algebras in programming - harrylaou · •a calculus for lazy functional programming based on...

Algebras in ProgrammingWhat do we mean by algerbra

Harry Laoulakos@harrylaou

• Senior Scala Consultant

• Functional Programming

• Check Cost vs Benefit

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For absolute beginners

• not necessary to know scala

• stay focussed on semantics

• build a vocabulary to communicate with fellow programmers

• "touches" mathematical concepts, but doesn't go deep

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Slideshttps://www.harrylaou.com/slides/AlgebrasInProgramming.pdf

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Areas of Mathematics• Elementary algebra

• Abstract algebra

• Linear algebra

• Boolean algebra

• Commutative algebra

• Homological algebra

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more areas• Universal algebra

• Algebraic number theory

• Algebraic geometry

• Algebraic combinatorics

• Relational algebra

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Algebras can also be mathematical structuresFor example in category theory

• F-algebra

• F-coalgebra

• T-algebra

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When we are talking about algebra, it is not necessarily one or even similar things.

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In Programming

• ADTs: algebraic data types

• GADTs: Generalized algebraic data types

• Tagless Final Algebra

• Free Μonads Algebra

• Recursion Schemes Algebra

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ADTAlgebraic data type

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ADTis a composite type of - Product Types (or Tagged Unions)- Sum Types (or Co-Product or Unions)

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Product Type

• case class

• tuple

• HList

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Product Type

case class User(id:Long, username:String, ...)

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Product Type

case class User(id:Long, username:String, ...)

Think AND

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Sum Type

• Either

• sealed trait

• Coproduct

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Sum Type

sealed trait Animalcase class Cat(name:String) extends Animalcase class Dog(name: String) extends Animal

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Sum Type

sealed trait Animalcase class Cat(name:String) extends Animalcase class Dog(name: String) extends Animal

Think OR

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Note

It is a different than inheritance.There are GADTs in Rust, which doesn't support inheritance.

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Product TypeRust

struct User { id: u64, name: String}

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Sum TypeRust

enum Animal { Cat(String), Dog(String)}

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Union TypeScala 3

case class Cat(name:String) case class Dog(name: String)type Animal = Cat | Dog

or even

def func(a: Cat | Dog ) : String = ???

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Enums as ADTScala 3

enum Color(val rgb: Int) { case Red extends Color(0xFF0000) case Green extends Color(0x00FF00) case Blue extends Color(0x0000FF) case Mix(mix: Int) extends Color(mix)}

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Very common ADTs

• Option

• Either

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Option

(more or less)

sealed trait Option[+A] final case class Some[+A](value: A) extends Option[A]case object None extends Option[Nothing]

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Either

(more or less)

sealed trait Either[+A, +B] extendsfinal case class Left[+A, +B] (value: A) extends Either[A, B] final case class Right[+A, +B]( value: B) extends Either[A, B]

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ADT• Parametrized : They can have type parameters

• Product Types: Think AND

• Sum Types: Think OR

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GADTGeneralized Algebraic Data Type

• A generalization of parametric algebraic data types.

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GADTExample

sealed trait Expr[T]case class Num(i: Int) extends Expr[Int]case class Bool(b: Boolean) extends Expr[Boolean]case class Add(x: Expr[Int], y: Expr[Int]) extends Expr[Int]case class Mult(x: Expr[Int], y: Expr[Int]) extends Expr[Int]case class Eq[A](x: Expr[A], y: Expr[A]) extends Expr[Boolean]

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GADTExample

def eval[T](e: Expr[T]): T = e match { case Num(i) => i case Bool(b) => b case Add(x,y) => eval(x)+eval(y) case Mult(x,y) => eval(x)*eval(y) case Eq(x,y) => eval(x) == eval(y) }

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GADTExample

val expr1 = Mult(Add(Num(1),Num(2)),Num(4))val expr2 = Num(12)val expr3 = Eq(expr1,expr2)

eval(expr3) true: Boolean

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GADT• Type-safe business logic

• Type-safe DSLs

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Coool, but...where the heck is the Algebra?

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Sum Type

sealed trait VehicleTypecase object Car extends VehicleTypecase object Moto extends VehicleType

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Sum Type

sealed trait VehicleTypecase object Car extends VehicleTypecase object Moto extends VehicleType

How many VehicleTypes?

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Sum Type

sealed trait Colour case object Red extends Colourcase object Yellow extends Colourcase object Blue extends Colour

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Sum Type

sealed trait Colour case object Red extends Colourcase object Yellow extends Colourcase object Blue extends Colour

How many Colours?

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Sum Type

sealed trait VehicleTypecase object Car extends VehicleTypecase object Moto extends VehicleType

2 VehicleTypes

sealed trait Colour case object Red extends Colourcase object Yellow extends Colourcase object Blue extends Colour

3 Colours

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Product Type

case class Vehicle (vehicleType:VehicleType, colour:Colour)

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Product Type

case class Vehicle (vehicleType:VehicleType, colour:Colour)

How many vehicles? If 2 VehicleTypes and 3 Colours?

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Product Type

case class Vehicle (vehicleType:VehicleType, colour:Colour)

If 2 VehicleTypes and 3 Colours?2 * 3 = 6 Vehicles

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Product Type

case class Vehicle (vehicleType:VehicleType, colour:Colour, isUsed:Boolean)

If 2 VehicleTypes, 3 Colours and 2 Boolean types?

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Product Type

case class Vehicle (vehicleType:VehicleType, colour:Colour, isSecondHand:Boolean)

If 2 VehicleTypes, 3 Colours and 2 Boolean types?2 * 3 * 2 = 12 Vehicles

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ADTAs in elementary algebra

• Sum is the result of addition

• Product is the result of multiplication.

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Function TypeFunction is a type in Scala

def like(colour:Colour) :Boolean

or

val like : Colour => Boolean

also

type Like = Colour => Boolean

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Function Type

type Like = Colour -> Boolean

Red -> {t,f}Yellow -> {t,f}Blue -> {t,f}

How many implementations?

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Function Type

type Like = Colour -> Boolean

Red -> {t,f}Yellow -> {t,f}Blue -> {t,f}

There can be 2 * 2 * 2 = 23 implementations

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Function Type

sealed trait Countrycase object France extends Country case object UK extends Countrycase object Germany extends Country case object Japan extends Country case object USA extends Country

5 countries

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Function Type

sealed trait Brand case object Brand1 extends Brand case object Brand2 extends Brand...case object Brand20 extends Brand

2O brands

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Function Type

def isProduced(brand:Brand) : Country

Brand1 -> {France, UK, Germany, Japan, USA}Brand2 -> {France, UK, Germany, Japan, USA} ...Brand20 -> {France, UK, Germany, Japan, USA}

How many implementations?

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Function Type

def isProduced(brand:Brand) : Country

Brand1 -> {France, UK, Germany, Japan, USA}Brand2 -> {France, UK, Germany, Japan, USA} ...Brand20 -> {France, UK, Germany, Japan, USA}

There can be 5 * 5 * ... * 5 (20 times) = 520 implementations

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ExponentialsIn Category Theory

In mathematical literature, the function object, or the internal hom-object between two objects a and b, is often called the exponential and denoted by ba.

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Exponentials

In simpler terms

A Function Type val f : A => B

corresponds to the exponential

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Practical example

From the book: Functional Programming for Mortals

• (A => C) => ((B => C) => C)

•

•

•

•

•

• (Either[A, B] => C) => C

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More Algebras

• Tagless Final Algebra

• Free Μonads Algebra

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Tagless Final vs Free Μonads

• The comparison is out of scope for this talk

• But let's try to find out what Algebra means

• A random blogpost: Free Monad vs Tagless Final

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Tagless Final Algebra

trait DatabaseAlgebra[F[_], T] { def create(t: T): F[Boolean] def read(id: Long): F[Either[DatabaseError, T]] def delete(id: Long): F[Either[DatabaseError, Unit]]}

... tagless final encoded algebras.

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Free Monad Algebra

sealed trait DBFreeAlgebraT[T]case class Create[T](t: T) extends DBFreeAlgebraT[Boolean]case class Read[T](id: Long) extends DBFreeAlgebraT[Either[DatabaseError, T]]case class Delete[T](id: Long) extends DBFreeAlgebraT[Either[DatabaseError, Unit]]

• ... We need to create some interpreters to interpret our algebra into actions'

• ... For this section we’ll create interpreters to execute...

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Free Monad and Tagless Final

• in essence the Interpreter pattern

• A grammar for a simple language should be defined

• in Free Monad and Tagless Final : the grammar is called "Algebra"

• the grammar/algebra is implemented by concrete implementations.

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Category Theory

• formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms)

• the mathematics of mathematics

• the science of composition and abstraction

• the mathematics behind functional programming

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Algebras in Category Theory• F-algebra

• F-coalgebra

• Free Algebra

• Cofree Algebra

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Algebra

• Set of A

• Some operator

• unary: A => A

• binary (A, A) => A

• n-ary (A,A,...,A ) => A

• What about Seq[A] => A ?

• or Option[A]=> A

• F[A] => A

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F-Algebra

• F[A] => A

• If the F is also a functor

• this is called F-algebra

• F-algebra generalization of normal algebra

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Co-

• Co-Algebra

• Co-Free

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Co-

• Co-Algebra

• Co-Free

What does co- mean?

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Duality in Category Theory

Duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop

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F-coalgebra

• if F[A] => A is an algebra

• If we reverse the arrows

• type Coalgebra[F[_], A] = A => F[A]

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Recursion Schemes

• a calculus for lazy functional programming based on recursion operators associated with data type definitions.

• in simple terms: is an advanced functional programming technique that moves the the recursion in the type level

• using F-Algebras and F- Coalgebras

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Recap

• (G)ADT analogy to elementary algebra

• Sum Type => arithmetic addition

• Product Type => arithmetic multiplication

• Function Type => exponential operation

• Free Monads Algebra - Tagless Final Algebra => Grammar in Interpreter Pattern

• F-algebra

• F-coalgebra

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References / Read more• https://en.wikipedia.org/wiki/Algebra

• What the F is a GADT?

• GADT support in Scala

• Function Types

• Functional Programming for Mortals

• Free Monad vs Tagless Final

• Recursion Schemes for Higher Algebras

• The Science Behind Functional Programming

• AST playground: recursion schemes and recursive data

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Questions?https://www.harrylaou.com/slides/AlgebrasInProgramming.pdf

Harry Laoulakos : @harrylaou

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