Traversing on Algebraic Datatype

Walking
insidea
datastructure1 — jaiyalas @ funth - 2017 oct 05

algebraicdatatype
2 — jaiyalas @ funth - 2017 oct 05

algebraicdatatype
review3 — jaiyalas @ funth - 2017 oct 05

· zero: 0
· unit: 1
· variable: a, b, x, ...
· product: (×)
· coproduct: (+)
· fixed-point: μ

1.basefunctor
2.typefunctor†
†
type functor = base functor + fixed-point

Unit = 1
Maybe a = 1 + a
Pair a b = a × b
Either a b = a + b
data () = ()
data Maybe a = Nothing | Just a
data a b = (a, b)
data Either a b = Left a | Right b

Unit = 1
Maybe a = 1 + a
Pair a b = a × b
Either a b = a + b
data () = ()
data a b = (a, b)

ListF a x = 1 + a × x
List a = μx . (1 + a × x)
BTreeF a x = 1 + a × x × x
BTree a = μx . (1 + a × x × x)
data ListF a x = [] | a : x
type List = Fix ListF
data BTreeF a x = Nil | Fork a x x
type BTree = Fix BTreeF

polynomialfunctor

advantage?13 — jaiyalas @ funth - 2017 oct 05

Beauty14 — jaiyalas @ funth - 2017 oct 05

Properties15 — jaiyalas @ funth - 2017 oct 05

disadvantage?

outermost
destruction17 — jaiyalas @ funth - 2017 oct 05

traversal18 — jaiyalas @ funth - 2017 oct 05

introducing..

zipper22 — jaiyalas @ funth - 2017 oct 05

alternative
presentation
for2-tree23 — jaiyalas @ funth - 2017 oct 05

spine+2-tree

data BT a = Leaf a
| Fork a (BT a) (BT a)
data Rib a = LRib a (Bt a)
| RRib a (Bt a)
data Zipper a = Zipper [Rib a] (BT a)

data BT a = Leaf a
| RRib a (Bt a)

complex
structure?32 — jaiyalas @ funth - 2017 oct 05

one-holecontext

data D a
= C0
| C1 a
| C2 a (D a)
| C3 (D a) Int (D a)
--
--

data D a
= C0
| C1 a
| C2 a (D a)
-- path: ?
-- left: ?

data D a
= C0
| C1 a
| C2 a (D a)
-- path: ?
-- left: 1

data D a
= C0
| C1 a
| C2 a (D a)
-- path: ?
-- left: 1 + (D a)

data D a
= C0
| C1 a
| C2 a (D a)
-- path: Int × (D a) + (D a) × Int
-- left: 1 + (D a)

mechanical
apprach?41 — jaiyalas @ funth - 2017 oct 05

the
Partial
Differentiation42 — jaiyalas @ funth - 2017 oct 05

basicrules

substitutionrule

fixed-pointrule

observation

1-holectx:
list47 — jaiyalas @ funth - 2017 oct 05

data List a = Nil
| Cons a (List a)
-- ListF a x = 1 + a × x
-- List a = μx.ListF a x

[a,b,c,..,m,n,o,..,x,y,z]

[a,b,c,..,m],n,[o,..,x,y,z]

1-holectx:
lambdaTerm52 — jaiyalas @ funth - 2017 oct 05

data Term a = Lit a
| Var Int
| Abs (Term a)
| App (Term a) (Term a)
-- F a x = a + Int + x + x × x
-- T a = μx.F a x

with
data TermStep a = Lambda
| AppFun (Term a)
| AppArg (Term a)

-- for a term:
App (Abs (Var 0))
(App (Abs (Lit 2))
(Lit 1))
-- one can reach a node
Lit 2
-- and leave a context:
([ AppArg (Abs (Var 0))
, AppFun (Lit 1)
, Lambda
], ())

1-holectx:
rosetree57 — jaiyalas @ funth - 2017 oct 05

data Rose a = Rose a [Rose a]
-- RoseF a x = a × (List x)
-- = a × (μy.ListF x y)
-- = a × (μy.1 + x × y)
-- Rose a = μx.RoseF a x

richmutually-recursivedatatype?

type Size = Int
type Name = String
type Attack = Int
type HP = Int
data Player = Player Name Bag HP
data Bag = Bag [Item]
data Item = Weapon Name Attack
| Potion Name PotionEffect
| SmallBag Size Bag
data PotionEffect = Poison Int | Healing Int

yougetblessed!allhealing-potionsget"hp+10"

bless :: Player -> Player
bless (Player n bag ss) = Player n (updateBag bag) ss
updateBag :: Bag -> Bag
updateBag (Bag xs) = Bag $ map updatePotion xs
updatePotion :: Item -> Item
updatePotion (Potion n pe) = Potion n (updateHealing pe)
updatePotion (SmallBag s bag) = SmallBag s (updateBag bag)
updatePotion x = x
updateHealing :: PotionEffect -> PotionEffect
updateHealing (Healing i) = Healing (i + 10)
updateHealing x = x

bless :: Player -> Player
bless (Player n bag ss) = Player n (updateBag bag) ss
updateBag :: Bag -> Bag
updateBag (Bag xs) = Bag $ map updatePotion xs
updatePotion :: Item -> Item
updatePotion (Potion n pe) = Potion n (updateHealing pe)
updatePotion (SmallBag s bag) = SmallBag s (updateBag bag)
updatePotion x = x
updateHealing x = x

non-recursivemaptrick

1.typeextension
2.generaltraversal

updateHealing x = x
bless' :: Player -> Player
bless' p = everywhere (mkT updateHealing) p
-- mkT = type extension
-- everywhere = general traversal

1.typeextension

{- Data.Typeable -}
-- abstract class
class Typeable a where
typeOf :: a -> TypeRep
-- type-safe cast operation
cast :: (Typeable a, Typeable b) => a -> Maybe b
cast = ... typeOf ...

mkT :: (Typeable a, Typeable b)
=> (b -> b) -> a -> a
mkT f = case cast f of
Just g -> g
Nothing -> id
updateHealing' :: Typeable a => a -> a
updateHealing' = mkT updateHealing

*Main> updateHealing' (HP 10)
HP 10
*Main> updateHealing' (Potion "A" (Healing 10))
Potion "A" (Healing 10)
*Main> updateHealing' (Healing 10)
Healing 20

deriving (Typeable)
deriving (Typeable)
| SmallBag Size Bag
deriving (Typeable)
data PotionEffect = Poison Int
| Healing Int
deriving (Typeable)

2.generaltraversal

one-layertraversal
{- Data.Data -}
class Typeable a => Data a where
gmapT :: (forall b. Data b => b -> b) -> a -> a
instance Typeable a => Data [a] where
gmapT f [] = []
gmapT f (x : xs) = (f x) : (f xs)
instance Data Player where
gmapT f (Player name bag hp) =
Player (f name) (f bag) (f hp)

recursivetraversal
-- bottom-up
everywhere :: Data a
=> (forall b. Data b => b -> b)
-> a -> a
everywhere f x = f (gmapT (everywhere f) x)
-- top-down
everywhere' :: Data a
=> (forall b. Data b => b -> b)
-> a -> a
everywhere' f x = gmapT (everywhere' f) (f x)

deriving (Typeable, Data)
| SmallBag Size Bag
data PotionEffect = Poison Int
| Healing Int

bless' p = everywhere (mkT updateHealing) p
jack = Player
"Jack"
(Bag [ Potion "A" (Healing 1)
, Potion "B" (Poison 5)
, SmallBag 1 (Bag [Potion "C" (Healing 5)])
]) 10
*Main> bless' jack
Player
"Jack"
(Bag [ Potion "A" (Healing 11)
, Potion "B" (Poison 5)
, SmallBag 1 (Bag [Potion "C" (Healing 15)])
]) 10

TL;DR79 — jaiyalas @ funth - 2017 oct 05

Zipper[1/3]80 — jaiyalas @ funth - 2017 oct 05

2-tree's
alternative
presentation81 — jaiyalas @ funth - 2017 oct 05

resumable
traversal82 — jaiyalas @ funth - 2017 oct 05

spine+2-tree

1-HoleCtx[2/3]

generalizedzipper

PartialDifferentiation
polynomialfunctor

resumable
structure87 — jaiyalas @ funth - 2017 oct 05

Non-RecursiveMap
[3/3]

richmutural-recursivestructure

1.typeextension
2.generaltraversal
3.haskellext+deriving90 — jaiyalas @ funth - 2017 oct 05

· zipper
· one-hole context
· non-recursive map
thankyou

appendix92 — jaiyalas @ funth - 2017 oct 05

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Traversing on Algebraic Datatype

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