free[F, B] {
    type A; def fa: Free[F, A]; def afb: A => Free[F, B]
  }
  private def flatMap[F[_], A0, B](fa0: Free[F, A0], afb0: A0 => Free[F, B]): Free[F, B] = new FlatMap[F, B] {
    type A = A0; def fa = fa0; def afb = afb0
  }
}

Free Monad Example

sealed trait ConsoleF[A]
final case class WriteLine(line: String) extends ConsoleF[Unit]
final case object ReadLine extends ConsoleF[String]

def writeLine(line: String): Free[ConsoleF, Unit] = WriteLine[ConsoleF, String](line)
def readLine: Free[ConsoleF, String] = Free.liftF[ConsoleF, String](ReadLine)

def example: Free[ConsoleF, Unit] = for {
  _    <- writeLine("What is your name?")
  name <- readLine
  _    <- writeLine("Hello, " + name + "!")
} yield ()

Deferring Effects

Pervasive use of type classes allows polymorphism in the implementation for an effect and supports painless large-scale refactoring. This pattern is associated with monad transformers (MTL) but it can be applied to free computations as well.

trait Logging[F[_]] {
  def log(v: String): F[Unit]
}
object Logging {
  def apply[F[_]](implicit f: Logging[F]): Logging[F] = f
  implicit def LoggingFree[F[_]: Logging]: Logging[Free[F, ?]] =
    new Logging[Free[F, ?]] {
      def log(v: String): Free[F, Unit] = Free.liftF(Logging[F].log(v))
    }
  }
}

def myFunctionThatLogs[F[_]: Logging: Monad] = ...

Free Applicative

A free applicative provides a way to record and playback a parallel tree of operations described by some F[_]. It is not as expressive as a free monad, but it can be introspected all the way to the leaves, without runtime interpretation.

sealed trait FreeAp[F[_], A] { self =>
  final def map[B](ab: A => B): FreeAp[F, B] = FreeAp.map(self, ab)
  final def ap[B](fab: FreeAp[F, A => B]): FreeAp[F, B] = FreeAp.ap(self, fab)

  final def interpret[G[_]: Applicative](fg: F ~> G): G[A] = self match {
    case FreeAp.Point(a0) => a0().point[G]
    case FreeAp.Effect(fa) => fg(fa)
    case fm : FreeAp.Map[F, A] => fm.fa.interpret[G](fg).map(fm.ab)
    case fap : FreeAp.Ap[F, A] =>
    	val ga0 = fap.fa.interpret[G](fg)
      ga0 <*> fap.fab.interpret[G](fg)
  }
}
object FreeAp {
  def point[F[_], A](a: => A): FreeAp[F, A] = Point[F, A](() => a)
  def liftF[F[_], A](fa: F[A]): FreeAp[F, A] = Effect[F, A](fa)
  private final case class Point[F[_], A](a0: () => A) extends FreeAp[F, A]
  private final case class Effect[F[_], A](fa: F[A]) extends FreeAp[F, A]
  private sealed trait Map[F[_], B] extends FreeAp[F, B] {
    type A; def fa: FreeAp[F, A]; def ab: A => B
  }
  private sealed trait Ap[F[_], B] extends FreeAp[F, B] {
    type A; def fa: FreeAp[F, A]; def fab: FreeAp[F, A => B]
  }
  private def ap[F[_], A0, B](fa0: FreeAp[F, A0], fa0b: FreeAp[F, A0 => B]): FreeAp[F, B] = new Ap[F, B] {
    type A = A0; def fa = fa0; def fab = fa0b
  }
  private def map[F[_], A0, B](fa0: FreeAp[F, A0], a0b: A0 => B): FreeAp[F, B] = new Map[F, B] {
    type A = A0; def fa = fa0; def ab = a0b
  }
}

Interpreters

Interpreters for reified computations often have the structure F ~> T[G, ?], where T describes the computational context (sequential or parallel, for example), F describes the source operations, and G describes the target operations.

type Interpreter[T[_[_], _], F[_], G[_]] = F ~> T[G, ?]

Horizontal Composition

Interpreters compose horizontally. You can compose Interpreter[T, F, G] and Interpreter[T, G, H] into Interpreter[T, F, H]. Stated differently, if you can interpret F to G in some context T, and G to H (in the same context), then you can interpret F to H.
Vertical Composition

Interpreters compose vertically. You can compose Interpreter[T, F, G] and Interpreter[T, F, H] into Interpreter[T, F, Product[G, H, ?]].
Monoidal Composition

Interpreters compose monoidally as per the operations supported by T[_[_], _]. For example, if T supports a notion of failure, then two T[F, A] operations can be appended together, such that the first computation is used if it succeeds, otherwise the second. Racing is another example of monoidal composition possible with parallel computations.

Computation Contexts

Computation contexts T[_[_], _] are higher-order abstractions that reify computation. Free is a sequential context for computation, and a higher-order monad. FreeAp is a parallel context for computation, and a higher-order applicative.

Applications

Reified computation allows for dynamic introspection and transformation of a program's structure, including weaving effects (by composing interpreters), optimizing program structure (for parallel computational contexts), and even purely-functional mocking.

Notices

Various code snippets and laws derive from Scalaz, Monocle, and Matryoshka. All rights are reserved via their respective copyright holders.

lupuszr/Applied-FP-with-Scala.md

Applied Functional Programming with Scala - Notes

1. Mastering Functions

Higher-Order Functions

Combinators

Polymorphic Functions

2. Mastering Types

Type Theory 101

Algebraic Data Types

Product Type

Case Classes

Sum Types

Sealed Traits

Subtyping

Supertyping

Universals

Type Constructors

Methods

Higher-Kinded Types

Type-Level Functions

Kinds

Higher-Order Kinds

Existentials

Skolemization

Type Lambdas

Kind Projector

3. Mastering Type Classes

Instances

Syntax

4. Mastering Functional Patterns

Option, Either, Validation

Semigroup, Monoid

Functors

Natural Transformations

Functor Composition

Natural

Product

Coproduct

Lifting

Apply

Applicative

Bind

Monad

Other Types of Functors

Functional Collections

Foldable

Traversable

5. Mastering Data

Data & Codata

Church Encoding

Natural Numbers

Optics

Lens

Prisms

Traversal

Fold

Getter

Setter

Iso

Optional

Optic Composition Table

Recursion Schemes

Fixed-Point Types

Type Classes

Algebras

Folds

Unfolds

Refolds

Transforms

6. Mastering Effects

IO

State

Reader

Combining Effects

Classes

Transformers

Common Transformers

7. Mastering Functional Architecture

Onion Architecture

Reified Computation