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Va1u3 Your TYPES! A !!Con West Adventure Eric Weinstein UC Santa Cruz, California, USA 24 February 2019

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eric@testdouble.com

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@ericqweinstein

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@ericqweinstein

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"queue"

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

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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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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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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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The Curry-Howard Correspondence!

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

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Aside! #

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# Aside!

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Aside! #

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Aside!

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Aside!

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Aside!

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∴ Aside!

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? ? ? Aside!

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Undecidable!

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Credit: The Princess Bride (1987)

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

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

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