Some examples and motivation for creating data structures from nothing but functions - Church Encoding! There's particular detail on how it can make free monads more efficient.

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

14. [a,b,c,d]
STRUCT
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

19. Church booleans
resultBool

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

24. TRUE
FALSE
or
result
result
result

25. ()
()
result
result
result

26. result
result
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

29. Natural numbers
0
1
2
3
4
…

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)

36. result
result
resultor
Zero
Succ(Nat)

37. Nat
()
result
result
result

38. Nat result
result
result

39. Nat result
result
result()
Nat
()
Nat
()
Nat
()
Nat

40. result
result
result
result

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

48. Nil
Cons(a, List a)
or
result
result
result

49. (a, List a) result
result
result
()

50. (a, ) result
result
result
result

51. a result
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

60. Free monad >>=
a
fmap

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

96. f result
result
result
a
result

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
∴