301AA - Advanced Programming [AP-2017]

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1 301AA - Advanced Programming [AP-2017] Lecturer: Andrea Corradini Tutor: Lillo GalleBa Department of Computer Science, Pisa Academic Year 2017/18 AP : Type inference

2 Type Inference Type inference: the process of associanng a type with each symbol of a program, if possible, ensuring type safety. Recalling several relevant defininons: (Data) types Type system Type error Type safety Dynamic vs. stanc type checking 2

3 Data types A (data) type is a homogeneous collecnon of values, effecnvely presented, equipped with a set of operanons which manipulate these values Various perspecnves: collecnon of values from a domain (the denotanonal approach) internal structure of a bunch of data, described down to the level of a small set of fundamental types (the structural approach) collecnon of well-defined operanons that can be applied to objects of that type (the abstracnon approach) 3

4 Type system A type system consists of 1. The set of predefined types of the language. 2. The mechanisms which permit the defini2on of new types. 3. The mechanisms for the control (checking) of types, which include: 1. Equivalence rules which specify when two formally different types correspond to the same type. 2. Compa2bility rules specifying when a value of a one type can be used in given context. 3. Rules and techniques for type inference which specify how the language assigns a type to a complex expression based on informanon about its components (and somenmes on the context). 4. The specificanon as to whether (or which) constraints are sta2cally or dynamically checked. 4

5 Type errors A type error occurs when a value is used in a way that is inconsistent with its defininon ImplementaNons can react in various ways Hardware interrupt, e.g. apply fp addi1on to non-legal bit configura1on OS excepnon, e.g. segmenta1on fault when dereferencing 0 in C ConNnue execunon possibly with wrong values Examples Array out of bounds access C/C++: runnme errors Java: dynamic type error Null pointer dereferencing C/C++: run-nme errors Java: dynamic type error Haskell/ML: pointers are hidden inside datatypes Null pointer dereferences would be incorrect use of these datatypes, therefore stanc type errors 5

6 Type safety A language is type safe (strongly typed) when no program can violate the disnncnons between types defined in its type system That is, when no program, during its execunon, can generate an unsignalled type error Also: if code accesses data, it is handled with the type associated with the creanon and previous manipulanon of that data 6

7 Type checking To prevent type errors, before any operanon is performed, its operands must be typechecked to ensure that they comply with the companbility rules of the type system E.g., indexing opera1on: check that the leg operand is an array, and that the right operand is a value of the array s index type. Sta2cally typed languages: (most) type checking is done during compilanon Dynamically typed languages: type checking is done at runnme 7

8 StaNc vs dynamic typing In a sta2cally typed PL: all variables and expressions have fixed types (either stated by the programmer or inferred by the compiler) most operands are type-checked at compile-1me. Most PLs are called stancally typed, including Ada, C, C++, Java, Haskell, even if some type-checking is done at run-nme (e.g. access to arrays) In a dynamically typed PL: values have fixed types, but variables and expressions do not operands must be type-checked when they are computed at run-1me. Some PLs and many scripnng languages are dynamically typed, including Smalltalk, Lisp, Prolog, Perl, Python. 8

9 Polymorphism in Haskell Coercion Type Inference Implicit Parametric Universal Explicit Bounded Polymorphism Inclusion Covariant Invariant Contravariant Ad hoc Overriding Overloading Type classes

10 Universal vs Ad-hoc Polymorphism Universal polymorphism Single algorithm may be given many types Type variables (implicit or explicit) may be replaced by any type (almost -> bounded polymorphism) head::[a] a thus head::[int] Int, head:[bool] Bool,... Ad-hoc polymorphism (overloading) A single symbol may refer to more than one algorithm. Each algorithm may have different type. Choice of algorithm determined by type context. + has types Int Int Int and Float Float Float, but not t t t for arbitrary t. 10

11 Type Checking vs Type Inference Standard type checking: int f(int x) { return x+1; }; int g(int y) { return f(y+1)*2; }; Examine body of each funcnon Use declared types to check agreement Type inference: int f(int x) { return x+1; }; int g(int y) { return f(y+1)*2; }; Examine code without type informanon. Infer the most general types that could have been declared. ML and Haskell are designed to make type inference feasible. 11

12 Why study type inference? Types and type checking Improved steadily since Algol 60 Eliminated sources of unsoundness. Become substannally more expressive. Important for modularity, reliability and compilanon Type inference Reduces syntacnc overhead of expressive types. Guaranteed to produce most general type. Increasingly used also in imperanve/oo languages. IllustraNve example of a flow-insensinve stanc analysis algorithm. 12

13 uhaskell Subset of Haskell to explain type inference. Will do not consider overloading now <decl> ::= <name> <pat> = <exp> <pat> ::= Id (<pat>, <pat>) <pat> : <pat> [] <exp> ::= Int Bool [] Id (<exp>) <exp> <op> <exp> <exp> <exp> (<exp>, <exp>) if <exp> then <exp> else <exp> 13

14 Type Inference: Basic Idea Example f x = 2 + x -- a simple declaration What is the type of f? + has type: Int Int Int (with overloading would be Num a => a a a) 2 has type: Int Since we are applying + to x we need x :: Int Therefore f x = 2 + x implies that f has type Int Int f x = 2 + x -- a simple declaration > f :: Int -> Int 14

15 Step 1: Parse Program Parse program text to construct parse tree. f x = 2 + x to represent application Ternary Fun-node for function definitions Infix operators are converted to Curried function application during parsing: 2 + x è (+) 2 x 15

16 Step 2: Assign type variables to nodes f x = 2 + x Variables are given same type as binding occurrence. 16

17 Constraints from ApplicaNon Nodes f x t_0 = t_1 -> t_2 FuncNon applicanon (apply f to x) Type of f (t_0 in figure) must be domain range. Domain of f must be type of argument x (t_1 in fig) Range of f must be result of applicanon (t_2 in fig) Constraint: t_0 = t_1 -> t_2 17

18 Constraints from AbstracNons f x = e t_0 = t_1 -> t_2 FuncNon declaranon: Type of f (t_0 in figure) must be domain range Domain is type of abstracted variable x (t_1 in fig) Range is type of funcnon body e (t_2 in fig) Constraint: t_0 = t_1 -> t_2 Idem for lambdas: f = \x -> e 18

19 Step 3: Add Constraints f x = 2 + x t_0 = t_1 -> t_6 t_4 = t_1 -> t_6 t_2 = t_3 -> t_4 t_2 = Int -> Int -> Int t_3 = Int 19

20 Step 4: Solve Constraints t_0 = t_1 -> t_6 t_4 = t_1 -> t_6 t_2 = t_3 -> t_4 t_2 = Int -> Int -> Int t_3 = Int t_0 = t_1 -> t_6 t_4 = t_1 -> t_6 t_4 = Int -> Int t_2 = Int -> Int -> Int t_3 = Int t_3 -> t_4 = Int -> (Int -> Int) t_3 = Int t_4 = Int -> Int t_1 -> t_6 = Int -> Int t_0 = Int -> Int t_1 = Int t_6 = Int t_4 = Int -> Int t_2 = Int -> Int -> Int t_3 = Int t_1 = Int t_6 = Int 20

21 Step 5: Determine type of declaranon t_0 = Int -> Int t_1 = Int t_6 = Int t_4 = Int -> Int t_2 = Int -> Int -> Int t_3 = Int f x = 2 + x > f :: Int -> Int 21

22 Type Inference Algorithm Parse program to build parse tree Assign type variables to nodes in tree Generate constraints: From environment: constants (2), built-in operators (+), known funcnons (tail). From form of parse tree: e.g., applicanon and abstracnon nodes. Solve constraints using Determine types of top-level declaranons 22

23 Inferring Polymorphic Types Example: Step 1: Build Parse Tree f g = g 2 > f :: (Int -> t_4) -> t_4 23

24 Inferring Polymorphic Types Example: Step 2: Assign type variables f g = g 2 > f :: (Int -> t_4) -> t_4 24

25 Inferring Polymorphic Types Example: Step 3: Generate constraints t_0 = t_1 -> t_4 t_1 = t_3 -> t_4 t_3 = Int f g = g 2 > f :: (Int -> t_4) -> t_4 25

26 Inferring Polymorphic Types Example: Step 4: Solve constraints t_0 = t_1 -> t_4 t_1 = t_3 -> t_4 t_3 = Int f g = g 2 > f :: (Int -> t_4) -> t_4 t_0 = (Int -> t_4) -> t_4 t_1 = Int -> t_4 t_3 = Int 26

27 Inferring Polymorphic Types Example: Step 5: Determine type of top-level declaranon Unconstrained type variables become polymorphic types. f g = g 2 > f :: (Int -> t_4) -> t_4 t_0 = (Int -> t_4) -> t_4 t_1 = Int -> t_4 t_3 = Int 27

28 Using Polymorphic FuncNons FuncNon: f g = g 2 > f :: (Int -> t_4) -> t_4 Possible applicanons: add x = 2 + x > add :: Int -> Int f add > 4 :: Int iseven x = mod (x, 2) == 0 > iseven:: Int -> Bool f iseven > True :: Int 28

29 Recognizing Type Errors FuncNon: f g = g 2 > f :: (Int -> t_4) -> t_4 Incorrect use not x = if x then True else False > not :: Bool -> Bool f not > Error: operator and operand don t agree operator domain: Int -> a operand: Bool -> Bool Type error: cannot unify Bool Bool and Int t 29

30 Polymorphic Datatypes FuncNons may have mulnple clauses, and may be recursive length [] = 0 length (x:rest) = 1 + (length rest) Type inference Infer separate type for each clause Add constraint that all clauses must have the same type Recursive calls: funcnon has same type as its defininon 30

31 Type Inference with Datatypes Example: Step 1: Build Parse Tree length (x:rest) = 1 + (length rest) 31

32 Type Inference with Datatypes Example: length (x:rest) = 1 + (length rest) Step 2: Assign type variables 32

33 Type Inference with Datatypes Example: length (x:rest) = 1 + (length rest) Step 3: Generate constraints t_0 = t_3 -> t_10 t_3 = t_2 t_3 = [t_1] t_6 = t_9 -> t_10 t_4 = t_5 -> t_6 t_4 = Int -> Int -> Int t_5 = Int t_0 = t_2 -> t_9 33

34 Type Inference with Datatypes Example: Step 3: Solve Constraints length (x:rest) = 1 + (length rest) t_0 = t_3 -> t_10 t_3 = t_2 t_3 = [t_1] t_6 = t_9 -> t_10 t_4 = t_5 -> t_6 t_4 = Int -> Int -> Int t_5 = Int t_0 = t_2 -> t_9 t_0 = [t_1] -> Int 34

35 MulNple Clauses FuncNon with mulnple clauses Infer type of each clause First clause: append ([],r) = r append (x:xs, r) = x : append (xs, r) > append :: ([t_1], t_2) -> t_2 Second clause: > append :: ([t_3], t_4) -> [t_3] Combine by equanng types of two clauses > append :: ([t_1], [t_1]) -> [t_1] 35

36 Most General Type Type inference produces the most general type map (f, [] ) = [] map (f, x:xs) = f x : map (f, xs) > map :: (t_1 -> t_2, [t_1]) -> [t_2] FuncNons may have many less general types > map :: (t_1 -> Int, [t_1]) -> [Int] > map :: (Bool -> t_2, [Bool]) -> [t_2] > map :: (Char -> Int, [Char]) -> [Int] Less general types are all instances of most general type, also called the principal type 36

37 History Original type inference algorithm Invented by Haskell Curry and Robert Feys for the simply typed lambda calculus in 1958 In 1969, Hindley extended the algorithm to a richer language and proved it always produced the most general type In 1978, Milner independently developed equivalent algorithm, called algorithm W, during his work designing ML. In 1982, Damas proved the algorithm was complete. Currently used in many languages: ML, Ada, Haskell, C# 3.0, F#, Visual Basic.Net 9.0. Have been plans for Fortress, Perl 6, C+ +0x, A bit also in Java. 37

38 Complexity of Type Inference Algorithm When Hindley/Milner type inference algorithm was developed, its complexity was unknown In 1989, Kanellakis, Mairson, and Mitchell proved that the problem was exponennal- Nme complete Usually linear in pracnce though Running Nme is exponennal in the depth of polymorphic declaranons 38

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