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Tom Schrijvers K.U.Leuven, Belgium with Manuel Chakravarty, Martin Sulzmann and Simon Peyton Jones

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Type-level functions Functional programming at the type level! Examples Usefulness Interaction Type Checking Eliminating functional dependencies Examples Usefulness Interaction Type Checking Eliminating functional dependencies

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Examples

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Example: 1 + 1 = 2? -- Peano numerals data Z data Succ n -- Abbreviations type One = Succ Z type Two = Succ One -- Type-level addition type family Sum m n type instance Sum Z n = n type instance Sum (Succ a) b = Succ (Sum a b) -- Peano numerals data Z data Succ n -- Abbreviations type One = Succ Z type Two = Succ One -- Type-level addition type family Sum m n type instance Sum Z n = n type instance Sum (Succ a) b = Succ (Sum a b) type function declaration Sum :: * -> * -> * type function declaration Sum :: * -> * -> * 1 st type function instance 2 nd type function instance

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Example: 1 + 1 = 2? -- Type-level addition type family Sum m n type instance Sum Z nat = nat type instance Sum (Succ a) b = Succ (Sum a b) -- Hypothesis hypthesis :: (Sum One One, Two) hypothesis = undefined -- Test test :: (a,a) -> Bool test _ = True -- Type-level addition type family Sum m n type instance Sum Z nat = nat type instance Sum (Succ a) b = Succ (Sum a b) -- Hypothesis hypthesis :: (Sum One One, Two) hypothesis = undefined -- Test test :: (a,a) -> Bool test _ = True

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Example: 1 + 1 = 2? -- Hypothesis hypthesis :: (Sum One One, Two) hypothesis = undefined -- Test test :: (a,a) -> Bool test _ = True -- Proof proof :: Bool proof = test hypothesis -- Hypothesis hypthesis :: (Sum One One, Two) hypothesis = undefined -- Test test :: (a,a) -> Bool test _ = True -- Proof proof :: Bool proof = test hypothesis static check: 1+1 = 2

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Example: Length-indexed Lists -- Length-indexed lists : vectors data Vec e l where VNil :: Vec e Z VCons :: e -> Vec e l -> Vec e (Succ l) -- Safe zip: matched lengths vzip :: Vec a l -> Vec b l -> Vec (a,b) l vzip VNil VNil = VNil vzip (VCons x xs) (VCons y ys) = VCons (x,y) (vzip xs ys) -- Length-indexed lists : vectors data Vec e l where VNil :: Vec e Z VCons :: e -> Vec e l -> Vec e (Succ l) -- Safe zip: matched lengths vzip :: Vec a l -> Vec b l -> Vec (a,b) l vzip VNil VNil = VNil vzip (VCons x xs) (VCons y ys) = VCons (x,y) (vzip xs ys)

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Example: Length-indexed Lists -- Safe zip: matched lengths vzip :: Vec a l -> Vec b l -> Vec (a,b) l vzip VNil VNil = VNil vzip (VCons x xs) (VCons y ys) = VCons (x,y) (vzip xs ys) -- Safe zip: matched lengths vzip :: Vec a l -> Vec b l -> Vec (a,b) l vzip VNil VNil = VNil vzip (VCons x xs) (VCons y ys) = VCons (x,y) (vzip xs ys) > let l1 = (VCons 1 VNil) l2 = (VCons a (VCons b VNil)) in vzip l1 l2 12

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Example: Length-indexed Lists -- Safe zip: matched lengths vzip :: Vec a l -> Vec b l -> Vec (a,b) l vzip VNil VNil = VNil vzip (VCons x xs) (VCons y ys) = VCons (x,y) (vzip xs ys) -- Safe zip: matched lengths vzip :: Vec a l -> Vec b l -> Vec (a,b) l vzip VNil VNil = VNil vzip (VCons x xs) (VCons y ys) = VCons (x,y) (vzip xs ys) > let l1 = (VCons 1 (VCons 2 VNil)) l2 = (VCons a (VCons b VNil)) in vzip l1 l2 VCons (1,a) (VCons (2,b) VNil) 2=2

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Example: Length-indexed lists -- concatenation vconc :: Vec a m -> Vec a n -> Vec a ??? vconc VNil VNil = VNil Vconc (VCons x xs) ys = VCons x (vconc xs ys) -- concatenation vconc :: Vec a m -> Vec a n -> Vec a ??? vconc VNil VNil = VNil Vconc (VCons x xs) ys = VCons x (vconc xs ys)

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Example: Length-indexed lists -- concatenation vconc :: Vec a m -> Vec a n -> Vec a (Sum m n) vconc VNil VNil = VNil Vconc (VCons x xs) ys = VCons x (vconc xs ys) -- concatenation vconc :: Vec a m -> Vec a n -> Vec a (Sum m n) vconc VNil VNil = VNil Vconc (VCons x xs) ys = VCons x (vconc xs ys) > let l1 = (VCons 1) l2 = (VCons a (VCons b VNil)) l3 = vconc l1 l1 in vzip l3 l2 VCons (1,a) (VCons (1,b) VNil) 1+1=2

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Example: Collections (1/4) class Collection c where type Elem c add :: Elem c -> c -> c list :: c -> [Elem c] elem :: c -> Elem c Associated type function Elem :: * -> * Associated value function elem :: c -> Elem c

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Example: Collections (2/4) instance Collection [e] where type Elem [e] = e instance Collection (Tree e) where type Elem (Tree e) = e instance Collection BitVector where type Elem BitVector = Bit type functions are open!

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instance Collection [e] where type Elem [e] = e add :: e -> [e] -> [e] add x xs = x : xs list :: [e] -> [ e ] list xs = xs Example: Collections (3/4) instance Collection [e] where type Elem [e] = e add :: Elem [e] -> [e] -> [e] add x xs = x : xs list :: [e] -> [Elem [e]] list xs = xs

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addAll :: c1 -> c2 -> c2 addAll xs ys = foldr add ys (list xs) Example: Collections (4/4) addAll :: Collection c1 => c1 -> c2 -> c2 addAll xs ys = foldr add ys (list xs) addAll :: Collection c1, Collection c2 => c1 -> c2 -> c2 addAll xs ys = foldr add ys (list xs) addAll :: Collection c1, Collection c2, Elem c1 ~ Elem c2=> c1 -> c2 -> c2 addAll xs ys = foldr add ys (list xs) > addAll [0,1,0,1,0] emptyBitVector context constraint, satisfied by caller context constraint, satisfied by caller Elem [Bit] ~ Elem BitVector

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Summary of Examples functions over types open definitions stand-alone associated to a type class equational constraints in signature contexts usefulness limited on their own really great with GADTs type classes

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Type Checking

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Compiler Overview Haskell Constraints Type Checker Coercions System F C

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Elem [Int] ~ Int ? γ Int :: Elem [Int] ~ Int ! Compiler Overview (elem [0]) (γ Int) == 0 elem [0] == 0γ : e.Elem [e] = e hypothesis axiom proof label witness cast

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Basic Hindley-Milner [Int] ~ [Int] syntactic equality (unification) Type Functions Elem [Int] ~ Int syntactic equality modulo equational theory Type Checking = Syntactical?

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Type Checking = Term Rewriting γ : e.Elem [e] = e Elem [Int] ~ Int Int γ Int Int Equational theory = given equations Toplevel equations: E t Context equations: E g Theorem proving Operational interpretation of theory = Term Rewriting System (Soundness!)

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Completeness TRS must be Strongly Normalizing (SN): Confluent: every type has a canonical form Terminating TRS must be Strongly Normalizing (SN): Confluent: every type has a canonical form Terminating Practical & Modular Conditions: Confluence: no overlap of rule heads Terminating: Decreasing recursive calls No nested calls Practical & Modular Conditions: Confluence: no overlap of rule heads Terminating: Decreasing recursive calls No nested calls

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Completeness Conditions Practical & Modular Conditions: Confluence: no overlap Elem [e] = e Elem [Int] = Bit Terminating: Decreasing recursive calls Elem [e] = Elem [[e]] No nested calls Elem [e] = Elem (F e) F e = [e] Practical & Modular Conditions: Confluence: no overlap Elem [e] = e Elem [Int] = Bit Terminating: Decreasing recursive calls Elem [e] = Elem [[e]] No nested calls Elem [e] = Elem (F e) F e = [e]

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Term Rewriting Conditions EtEt EgEg E g We can do better: E t : satisfy conditions E g : free form(but no schema variabels) We can do better: E t : satisfy conditions E g : free form(but no schema variabels) EtEt Completion!

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Type Checking Summary EtEt EgEg t1 ~ t2 E g t ~ t 1.Completion 2.Rewriting 3.Syntactic Equality Its all been implemented in GHC! Its all been implemented in GHC!

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Eliminating Functional Dependencies

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Functional Dependencies Using relations as functions Logic Programming! class Collection c e | c -> e where add :: e -> c -> c list :: c -> [e] elem :: c -> [e] instance Collection [e] e where...

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FDs: Two Problems Haskell implementors are still debugging FD type checking Haskell programmers know Functional Programming FDs = Logic Programming

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FDs: Solution Functional Dependencies Type Functions

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FDs: Solution 2 Backward Compatibility??? Expressiveness lost??? automatic transformation

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class FD c ~ e => Collection c e where type FD c add :: e -> c -> c list :: c -> [e] elem :: c -> e instance Collection [e] e where... class FD c ~ e => Collection c e where type FD c add :: e -> c -> c list :: c -> [e] elem :: c -> e instance Collection [e] e where type FD [e] = e class Collection c e | c -> e where add :: e -> c -> c list :: c -> [e] elem :: c -> [e] instance Collection [e] e where... class Collection c e | c -> e where type FD c add :: e -> c -> c list :: c -> [e] elem :: c -> e instance Collection [e] e where... FDs: Simple Transformation Minimal Impact: class and instance declarations only

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class FD c ~ e => Collection c where type FD c add :: e -> c -> c list :: c -> [e] elem :: c -> e instance Collection [e] e where type FD [e] = e class Collection c where type FD c add :: FD c -> c -> c list :: c -> [FD c] elem :: c -> FD c instance Collection [e] where type FD [e] = e FDs: Advanced Transformation Drop dependent parameters class FD c ~ e => Collection c e where type FD c add :: e -> c -> c list :: c -> [e] elem :: c -> e instance Collection [e] e where type FD [e] = e

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class FD c ~ e => Collection c e where type FD c addAll :: Collection c1 e, Collection c2 e =>... FDs: Advanced Transformation Bigger Impact: signature contexts too class Collection c where type FD c addAll :: Collection c1 e, Collection c2 e =>... class Collection c where type FD c addAll :: Collection c1, Collection c2, Elem c1 ~ Elem c2 =>...

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Conclusion

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Type checking with type functions = Term Rewriting sound, complete and terminating completion algorithm Seemless integration in Haskells rich type system Type classes GADTs Understanding functional dependencies better Type checking with type functions = Term Rewriting sound, complete and terminating completion algorithm Seemless integration in Haskells rich type system Type classes GADTs Understanding functional dependencies better

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Future Work Improvements and variations Weaker termination conditions Closed type functions Rational tree types Efficiency Applications: Port libraries Revisit type hacks Write some papers! Improvements and variations Weaker termination conditions Closed type functions Rational tree types Efficiency Applications: Port libraries Revisit type hacks Write some papers!

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Extra Slide

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Rational tree types -- Non-recursive list data List e t = Nil | Cons e t -- Fixedpoint operator type family Fix (k :: *->*) type instance Fix k = k (Fix k) -- Tie the knot type List e = Fix (List e) = List e (List e (List e …))

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