Value Your Types!

You’re probably familiar with types in programming languages, such as “integer” or “list of integers.” But what if your type system were powerful enough to express types like “non-negative integer” or “list of strings where each string is at least eight characters long”? Welcome to the world of dependent types! In this talk, we’ll explore what dependent types are, how they work, and the relationship between type theory and predicate logic, all using the Idris programming language. We’ll even see when type checking can become (gasp) undecidable!

Transcript

1. Va1u3 Your TYPES! A !!Con West Adventure Eric Weinstein UC Santa Cruz, California, USA 24 February 2019
2. eric@testdouble.com
3. @ericqweinstein
4. @ericqweinstein
5. "queue"
6. Dependent Types! A type whose definition depends on a value "List of integers" is a type; "list of integers where each successive value is strictly larger than its predecessor" is a dependent type!
7. History! 1934: Haskell Curry notices similarities between the typed lambda calculus and propositional logic 1960s: William Howard and Nicolaas De Bruijn extend Curry’s work to predicate logic
8. History! Credit: https://manishearth.github.io/blog/2017/03/04/what-are-sum-product-and-pi-types/
9. History! Credit: https://manishearth.github.io/blog/2017/03/04/what-are-sum-product-and-pi-types/ ∀x: fn() → Array<bool, x> "For all x, we can construct this array!" ∃x: fn() → Array<bool, x> "There exists an x such that we can construct this array!"
10. History! 1934: Haskell Curry notices similarities between the typed lambda calculus and propositional logic 1960s: William Howard and Nicolaas De Bruijn extend Curry’s work to predicate logic
11. History! 1934: Haskell Curry notices similarities between the typed lambda calculus and propositional logic 1960s: William Howard and Nicolaas De Bruijn extend Curry’s work to predicate logic
12. The Curry-Howard Correspondence!
14. Idris! data Vect : Nat -> Type -> Type where Nil : Vect 0 a (::) : (x : a) -> (xs : Vect n a) -> Vect (n + 1) a Credit: https://en.wikipedia.org/wiki/Idris_(programming_language)#Dependent_types
15. Idris! data Vect : Nat -> Type -> Type where Nil : Vect Z a (::) : a -> Vect n a -> Vect (S n) a Credit: https://www.idris-lang.org/example/
16. Idris! total pairAdd : Num a => Vect n a -> Vect n a -> Vect n a pairAdd Nil Nil = Nil pairAdd (x :: xs) (y :: ys) = x + y :: pairAdd xs ys Credit: https://en.wikipedia.org/wiki/Idris_(programming_language)#Dependent_types
17. Idris! total append : Vect n a => Vect m a -> Vect (n + m) a append Nil ys = ys append (x :: xs) ys = x :: append xs ys Credit: https://en.wikipedia.org/wiki/Idris_(programming_language)#Dependent_types
18. Idris! total append : Vect n a => Vect m a -> Vect (n + m) a append Nil ys = ys append (x :: xs) ys = x :: append xs ys Credit: https://en.wikipedia.org/wiki/Idris_(programming_language)#Dependent_types
19. Idris! Credit: https://github.com/HoTT/book/issues/743
20. Aside! #
21. # Aside!
22. Aside! #
23. Aside!
24. Aside!
25. Aside!
26. ∴ Aside!
27. ? ? ? Aside!
28. Undecidable!
29. Credit: The Princess Bride (1987)
30. Undecidable! When types contain values, they become little programs... ...and general program equivalence is undecidable! !!
31. Undecidable! "At compile-time it will only evaluate things which it knows to be total (i.e. terminating and covering all possible inputs) in order to keep type checking decidable." !! Credit: https://docs.idris-lang.org/en/latest/faq/faq.html
33. Papers! http://www.cse.chalmers.se/~peterd/papers/ DependentTypesAtWork.pdf https://dl.acm.org/citation.cfm?id=671431 http://www.cis.upenn.edu/~plclub/lambda-eek/lambda-eek.pdf http://www.kb.ecei.tohoku.ac.jp/~terauchi/papers/popl10- depceg.pdf
34. TL;DPA Dependent types are amazing! Idris is an amazing language and you should try it!!
35. Credit: DefendingDRealm (https://www.redbubble.com/)