The Math Every Programmer Needs

This talk was given at RailsConf 2023

Leave your fear of math behind along with the equations and symbols, we're headed to Ruby-land! On this whirlwind tour, we’ll explore how to write better conditionals with Boolean algebra, figure out how many test cases we need with combinatorics, a mental model for breaking down large tasks with graph theory, and more.

You’ll leave this talk equipped to better do your job with some ideas, techniques, and mental models from the wonderfully practical world of discrete math.

Transcript

1. The Math Every Programmer Needs
2. You must be really good at math!
3. Not "good at math" Joël Quenneville @joelquen Principal developer at thoughtbot
4. I mostly use basic arithmetic
5. What is the most useful CS class for the working Rails developer?
6. Not Data Structures and Algorithms
7. Discrete Math I use this daily
8. Discrete Math Propositional Logic Boolean Algebra Graph Theory Set Theory Combinatorics Probability Predicate Logic
9. Practical problems not theory
10. BOOLEAN ALGEBRA Unnecessary Conditionals
12. def can_edit? if admin? true else false end end Written code like this
13. def can_edit? admin? end Can be simplified to this
14. LESSON FROM BOOLEAN ALGEBRA Operators
15. Boolean operators identity (admin?) negation (!admin?) and (admin? && moderator?) or (admin? || moderator?)
16. admin? Identity
17. if admin? false else true end Negation
18. !admin? Negation
19. def can_edit_admin_post? return owner? unless admin? true end Two variables
20. def can_edit_admin_post? owner? || admin? end Two variables
21. Use operators!
22. PROPOSITIONAL LOGIC Difficult code review
23. def can_edit_admin_post? return owner? unless admin? true end Complex condition
25. TOOL FROM PROPOSITIONAL LOGIC Truth Table
26. Truth Table owner? admin? can_edit_admin_post? true true true true false true false true true false false false
27. Two expressions are equal if their truth tables match
28. Truth Table owner? admin? can_edit_admin_post? true true true true false true false true true false false false owner? || admin? true true true false
29. Use truth tables in GitHub comments
30. Use truth tables in GitHub comments
31. Understand
32. Improve communication
33. BOOLEAN ALGEBRA Broken Authorization Logic
34. Forbid signed out users and untrusted IPs
35. def allow_access_to_site? !(signed_out? && untrusted_ip?) end Confusing negated code
36. def allow_access_to_site? !signed_out? && !untrusted_ip? end Individual negation: cleaner?
37. Broken
38. Expected behavior signed_out? untrusted_ip? Correct true true false true false true false true true false false true
39. Individual negation signed_out? untrusted_ip? Correct true true false true false true false true true false false true individual negation false false false true
40. LESSON FROM BOOLEAN ALGEBRA DeMorgan's laws
41. DeMorgan's laws To negate compound conditions Negate each clause Invert operator 1 2
42. DeMorgan's laws To negate compound conditions !(a && b) == !a || !b !(a || b) == !a && !b 1 2
43. def allow_access_to_site? !(signed_out? && untrusted_ip?) end Applying DeMorgan: Original
44. def allow_access_to_site? !signed_out? && !untrusted_ip? end Applying DeMorgan: Step 1 NEGATE INDIVIDUAL CLAUSES
45. def allow_access_to_site? !signed_out? || !untrusted_ip? end Applying DeMorgan: Step 2 INVERT OPERATOR
46. DeMorgan signed_out? untrusted_ip? Correct true true false true false true false true true false false true DeMorgan false true true true
47. def allow_access_to_site? !signed_out? || !untrusted_ip? end DeMorganized code
48. def allow_access_to_site? signed_in? || trusted_ip? end Alias methods
49. Make code more readable
50. Catch bugs
51. unless signed_out? && untrusted_ip? render :not_authorized end Unless is also negation
52. COMBINATORICS How many test cases do I need?
53. def can_read? article.access_policy&.public? || user.admin? end Policy Object
54. describe "#can_read?" do it "allows admin to access public article" it "allows admin to access private article" it "allows non-admin to access public article" it "disallows non-admin from accessing private article" end Enough tests?
55. 2 inputs User is admin? Article is public? 1 2
56. 2 states user.admin? true false 1 2
57. 3 States article.access_policy&.public? true false nil 1 2 3
58. LESSON FROM COMBINATORICS Compound states multiply
59. 2 states x 3 states = 6 states
60. Modified truth table Article is public? User is admin? Can read? true true true true false true false true true false false false nil true true nil false false
61. class ArticleQuery def initialize(limit: nil, scope: Article.all) # ... end end Default parameters
62. Optional params add 2 states to your signature
63. 2 states x 2 states = 4 states
64. Ways of calling constructor new new(limit: 10) new(scope: Article.published) new(limit: 10, scope: Article.published) 1 2 3 4
65. TOOL FROM SET THEORY Venn diagrams
66. Set of tests is smaller than set of states Tests States
67. Grow set of tests Tests States
68. Shrink set of states Tests States
69. Solution: Add more tests or reduce your states
70. GRAPH THEORY Which RSpec let blocks actually get invoked?
71. describe User do let(:organization) { create(:organization) } let(:user) { create(:user, organization: organization) } let!(:admin) { create(:user, admin: true, organization: organization) } # lots of other tests describe "Invoices" do let(:product) { create(:product) } let(:invoice) { create(:invoice, owner: user, items: [product] } it "is complete" do expect(invoice).to be_complete end it "is an active product" do expect(product).to be_active end end end
72. Visual Model that I like
73. List lets organization user admin product invoice
74. Connect boxes organization user admin product invoice
75. Dependency Graph Terminology
76. Mark let! organization user admin product invoice
77. LESSON FROM GRAPH THEORY Evaluating a node also requires evaluating all downstream nodes
78. Follow dependencies organization user admin product invoice
79. Basic State All tests start with this
80. it "is complete" do expect(invoice).to be_complete end
81. Mark items referenced in test organization user admin product invoice
82. Follow dependencies organization user admin product invoice
83. it "is an active product" do expect(product).to be_active end
84. Mark items referenced in test organization user admin product invoice
85. GRAPH THEORY Working incrementally
86. Linear list Step1 Step 2 Step 3 Step 4 Step 5
87. Actual dependencies Step 1 Step2 Step 3 Step 4 Step 5
88. Actual dependencies Step 1 Step2 Step 3 Step 4 Step 5
89. LESSON FROM GRAPH THEORY Evaluating a node also requires evaluating all downstream nodes
90. Actual dependencies Step 1 Step2 Step 3 Step 4 Step 5
91. Doing tasks with no dependencies Step 1 Step2 Step 3 Step 4 Step 5
92. Doing tasks with no dependencies Step 1 Step2 Step 3 Step 4 Step 5
93. Doing tasks with no dependencies Step 1 Step2 Step 3 Step 4 Step 5
94. Doing tasks with no dependencies Step 1 Step2 Step 3 Step 4 Step 5
95. Doing tasks with no dependencies Step 1 Step2 Step 3 Step 4 Step 5
96. LESSON FROM GRAPH THEORY Dependency graphs must be evaluated bottom-up
97. Topological Sort Terminology
98. Replace image processing gem Change the Gemfile Update User model Update Article model
99. Replace image processing gem Change image processing gem Update User model Update Article model
100. Cycles Change image processing gem Update User model Update Article model
101. LESSON FROM GRAPH THEORY Cycles cannot be evaluated incrementally
102. Breaking Cycles Change image processing gem Update User model Introduce adapter Update Article model
103. Directed Acyclic Graph (DAG) Terminology
104. Strangler Fig Pattern
105. COMBINATORICS What are the risks of a collision?
106. SET THEORY, COMBINATORICS Designing a DB schema
107. PREDICATE LOGIC Why do all? and any? behave like that on an empty array?
108. PREDICATE LOGIC, SET THEORY Improve your communication
109. Ideas from Discrete Math Break cycles in your dependency graphs Compound states multiply DeMorgan laws Truth tables Use Boolean operators Evaluate dependency graph bottom- up Evaluating a graph node requires evaluating downstream nodes
110. Not "good at math" Joël Quenneville @joelquen Principal developer at thoughtbot