Data made out of functions - YOW! Conferences...data made out of functions #ylj2016 @KenScambler...
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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!
Transcript of Data made out of functions - YOW! Conferences...data made out of functions #ylj2016 @KenScambler...
data made out of
functions
#ylj2016
@KenScambler
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For faster monads!
Diogenes of Sinope412 – 323 BC
Diogenes of Sinope412 – 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 Church1903 - 1995
λa -> b
Lambda calculus
λa -> b
Alonzo Church1903 - 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
FALSEor
TRUE
FALSEor
result
“What we do if it’s true”
“What we do if it’s false”
TRUE
FALSEor
result
TRUE
FALSEor
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 :: CBoolcTrue x y = x
cFalse :: CBoolcFalse x y = y
cNot :: CBool -> CBoolcNot cb = cb cFalse cTrue
cAnd :: CBool -> CBool -> CBoolcAnd cb1 cb2 = cb1 cb2 cFalse
cOr :: CBool -> CBool -> CBoolcOr cb1 cb2 = cb1 cTrue cb2
Natural numbers0
1
2
3
4
…
Natural numbers0
0 +1
0 +1 +1
0 +1 +1 +1
0 +1 +1 +1 +1
…
Natural numbers0
0 +1
0 +1 +1
0 +1 +1 +1
0 +1 +1 +1 +1
…Giuseppe Peano1858 - 1932
Natural numbers form
a data structure!
Zero
Succ(Nat)or
Nat =
Natural Peano numbers
Giuseppe Peano1858 - 1932
orNat =
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 = zc1 f z = f zc2 f z = f (f z)c3 f z = f (f (f z))c4 f z = f (f (f (f z)))
cSucc :: CNat -> CNatcSucc cn f = f . cn f
cPlus :: CNat -> CNat -> CNatcPlus cn1 cn2 f = cn1 f . cn2 f
cMult :: CNat -> CNat -> CNatcMult cn1 cn2 = cn1 . cn2
type CNat = forall r. (r -> r) -> r -> r
c0, c1, c2, c3, c4 :: CNat
c0 f = idc1 f = fc2 f = f . fc3 f = f . f . fc4 f = f . f . f . f
cSucc :: CNat -> CNatcSucc cn f = f . cn f
cPlus :: CNat -> CNat -> CNatcPlus cn1 cn2 f = cn1 f . cn2 f
cMult :: CNat -> CNat -> CNatcMult cn1 cn2 = cn1 . cn2
Performance
Native ints Peano numbers Church numbers
addition
O(n)
O(n2)
multiplication
O(n) O(n)
O(1)
O(1)
Performance
Native ints Peano numbers Church numbers
addition
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 >>=
afmap
Free monad >>=
f a
Free monad >>=
f afmap
Free monad >>=
f afmap
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
∴
