Type Theory

Part I - Logic

GIF

Why Type Theory??

When we write code

How do we guarantee our program meets the spec?

Most of the time is spent with "I think..."

Only formal proofs could guarantee something more precise

Formal proofs => Logic?

What's the relationship between programming and logic?

Programing

Look at the compiler or lexical structure of a language

It's a bunch of logical statements

Programming language can be seen as a logical system

Propositions and types become identical

How accurate are you with programming specs?

We usually do the reverse that we do in Logic

We write some code and after that we try to bring up a "proof"

Would it be possible to have a way to "pre-prove" our code?

Maybe some formalism could help

Can you think about strong typed languages and the opposite?

If you have ensured types, doing this proof would require less steps

Let's talk about ensured types, quickly

p:P

WTF???

• This is not an emoji
• "The value p has the type P"
• And so what? Then we analyse the functions, implications and mathematical constructions.

p:P

• Look carefully and think about functions (form Algebra)
• D(f):{2,9} or 
• Value and Types could express domains and ranges in functions!

And this is a bit about Type Theory

Type Theory

Logic

Type Theory

λ-calculus

Functional Programming

Mathematical Constructions

Logic

Nice course

https://www.coursera.org/course/maththink

@profkeithdevlin

About Logic

"If it's contradictory for no object x has the property P(x), then there's an object x with the property P(x)."

Concept #1

Constructive vs non-constructive

Constructive Logic

What is love ?

Logic

todo add gif

Science of the argument

What is correct?

• Write arguments that can be checked mechanically
• Study properties
• Elaborate semantic meanings

Concept #2

Types of Logic (Type(type(type))

Propositional Logic

and, or, not

https://github.com/dyokomizo/type-algebra-slides

Type Algebras

Predicate Logic

Relating with Functions

Next steps: lamba calculus

References and special thanks

• GIRARD, Jean-Yves. Proofs and Types.
• @lafp, @romulostotel, @pedrofelipee: GIFs
• @klauss_weyl, for showing me the importance of logic and Type Theory
• Henrique

Thank you :)

Questions?

hannelita@gmail.com

@hannelita