Using the work of Dr. Graham Hutton as our guide, we'll look at how to satisfy all of your list processing needs with one function, fold. First we'll start off simple by finding the length of a list, then we'll reverse a list, followed by and-ing and or-ing a list; all using fold. Next we'll look at implementing the higher order functions of: map, filter, and zip. Last we'll look at fold in action by using it on the Coin Changer kata. You'll never look at fold the same way again.

1. All You Need is Fold Mike Harris

2. All You Need
Is FOLD

3. All You Need Is
FOLD

4. All you need
is FOLD

5. FOLD

6. Fold is All You
Need!

7. Tour
• Backstage
• Concert
• Farewell

8. Backstage

9. f
g
Function

10. f g
f(g)
Function

11. Fold
a1
a2
an
b

12. Fold
foldr :: (α → β → β) → β → ([α] → β)
foldl :: (β → α→ β) → β → ([α] → β)
SeedFunc
a1
a2
an
Fold b
Given
We
Func
Seed

13. f anf a2f Seed a1

14. f anf a2f(Seed, a1)

15. f anf(f(Seed, a1), a2)

16. b

17. Fold
a1
a2
an
b

18. — Paul Feyerabend
Against Method
Chapter 16
“Fundamental conceptual change …
presupposes new world-views and
new languages capable of expressing
them.”

19. Concert
All You Need is Fold

20. Set List
• And
• Or
• Length
• Reverse
• Map
• Filter
• Zip

21. Set List
• And
• Or
• Length
• Reverse
• Map
• Filter
• Zip

22. And
T/F
T/F
T/F
T/F

23. And
and :: [Bool] → Bool
and = fold (∧) True
TrueAnd
f1
f2
fn
Fold f
Given
We
And
True

24. And
True And

25. foldAnd = (...facts) => {
_.foldl(
facts,
(m, x) => m && x,
true);
};
And True

26. foldAnd(false, true, false);

27. And
True
False True False
And

28. And
True
False True False
M
X
result False
And

29. And
False
True False
M
X
result False
And

30. And
False
False
M
X
result False
And

31. And
True
False True False
result False
And

32. foldAnd = (...facts) => {
_.foldl(
facts,
(m, x) => m && x,
true);
};
And True

33. Set List
• And
• Or
• Length
• Reverse
• Map
• Filter
• Zip

34. Or
T/F
T/F
T/F
T/F

35. Or
or :: [Bool] → Bool
or = fold (∨) False
FalseOr
f1
f2
fn
Fold f
Given
We
Or
False

36. Or
False Or

37. foldOr = (...facts) => {
_.foldl(
facts,
(m, x) => m || x,
false);
};
Or False

38. foldOr(false, true, false);

39. Or
False
False True False
Or

40. Or
False
False True False
M
X
result False
Or

41. Or
False
True False
M
X
result True
Or

42. Or
True
False
M
X
result True
Or

43. Or
False
False True False
result True
Or

44. foldOr = (...facts) => {
_.foldl(
facts,
(m, x) => m || x,
false);
};
Or False

45. Set List
• And
• Or
• Length
• Reverse
• Map
• Filter
• Zip

46. Length size

47. Length
length :: [α] → Int
length = fold (λx n → 1 + n) 0
0+ 1
x1
x2
xn
Fold #
Given
We
+ 1
0

48. + 1
0 Length

49. foldLength = (coll) => {
_.foldl(
coll,
m => m + 1,
0);
};
+ 1 0

50. foldLength([…'Mike']);

51. + 1
0
M i k e
Length

52. + 1
0M
result 1
M i k e
Length

53. + 1
1M
result 2
i k e
Length

54. + 1
2M
result 3
k e
Length

55. + 1
3M
result 4
e
Length

56. + 1
0
result 4
M i k e
Length

57. foldLength = (coll) => {
_.foldl(
coll,
m => m + 1,
0);
};
+ 1 0

58. Set List
• And
• Or
• Length
• Reverse
• Map
• Filter
• Zip

59. Reverse
x1
x2
xn
xn
x2
x1

60. Reverse
reverse :: [α] → [α]
reverse = fold (λx m → m ++ [x]) [ ]
[ ]Cons
x1
x2
xn
Fold [x]
Given
We
Cons
[ ]

61. Cons
[ ] Reverse

62. foldReverse = (coll) => {
_.foldl(
coll,
(m, x) => {
m.unshift(x);
return m; },
[]);
}; Cons [ ]

63. foldReverse([…'Mike']);

64. Cons
[ ]
M i k e
Reverse

65. Cons
[ ]
M i k
M
X
result M
e
Reverse

66. Cons
M
i k e
M
X
result iM
Reverse

67. Cons
iM
k e
M
X
result kiM
Reverse

68. Cons
kiM
e
M
X
result ekiM
Reverse

69. Cons
[ ]
result ekiM
M i k e
Reverse

70. foldReverse = (coll) => {
_.foldl(
coll,
(m, x) => {
m.unshift(x);
return m; },
[]);
}; Cons [ ]

71. Set List
• And
• Or
• Length
• Reverse
• Map
• Filter
• Zip

72. Map
x1
x2
xn
z1
z2
zn
func

73. Map
map :: (α → β) → ([α] → [β])
map f = fold (λx xs → f x : xs) [ ]
[ ]Conj
x1
x2
xn
Fold [z]
Given
We
Conj
[ ]

74. Conj
[ ]
func
Map

75. foldMap = (f, coll) => {
_.foldl(
coll,
(m, x) => {
m.push(f(x));
return m; },
[]);
}; Conj [ ]

76. foldMap(
s => s.toUpperCase(),
[…'Mike']);

77. Conj
[ ]
M i k e
upper
Map

78. Conj
[ ]
M i k
M
X
result M
e
upper
Map

79. Conj
M
i k e
M
X
result MIupper
Map

80. Conj
MI
k e
M
X
result MIKupper
Map

81. Conj
MIK
e
M
X
result MIKEupper
Map

82. Conj
[ ]
M i k
result
e
upper MIKE
Map

83. foldMap = (f, coll) => {
_.foldl(
coll,
(m, x) => {
m.push(f(x));
return m; },
[]);
}; Conj [ ]

84. Set List
• And
• Or
• Length
• Reverse
• Map
• Filter
• Zip

85. Filter
x1
x2
xn
x1
xn
pred

86. Filter
filter :: (α → Bool) → ([α] → [α])
filter p = fold (λx xs →
if p x then x : xs else xs) [ ]
if
Conj
[ ]
x1
x2
xn
Fold [x]
Given
We
[ ]
if
Conj

87. if
[ ]
pred Conj
Filter

88. foldFilter = (pred, coll) => {
_.foldl(
coll,
(m, x) => {
if (pred(x)) m.push(x);
return m; },
[]);
}; [ ]if Conj

89. foldFilter(
s => /^[a-z]*$/.test(s),
[…'Mike']);

90. if
[ ]
M i k e
isLower Conj
Filter

91. if
[ ]
M i k
M
X
result [ ]
e
isLower Conj
Filter

92. if
[ ]
i k e
M
X
isLower result iConj
Filter

93. if
i
k e
M
X
isLower result ikConj
Filter

94. if
ik
e
M
X
isLower result ikeConj
Filter

95. if
[ ]
isLower
M i k e
result ikeConj
Filter

96. foldFilter = (pred, coll) => {
_.foldl(
coll,
(m, x) => {
if (pred(x)) m.push(x);
return m; },
[]);
}; [ ]if Conj

97. Set List
• And
• Or
• Length
• Reverse
• Map
• Filter
• Zip

98. Zip
y1
y2
yj
z1
z2
zk
func
x1
x2
xi

99. Zip
zip2 :: (α → β → γ) →
([α] → [β] → [γ])
zip2 f = fold (λx y xs ys →
f x y : xs ys) [ ]
[ ]Conj
y1
y2
ym
Fold [z]
Given
We
Conj
[ ]
x1
x2
xn

100. Conj
[ ]
func
Zip

101. foldZip = (f, xs, ys) => {
_.foldl(
_.zip(xs, ys),
(m, [x, y]) => {
m.push(f(x, y));
return m; },
[]);
}; Conj [ ]

102. foldZip(
_.add, [1, 2], [10, 20]);

103. Conj
[ ]
1 2
+
10 20
Zip

104. Conj
[ ]
1 2
+
10 20
M
result 11
X
Y
Zip

105. Conj
11
2
+
20
M
result 11 22
X
Y
Zip

106. Conj
[ ]
+ result 11 22
1 2
10 20
Zip

107. foldZip = (f, xs, ys) => {
_.foldl(
_.zip(xs, ys),
(m, [x, y]) => {
m.push(f(x, y));
return m; },
[]);
}; Conj [ ]

108. Encore
Coin Changer

109. Change
coin1
coin2
coinn
c1
c2
cn
amount

110. Coin Changer kata
changeFor :: [α] → α → [α]
changeFor a =
fold (λa x xs → (a % x, a / x) : xs
(a, [ ])
Conj
Changer
[ ]
amount
c1
c2
cn
Fold
Given
We
[c]
0
Conj
Changer
[ ]
amount

111. Conj
amount [ ]
Changer
Change

112. foldChange = (coins, amount) => {
_.foldl(coins,
(m, coin) => {
m.result.push(
_.floor(m.result / coin));
m.amount %= coin;
return m;
},{amount: amount, result: []}
).result;
};
Conj [ ]
Changer amount

113. foldChange(
[25, 10, 5, 1],
99);

114. Conj
amount
25 10 5 1
[ ]
Changer
Change

115. Conj
99
25 10 5 1
[ ]M
X
result 24 3Changer
Change

116. Conj
24
10 5 1
3M
X
result 4 3 2Changer
Change

117. Conj
4
5 1
3 2M
X
result 4 3 2 0Changer
Change

118. Conj
4
1
3 2 0M
X
result 0 3 2 0 4Changer
Change

119. Conj result 0 3 2 0 4Changer
amount [ ]
25 10 5 1
Change

120. foldChange = (coins, amount) => {
_.foldl(coins,
(m, coin) => {
m.result.push(
_.floor(m.result / coin));
m.amount %= coin;
return m;
},{amount: amount, result: []}
).result;
};
Conj [ ]
Changer amount

121. — Paul Feyerabend
Against Method
Chapter 18
“[A]ll methodologies, even the most
obvious ones, have their limits”

125. Thank you!
Mike Harris
@MikeMKH
http://comp-phil.blogspot.com/
Remember, all you need is Fold!

126. Talks
• Rich Hickey - “Reducers”, EuroClojure 2012.
https://vimeo.com/45561411
• Rich Hickey - “Transducers”, StrangeLoop 2014.
https://www.youtube.com/watch?v=6mTbuzafcII
• Tomas Petricek - “History and Philosophy of Types”,
StrangeLoop 2015.
http://tpetricek.github.io/Talks/2015/philosophy-of-types/
#/

127. Books
• Richard Bird and Phil Wadler - Introduction to Functional
Programming. Prentice-Hall,1988.
http://usi-pl.github.io/lc/sp-2015/doc/Bird_Wadler.
%20Introduction%20to%20Functional%20Programming.
1ed.pdf
• Paul Feyerabend - Against Method. Verso, 2010.
• Michael Fogus - Functional JavaScript: Introducing
Functional Programming with Underscore.js. O’Reilly,
2013.
http://functionaljavascript.com/

128. Papers
• Erik Meijer, Maarten Fokkinga, and Ross Paterson -
“Functional Programming with Bananas, Lenses,
Envelopes and Barbed Wire”.
http://eprints.eemcs.utwente.nl/7281/01/db-
utwente-40501F46.pdf
• Graham Hutton - “A tutorial on the universality and
expressiveness of fold”.
http://www.cs.nott.ac.uk/~gmh/fold.pdf

129. Blog Posts
• Brian McNamara - “Catamorphisms, part one”
https://lorgonblog.wordpress.com/2008/04/05/
catamorphisms-part-one/
• Edward Kmett - “Catamorphisms”
https://www.fpcomplete.com/user/edwardk/recursion-
schemes/catamorphisms

130. Websites
• ClojureDocs - https://clojuredocs.org/
• HaskellWiki - https://wiki.haskell.org/Haskell
• Inside F# - https://lorgonblog.wordpress.com/
• lodash - https://lodash.com/docs
• MDN - https://developer.mozilla.org/en-US/
• ReactiveX - http://reactivex.io/

131. Images (in order used)
• "Abbey Rd Studios" by Misterweiss at en.wikipedia - Transfered from en.wikipedia Transfer was stated to be made by User:Blast..
Licensed under Public Domain via Commons - https://commons.wikimedia.org/wiki/File:Abbey_Rd_Studios.jpg#/media/
File:Abbey_Rd_Studios.jpg
• "60s wallpaper" by Domincspics - Flickr. Licensed under CC BY 2.0 via Commons - https://commons.wikimedia.org/wiki/File:
60s_wallpaper.jpg#/media/File:60s_wallpaper.jpg
• "70's Wallpaper" by Liz Sullivan - Own work. Licensed under CC BY-SA 4.0 via Wikimedia Commons - https://
commons.wikimedia.org/wiki/File:70%27s_Wallpaper.JPG#/media/File:70%27s_Wallpaper.JPG
• "Flickr - ronsaunders47 - The Beatles-Sgt Pepper backdrop.". Licensed under CC BY-SA 2.0 via Wikimedia Commons - https://
commons.wikimedia.org/wiki/File:Flickr_-_ronsaunders47_-_The_Beatles-Sgt_Pepper_backdrop..jpg#/media/File:Flickr_-
_ronsaunders47_-_The_Beatles-Sgt_Pepper_backdrop..jpg
• "All you need is love graffiti Osijek" by Objavljeno - Own work. Licensed under CC BY-SA 3.0 via Wikimedia Commons - https://
commons.wikimedia.org/wiki/File:All_you_need_is_love_graffiti_Osijek.JPG#/media/File:All_you_need_is_love_graffiti_Osijek.JPG
• "Illumined Pleasure Salvador Dali" by Salvador Dalí - Own workMrArifnajafov (Photography from the original). Licensed under CC
BY-SA 3.0 via Wikimedia Commons - https://commons.wikimedia.org/wiki/File:Illumined_Pleasure_Salvador_Dali.JPG#/media/
File:Illumined_Pleasure_Salvador_Dali.JPG
• "3 Savile Row" by Original uploader was Misterweiss at en.wikipedia - Transfered from en.wikipedia Transfer was stated to be
made by User:Vinhtantran.. Licensed under Public Domain via Commons - https://commons.wikimedia.org/wiki/File:
3_Savile_Row.jpg#/media/File:3_Savile_Row.jpg
• "Paul Feyerabend 2" by Grazia Borrini-Feyerabend. Licensed under Attribution via Commons - https://commons.wikimedia.org/
wiki/File:Paul_Feyerabend_2.jpg#/media/File:Paul_Feyerabend_2.jpg

132. Images (in order used)
• "The Beatles on Green Hill in Almaty, Kazakhstan" by Ken and Nyetta - Flickr: The Beatles on
Green Hill in Almaty. Licensed under CC BY 2.0 via Commons - https://
commons.wikimedia.org/wiki/File:The_Beatles_on_Green_Hill_in_Almaty,_Kazakhstan.jpg#/
media/File:The_Beatles_on_Green_Hill_in_Almaty,_Kazakhstan.jpg
• "The Beatles in America" by United Press International, photographer unknown - This image is
available from the United States Library of Congress's Prints and Photographs division under
the digital ID cph.3c11094.This tag does not indicate the copyright status of the attached work.
• "The Beatles rooftop concert" by Source. Licensed under Fair use via Wikipedia - https://
en.wikipedia.org/wiki/File:The_Beatles_rooftop_concert.jpg#/media/
File:The_Beatles_rooftop_concert.jpg
• "Paul Feyerabend 2" by Grazia Borrini-Feyerabend. Licensed under Attribution via Commons -
https://commons.wikimedia.org/wiki/File:Paul_Feyerabend_2.jpg#/media/
File:Paul_Feyerabend_2.jpg
• Myself taken at StrangeLoop 2014 by my wife, Kelsey Harris