[PPT] - data made out of PowerPoint Presentation

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data made

ut 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!

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Diogenes of Sinope 412 – 323 BC

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Diogenes of Sinope 412 – 323 BC

Simplest man of all time
Obnoxious hobo
Lived in a barrel

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I’ve been using this bowl like a sucker!

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Um…. what

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

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

Strings are pretty much arrays

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

Recursion can do loops

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

Recursive data structures can do lists

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

Ints can do bools

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

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[a,b,c,d] STRUCT

λa -> b

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Alonzo Church 1903 - 1995

λa -> b Lambda calculus

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λa -> b

Alonzo Church 1903 - 1995

Lambda calculus

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We can make any data structure out of functions!

Church encoding

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Booleans

Bool

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Church booleans

result

Bool

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Church booleans

result

Bool

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

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TRUE FALSE

r

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TRUE FALSE

r

result “What we do if it’s true”

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“What we do if it’s false”

TRUE FALSE

r

result

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TRUE FALSE

r

result result result

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() ()

result result result

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result result result

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r r r

The Church encoding

f a boolean is:

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type pe CBool = forall forall r. r -> r -> r cT cTrue ue :: :: CBoo CBool cTrue x y = x cFals alse :: : CBo CBool

l

cFalse x y = y cNot

t ::

:: CBool CBool -> > CBool

ol

cNot cb = cb cFalse cTrue cAnd nd :: :: CBool CBool -> > CBool

ol ->

> CBo Bool cAnd cb1 cb2 = cb1 cb2 cFalse cOr cOr :: : CBool Bool -> > CBo CBool

l ->

> CBool

ol

cOr cb1 cb2 = cb1 cTrue cb2

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Natural numbers

1 2 3 4 …

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Natural numbers

0 +1 0 +1 +1 0 +1 +1 +1 0 +1 +1 +1 +1 …

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Natural numbers

0 +1 0 +1 +1 0 +1 +1 +1 0 +1 +1 +1 +1 …

Giuseppe Peano 1858 - 1932

Natural numbers form a data structure!

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Zero Succ(Nat)

r

Nat = Natural Peano numbers

Giuseppe Peano 1858 - 1932

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r

Nat =

Now lets turn it into functions!

Zero Succ(Nat)

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Zero Succ(Nat)

r

result “If it’s a successor”

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r

“If it’s zero” result

Zero Succ(Nat)

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result result result

r

Zero Succ(Nat)

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Nat ()

result result result

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Nat

result result result

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Nat

result result result () Nat

() Nat

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result result result result

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(r r) r

The Church encoding

f natural numbers is:

r

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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

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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

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Performance

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

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Performance

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

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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

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Nil Cons(a, List a)

r

List a = Cons lists

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Nil Cons(a, List a)

r

result result result

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(a, List a)

result result result

()

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(a, )

result result result result

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a

result result result result

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r r

The Church encoding

f lists is:

r (a ) r

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r r

The Church encoding

f lists is:

r (a ) r

AKA: foldr

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Functors

a

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Functors

f a a

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Functors

f (f a) They compose! f a a

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Functors

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

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Free monads

f (f (f (f a))) f (f (f a)) f (f a) f a a

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Free monad >>=

a

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Free monad >>=

a

fmap

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Free monad >>=

f a

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Free monad >>=

f a

fmap

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Free monad >>=

f a

fmap

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Free monad >>=

f (f a)

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Free monad >>=

f (f a)

fmap

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Free monad >>=

f (f a)

fmap

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Free monad >>=

f (f a)

fmap

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Free monad >>=

f (f (f a))

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Free monad >>=

f (f (f a))

fmap

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Free monad >>=

f (f (f a))

fmap

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Free monad >>=

f (f (f a))

fmap

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Free monad >>=

f (f (f a))

fmap

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λn  [n+1, n*2]

3

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λn  [n+1, n*2]

4 6

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λn  [n+1, n*2]

4 6

fmap

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λn  [n+1, n*2]

5 8 7 12

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λn  [n+1, n*2]

5 8 7 12

fmap

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λn  [n+1, n*2]

5 8 7 12

fmap

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λn  [n+1, n*2]