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!

5.
Um….
what

6.
"abcd"
IF x THEN y ELSE z
WHILE cond {…}
[a, b, c, d]
BOOL
INT
STRUCT {
fields…
}
λa -> b

7.
IF x THEN y ELSE z
WHILE cond {…}
[a, b, c, d]
BOOL
INT
STRUCT {
fields…
}
λa -> b
Strings are
pretty much
arrays

8.
IF x THEN y ELSE z
[a, b, c, d]
BOOL
INT
STRUCT {
fields…
}
λa -> b
Recursion can
do loops

9.
IF x THEN y ELSE z
[a,b,c,d]
BOOL
INT
STRUCT {
fields…
}
λa -> b
Recursive data
structures can do
lists

10.
IF x THEN y ELSE z
[a,b,c,d]
INT
STRUCT {
fields…
}
λa -> b
Ints can do bools

11.
IF x THEN y ELSE z
[a,b,c,d]
INT
STRUCT {
fields…
}
λa -> b

12.
[a,b,c,d]
STRUCT
λa -> b

13.
Alonzo Church
1903 - 1995
λa -> b
Lambda calculus

14.
λa -> b
Alonzo Church
1903 - 1995
Lambda calculus

15.
We can make any
data structure out of
functions!
Church encoding

16.
Booleans
Bool

17.
Church booleans
resultBool

18.
Church booleans
resultBool
If we define everything you
can do with a structure, isn’t
that the same as defining
the structure itself?

19.
TRUE
FALSE
or

20.
TRUE
FALSE
or
result
“What we do if
it’s true”

21.
“What we do if
it’s false”
TRUE
FALSE
or
result

22.
TRUE
FALSE
or
result
result
result

23.
()
()
result
result
result

24.
result
result
result

25.
r r r
The Church encoding
of a boolean is:

26.
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

27.
Natural numbers
0
1
2
3
4
…

28.
Natural numbers
0
0 +1
0 +1 +1
0 +1 +1 +1
0 +1 +1 +1 +1
…

29.
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!

30.
Zero
Succ(Nat)
or
Nat =
Natural Peano numbers
Giuseppe Peano
1858 - 1932

31.
or
Nat =
Now lets turn it
into functions!
Zero
Succ(Nat)

32.
Zero
Succ(Nat)
or
result
“If it’s a successor”

33.
or “If it’s zero”
resultZero
Succ(Nat)

34.
result
result
resultor
Zero
Succ(Nat)

35.
Nat
()
result
result
result

36.
Nat result
result
result

37.
Nat result
result
result()
Nat
()
Nat
()
Nat
()
Nat

38.
result
result
result
result

39.
(r r) r
The Church encoding
of natural numbers is:
r

40.
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

41.
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

42.
Performance
Native ints Peano numbers Church numbers
addition
print
O(n)
O(n2)
multiplication
O(n) O(n)
O(1)
O(1)

43.
Performance
Native ints Peano numbers Church numbers
addition
print
O(n)
O(n2)
multiplication
O(n) O(n)
O(1)
O(1)

44.
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

45.
Nil
Cons(a, List a)
or
List a =
Cons lists

46.
Nil
Cons(a, List a)
or
result
result
result

47.
(a, List a) result
result
result
()

48.
(a, ) result
result
result
result

49.
a result
result
result
result

50.
r r
The Church encoding
of lists is:
r(a ) r

51.
r r
The Church encoding
of lists is:
r(a ) r
AKA: foldr

52.
Functors
a

53.
Functors
f a
a

54.
Functors
f (f a)
They compose!
f a
a

55.
Functors
f (f (f a))
What if we make a
“Church numeral” out of
them?
f (f a)
f a
a

56.
Free monads
f (f (f (f a)))
f (f (f a))
f (f a)
f a
a

57.
Free monad >>=
a

58.
Free monad >>=
a
fmap

59.
Free monad >>=
f a

60.
Free monad >>=
f a
fmap

61.
Free monad >>=
f a
fmap

62.
Free monad >>=
f (f a)

63.
Free monad >>=
f (f a)
fmap

64.
Free monad >>=
f (f a)
fmap

65.
Free monad >>=
f (f a)
fmap

66.
Free monad >>=
f (f (f a))

67.
Free monad >>=
f (f (f a))
fmap

68.
Free monad >>=
f (f (f a))
fmap

69.
Free monad >>=
f (f (f a))
fmap

70.
Free monad >>=
f (f (f a))
fmap

71.
λn  [n+1, n*2]
3

72.
λn  [n+1, n*2]
4 6

73.
λn  [n+1, n*2]
4 6 fmap

74.
λn  [n+1, n*2]
5 8 7 12

75.
λn  [n+1, n*2]
5 8 7 12
fmap

76.
λn  [n+1, n*2]
5 8 7 12 fmap

77.
λn  [n+1, n*2]
6 10 9 16 8 14 13 24

78.
λn  Wrap [Pure (n+1), Pure (n*2)]
3

79.
λn  Wrap [Pure (n+1), Pure (n*2)]
>>=3

80.
4 6
λn  Wrap [Pure (n+1), Pure (n*2)]

81.
4 6
λn  Wrap [Pure (n+1), Pure (n*2)]
>>=

82.
4 6
λn  Wrap [Pure (n+1), Pure (n*2)]
fmap

83.
4 6
λn  Wrap [Pure (n+1), Pure (n*2)]
>>=

84.
λn  Wrap [Pure (n+1), Pure (n*2)]
5 8 7 12

85.
λn  Wrap [Pure (n+1), Pure (n*2)]
5 8 7 12
>>=

86.
λn  Wrap [Pure (n+1), Pure (n*2)]
5 8 7 12
fmap

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)]
6 10 9 16 8 14 13 24

91.
Pure a
Wrap f (Free f a)
or
Free a =
Free monads

92.
Pure a
Wrap f (Free f a)
or
result
result
result

93.
f (Free f a) result
result
result
a

94.
f result
result
result
a
result

95.
r r
The Church encoding
of free monads is:
(f ) rr(a )

96.
r r(f ) rr(a )
>>=
CFree f b
Bind is constant time!

97.
λa -> b

98.
λa -> b
∴

99.
λa -> b
∴