Are you tired of forgetting which keys go in which maps? Are your data transformation pipelines reaching trans-continental proportions? A smidgen of high school algebra may go a long way to eliminating your deeply nested headaches. In this talk, we will explore several functional programming concepts and techniques, lifted right out of high school algebra, that can deepen your functional programming skills and get you slicing your problems along new dimensions.

1. All I needed for FP I
learned in High
School Algebra
Eric Normand

2. 1 2 3 4 5 6

4. •Sum up all the rocks for the year
•Average # of rocks per day
•Biggest week
•Smallest month

5. For the video and transcript
of this presentation,
click here:
https://lispcast.com/all-i-needed-for-fp-i-learned-in-high-s
chool-algebra/

6. What makes numbers,
an abstract idea, so
useful for modeling real
piles of rocks?

7. Correspondence
of
Properties

8. Information System
Distributed and Concurrent

9. Parallelization/
Distributed work
In distributed/parallel work, work comes
back out of order

10. Order doesn’t matter

11. 4
a
2
b+

12. 6
a b+

13. 2
b
4
a+

14. 6
b a+

15. a + b = b + a

16. (f a b)

17. (f a b)
(f b a)

18. (= (f a b)
(f b a))

19. (f a)

20. (g (f a))

21. (g (f a))
(g a)

22. (g (f a))
(f (g a))

23. (= (g (f a))
(f (g a)))

24. Parallelization/
Distributed work
Need to break up task to give to workers
Need to combine groups of answers
Needs to be cheap to break up and recombine groups

25. Grouping doesn’t matter

26. 15 3
a b c+ +

27. 6 3
(a + b) c+

28. 9
a + b + c

29. 15 3
a b c+ +

30. 5 4
a (b + c)+

31. 9
a + b + c

32. (a + b) + c = a + (b + c)

33. a b c
a b c

34. (f a b) c
a b c

35. (f a b) c
a (f b c)

36. (f (f a b) c)
(f a (f b c))

37. (f (f a b) c)
(f a (f b c))

38. (= (f (f a b) c)
(f a (f b c)))

39. Types
return value of f and its two
arguments need to be the same type
(= (f (f a b) c)
(f a (f b c)))

40. Whole Values
Combining two piles makes a new pile
Concatenating two lists makes a new list
Self-contained

41. (defn average [a b]
(/ (+ a b) 2))

42. (= (average a b) (average b a))
a = 10, b = 4
Order doesn’t
matter
(average 10 4) => 7
(average 4 10) => 7

43. (= (average (average a b) c)
(average a (average b c)))
a = 10, b = 4, c = 6
(average 10 4) => 7
(average 7 6) => 6.5
(average 4 6) => 5
(average 10 5) => 7.5
Does grouping matter?

44. function average(numbers) {
var sum = 0;
var count = 0;
for(i = 0; i < numbers.length; i++) {
sum += numbers[i];
count += 1;
}
if(count === 0) {
return null;
}
return sum / count;
}

45. function average(numbers) {
var sum = 0;
var count = 0;
for(i = 0; i < numbers.length; i++) {
sum += numbers[i];
count += 1;
}
if(count === 0) {
return null;
}
return sum / count;
}

46. (defn combine [[sum1 count1] [sum2 count2]]
[(+ sum1 sum2) (+ count1 count2)])
(defn ->average [number]
[number 1])
(defn average [numbers]
(reduce combine (map ->average numbers)))?

47. Where do you start a computation?

48. a + 0 = a

49. (f a i)

50. (= (f a i) a)

51. (defn combine [[sum1 count1] [sum2 count2]]
[(+ sum1 sum2) (+ count1 count2)])
(defn ->average [number]
[number 1])
(defn average [numbers]
(reduce combine (map ->average numbers)))?

52. function average(numbers) {
var sum = 0;
var count = 0;
for(i = 0; i < numbers.length; i++) {
sum += numbers[i];
count += 1;
}
if(count === 0) {
return null;
}
return sum / count;
}

53. (defn combine [[sum1 count1] [sum2 count2]]
[(+ sum1 sum2) (+ count1 count2)])
(defn ->average [number]
[number 1])
(defn average [numbers]
(reduce combine [0 0] (map ->average numbers)))

55. Going back and forth matters

56. Great for moving into a new space, doing
a calculation, then moving back

57. {:name “Eric”
:birthday #inst “1981-07-18”}
"{:name ”Eric”,…}"
pr-str
send
"{:name ”Eric”,…}"
{:name “Eric”
:birthday #inst “1981-07-18”}
read-string

58. (f a)

59. (g (f a))

60. (= (g (f a)) a)

61. (defn combine [[sum1 count1] [sum2 count2]]
[(+ sum1 sum2) (+ count1 count2)])
(defn ->average [number]
[number 1])
(defn average [numbers]
(reduce combine [0 0]
(map ->average numbers)))

62. (defn combine [[sum1 count1] [sum2 count2]]
[(+ sum1 sum2) (+ count1 count2)])
(defn ->average [number]
[number 1])
(defn average-> [[sum count]]
(/ sum count))
(defn average [numbers]
(->> numbers
(map ->average)
(reduce combine [0 0])
average->))

63. Distributed
Messages arrive one or more times

64. Distributed
Independent workers have to coordinate to avoid
duplicate work

65. Duplicates don’t matter

67. (= (-> m
(assoc :a “hello”)
(assoc :a “hello”))
(-> m
(assoc :a “hello”))

68. (= (f a)
(f a))

69. (= (f (f a))
(f a))

70. (def button-state (atom {}))
(defn press! [button-id]
(swap! button-state assoc button-id true))
…
(press! :3rd-floor-north-up)
(press! :3rd-floor-north-up)
(press! :3rd-floor-north-up)

71. Nothing else matters
Know when to end
Circuit-breaking

72. a * b * c * 0 * d * e * f

73. a * 0 = 0

74. (f a z)

75. (= (f a z) z)

79. Splitting up work and recombining it matters
Great for arranging and rearranging work in a pipeline
Composing transducers

80. (= (->> cars
(map add-back-wheel)
(map add-front-wheel))
(->> cars
(map (comp
add-front-wheel
add-back-wheel))))

81. (= (map identity a) a)

82. (= (map identity a) a)
(map g a)

83. (= (map identity a) a)
(map f (map g a))

84. (= (map identity a) a)
(map f (map g a))
(comp f g)

85. (= (map identity a) a)
(map f (map g a))
(map (comp f g) a)

86. (= (map identity a) a)
(= (map f (map g a))
(map (comp f g) a))

87. Conclusions

88. Commutative Order doesn’t matter (= (f a b) (f b a))
Associative Grouping doesn’t matter
(= (f (f a b) c)
(f a (f b c)))
Identity value Where to start (= (f a i) a)
Zero value When to stop (= (f a z) z)
Idempotence Duplicates don’t matter (= (f (f a)) (f a))
Reversibility Going back and forth (= (g (f a)) a)
Structure
Preservation
Rearranging work
(= (m identity a) a)
(= (m (comp f g) a) (m f (m g
a)))

89. These properties are
what allow us to do
our work

90. (= (f a b)
(f b a)))

91. (prop/for-all [a S
b S]
(= (f a b)
(f b a)))

92. (prop/for-all [a gen/int
b gen/int]
(= (* a b)
(* b a)))

93. Algebraic properties
make great
test.check
properties

94. Eric Normand
Follow Eric on:
Eric Normand @EricNormand
eric@lispcast.comlispcast.com