1. Zipper 2. One-Hole Context 3. Non-recursive Map

1. Walking
insidea
datastructure1 — jaiyalas @ funth - 2017 oct 05

2. algebraicdatatype
2 — jaiyalas @ funth - 2017 oct 05

3. algebraicdatatype
review3 — jaiyalas @ funth - 2017 oct 05

4. · zero: 0
· unit: 1
· variable: a, b, x, ...
· product: (×)
· coproduct: (+)
· fixed-point: μ
4 — jaiyalas @ funth - 2017 oct 05

5. 1.basefunctor
2.typefunctor†
†
type functor = base functor + fixed-point
5 — jaiyalas @ funth - 2017 oct 05

6. 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
6 — jaiyalas @ funth - 2017 oct 05

7. 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
7 — jaiyalas @ funth - 2017 oct 05

8. 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
8 — jaiyalas @ funth - 2017 oct 05

9. 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
9 — jaiyalas @ funth - 2017 oct 05

10. 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
10 — jaiyalas @ funth - 2017 oct 05

11. 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
11 — jaiyalas @ funth - 2017 oct 05

12. polynomialfunctor
12 — jaiyalas @ funth - 2017 oct 05

13. advantage?13 — jaiyalas @ funth - 2017 oct 05

14. Beauty14 — jaiyalas @ funth - 2017 oct 05

15. Properties15 — jaiyalas @ funth - 2017 oct 05

16. disadvantage?
16 — jaiyalas @ funth - 2017 oct 05

17. outermost
destruction17 — jaiyalas @ funth - 2017 oct 05

18. traversal18 — jaiyalas @ funth - 2017 oct 05

19. 19 — jaiyalas @ funth - 2017 oct 05

20. 20 — jaiyalas @ funth - 2017 oct 05

21. introducing..
21 — jaiyalas @ funth - 2017 oct 05

22. zipper22 — jaiyalas @ funth - 2017 oct 05

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

24. 24 — jaiyalas @ funth - 2017 oct 05

25. 25 — jaiyalas @ funth - 2017 oct 05

26. 26 — jaiyalas @ funth - 2017 oct 05

27. 27 — jaiyalas @ funth - 2017 oct 05

28. spine+2-tree
28 — jaiyalas @ funth - 2017 oct 05

29. 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)
29 — jaiyalas @ funth - 2017 oct 05

30. 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)
30 — jaiyalas @ funth - 2017 oct 05

31. 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)
31 — jaiyalas @ funth - 2017 oct 05

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

33. one-holecontext
33 — jaiyalas @ funth - 2017 oct 05

34. 34 — jaiyalas @ funth - 2017 oct 05

35. 35 — jaiyalas @ funth - 2017 oct 05

36. data D a
= C0
| C1 a
| C2 a (D a)
| C3 (D a) Int (D a)
--
--
36 — jaiyalas @ funth - 2017 oct 05

37. data D a
= C0
| C1 a
| C2 a (D a)
| C3 (D a) Int (D a)
-- path: ?
-- left: ?
37 — jaiyalas @ funth - 2017 oct 05

38. data D a
= C0
| C1 a
| C2 a (D a)
| C3 (D a) Int (D a)
-- path: ?
-- left: 1
38 — jaiyalas @ funth - 2017 oct 05

39. data D a
= C0
| C1 a
| C2 a (D a)
| C3 (D a) Int (D a)
-- path: ?
-- left: 1 + (D a)
39 — jaiyalas @ funth - 2017 oct 05

40. data D a
= C0
| C1 a
| C2 a (D a)
| C3 (D a) Int (D a)
-- path: Int × (D a) + (D a) × Int
-- left: 1 + (D a)
40 — jaiyalas @ funth - 2017 oct 05

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

42. the
Partial
Differentiation42 — jaiyalas @ funth - 2017 oct 05

43. basicrules
43 — jaiyalas @ funth - 2017 oct 05

44. substitutionrule
44 — jaiyalas @ funth - 2017 oct 05

45. fixed-pointrule
45 — jaiyalas @ funth - 2017 oct 05

46. observation
46 — jaiyalas @ funth - 2017 oct 05

47. 1-holectx:
list47 — jaiyalas @ funth - 2017 oct 05

48. data List a = Nil
| Cons a (List a)
-- ListF a x = 1 + a × x
-- List a = μx.ListF a x
48 — jaiyalas @ funth - 2017 oct 05

49. 49 — jaiyalas @ funth - 2017 oct 05

50. [a,b,c,..,m,n,o,..,x,y,z]
50 — jaiyalas @ funth - 2017 oct 05

51. [a,b,c,..,m],n,[o,..,x,y,z]
51 — jaiyalas @ funth - 2017 oct 05

52. 1-holectx:
lambdaTerm52 — jaiyalas @ funth - 2017 oct 05

53. 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
53 — jaiyalas @ funth - 2017 oct 05

54. 54 — jaiyalas @ funth - 2017 oct 05

55. with
data TermStep a = Lambda
| AppFun (Term a)
| AppArg (Term a)
55 — jaiyalas @ funth - 2017 oct 05

56. -- 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
], ())
56 — jaiyalas @ funth - 2017 oct 05

57. 1-holectx:
rosetree57 — jaiyalas @ funth - 2017 oct 05

58. 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
58 — jaiyalas @ funth - 2017 oct 05

59. 59 — jaiyalas @ funth - 2017 oct 05

60. richmutually-recursivedatatype?
60 — jaiyalas @ funth - 2017 oct 05

61. 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
61 — jaiyalas @ funth - 2017 oct 05

62. yougetblessed!allhealing-potionsget"hp+10"
62 — jaiyalas @ funth - 2017 oct 05

63. 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
63 — jaiyalas @ funth - 2017 oct 05

64. 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
64 — jaiyalas @ funth - 2017 oct 05

65. 65 — jaiyalas @ funth - 2017 oct 05

66. non-recursivemaptrick
66 — jaiyalas @ funth - 2017 oct 05

67. 1.typeextension
2.generaltraversal
67 — jaiyalas @ funth - 2017 oct 05

68. updateHealing :: PotionEffect -> PotionEffect
updateHealing (Healing i) = Healing (i + 10)
updateHealing x = x
bless' :: Player -> Player
bless' p = everywhere (mkT updateHealing) p
-- mkT = type extension
-- everywhere = general traversal
68 — jaiyalas @ funth - 2017 oct 05

69. 1.typeextension
69 — jaiyalas @ funth - 2017 oct 05

70. {- 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 ...
70 — jaiyalas @ funth - 2017 oct 05

71. 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
71 — jaiyalas @ funth - 2017 oct 05

72. *Main> updateHealing' (HP 10)
HP 10
*Main> updateHealing' (Potion "A" (Healing 10))
Potion "A" (Healing 10)
*Main> updateHealing' (Healing 10)
Healing 20
72 — jaiyalas @ funth - 2017 oct 05

73. data Player = Player Name Bag HP
deriving (Typeable)
data Bag = Bag [Item]
deriving (Typeable)
data Item = Weapon Name Attack
| Potion Name PotionEffect
| SmallBag Size Bag
deriving (Typeable)
data PotionEffect = Poison Int
| Healing Int
deriving (Typeable)
73 — jaiyalas @ funth - 2017 oct 05

74. 2.generaltraversal
74 — jaiyalas @ funth - 2017 oct 05

75. 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)
75 — jaiyalas @ funth - 2017 oct 05

76. 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)
76 — jaiyalas @ funth - 2017 oct 05

77. data Player = Player Name Bag HP
deriving (Typeable, Data)
data Bag = Bag [Item]
deriving (Typeable, Data)
data Item = Weapon Name Attack
| Potion Name PotionEffect
| SmallBag Size Bag
deriving (Typeable, Data)
data PotionEffect = Poison Int
| Healing Int
deriving (Typeable, Data)
77 — jaiyalas @ funth - 2017 oct 05

78. 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
78 — jaiyalas @ funth - 2017 oct 05

79. TL;DR79 — jaiyalas @ funth - 2017 oct 05

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

81. 2-tree's
alternative
presentation81 — jaiyalas @ funth - 2017 oct 05

82. resumable
traversal82 — jaiyalas @ funth - 2017 oct 05

83. spine+2-tree
83 — jaiyalas @ funth - 2017 oct 05

84. 1-HoleCtx[2/3]
84 — jaiyalas @ funth - 2017 oct 05

85. generalizedzipper
85 — jaiyalas @ funth - 2017 oct 05

86. PartialDifferentiation
polynomialfunctor
86 — jaiyalas @ funth - 2017 oct 05

87. resumable
structure87 — jaiyalas @ funth - 2017 oct 05

88. Non-RecursiveMap
[3/3]
88 — jaiyalas @ funth - 2017 oct 05

89. richmutural-recursivestructure
89 — jaiyalas @ funth - 2017 oct 05

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

91. · zipper
· one-hole context
· non-recursive map
thankyou
91 — jaiyalas @ funth - 2017 oct 05

92. appendix92 — jaiyalas @ funth - 2017 oct 05

93. about
93 — jaiyalas @ funth - 2017 oct 05