FP in Scala 106: Set Category

Set category
Function programing aims at build large scale programs based on small composed functions. Those smaller functions are the atomic capabilities of the programs. Once composed with each other, they form a bigger capability, yet testable and still composable.

We've already seen functions compositions in previous a session.

We've seen that some abstractions make possible a higher level for programing for a better and safer code re-usage. Functors and Applicatives and Monads are the most famous ones that we've already studies together.

What is interesting is that all of this is supported by a branch of mathematical study, Cateogry Theory. Another branch study a complementary field, the Sets. Let's have a look together about this today. We'll talk about weird structures such SemiGroup, Monoid, some mathematical properties such as associativity or commutativity.

Those categories will also be implemented with the Type Class pattern.

Transcript

1. SET CATEGORY FP 106
2. FP 101: FUNCTION A B f : =>
3. FP 102: FUNCTOR A => B map f : A B f : =>
4. FP 102: APPLICATIVE FUNCTOR A B f : => A B ap f : => C => C
5. FP 103: MONAD A B flatMap f : => A B f : =>
6. FP 104: FUNCTION COMPOSITION C B g : => C f o g : => A A B f : =>
7. CATEGORY THEORY C B g : => C f o g : => A A B f : =>
8. SET CATEGORY C B g : => C f o g : => A B f : => A
9. ALGEBRA
10. DESIGN PATTERNS
11. "Being abstract is something profoundly different from being vague … The purpose of abstraction is not to be vague, but to create a new semantic level in which one can be absolutely precise." Ew Dijkstra TEXTE
12. SET A
13. SET A x x x x x x x x x x x x x x x x x x
14. SET EXAMPLE: BOOLEAN BOOLEAN TRUE FALSE
15. SET EXAMPLE: BOOLEAN BOOLEAN TRUE FALSE f : ( Boolean , Boolean) => Boolean
16. SET EXAMPLE: BOOLEAN BOOLEAN TRUE FALSE f : ( Boolean , Boolean) => Boolean
17. SETS (WITH HTTP://TYPELEVEL.ORG/CATS)
18. SETS (WITH HTTP://TYPELEVEL.ORG/CATS) Semigroup
19. SETS (WITH HTTP://TYPELEVEL.ORG/CATS) Semigroup Monoid
20. SETS (WITH HTTP://TYPELEVEL.ORG/CATS) Semigroup Monoid Commutative
21. SETS (WITH HTTP://TYPELEVEL.ORG/CATS) Semigroup Monoid Commutative Additive
22. SETS (WITH HTTP://TYPELEVEL.ORG/CATS) Semigroup Monoid Commutative Additive Multiplicative
23. SEMI GROUP
24. SEMI GROUP A semigroup is a set S together with a binary operation "•" that satisfies the associative property: For all a,b,c in S, the equation (a • b) • c = a • (b • c) holds. HTTPS://EN.WIKIPEDIA.ORG/WIKI/SEMIGROUP
25. SEMI GROUP A x x x x x x x x x x x x x x x x x x A A f : => => A
26. SEMI GROUP f ( a , b ) = d
27. SEMI GROUP f ( a , b ) = d val d = f ( a , b )
28. SEMI GROUP x == y val x = f ( a , f (b , c) ) val y = f ( f ( a , b ) , c)
29. SEMI GROUP a b c
30. SEMI GROUP a b c d c f
31. SEMI GROUP a b c d c x f f
32. SEMI GROUP a b c
33. SEMI GROUP a b c e a f
34. SEMI GROUP x a b c e a f f
35. SEMI GROUP a b c d c x a b c e a f f f f
36. SEMI GROUP a b c d c x f f
37. SEMI GROUP a b c d c x f f
38. SEMI GROUP a b c d c x a b c e a f f f f
39. SEMI GROUP a b c d c x a b c e a f f f f FOLDLEFT FOLDRIGHT
40. SEMI GROUP a b d c x f f
41. SEMI GROUP a b d c x f f y f z
42. SEMI GROUP a b c y z f REDUCE
43. ACTION SEMI GROUP IN
44. SETS (WITH HTTP://TYPELEVEL.ORG/CATS) Semigroup
45. SETS (WITH HTTP://TYPELEVEL.ORG/CATS) Semigroup Monoid
46. MONOID
47. MONOID A monoid is a semigroup with an identity element. Identity element There exists an element e in S such that for every element a in S, the equations e • a = a • e = a hold. HTTPS://EN.WIKIPEDIA.ORG/WIKI/MONOID
48. MONOID A x x x x x x x x x x x x x x x x x x A A f : => => A
49. MONOID A x x x x x x x x x x x x x x x x x x A A f : => => A id
50. MONOID val x = f ( a , id ) => x == a val y = f ( id , b ) => y == b f ( id , id ) = id
51. MONOID a b c d c x f f id
52. ACTION MONOID IN
53. SETS (WITH HTTP://TYPELEVEL.ORG/CATS) Semigroup Monoid
54. SETS (WITH HTTP://TYPELEVEL.ORG/CATS) Semigroup Monoid Commutative
55. COMMUTATIVITY
56. COMMUTATIVITY A binary operation * on a set S is called commutative if: x * y = y * x for all x,y in S HTTPS://EN.WIKIPEDIA.ORG/WIKI/COMMUTATIVE_PROPERTY
57. a b c d COMMUTATIVITY
58. a b c d x f COMMUTATIVITY
59. c d x f y COMMUTATIVITY a b
60. d z f y COMMUTATIVITY c a b
61. z COMMUTATIVITY d c a b
62. a b c d COMMUTATIVITY
63. a b c d COMMUTATIVITY z f
64. a b c d COMMUTATIVITY z f Even in parallel, order of processing in f is not important
65. ACID 2.0 * ASSOCIATIVE * COMMUTATIVE * IDEMPOTENT * DISTRIBUTED Jonas Bonér HTTP://FR.SLIDESHARE.NET/JBONER/LIFE-BEYOND-THE-ILLUSION-OF-PRESENT COMMUTATIVITY
66. CRDT Conflict-free Replicated Data Types HTTP://WWW.INFOQ.COM/PRESENTATIONS/CRDT
67. CRDT HTTP://WWW.INFOQ.COM/PRESENTATIONS/CRDT
68. SETS (WITH HTTP://TYPELEVEL.ORG/CATS) Semigroup Monoid Commutative
69. SETS (WITH HTTP://TYPELEVEL.ORG/CATS) Semigroup Monoid Commutative Additive Multiplicative
70. ADDITIVE OR MULTIPLICATIVE
71. ADDITIVITY OR MULTIPLICITY BOOLEAN TRUE FALSE f = OR MULTIPLICATIVE MONOID ADDITIVE MONOID id = false (zero) f = AND id = true (one)
72. ACTION ADDITIVITY IN
73. LEVEL UP
74. KIND
75. KINDS val i: Int = 0
76. KINDS val i: Int = 0 KIND 0: VALUE
77. KINDS val i: Int = 0
78. KINDS val i: Int = 0 KIND 1: TYPE
79. KINDS val ints: List[Int] = List(0)
80. KINDS val ints: List[Int] = List(0) KIND 2: HIGHER KINDED TYPE
81. KINDS Semigroup[Boolean]
82. KINDS Semigroup[Boolean] SEMIGROUP FOR A TYPE
83. KINDS SemigroupK[Set[_]]
84. KINDS SemigroupK[Set[_]] SEMIGROUP FOR A HIGHER KINDED TYPE
85. ACTION SEMIGROUPK IN
86. CONCLUSION
87. SETS (WITH HTTP://TYPELEVEL.ORG/CATS) Semigroup Monoid Commutative Additive Multiplicative
88. What Is A Good Language for Design? One that helps discovering great designs. Pattern ꔄ Abstractions Constraints ꔄ Types Martin Odersky BLOG.XEBIA.FR/2014/09/17/TROUBLES-WITH-TYPES/ COMMUTATIVITY
89. LINKS ▸ Martin Odersky, Troubles with Types ▸ ▸ Jonas Bonér, Beyond the illusion of present ▸ ▸ Functional Programing in Scala ▸ ▸ Function ▸ BLOG.XEBIA.FR/2014/09/17/TROUBLES-WITH-TYPES/ HTTPS://VIMEO.COM/130759483 HTTPS://WWW.MANNING.COM/BOOKS/FUNCTIONAL-PROGRAMMING-IN-SCALA HTTPS://WWW.MANNING.COM/BOOKS/FUNCTIONAL-AND-REACTIVE-DOMAIN- MODELING
90. LINKS ▸ Code, powered by Scala && Typelevel Cats ▸ GITHUB.COM/XEBIA-FRANCE/FP_IN_SCALA_6