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The Math Every
Programmer Needs
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You must be
really good
at math!
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Not "good at math"
Joël Quenneville @joelquen Principal developer at
thoughtbot
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I mostly use basic arithmetic
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What is the most
useful CS class
for the working
Rails developer?
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Not Data
Structures and
Algorithms
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Discrete Math
I use this daily
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Discrete Math
Propositional Logic
Boolean Algebra
Graph Theory
Set Theory
Combinatorics Probability Predicate Logic
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Practical
problems
not theory
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BOOLEAN ALGEBRA
Unnecessary Conditionals
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No content
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def can_edit?
if admin?
true
else
false
end
end
Written code like this
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def can_edit?
admin?
end
Can be simplified to this
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LESSON FROM BOOLEAN ALGEBRA
Operators
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Boolean operators
identity (admin?)
negation (!admin?)
and (admin? && moderator?)
or (admin? || moderator?)
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admin?
Identity
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if admin?
false
else
true
end
Negation
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!admin?
Negation
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def can_edit_admin_post?
return owner? unless admin?
true
end
Two variables
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def can_edit_admin_post?
owner? || admin?
end
Two variables
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Use operators!
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PROPOSITIONAL LOGIC
Difficult code review
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def can_edit_admin_post?
return owner? unless admin?
true
end
Complex condition
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No content
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TOOL FROM PROPOSITIONAL LOGIC
Truth Table
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Truth Table
owner? admin? can_edit_admin_post?
true true true
true false true
false true true
false false false
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Two expressions are equal
if their truth tables match
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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
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Use truth tables in GitHub comments
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Use truth tables in GitHub comments
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Understand
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Improve
communication
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BOOLEAN ALGEBRA
Broken Authorization Logic
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Forbid signed out users
and untrusted IPs
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def allow_access_to_site?
!(signed_out? && untrusted_ip?)
end
Confusing negated code
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def allow_access_to_site?
!signed_out? && !untrusted_ip?
end
Individual negation: cleaner?
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Broken
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Expected behavior
signed_out? untrusted_ip?
Correct
true true false
true false true
false true true
false false
true
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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
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LESSON FROM BOOLEAN ALGEBRA
DeMorgan's laws
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DeMorgan's laws
To negate compound conditions
Negate each clause
Invert operator
1
2
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DeMorgan's laws
To negate compound conditions
!(a && b) == !a || !b
!(a || b) == !a && !b
1
2
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def allow_access_to_site?
!(signed_out? && untrusted_ip?)
end
Applying DeMorgan: Original
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def allow_access_to_site?
!signed_out? && !untrusted_ip?
end
Applying DeMorgan: Step 1
NEGATE INDIVIDUAL CLAUSES
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def allow_access_to_site?
!signed_out? || !untrusted_ip?
end
Applying DeMorgan: Step 2
INVERT OPERATOR
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DeMorgan
signed_out? untrusted_ip?
Correct
true true false
true false true
false true true
false false
true
DeMorgan
false
true
true
true
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def allow_access_to_site?
!signed_out? || !untrusted_ip?
end
DeMorganized code
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def allow_access_to_site?
signed_in? || trusted_ip?
end
Alias methods
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Make code more
readable
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Catch bugs
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unless signed_out? && untrusted_ip?
render :not_authorized
end
Unless is also negation
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COMBINATORICS
How many test
cases do I need?
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def can_read?
article.access_policy&.public? || user.admin?
end
Policy Object
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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?
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2 inputs
User is admin?
Article is public?
1
2
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2 states
user.admin?
true
false
1
2
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3 States
article.access_policy&.public?
true
false
nil
1
2
3
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LESSON FROM COMBINATORICS
Compound
states multiply
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2 states x 3 states = 6 states
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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
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class ArticleQuery
def initialize(limit: nil, scope: Article.all)
# ...
end
end
Default parameters
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Optional params add 2
states to your signature
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2 states x 2 states = 4 states
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Ways of calling constructor
new
new(limit: 10)
new(scope: Article.published)
new(limit: 10, scope:
Article.published)
1
2
3
4
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TOOL FROM SET THEORY
Venn diagrams
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Set of tests is smaller than set of states
Tests
States
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Grow set of tests
Tests
States
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Shrink set of states
Tests
States
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Solution: Add more tests
or reduce your states
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GRAPH THEORY
Which RSpec let blocks
actually get invoked?
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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
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Visual Model
that I like
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List lets
organization user admin product invoice
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Connect boxes
organization
user admin product
invoice
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Dependency Graph
Terminology
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Mark let!
organization
user admin product
invoice
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LESSON FROM GRAPH THEORY
Evaluating a node
also requires
evaluating all
downstream
nodes
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Follow dependencies
organization
user admin product
invoice
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Basic State
All tests start with this
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it "is complete" do
expect(invoice).to be_complete
end
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Mark items referenced in test
organization
user admin product
invoice
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Follow dependencies
organization
user admin product
invoice
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it "is an active product" do
expect(product).to be_active
end
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Mark items referenced in test
organization
user admin product
invoice
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GRAPH THEORY
Working incrementally
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Linear list
Step1
Step 2
Step 3
Step 4
Step 5
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Actual dependencies
Step 1
Step2 Step 3
Step 4
Step 5
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Actual dependencies
Step 1
Step2 Step 3
Step 4
Step 5
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LESSON FROM GRAPH THEORY
Evaluating a node
also requires
evaluating all
downstream
nodes
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Actual dependencies
Step 1
Step2 Step 3
Step 4
Step 5
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Doing tasks with no dependencies
Step 1
Step2 Step 3
Step 4
Step 5
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Doing tasks with no dependencies
Step 1
Step2 Step 3
Step 4
Step 5
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Doing tasks with no dependencies
Step 1
Step2 Step 3
Step 4
Step 5
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Doing tasks with no dependencies
Step 1
Step2 Step 3
Step 4
Step 5
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Doing tasks with no dependencies
Step 1
Step2 Step 3
Step 4
Step 5
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LESSON FROM GRAPH THEORY
Dependency
graphs must
be evaluated
bottom-up
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Topological Sort
Terminology
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Replace image processing gem
Change the
Gemfile
Update User
model
Update Article model
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Replace image processing gem
Change image
processing gem
Update User
model
Update Article model
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Cycles
Change image
processing gem
Update User
model
Update Article model
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LESSON FROM GRAPH THEORY
Cycles cannot
be evaluated
incrementally
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Breaking Cycles
Change image
processing gem
Update User
model
Introduce
adapter
Update Article model
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Directed Acyclic
Graph (DAG)
Terminology
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Strangler
Fig Pattern
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COMBINATORICS
What are the risks
of a collision?
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SET THEORY, COMBINATORICS
Designing a DB schema
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PREDICATE LOGIC
Why do all? and any? behave
like that on an empty array?
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PREDICATE LOGIC, SET THEORY
Improve your
communication
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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
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Not "good at math"
Joël Quenneville @joelquen Principal developer at
thoughtbot