data made out of - PowerPoint PPT Presentation

λλλλλ data made λλ λλ λ λ λ λ out of λ λ λ λ λ λ functions λ λ λ λ λ λλλλ λ λ λ λλ λλ λλλλλ #ylj2016 For faster @KenScambler monads!

Diogenes of Sinope 412 – 323 BC

• Simplest man of all time • Obnoxious hobo • Lived in a barrel Diogenes of Sinope 412 – 323 BC

I’ve been using this bowl like a sucker!

Um…. what

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

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

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

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

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

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

λ a -> b [a,b,c,d] STRUCT

λ a -> b Lambda calculus Alonzo Church 1903 - 1995

λ a -> b Lambda calculus Alonzo Church 1903 - 1995

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

Booleans Bool

Church booleans Bool result

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

TRUE or FALSE

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

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

TRUE result or result FALSE result

() result result () result

result result result

The Church encoding of a boolean is: r r r

type pe CBool = forall forall r. r -> r -> r cT cTrue ue :: :: CBoo CBool cTrue x y = x cFals alse :: : CBo CBool ol cFalse x y = y cNot ot :: :: CBool CBool -> > CBool ool cNot cb = cb cFalse cTrue cAnd nd :: :: CBool CBool -> > CBool ool -> > CBo Bool cAnd cb1 cb2 = cb1 cb2 cFalse cOr cOr :: : CBool Bool -> > CBo CBool ol -> > CBool ool 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 Natural 0 numbers form 0 +1 a data 0 +1 +1 structure! 0 +1 +1 +1 0 +1 +1 +1 +1 … Giuseppe Peano 1858 - 1932

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

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

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

Succ(Nat) or “If it’s zero” Zero result

Succ(Nat) result result or Zero result

Nat result result () result

Nat result result result

Nat result () result Nat result () Nat () Nat () Nat

result result result result

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

type pe CNat = forall ll r. (r -> r) -> r -> r c0 c0, c c1, , c2, c 2, c3, , c4 4 :: : CN 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 ucc :: :: CNat Nat -> > CNat at cSucc cn f = f . cn f cPlus lus :: :: CNat Nat -> > CNat at -> > CNat at cPlus cn1 cn2 f = cn1 f . cn2 f cMult ult :: :: CNat Nat -> > CNat at -> > CNat at cMult cn1 cn2 = cn1 . cn2

type pe CNat = forall ll r. (r -> r) -> r -> r c0 c0, c c1, , c2, c 2, c3, , c4 4 :: : CN CNat c0 f = id c1 f = f c2 f = f . f c3 f = f . f . f c4 f = f . f . f . f cSucc ucc :: :: CNat Nat -> > CNat at cSucc cn f = f . cn f cPlus lus :: :: CNat Nat -> > CNat at -> > CNat at cPlus cn1 cn2 f = cn1 f . cn2 f cMult ult :: :: CNat Nat -> > CNat at -> > CNat at cMult cn1 cn2 = cn1 . cn2

Performance Native ints Peano numbers Church numbers O(n) O(1) addition O(1) O(n 2 ) multiplication O(n) O(n) print

Performance Native ints Peano numbers Church numbers O(n) O(1) addition O(1) O(n 2 ) multiplication O(n) O(n) print

Church encoding cheat sheet A a r A | B (a r) (b r) r (A, B) r) r (a b Singleton r Recursion r r

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

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

(a, List a) result result () result

(a, ) result result result result

a result result result result

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

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

Functors a

Functors a f a

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

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

Free monads a f a f (f a) f (f (f a)) f (f (f (f 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

λ n  [n+1, n*2] 3

λ n  [n+1, n*2] 4 6

λ n  [n+1, n*2] 4 6 fmap

λ n  [n+1, n*2] 7 12 5 8

λ n  [n+1, n*2] fmap 7 12 5 8

λ n  [n+1, n*2] 7 12 5 8 fmap

λ n  [n+1, n*2] 8 14 6 10 9 16 13 24

λ n  Wrap [Pure (n+1), Pure (n*2)] 3

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

λ n  Wrap [Pure (n+1), Pure (n*2)] 6 4

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

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

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

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

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

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

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

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

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

λ n  Wrap [Pure (n+1), Pure (n*2)] 9 16 8 14 6 10 13 24

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

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

f (Free f a) result result a result

f result result result a result

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

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

λ a -> b

λ a -> b ∴