1. Type Inference: CIS Seminar, 11/3/2009 Type inference: Inside the Type Checker. A presentation by: Daniel Tuck

2. Type Inference: CIS Seminar, 11/3/2009 Types in Computer Languages Dynamic Languages –(p*, Ruby, Lisp) – Provide no compile time type checking: type errors occur at runtime. Static Languages –Eliminates most runtime errors – type conflicts recognized at compile time –Explicit (Java, C++) – Types must be declared for every name –Implicit (Haskell, ML) – Types don’t have to be declared as they are inferred. However types may be declared to improve readability, or remove ambiguity.

3. Type Inference: CIS Seminar, 11/3/2009 Type Inference Wikipedia - Type inference refers to the ability to automatically either partially or fully deduce the type of the value derived from the eventual evaluation of an expression. A type expresses a common property of all values an expression might assume.

4. Type Inference: CIS Seminar, 11/3/2009 History Type checking has traditionally been done "bottom up" – if you know the types of all arguments to a function you know the type of the result. 1958: Haskell Curry and Robert Feys develop a type inference algorithm for the simply typed lambda calculus. 1969: Roger Hindley extends this work and proves his algorithm infers the most general type. 1978: Robin Milner, independently of Hindley's work, develops equivalent algorithm 2004: Java 5 Adopts the H-M algorithm and type inference becomes respectable

5. Type Inference: CIS Seminar, 11/3/2009 World's geekiest T-shirt. This is the entire H-M algorithm expressed as logic.

6. Type Inference: CIS Seminar, 11/3/2009 Haskell vs Java Everything we'll do works in both Haskell and Java. We'll use Haskell because the notations are much simpler.

7. Type Inference: CIS Seminar, 11/3/2009 Haskell Expressions Haskell uses spaces instead of (,, ) for arguments to a function. add 1 4 instead of add(1,4) Lists in Haskell are compromised of two brackets with commas as delimiters: [1,4] Add an element to a list with ":": x:xs Function definition uses pattern matching append [] x = x append (y:ys) x = y : append ys x

8. Type Inference: CIS Seminar, 11/3/2009 Haskell Expressions Haskell uses spaces instead of (,, ) for arguments to a function. add 1 4 instead of add(1,4) Lists in Haskell are compromised of two brackets with commas as delimiters: [1,4] Add an element to a list with ":": x:xs Function definition uses pattern matching append [] x = x append (y:ys) x = y : append ys x Two definitions of append – Haskell tries these in order

9. Type Inference: CIS Seminar, 11/3/2009 Haskell Expressions Haskell uses spaces instead of (,, ) for arguments to a function. add 1 4 instead of add(1,4) Lists in Haskell are compromised of two brackets with commas as delimiters: [1,4] Add an element to a list with ":": x:xs Function definition uses pattern matching append [] x = x append (y:ys) x = y : append ys x Match first argument against the empty list

10. Type Inference: CIS Seminar, 11/3/2009 Haskell Expressions Haskell uses spaces instead of (,, ) for arguments to a function. add 1 4 instead of add(1,4) Lists in Haskell are compromised of two brackets with commas as delimiters: [1,4] Add an element to a list with ":": x:xs Function definition uses pattern matching append [] x = x append (y:ys) x = y : append ys x This takes a list apart into the head (y) and tail (ys)

11. Type Inference: CIS Seminar, 11/3/2009 Haskell Expressions Haskell uses spaces instead of (,, ) for arguments to a function. add 1 4 instead of add(1,4) Lists in Haskell are compromised of two brackets with commas as delimiters: [1,4] Add an element to a list with ":": x:xs Function definition uses pattern matching append [] x = x append (y:ys) x = y : append ys x Since this isn't in a pattern it adds y to the front of the list

12. Type Inference: CIS Seminar, 11/3/2009 Haskell Types Primitive types: Integer, String, Bool Type declaration: f :: Integer Structured types: –Function: g :: Integer -> String –Multiple argument functions: h :: String -> Integer -> Bool –List: x :: [Integer] –Functional arguments: f1 :: [Integer] -> (Integer -> Integer) -> Integer

13. Type Inference: CIS Seminar, 11/3/2009 Polymorphism The Haskell map function takes 2 arguments. The first being a function, and the second being a list of inputs the function should be applied to. The output is a list of result values. Input: map abs [-1,-3,4,-12] Output: [1,3,4,12] Input: map reverse ["abc","cda","1234"] Output: ["cba","adc","4321"] Input: map (3*) [1,2,3,4] Output: [3,6,9,12]

14. Type Inference: CIS Seminar, 11/3/2009 Type Variables Haskell determines if a type name is an actual type (like Integer ) or parameter type (type variable) by the case of the first character. append :: [a] -> [a] -> [a] Type variable – means that any type can be in the list Reuse of "a" means that both parameters to append must share the same type Java: public List append(List x, List y)

15. Type Inference: CIS Seminar, 11/3/2009 Type inference We can infer the type of the following definition using the H-M algorithm: map f [] = [] map f (x:xs) = f x : map f xs

16. Type Inference: CIS Seminar, 11/3/2009 Type inference We can infer the type of the following definition using the H-M algorithm: map f [] = [] map f (x:xs) = f x : map f xs map :: a -> b -> c #1: Map has two arguments (observed from the pattern matching)

17. Type Inference: CIS Seminar, 11/3/2009 Type inference We can infer the type of the following definition using the H-M algorithm: map f [] = [] map f (x:xs) = f x : map f xs map :: a -> [b] -> [c] #2: The second argument and the returned value must be a list (observing the first clause and the [ ]s)

18. Type Inference: CIS Seminar, 11/3/2009 Type inference We can infer the type of the following definition using the H-M algorithm: map f [] = [] map f (x:xs) = f x : map f xs map :: (d -> e) -> [b] -> [c] #3: The first argument to map is used as a function

19. Type Inference: CIS Seminar, 11/3/2009 Type inference We can infer the type of the following definition using the H-M algorithm: map f [] = [] map f (x:xs) = f x : map f xs map :: (b -> e) -> [b] -> [c] #4: The input to the function is taken from the second argument list

20. Type Inference: CIS Seminar, 11/3/2009 Type inference We can infer the type of the following definition using the H-M algorithm: map f [] = [] map f (x:xs) = f x : map f xs map :: (b -> c) -> [b] -> [c] #5: The output of the function is added to the result list

21. Type Inference: CIS Seminar, 11/3/2009 Type inference We can infer the type of the following definition using the H-M algorithm: map f [] = [] map f (x:xs) = f x : map f xs map :: (b -> c) -> [b] -> [c] #6: This type is consistent with all other parts of map so it is the inferred type of map

22. Type Inference: CIS Seminar, 11/3/2009 Your turn. Easy What is the type of the following function inc n = n + 1 Java - Public int inc(int n){return n+1} This is a function that takes a Integer and results in an Integer (the type of "+" determines this) inc : Integer -> Integer

23. Type Inference: CIS Seminar, 11/3/2009 Your turn. Medium What is the type of the following function f n = if n==0 then 1 else n*f(n-1) This is again a function that takes an Integer and results in an Integer. inc : Integer -> Integer

24. Type Inference: CIS Seminar, 11/3/2009 Type Constraints f n = if n==0 then 1 else n*f(n-1) N must be an Integer because of comparison with 0. The same applies here. EVERY one of these constraints must be satisfied or the inference process will yield a type error.

25. Type Inference: CIS Seminar, 11/3/2009 Your turn. Hard What is the following function’s type? length [] = 0 length (x:xs) = 1 + length xs The answer is that the input must be a list but since we never use the elements of the list it can contain anything. The result is a integer. length :: [a] -> Integer

26. Type Inference: CIS Seminar, 11/3/2009 Polymorphism Here's the BIG IDEA behind H-M: When you infer a type and parts of the type are "unconstrained", you can use ANY type for these unconstrained values. Type variables in signatures represent these unconstrained type components. When the result of type inference contains a type variable, we can reuse the function for any type of argument. a = length [1,2,3] b = length [True, False, True]

27. Type Inference: CIS Seminar, 11/3/2009 Haskell/Java Comparison Haskell: length [] = 0 length (x:xs) = 1 + length xs Java: static int length(List x){ if (null x) return 0; return 1 + length(x.next);}

28. Type Inference: CIS Seminar, 11/3/2009 Implementing H-M Typechecking How do we actually implement H-M typechecking? Logic programming to the rescue! We represent unknown types with logic variables. Constraints are represented by unifications Unlike Prolog, if unification fails you immediately quit and complain about the code you're checking.

29. Type Inference: CIS Seminar, 11/3/2009 A H-M Typechecker To typecheck a function: a) Assign a new type variable (logic variable) to each name being defined. b) Every occurrence of known function / variable requires a fresh copy (new logic variables) of the type of the type of that function. c) Every syntactic construct adds type constraints (unification) d) When done, if a logic variable is unbound this will turn into a type variable.

30. Type Inference: CIS Seminar, 11/3/2009 Syntax Type Constraints Function call: f x The type of f must be a -> b The type of x must be a The resulting type is b If – then – else: The type of the test must be Bool The then clause and else clause must have the same type. List construction: x:y The type of x is "a", the type of y must be [a], and the result is type [a]. Operators like + or – constrain all types to Integer

31. Type Inference: CIS Seminar, 11/3/2009 H-M In Action length [ ] = 0 length (first:rest) = 1 + length rest α0 α1 α0 α2 Integer Integer -> Integer -> Integer Assign types to all names in the code

32. Type Inference: CIS Seminar, 11/3/2009 H-M In Action length [ ] = 0 length (first:rest) = 1 + length rest α0 α1 α0 α2 Integer Integer -> Integer -> Integer Function definition of length: a0  a3 -> a4 Unification: All occurrences of a0 become a3 -> a4 a0  a3 -> a4

33. Type Inference: CIS Seminar, 11/3/2009 H-M In Action length [ ] = 0 length (first:rest) = 1 + length rest α0 α1 α0 α2 Integer Integer -> Integer -> Integer [ ] argument to length: a3  [a5] The empty list is polymorphic Manufacture new logic variables! a0  a3 -> a4 a3  [a5]

34. Type Inference: CIS Seminar, 11/3/2009 H-M In Action length [ ] = 0 length (first:rest) = 1 + length rest α0 α1 α0 α2 Integer Integer -> Integer -> Integer Right hand side of definition: a4  Integer a0  a3 -> a4 a3  [a5] a4  Integer

35. Type Inference: CIS Seminar, 11/3/2009 H-M In Action length [ ] = 0 length (first:rest) = 1 + length rest α0 α1 α0 α2 Integer Integer -> Integer -> Integer Type of first:rest a2  [a6] a1  a6 a3  [a6] Because a3 is the first argument to length a0  a3 -> a4 a3  [a5] a4  Integer a2  [a6] a1  a6 a3  [a6]

36. Type Inference: CIS Seminar, 11/3/2009 H-M In Action length [ ] = 0 length (first:rest) = 1 + length rest α0 α1 α0 α2 Integer Integer -> Integer -> Integer The "+" adds the following constraints: Integer  Integer (1 st argument) Integer  a4 (result of length, 2 nd argument) Integer  a4 (result of length) a0  a3 -> a4 a3  [a5] a4  Integer a2  [a6] a1  a6 a3  [a6] Integer  a4

37. Type Inference: CIS Seminar, 11/3/2009 H-M In Action length [ ] = 0 length (first:rest) = 1 + length rest α0 α1 α0 α2 Integer Integer -> Integer -> Integer In the end, a0 contains the final type: length :: [a2] -> Integer This is generalized as length :: [a] -> Integer (or a: a -> Integer) A a0  a3 -> a4 a3  [a5] a4  Integer a2  [a6] a1  a6 a3  [a6] Integer  a4 length :: [a2] -> Integer

38. Type Inference: CIS Seminar, 11/3/2009 Inside H-M The order in which constraints are added to the type environment doesn't matter – all the algorithm has to do is ensure that every constraint is accounted for. Note that inside "length" every occurrence of length is represented by the same type variable. After generalization, each call to length generates a fresh type variable. This is the key insight of the H-M algorithm.

39. Type Inference: CIS Seminar, 11/3/2009 Why? Polymorphic functions are extremely useful but writing out their types is tedious. H-M can check type consistency (Java) or infer types from scratch (Haskell). Q: Why don’t all languages have generics? –A: As a matter of fact many languages are adding Generics to their suite of tools. Even Visual Basic now has this! Q: Why aren’t there implicit types in Java? –A: H-M doesn't "play nice" with the object-oriented part of the Java type system. Q: What good is all of this for a Java programmer? –A: Generics avoid casting to Object – something that can be unsafe in Java.

40. Type Inference: CIS Seminar, 11/3/2009 Conclusions As more languages adopt generic typing, the H-M algorithm has become important to understanding how languages work. H-M is simple (sort of!) Many extensions to H-M have been developed to allow more precise typing (catch more errors at compile time). Type checking can address security, safety (null pointers, array out of bounds errors), and many other important program properties. Types give strong guarantees of program correctness. Everyone should learn more math (logic!)