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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!
Diogenes of Sinope
412 – 323 BC
Diogenes of Sinope
412 – 323 BC
• Simplest man of all time
• Obnoxious hobo
• Lived in a barrel
I’ve been using this bowl
like a sucker!
Um….
what
"abcd"
IF x THEN y ELSE z
WHILE cond {…}
[a, b, c, d]
BOOL
INT
STRUCT {
fields…
}
λa -> b
IF x THEN y ELSE z
WHILE cond {…}
[a, b, c, d]
BOOL
INT
STRUCT {
fields…
}
λa -> b
Strings are
pretty much
arrays
IF x THEN y ELSE z
[a, b, c, d]
BOOL
INT
STRUCT {
fields…
}
λa -> b
Recursion can
do loops
IF x THEN y ELSE z
[a,b,c,d]
BOOL
INT
STRUCT {
fields…
}
λa -> b
Recursive data
structures can do
lists
IF x THEN y ELSE z
[a,b,c,d]
INT
STRUCT {
fields…
}
λa -> b
Ints can do bools
IF x THEN y ELSE z
[a,b,c,d]
INT
STRUCT {
fields…
}
λa -> b
[a,b,c,d]
STRUCT
λa -> b
Alonzo Church
1903 - 1995
λa -> b
Lambda calculus
λa -> b
Alonzo Church
1903 - 1995
Lambda calculus
We can make any
data structure out of
functions!
Church encoding
Booleans
Bool
Church booleans
resultBool
Church booleans
resultBool
If we define everything you
can do with a structure, isn’t
that the same as defining
the structure itself?
TRUE
FALSE
or
TRUE
FALSE
or
result
“What we do if
it’s true”
“What we do if
it’s false”
TRUE
FALSE
or
result
TRUE
FALSE
or
result
result
result
()
()
result
result
result
result
result
result
r r r
The Church encoding
of a boolean is:
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
Natural numbers
0
1
2
3
4
…
Natural numbers
0
0 +1
0 +1 +1
0 +1 +1 +1
0 +1 +1 +1 +1
…
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!
Zero
Succ(Nat)
or
Nat =
Natural Peano numbers
Giuseppe Peano
1858 - 1932
or
Nat =
Now lets turn it
into functions!
Zero
Succ(Nat)
Zero
Succ(Nat)
or
result
“If it’s a successor”
or “If it’s zero”
resultZero
Succ(Nat)
result
result
resultor
Zero
Succ(Nat)
Nat
()
result
result
result
Nat result
result
result
Nat result
result
result()
Nat
()
Nat
()
Nat
()
Nat
result
result
result
result
(r r) r
The Church encoding
of natural numbers is:
r
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
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
Performance
Native ints Peano numbers Church numbers
addition
print
O(n)
O(n2)
multiplication
O(n) O(n)
O(1)
O(1)
Performance
Native ints Peano numbers Church numbers
addition
print
O(n)
O(n2)
multiplication
O(n) O(n)
O(1)
O(1)
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
Nil
Cons(a, List a)
or
List a =
Cons lists
Nil
Cons(a, List a)
or
result
result
result
(a, List a) result
result
result
()
(a, ) result
result
result
result
a result
result
result
result
r r
The Church encoding
of lists is:
r(a ) r
r r
The Church encoding
of lists is:
r(a ) r
AKA: foldr
Functors
a
Functors
f a
a
Functors
f (f a)
They compose!
f a
a
Functors
f (f (f a))
What if we make a
“Church numeral” out of
them?
f (f a)
f a
a
Free monads
f (f (f (f a)))
f (f (f a))
f (f a)
f a
a
Free monad >>=
a
Free monad >>=
a
fmap
Free monad >>=
f a
Free monad >>=
f a
fmap
Free monad >>=
f a
fmap
Free monad >>=
f (f a)
Free monad >>=
f (f a)
fmap
Free monad >>=
f (f a)
fmap
Free monad >>=
f (f a)
fmap
Free monad >>=
f (f (f a))
Free monad >>=
f (f (f a))
fmap
Free monad >>=
f (f (f a))
fmap
Free monad >>=
f (f (f a))
fmap
Free monad >>=
f (f (f a))
fmap
λn  [n+1, n*2]
3
λn  [n+1, n*2]
4 6
λn  [n+1, n*2]
4 6 fmap
λn  [n+1, n*2]
5 8 7 12
λn  [n+1, n*2]
5 8 7 12
fmap
λn  [n+1, n*2]
5 8 7 12 fmap
λn  [n+1, n*2]
6 10 9 16 8 14 13 24
λn  Wrap [Pure (n+1), Pure (n*2)]
3
λn  Wrap [Pure (n+1), Pure (n*2)]
>>=3
4 6
λn  Wrap [Pure (n+1), Pure (n*2)]
4 6
λn  Wrap [Pure (n+1), Pure (n*2)]
>>=
4 6
λn  Wrap [Pure (n+1), Pure (n*2)]
fmap
4 6
λn  Wrap [Pure (n+1), Pure (n*2)]
>>=
λn  Wrap [Pure (n+1), Pure (n*2)]
5 8 7 12
λn  Wrap [Pure (n+1), Pure (n*2)]
5 8 7 12
>>=
λn  Wrap [Pure (n+1), Pure (n*2)]
5 8 7 12
fmap
λn  Wrap [Pure (n+1), Pure (n*2)]
5 8 7 12 >>=
λn  Wrap [Pure (n+1), Pure (n*2)]
5 8 7 12 fmap
λn  Wrap [Pure (n+1), Pure (n*2)]
>>=5 8 7 12
λn  Wrap [Pure (n+1), Pure (n*2)]
6 10 9 16 8 14 13 24
Pure a
Wrap f (Free f a)
or
Free a =
Free monads
Pure a
Wrap f (Free f a)
or
result
result
result
f (Free f a) result
result
result
a
f result
result
result
a
result
r r
The Church encoding
of free monads is:
(f ) rr(a )
r r(f ) rr(a )
>>=
CFree f b
Bind is constant time!
λa -> b
λa -> b
∴
λa -> b
∴

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