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
   

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2. eric@testdouble.com
   

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3. @ericqweinstein
   

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4. @ericqweinstein
   

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5. "queue"
   

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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!
   

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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
   

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8. History!
   Credit: https://manishearth.github.io/blog/2017/03/04/what-are-sum-product-and-pi-types/
   

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9. History!
   Credit: https://manishearth.github.io/blog/2017/03/04/what-are-sum-product-and-pi-types/
   ∀x: fn() → Array
   "For all x, we can construct this array!"
   ∃x: fn() → Array
   "There exists an x such that we can construct this array!"
   

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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
    

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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
    

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12. The Curry-Howard
    Correspondence!
    

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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
    

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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/
    

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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
    

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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
    

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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
    

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19. Idris!
    Credit: https://github.com/HoTT/book/issues/743
    

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20. Aside!
    #
    

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21. #
    Aside!
    

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22. Aside!
    #
    

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23. 
    Aside!
    

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24. 
    Aside!
    

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25. 
    Aside!
    

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26. ∴
    Aside!
    

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27. ? ? ?
    Aside!
    

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28. Undecidable!
    

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29. Credit: The Princess Bride (1987)
    

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30. Undecidable!
    When types contain values, they become little programs...
    ...and general program equivalence is undecidable!
    !!
    

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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
    

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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
    

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34. TL;DPA
    Dependent types are amazing!
    Idris is an amazing language and you should try it!!
    

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35. Credit: DefendingDRealm (https://www.redbubble.com/)
    

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