In the functional programming world, we rely on languages with type systems that help us write, extend and maintain our software. These type systems, such as the one in Haskell, are based on solid type theory. While we usually and easily see the benefits of having a static type system on simple types such as integers, we might fail to see the benefits of applying the same principles to user-defined data types. Continue reading

# Tag Archives: type theory

# Boeing and Deaths due to Software Bugs

The IT field has been aware for decades that software can kill people and so we are witness of this fact again with Boeing and its two airplanes 737 Max crashing and killing everyone on board due to erroneous software in the accompanying MCAS system. Continue reading

## Notice of AMS: Vladimir Voevodsky

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The American Mathematical Society wrote a memorial notice on Vladimir Voevodsky where, among others, his contributions to type theory are considered.

## Gentle Introduction to Homotopy Type Theory

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If you’re a programmer and you want to understand what is homotopy type theory about, Dan Licata has a talk on the topic.

## Interactive Theorem Proving

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Interactive theorem proving such as one provided by Agda is so empowering. The help provided by the compiler while interactively developing a proof is invaluable. Types guide you to a solution at every single step.

# Giving a Talk on Propositions as Types

On Tuesday, March 26, I’ll be giving a talk on propositions as types to the Lambda Zagreb Meetup group. The talk will be in Croatian and it is titled *¬ sve za ∀ (no ∃ uvod u propozicije kao tipove)*, which can be translated as *¬ everything is for ∀ (but there ∃ an introduction to propositions as types)*. Obviously, I incorporated logic operators into the title with the goal to make it catchy and to attract more audience with an unusual title. Given that only about 10 people RSVPed so far and that there are only three days left until the talk, it looks like I’ve achieved quite the opposite!

# Insisting on a Three-valued Boolean Type

If your program can have only two legal states, what sense does it make to extend a Boolean type with a third data constructor to account for the possibility of an “illegal program state”? Use a coproduct type with a Boolean type on the right instead. Make illegal program states irrepresentable in your types. I’m baffled by folks that insist on having their three-valued Boolean type.

## Comparison of Type Theory and Set Theory

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If you ever wondered how are types different from sets or more generally what is the difference between type theory and set theory, I recommend watching a talk by Thorsten Altenkirch at Foundations of Mathematics: Univalent Foundations and Set Theory 2016.

# Isomorphism and Embedding

To me as a programmer, to write mathematical proofs that are mechanically checked by a computer feels empowering. To have these proofs as executable programs feels even more empowering. Therefore, our proofs have a computational aspect and vice-versa: our programs have a logical aspect. To be able to get an instant feedback while proving a theorem is amazing. With Agda, a dependently typed functional programming language, one can interactively write a proof by getting guidance from Agda as to what is left to prove. Furthermore, Agda checks the correctness of proofs by following a set of rules. Unlike with pen and paper proofs, proofs in Agda are much more rigorous because there is no room for hand-waving nor unwarranted claims for something to be trivial. An uninformed mathematician will likely find this comparison to Agda hard to believe.

# Gave a Talk on Function Totality

As announced last week, yesterday I gave a talk on function totality at the Institute of Electrical and Electronics Engineers, Croatian Section. The talk was located at the University of Zagreb at the Faculty of Electrical Engineering and Computing. Slides used during the talk are freely available. The talk was in Croatian.