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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.
Functional Structures in PHP
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- 16. @_md#phpbnl17 combine(1, 2) ==3; combine("a", "b") == "ab"; combine(true, false) == false;
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- 18. @_md#phpbnl17 $f = function(string$a): int { return strlen($a); };
- 19. @_md#phpbnl17 $f = function(string$a): int { return strlen($a); }; $g = function(int $b): bool { return $b % 2 === 0; };
- 20. @_md#phpbnl17 $f = function(string$a): int { return strlen($a); }; $g = function(int $b): bool { return $b % 2 === 0; }; $h = combine($f, g); $h("PHPBenelux") == true;
- 21. @_md#phpbnl17 combine($f, g) ==compose ($f, $g) // if $f and $g are functions
- 22. @_md#phpbnl17 compose ("strlen", odd,Option, ...)
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- 24. @_md#phpbnl17 combine(combine(1, 2), 3)== combine(1, combine(2, 3)) associativity
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- 26. @_md#phpbnl17 $identity = function($x) { return $x; }; A "PHPBenelux" i
- 27. @_md#phpbnl17 zero(42) == 0; zero("anystring") == ""; zero(true, false) == true;
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- 29. @_md#phpbnl17 combine(combine(1, 2), 3)== combine(1, combine(2, 3)) combine(zero($x), $x) == $x combine($x, zero($x)) == $x associativity (left and right) identity
- 30. @_md#phpbnl17 interface Monoid extendsSemigroup { public function zero(); // inherited from Semigroup public function combine($another); }
- 31. @_md#phpbnl17 interface Monoid extendsSemigroup { public function zero(); // inherited from Semigroup public function combine($another); } class Balance implements Monoid { // ... }
- 32. @_md#phpbnl17 $deposit100bucks = function(Balance$b) { return $b->plus(100); } $listOfBalances->foldMap($deposit100bucks);
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- 44. @_md#phpbnl17 ImmList(1,2,3) // List<Int> ImmList("a thing","another thing") // List<String>
- 45. @_md#phpbnl17 ImmList(1,2,3) // List<Int> ImmList("a thing","another thing") // List<String>
- 46. @_md#phpbnl17 abstract class Option{} class Some extends Option {} class None extends Option {}
- 47. @_md#phpbnl17 phunkie > Option(42) $var0:Option<Int> = Some(42) phunkie > Option(null) $var1: None = None
- 48. @_md#phpbnl17 phunkie > $f= compose(mayBeRubish, Option) phunkie > $f(42) $var0: None = None
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- 50. @_md#phpbnl17 function fmap(Mappable $mappable,callable $f) { return $mappable->map($f); }
- 51. @_md#phpbnl17 $listOfWords = ImmList("guacamole","nose", "penguin"); $lengths = fmap ("strlen") ($listOfWords); // List (9, 4, 7)
- 52. @_md#phpbnl17 $maybeSomeGuaca = Option("guacamole"); $length= fmap ("strlen") ($maybeSomeGuaca); // Some (9)
- 53. @_md#phpbnl17 $maybeSomeGuaca = Option("guacamole"); $length= fmap ("strlen") ($maybeSomeGuaca); // Some (9) $maybeSomeGuaca = Option(null); $length = fmap ("strlen") ($maybeSomeGuaca); // None
- 54. @_md#phpbnl17 // Already Mappablein Phunkie ImmList Option Function1 Validations (Either) // Planned ImmMap ImmSet Tuples (inc. Pair)
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- 56. @_md#phpbnl17 $fa == fmap(identity)($fa) fmap(compose($f,$g))($fa) == fmap($g)(fmap($f)($fa)) covariant identity covariant composition
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- 58. @_md#phpbnl17 interface Apply extendsFunctor { public function apply($ff); public function map2($fb, callable $f); } interface Applicative extends Apply { public function pure($a); }
- 59. @_md#phpbnl17 /** * @param A$a * @return FirstOrderKind<A> */ function pure($a)
- 60. @_md#phpbnl17 /** * @param A$a * @return FirstOrderKind<A> */ function pure($a) Option()->pure(42); // Some(42)
- 61. @_md#phpbnl17 /** * @param A$a * @return FirstOrderKind<A> */ function pure($a) Option()->pure(42); // Some(42) ImmList()->pure(42); // ImmList(42)
- 62. @_md#phpbnl17 /** * @param FirstOrderKind<callable<A,B>>$ff * @return FirstOrderKind<B> */ function apply($ff)
- 63. @_md#phpbnl17 /** * @param FirstOrderKind<callable<A,B>>$ff * @return FirstOrderKind<B> */ function apply($ff) $increment = function($x){ return $x + 1;};
- 64. @_md#phpbnl17 /** * @param FirstOrderKind<callable<A,B>>$ff * @return FirstOrderKind<B> */ function apply($ff) Some(1)->apply(Some($increment));
- 65. @_md#phpbnl17 /** * @param FirstOrderKind<callable<A,B>>$ff * @return FirstOrderKind<B> */ function apply($ff) Some(1)->apply(Some($increment)); // Some(2)
- 66. @_md#phpbnl17 /** * @param FirstOrderKind<callable<A,B>>$ff * @return FirstOrderKind<B> */ function apply($ff) Some(1)->apply(Some($increment)); // Some(2) None()->apply(Some($increment));
- 67. @_md#phpbnl17 /** * @param FirstOrderKind<callable<A,B>>$ff * @return FirstOrderKind<B> */ function apply($ff) Some(1)->apply(Some($increment)); // Some(2) None()->apply(Some($increment)); // None
- 68. @_md#phpbnl17 /** * @param FirstOrderKind<callable<A,B>>$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));
- 69. @_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($increment)); // ImmList(2,3,4)
- 70. @_md#phpbnl17 /** * @param FirstOrderKind<B>$fb * @param (A, B) => C $ * @return FirstOrderKind<C> */ function map2($fb, callable $f)
- 71. @_md#phpbnl17 /** * @param FirstOrderKind<B>$fb * @param (A, B) => C $ * @return FirstOrderKind<C> */ function map2($fb, callable $f) Some(1)->map2(Some(2), function($x, $y) { return $x + $y; });;
- 72. @_md#phpbnl17 /** * @param FirstOrderKind<B>$fb * @param (A, B) => C $ * @return FirstOrderKind<C> */ function map2($fb, callable $f) Some(1)->map2(Some(2), function($x, $y) { return $x + $y; });;
- 73. @_md#phpbnl17 /** * @param FirstOrderKind<B>$fb * @param (A, B) => C $ * @return FirstOrderKind<C> */ function map2($fb, callable $f) Some(1)->map2(Some(2), function($x, $y) { return $x + $y; });; // Some(3)
- 74. @_md#phpbnl17 /** * @param FirstOrderKind<B>$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 + $y; });
- 75. @_md#phpbnl17 /** * @param FirstOrderKind<B>$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)
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- 77. @_md#phpbnl17 $fa->apply($fa->pure(identity)) == $fa $fa->pure($a)->fa->apply($fa->pure($f))== $fa->pure($f($a)) identity homomorphism
- 78. @_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
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- 81. @_md#phpbnl17 $xs = fmap(ImmList(1,2,3)) ($f); echo $xs->length;
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- 83. @_md#phpbnl17 interface FlatMap extendsFunctor { public function flatMap(callable $f); } interface Monad extends FlatMap { public function flatten(); }
- 84. @_md#phpbnl17 /** * @param (A)-> FirstOrderKind<B> $a * @return FirstOrderKind<B> */ function flatMap($a)
- 85. @_md#phpbnl17 /** * @param (A)-> FirstOrderKind<B> $a * @return FirstOrderKind<B> */ function flatMap($a) Option(42)->flatMap(function($x) { return Some($x + 1); });
- 86. @_md#phpbnl17 /** * @param (A)-> FirstOrderKind<B> $a * @return FirstOrderKind<B> */ function flatMap($a) Option(42)->flatMap(function($x) { return Some($x + 1); });
- 87. @_md#phpbnl17 /** * @param (A)-> FirstOrderKind<B> $a * @return FirstOrderKind<B> */ function flatMap($a) Option(42)->flatMap(function($x) { return Some($x + 1); });
- 88. @_md#phpbnl17 /** * @param (A)-> FirstOrderKind<B> $a * @return FirstOrderKind<B> */ function flatMap($a) Option(42)->flatMap(function($x) { return Some($x + 1); }); // Some(43)
- 89. @_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)
- 90. @_md#phpbnl17 /** * @return FirstOrderKind<B> */ functionflatten() Some(Some(42))->flatten(); // Some(42)
- 91. @_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)
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- 93. @_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); }) == $fa;
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- 95. @_md#phpbnl17 function repl($state) { return read()-> flatMap(evaluate)-> flatMap(andPrint)-> flatMap(loop) ->run($state); } defrepl(state) = for { (s,input) <- read (s,result)<- evaluate s <- andPrint s <- loop } yield s
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- 97. @_md#phpbnl17 f(A $a): M<B> g(B$b): M<C>
- 98. f(a) ~= M<B> f(a)map g ~= M<M<C>>
- 99. f(a) ~= M<B> f(a)map g ~= M<M<C>> urgh!
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- 101. flatten f(a) mapg ~= M<C>
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- 103. @_md#phpbnl17 abstract class Validation{} class Success extends Validation {} class Failure extends Validation {}
- 104. @_md#phpbnl17 phunkie > Success(42) $var0:Validation<E, Int> = Success(42) phunkie > Failure("nay") $var0: Validation<String, A> = Failure("nay")
- 105. @_md#phpbnl17 phunkie > Either("nay")(42) $var0:Validation<E, Int> = Success(42) phunkie > Either("nay")(null) $var0: Validation<String, A> = Failure("nay")
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