!["abcd"
IF x THEN y ELSE z
WHILE cond {…}
[a, b, c, d]
BOOL
INT
STRUCT {
fields…
}
λa -> b](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-8-320.jpg)
![IF x THEN y ELSE z
WHILE cond {…}
[a, b, c, d]
BOOL
INT
STRUCT {
fields…
}
λa -> b
Strings are
pretty much
arrays](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-9-320.jpg)
![IF x THEN y ELSE z
[a, b, c, d]
BOOL
INT
STRUCT {
fields…
}
λa -> b
Recursion can
do loops](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-10-320.jpg)
![IF x THEN y ELSE z
[a,b,c,d]
BOOL
INT
STRUCT {
fields…
}
λa -> b
Recursive data
structures can do
lists](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-11-320.jpg)
![IF x THEN y ELSE z
[a,b,c,d]
INT
STRUCT {
fields…
}
λa -> b
Ints can do bools](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-12-320.jpg)
![IF x THEN y ELSE z
[a,b,c,d]
INT
STRUCT {
fields…
}
λa -> b](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-13-320.jpg)
![[a,b,c,d]
STRUCT
λa -> b](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-14-320.jpg)
![λn [n+1, n*2]
3](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-73-320.jpg)
![λn [n+1, n*2]
4 6](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-74-320.jpg)
![λn [n+1, n*2]
4 6 fmap](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-75-320.jpg)
![λn [n+1, n*2]
5 8 7 12](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-76-320.jpg)
![λn [n+1, n*2]
5 8 7 12
fmap](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-77-320.jpg)
![λn [n+1, n*2]
5 8 7 12 fmap](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-78-320.jpg)
![λn [n+1, n*2]
6 10 9 16 8 14 13 24](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-79-320.jpg)
![λn Wrap [Pure (n+1), Pure (n*2)]
3](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-80-320.jpg)
![λn Wrap [Pure (n+1), Pure (n*2)]
>>=3](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-81-320.jpg)
![4 6
λn Wrap [Pure (n+1), Pure (n*2)]](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-82-320.jpg)
![4 6
λn Wrap [Pure (n+1), Pure (n*2)]
>>=](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-83-320.jpg)
![4 6
λn Wrap [Pure (n+1), Pure (n*2)]
fmap](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-84-320.jpg)
![4 6
λn Wrap [Pure (n+1), Pure (n*2)]
>>=](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-85-320.jpg)
![λn Wrap [Pure (n+1), Pure (n*2)]
5 8 7 12](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-86-320.jpg)
![λn Wrap [Pure (n+1), Pure (n*2)]
5 8 7 12
>>=](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-87-320.jpg)
![λn Wrap [Pure (n+1), Pure (n*2)]
5 8 7 12
fmap](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-88-320.jpg)
![λn Wrap [Pure (n+1), Pure (n*2)]
5 8 7 12 >>=](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-89-320.jpg)
![λn Wrap [Pure (n+1), Pure (n*2)]
5 8 7 12 fmap](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-90-320.jpg)
![λn Wrap [Pure (n+1), Pure (n*2)]
>>=5 8 7 12](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-91-320.jpg)
![λn Wrap [Pure (n+1), Pure (n*2)]
6 10 9 16 8 14 13 24](https://image.slidesharecdn.com/datamadeoutoffunctions-160201105803/85/Data-made-out-of-functions-92-320.jpg)
Data made out of functions
- 1. data made out of functions #ylj2016 @KenScambler λλλλλ λλ λλ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λλλλ λ λ λ λλ λλ λλλλλ For faster monads!
- 2. Diogenes of Sinope 412 – 323 BC
- 3. Diogenes of Sinope 412 – 323 BC • Simplest man of all time • Obnoxious hobo • Lived in a barrel
- 4. I’ve been using this bowl like a sucker!
- 7. Um…. what
- 8. "abcd" IF x THEN y ELSE z WHILE cond {…} [a, b, c, d] BOOL INT STRUCT { fields… } λa -> b
- 9. IF x THEN y ELSE z WHILE cond {…} [a, b, c, d] BOOL INT STRUCT { fields… } λa -> b Strings are pretty much arrays
- 10. IF x THEN y ELSE z [a, b, c, d] BOOL INT STRUCT { fields… } λa -> b Recursion can do loops
- 11. IF x THEN y ELSE z [a,b,c,d] BOOL INT STRUCT { fields… } λa -> b Recursive data structures can do lists
- 12. IF x THEN y ELSE z [a,b,c,d] INT STRUCT { fields… } λa -> b Ints can do bools
- 13. IF x THEN y ELSE z [a,b,c,d] INT STRUCT { fields… } λa -> b
- 15. Alonzo Church 1903 - 1995 λa -> b Lambda calculus
- 16. λa -> b Alonzo Church 1903 - 1995 Lambda calculus
- 17. We can make any data structure out of functions! Church encoding
- 18. Booleans Bool
- 20. Church booleans resultBool If we define everything you can do with a structure, isn’t that the same as defining the structure itself?
- 21. TRUE FALSE or
- 22. TRUE FALSE or result “What we do if it’s true”
- 23. “What we do if it’s false” TRUE FALSE or result
- 27. r r r The Church encoding of a boolean is:
- 28. type CBool = forall r. r -> r -> r cTrue :: CBool cTrue x y = x cFalse :: CBool cFalse x y = y cNot :: CBool -> CBool cNot cb = cb cFalse cTrue cAnd :: CBool -> CBool -> CBool cAnd cb1 cb2 = cb1 cb2 cFalse cOr :: CBool -> CBool -> CBool cOr cb1 cb2 = cb1 cTrue cb2
- 30. Natural numbers 0 0 +1 0 +1 +1 0 +1 +1 +1 0 +1 +1 +1 +1 …
- 31. Natural numbers 0 0 +1 0 +1 +1 0 +1 +1 +1 0 +1 +1 +1 +1 … Giuseppe Peano 1858 - 1932 Natural numbers form a data structure!
- 32. Zero Succ(Nat) or Nat = Natural Peano numbers Giuseppe Peano 1858 - 1932
- 33. or Nat = Now lets turn it into functions! Zero Succ(Nat)
- 34. Zero Succ(Nat) or result “If it’s a successor”
- 35. or “If it’s zero” resultZero Succ(Nat)
- 41. (r r) r The Church encoding of natural numbers is: r
- 42. type CNat = forall r. (r -> r) -> r -> r c0, c1, c2, c3, c4 :: CNat c0 f z = z c1 f z = f z c2 f z = f (f z) c3 f z = f (f (f z)) c4 f z = f (f (f (f z))) cSucc :: CNat -> CNat cSucc cn f = f . cn f cPlus :: CNat -> CNat -> CNat cPlus cn1 cn2 f = cn1 f . cn2 f cMult :: CNat -> CNat -> CNat cMult cn1 cn2 = cn1 . cn2
- 43. type CNat = forall r. (r -> r) -> r -> r c0, c1, c2, c3, c4 :: CNat c0 f = id c1 f = f c2 f = f . f c3 f = f . f . f c4 f = f . f . f . f cSucc :: CNat -> CNat cSucc cn f = f . cn f cPlus :: CNat -> CNat -> CNat cPlus cn1 cn2 f = cn1 f . cn2 f cMult :: CNat -> CNat -> CNat cMult cn1 cn2 = cn1 . cn2
- 44. Performance Native ints Peano numbers Church numbers addition print O(n) O(n2) multiplication O(n) O(n) O(1) O(1)
- 45. Performance Native ints Peano numbers Church numbers addition print O(n) O(n2) multiplication O(n) O(n) O(1) O(1)
- 46. Church encoding cheat sheet A | B (A, B) Singleton Recursion (a r) (b r) r (a r)b r r r r A a r
- 47. Nil Cons(a, List a) or List a = Cons lists
- 49. (a, List a) result result result ()
- 52. r r The Church encoding of lists is: r(a ) r
- 53. r r The Church encoding of lists is: r(a ) r AKA: foldr
- 54. Functors a
- 55. Functors f a a
- 56. Functors f (f a) They compose! f a a
- 57. Functors f (f (f a)) What if we make a “Church numeral” out of them? f (f a) f a a
- 58. Free monads f (f (f (f a))) f (f (f a)) f (f a) f a a
- 59. Free monad >>= a
- 61. Free monad >>= f a
- 62. Free monad >>= f a fmap
- 63. Free monad >>= f a fmap
- 64. Free monad >>= f (f a)
- 65. Free monad >>= f (f a) fmap
- 66. Free monad >>= f (f a) fmap
- 67. Free monad >>= f (f a) fmap
- 68. Free monad >>= f (f (f a))
- 69. Free monad >>= f (f (f a)) fmap
- 70. Free monad >>= f (f (f a)) fmap
- 71. Free monad >>= f (f (f a)) fmap
- 72. Free monad >>= f (f (f a)) fmap
- 73. λn [n+1, n*2] 3
- 74. λn [n+1, n*2] 4 6
- 75. λn [n+1, n*2] 4 6 fmap
- 76. λn [n+1, n*2] 5 8 7 12
- 77. λn [n+1, n*2] 5 8 7 12 fmap
- 78. λn [n+1, n*2] 5 8 7 12 fmap
- 79. λn [n+1, n*2] 6 10 9 16 8 14 13 24
- 80. λn Wrap [Pure (n+1), Pure (n*2)] 3
- 81. λn Wrap [Pure (n+1), Pure (n*2)] >>=3
- 82. 4 6 λn Wrap [Pure (n+1), Pure (n*2)]
- 83. 4 6 λn Wrap [Pure (n+1), Pure (n*2)] >>=
- 84. 4 6 λn Wrap [Pure (n+1), Pure (n*2)] fmap
- 85. 4 6 λn Wrap [Pure (n+1), Pure (n*2)] >>=
- 86. λn Wrap [Pure (n+1), Pure (n*2)] 5 8 7 12
- 87. λn Wrap [Pure (n+1), Pure (n*2)] 5 8 7 12 >>=
- 88. λn Wrap [Pure (n+1), Pure (n*2)] 5 8 7 12 fmap
- 89. λn Wrap [Pure (n+1), Pure (n*2)] 5 8 7 12 >>=
- 90. λn Wrap [Pure (n+1), Pure (n*2)] 5 8 7 12 fmap
- 91. λn Wrap [Pure (n+1), Pure (n*2)] >>=5 8 7 12
- 92. λn Wrap [Pure (n+1), Pure (n*2)] 6 10 9 16 8 14 13 24
- 93. Pure a Wrap f (Free f a) or Free a = Free monads
- 94. Pure a Wrap f (Free f a) or result result result
- 95. f (Free f a) result result result a
- 97. r r The Church encoding of free monads is: (f ) rr(a )
- 98. r r(f ) rr(a ) >>= CFree f b Bind is constant time!
- 99. λa -> b
- 100. λa -> b ∴
- 101. λa -> b ∴