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
in the Key of C#
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. public bool And(
params bool[] facts)
{
return facts.Aggregate(
true,
(m, x) => m && x);
}
And True

26. var sut = new Folder();
sut.And(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. public bool And(
params bool[] facts)
{
return facts.Aggregate(
true,
(m, x) => m && x);
}
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. public bool Or(
params bool[] facts)
{
return facts.Aggregate(
false,
(m, x) => m || x);
}
Or False

38. var sut = new Folder();
sut.Or(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. public bool Or(
params bool[] facts)
{
return facts.Aggregate(
false,
(m, x) => m || x);
}
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. public int Length(ICollection<T> coll)
{
return coll.Aggregate(
0,
(m, _) => m + 1);
}
+ 1 0

50. var sut = new Folder();
sut.Length("Mike".ToCharArray());

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. public int Length(ICollection<T> coll)
{
return coll.Aggregate(
0,
(m, _) => m + 1);
}
+ 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. public IEnumerable<T>
Reverse(IEnumerable<T> coll)
{
return coll.Aggregate(
new List<T>(),
(m, x) => new List<T> {x}
.Concat(m).ToList()););
}
Cons [ ]

63. var sut = new Folder();
sut.Reverse("Mike".ToCharArray());

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. public IEnumerable<T>
Reverse(IEnumerable<T> coll)
{
return coll.Aggregate(
new List<T>(),
(m, x) => new List<T> {x}
.Concat(m).ToList()););
}
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. public IEnumerable<U>
Map<T, U>( IEnumerable<T> coll,
Func<T, U> func)
{
return coll.Aggregate(
new List<U>(),
(m, x) => {
m.Add(func(x));
return m;});
}
Conj [ ]

76. var sut = new Folder();
sut.Map(
“Mike".ToCharArray(),
s => s.ToUpper());

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. public IEnumerable<U>
Map<T, U>( IEnumerable<T> coll,
Func<T, U> func)
{
return coll.Aggregate(
new List<U>(),
(m, x) => {
m.Add(func(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. public IEnumerable<T>
Filter<T>( IEnumerable<T> coll,
Func<T, bool> pred)
{
return coll.Aggregate(
new List<T>(),
(m, x) => {
if(pred(x)) m.Add(x);
return m;});
}
[ ]if Conj

89. var sut = new Folder();
sut.Filter(
“Mike".ToCharArray(),
s => s.All(char.IsLower));

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. public IEnumerable<T>
Filter<T>( IEnumerable<T> coll,
Func<T, bool> pred)
{
return coll.Aggregate(
new List<T>(),
(m, x) => {
if(pred(x)) m.Add(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. public ICollection<TZ>
Zip<TX, TY, TZ>(
ICollection<TX> xs, ICollection<TY> ys,
Func<TX, TY, TZ> func)
{
return xs.Zip(ys,(x, y) => new Tuple<TX, TY>(x, y))
.Aggregate(
new List<TZ>(),
(m, p) => {
m.Add(func.Invoke(p.Item1, p.Item2));
return m; });
}
Conj [ ]

102. return xs.Zip(ys, (x, y) => new Tuple<TX, TY>(x, y))
.Aggregate(
new List<TZ>(),
(m, p) => {
m.Add(
func.Invoke(p.Item1, p.Item2));
return m; });
Conj [ ]

103. var sut = new Folder();
sut.Zip(
new[] {1, 2},
new[] {10, 20},
(x, y) => x + y);

104. Conj
[ ]
1 2
+
10 20
Zip

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

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

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

108. public ICollection<TZ>
Zip<TX, TY, TZ>(
ICollection<TX> xs, ICollection<TY> ys,
Func<TX, TY, TZ> func)
{
return xs.Zip(ys,(x, y) => new Tuple<TX, TY>(x, y))
.Aggregate(
new List<TZ>(),
(m, p) => {
m.Add(func.Invoke(p.Item1, p.Item2));
return m; });
}
Conj [ ]

109. Encore
Coin Changer

110. Change
coin1
coin2
coinn
c1
c2
cn
amount

111. 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

112. Conj
amount [ ]
Changer
Change

113. public class CoinChanger
{
public IEnumerable<int> Coins { get; set; }
public IEnumerable<int> ChangeFor(int amount)
{
return Coins.Aggregate(
new Tuple<int, List<int>>(amount, new List<int>()),
(m, coin) =>
{
m.Item2.Add(m.Item1/coin);
return new Tuple<int, List<int>>(
m.Item1%coin, m.Item2);
}).Item2;
}
}
Conj [ ]
Changer amount

114. public IEnumerable<int>
ChangeFor(int amount)
{
return Coins.Aggregate(
new Tuple<int, List<int>>(amount, new List<int>()),
(m, coin) =>
{
m.Item2.Add(m.Item1/coin);
return new Tuple<int, List<int>>(
m.Item1%coin, m.Item2);
}).Item2;
}
Conj [ ]
Changer amount

115. var sut = new CoinChanger
{
Coins = new[] {25, 10, 5, 1}
};
sut.ChangeFor(99);

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

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

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

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

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

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

122. public class CoinChanger
{
public IEnumerable<int> Coins { get; set; }
public IEnumerable<int> ChangeFor(int amount)
{
return Coins.Aggregate(
new Tuple<int, List<int>>(amount, new List<int>()),
(m, coin) =>
{
m.Item2.Add(m.Item1/coin);
return new Tuple<int, List<int>>(
m.Item1%coin, m.Item2);
}).Item2;
}
}
Conj [ ]
Changer amount

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

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

128. 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/
#/

129. 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/

130. 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

131. 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

132. 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/

133. 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

134. 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