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Functional Structures in PHP

The document discusses various concepts in functional programming, particularly within the context of PHP, including categories, monoids, functors, and monads. It explores implementations and laws governing these structures, providing examples of function composition, mapping, and validations. The material aims to illustrate the underlying mathematical principles and practical applications of these functional programming concepts.

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{unctionalStructures byMarcelloDuarte @_md f inPHP
@_md#phpbnl17 Expressinganything
@_md#phpbnl17
@_md#phpbnl17 categorytheory
@_md#phpbnl17 categorytheory somecoolstructures somearrows somelaws {
@_md#phpbnl17 A B
@_md#phpbnl17 A B "PHPBenelux" 10
@_md#phpbnl17 A B "PHPBenelux" 10
@_md#phpbnl17 A B "PHPBenelux" f 10
@_md#phpbnl17 B 10 C true
@_md#phpbnl17 B 10 C true
@_md#phpbnl17 B 10 C true g
@_md#phpbnl17
@_md#phpbnl17 github.com/phunkie/phunkie
@_md#phpbnl17 SEMIGROUPS @_md#phpbnl17 https://upload.wikimedia.org/wikipedia/commons/c/c2/Falkland_Islands_Penguins_40.jpg
@_md#phpbnl17 combine(1, 2) == 3; combine("a", "b") == "ab"; combine(true, false) == false;
@_md#phpbnl17 h
@_md#phpbnl17 $f = function(string $a): int { return strlen($a); };
@_md#phpbnl17 $f = function(string $a): int { return strlen($a); }; $g = function(int $b): bool { return $b % 2 === 0; };
@_md#phpbnl17 $f = function(string $a): int { return strlen($a); }; $g = function(int $b): bool { return $b % 2 === 0; }; $h = combine($f, g); $h("PHPBenelux") == true;
@_md#phpbnl17 combine($f, g) == compose ($f, $g) // if $f and $g are functions
@_md#phpbnl17 compose ("strlen", odd, Option, ...)
@_md#phpbnl17 laws
@_md#phpbnl17 combine(combine(1, 2), 3) == combine(1, combine(2, 3)) associativity
@_md#phpbnl17 MONOIDS @_md#phpbnl17 https://upload.wikimedia.org/wikipedia/commons/c/c2/Falkland_Islands_Penguins_40.jpg
@_md#phpbnl17 $identity = function ($x) { return $x; }; A "PHPBenelux" i
@_md#phpbnl17 zero(42) == 0; zero("any string") == ""; zero(true, false) == true;
@_md#phpbnl17 laws
@_md#phpbnl17 combine(combine(1, 2), 3) == combine(1, combine(2, 3)) combine(zero($x), $x) == $x combine($x, zero($x)) == $x associativity (left and right) identity
@_md#phpbnl17 interface Monoid extends Semigroup {
 public function zero(); // inherited from Semigroup public function combine($another); }
@_md#phpbnl17 interface Monoid extends Semigroup {
 public function zero(); // inherited from Semigroup public function combine($another); } class Balance implements Monoid { // ... }
@_md#phpbnl17 $deposit100bucks = function(Balance $b) { return $b->plus(100); }
 
 $listOfBalances->foldMap($deposit100bucks);
@_md#phpbnl17 FUNCTORS @_md#phpbnl17 https://upload.wikimedia.org/wikipedia/commons/3/3b/World_Map_1689.JPG
@_md#phpbnl17 kinds
@_md#phpbnl17 proper {
@_md#phpbnl17 givemeanumber
@_md#phpbnl17 42givemeanumber
@_md#phpbnl17 givemeaword
@_md#phpbnl17 givemeaword "cabuzle"
@_md#phpbnl17 proper first-order{
@_md#phpbnl17 givemealist
@_md#phpbnl17 givemealist …
@_md#phpbnl17 givemealist alistofwhat?
@_md#phpbnl17 ImmList(1,2,3) // List<Int>
 ImmList("a thing", "another thing") // List<String>

@_md#phpbnl17 ImmList(1,2,3) // List<Int>
 ImmList("a thing", "another thing") // List<String>
@_md#phpbnl17 abstract class Option {} class Some extends Option {} class None extends Option {}
@_md#phpbnl17 phunkie > Option(42) $var0: Option<Int> = Some(42) phunkie > Option(null) $var1: None = None 
 

@_md#phpbnl17 phunkie > $f = compose(mayBeRubish, Option)
 phunkie > $f(42) $var0: None = None 
 

@_md#phpbnl17 proper first-order higherorder {
@_md#phpbnl17 function fmap(Mappable $mappable, callable $f) { return $mappable->map($f); }
@_md#phpbnl17 $listOfWords = ImmList("guacamole", "nose", "penguin"); $lengths = fmap ("strlen") ($listOfWords); // List (9, 4, 7)
@_md#phpbnl17 $maybeSomeGuaca = Option("guacamole"); $length = fmap ("strlen") ($maybeSomeGuaca); // Some (9)
@_md#phpbnl17 $maybeSomeGuaca = Option("guacamole"); $length = fmap ("strlen") ($maybeSomeGuaca); // Some (9) $maybeSomeGuaca = Option(null); $length = fmap ("strlen") ($maybeSomeGuaca);
 
 // None
@_md#phpbnl17 // Already Mappable in Phunkie ImmList Option Function1 Validations (Either) // Planned ImmMap ImmSet Tuples (inc. Pair)
@_md#phpbnl17 laws
@_md#phpbnl17 $fa == fmap(identity)($fa) fmap(compose($f, $g))($fa) == fmap($g)(fmap($f)($fa)) covariant identity covariant composition
@_md#phpbnl17 APPLICATIVE @_md#phpbnl17 https://upload.wikimedia.org/wikipedia/commons/5/5c/Brick_and_block_laying.jpg
@_md#phpbnl17 interface Apply extends Functor
 { public function apply($ff);
 public function map2($fb, callable $f);
 } interface Applicative extends Apply
 {
 public function pure($a);
 }
@_md#phpbnl17 /** * @param A $a * @return FirstOrderKind<A> */ function pure($a)
@_md#phpbnl17 /** * @param A $a * @return FirstOrderKind<A> */ function pure($a) Option()->pure(42); // Some(42)
@_md#phpbnl17 /** * @param A $a * @return FirstOrderKind<A> */ function pure($a) Option()->pure(42); // Some(42) ImmList()->pure(42); // ImmList(42)
@_md#phpbnl17 /** * @param FirstOrderKind<callable<A,B>> $ff * @return FirstOrderKind<B> */ function apply($ff)
@_md#phpbnl17 /** * @param FirstOrderKind<callable<A,B>> $ff * @return FirstOrderKind<B> */ function apply($ff) $increment = function($x){ return $x + 1;}; 

@_md#phpbnl17 /** * @param FirstOrderKind<callable<A,B>> $ff * @return FirstOrderKind<B> */ function apply($ff) Some(1)->apply(Some($increment));
@_md#phpbnl17 /** * @param FirstOrderKind<callable<A,B>> $ff * @return FirstOrderKind<B> */ function apply($ff) Some(1)->apply(Some($increment)); // Some(2)

@_md#phpbnl17 /** * @param FirstOrderKind<callable<A,B>> $ff * @return FirstOrderKind<B> */ function apply($ff) Some(1)->apply(Some($increment)); // Some(2)
 None()->apply(Some($increment));

@_md#phpbnl17 /** * @param FirstOrderKind<callable<A,B>> $ff * @return FirstOrderKind<B> */ function apply($ff) Some(1)->apply(Some($increment)); // Some(2)
 None()->apply(Some($increment));
 // None
@_md#phpbnl17 /** * @param FirstOrderKind<callable<A,B>> $ff * @return FirstOrderKind<B> */ function apply($ff) Some(1)->apply(Some($increment)); // Some(2)
 None()->apply(Some($increment));
 // None ImmList(1,2,3)->apply(ImmList($increment));

@_md#phpbnl17 /** * @param FirstOrderKind<callable<A,B>> $ff * @return FirstOrderKind<B> */ function apply($ff) Some(1)->apply(Some($increment)); // Some(2)
 None()->apply(Some($increment));
 // None ImmList(1,2,3)->apply(ImmList($increment));
 // ImmList(2,3,4)
@_md#phpbnl17 /** * @param FirstOrderKind<B> $fb * @param (A, B) => C $ * @return FirstOrderKind<C> */ function map2($fb, callable $f)
@_md#phpbnl17 /** * @param FirstOrderKind<B> $fb * @param (A, B) => C $ * @return FirstOrderKind<C> */ function map2($fb, callable $f) Some(1)->map2(Some(2), function($x, $y) { return $x + $y; });;
@_md#phpbnl17 /** * @param FirstOrderKind<B> $fb * @param (A, B) => C $ * @return FirstOrderKind<C> */ function map2($fb, callable $f) Some(1)->map2(Some(2), function($x, $y) { return $x + $y; });;
@_md#phpbnl17 /** * @param FirstOrderKind<B> $fb * @param (A, B) => C $ * @return FirstOrderKind<C> */ function map2($fb, callable $f) Some(1)->map2(Some(2), function($x, $y) { return $x + $y; });; // Some(3)

@_md#phpbnl17 /** * @param FirstOrderKind<B> $fb * @param (A, B) => C $ * @return FirstOrderKind<C> */ function map2($fb, callable $f) Some(1)->map2(Some(2), function($x, $y) { return $x + $y; });; // Some(3)
 ImmList(1,2)->map2(ImmList(4,5), function($x, $y) { return $x + $y; });

@_md#phpbnl17 /** * @param FirstOrderKind<B> $fb * @param (A, B) => C $ * @return FirstOrderKind<C> */ function map2($fb, callable $f) Some(1)->map2(Some(2), function($x, $y) { return $x + $y; });; // Some(3)
 ImmList(1,2)->map2(ImmList(4,5), function($x, $y) { return $x + $y; });
 // ImmList(5,6,6,7)
@_md#phpbnl17 laws
@_md#phpbnl17 $fa->apply($fa->pure(identity)) == $fa $fa->pure($a)->fa->apply($fa->pure($f)) ==
 $fa->pure($f($a)) identity homomorphism
@_md#phpbnl17 $fa->pure($a)->apply ==
 $fab->apply($fa->pure(function($f)use($a){return $f($a)})) $fa->map($f) == $fa->apply($fa->pure($f)) interchange map
@_md#phpbnl17 whywouldyoueveruseafunctor?
@_md#phpbnl17 weakertypesaremorepredictable
@_md#phpbnl17 $xs = fmap (ImmList(1,2,3)) ($f); echo $xs->length;
@_md#phpbnl17 MONADS @_md#phpbnl17 https://upload.wikimedia.org/wikipedia/commons/b/bd/Golden_tabby_and_white_kitten_n01.jpg
@_md#phpbnl17 interface FlatMap extends Functor
 { public function flatMap(callable $f);
 } interface Monad extends FlatMap
 {
 public function flatten();
 }
@_md#phpbnl17 /** * @param (A) -> FirstOrderKind<B> $a * @return FirstOrderKind<B> */ function flatMap($a)
@_md#phpbnl17 /** * @param (A) -> FirstOrderKind<B> $a * @return FirstOrderKind<B> */ function flatMap($a) Option(42)->flatMap(function($x) { return Some($x + 1); });
@_md#phpbnl17 /** * @param (A) -> FirstOrderKind<B> $a * @return FirstOrderKind<B> */ function flatMap($a) Option(42)->flatMap(function($x) { return Some($x + 1); });
@_md#phpbnl17 /** * @param (A) -> FirstOrderKind<B> $a * @return FirstOrderKind<B> */ function flatMap($a) Option(42)->flatMap(function($x) { return Some($x + 1); });
@_md#phpbnl17 /** * @param (A) -> FirstOrderKind<B> $a * @return FirstOrderKind<B> */ function flatMap($a) Option(42)->flatMap(function($x) { return Some($x + 1); }); // Some(43)
@_md#phpbnl17 /** * @param (A) -> FirstOrderKind<B> $a * @return FirstOrderKind<B> */ function flatMap($a) Option(42)->flatMap(function($x) { return Some($x + 1); }); // Some(43) ImmList(1,2,3)->flatMap(function($x) { return Option($x % 2 === 0 ? null : $x); }); // ImmList(1,3)
@_md#phpbnl17 /** * @return FirstOrderKind<B> */ function flatten() Some(Some(42))->flatten(); // Some(42)
@_md#phpbnl17 /** * @param (A) -> FirstOrderKind<B> $a * @return FirstOrderKind<B> */ function flatMap($a) Option(42)->flatMap(function($x) { return Some($x + 1); }); // Some(43) ImmList(1,2,3)->flatMap(function($x) { return Option($x % 2 === 0 ? null : $x); }); // ImmList(1,3)
@_md#phpbnl17 laws
@_md#phpbnl17 $fa->flatMap($f)->flatMap($g) == $fa->flatMap(function($a) use ($f,$g) { return $f($a)->flatMap( function($b) use ($g) { return $g($b); } ) ;}) $fa->pure($a)->flatMap($f) == $f($a) associativity right and left identity $fa->flatMap(function($a) use ($fa) { return $fa- >pure($a); }) == $fa;
@_md#phpbnl17 monadsyntaxsugar
@_md#phpbnl17 function repl($state)
 {
 return read()->
 flatMap(evaluate)->
 flatMap(andPrint)->
 flatMap(loop)
 ->run($state);
 } def repl(state) =
 for { (s,input) <- read
 (s,result)<- evaluate
 s <- andPrint
 s <- loop
 } yield s

@_md#phpbnl17 monadcomposability
@_md#phpbnl17 f(A $a): M<B>
 
 g(B $b): M<C>
f(a) ~= M<B>
 f(a) map g ~= M<M<C>> 

f(a) ~= M<B>
 f(a) map g ~= M<M<C>> 
 urgh!
@_md#phpbnl17 butyoucancomposethemonad’sfunctions!

 flatten f(a) map g ~= M<C> 

@_md#phpbnl17 VALIDATIONS @_md#phpbnl17 http://www.sbs.com.au/topics/sites/sbs.com.au.topics/files/gettyimages-470309868.jpg
@_md#phpbnl17 abstract class Validation {} class Success extends Validation {} class Failure extends Validation {}
@_md#phpbnl17 phunkie > Success(42) $var0: Validation<E, Int> = Success(42) phunkie > Failure("nay") $var0: Validation<String, A> = Failure("nay") 
 

@_md#phpbnl17 phunkie > Either("nay")(42) $var0: Validation<E, Int> = Success(42) phunkie > Either("nay")(null) $var0: Validation<String, A> = Failure("nay") 
 

@_md#phpbnl17 marcelloduarte @PhunkiePhp {
@_md#phpbnl17 thanks {bit.ly/inviqa-contact bit.ly/inviqa-careers you!
@_md#phpbnl17 credits https://upload.wikimedia.org/wikipedia/commons/c/c2/Falkland_Islands_Penguins_40.jpg {https://upload.wikimedia.org/wikipedia/commons/3/3b/World_Map_1689.JPG https://upload.wikimedia.org/wikipedia/commons/5/5c/Brick_and_block_laying.jpg https://upload.wikimedia.org/wikipedia/commons/b/bd/Golden_tabby_and_white_kitten_n01.jpg
@_md#phpbnl17 Questions? 
 joind.in/talk/75f65 ?

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