Type Checking and Type Inference

cs7120 (Prasad) L2-TypeInference 1

Type Checking and Type Inference


Motivation• Application Programmers

– Reliability• Logical and typographical errors manifest themselves as

type errors that can be caught mechanically, thereby increasing our confidence in the code execution.

• Language Implementers– Storage Allocation (temporaries)

– Generating coercion code

– Optimizations


Evolution of Type System• Typeless

– Assembly language• Any instruction can be

run on any data “bit pattern”

• Implicit typing and coercion

– FORTRAN

• Explicit type declarations

– Pascal• Type equivalence

• Weak typing– C

• Arrays (bounds not checked), Union type

• Actuals not checked against formals.

• Data Abstraction– Ada, CLU

• Type is independent of representation details.

• Generic Types– Ada, Java/C++/C#

• Compile-time facility for “container” classes.

• Reduces source code duplication.


• Languages– Strongly typed (“Type errors always caught.”)

• Statically typed (e.g., Ada, COOL and Eiffel)– Compile-time type checking : Efficient.

• Dynamically typed (e.g., Scheme and Smalltalk)– Run-time type checking : Flexible.

– Weakly typed (e.g., C)– Unreliable Casts (int to/from pointer).

– Object-Oriented Languages such as C# and Java impose restrictions that guarantee type safety and efficiency, but bind the code to function names at run-time.


Type inference is abstract interpretation.

( 1 + 4 ) / 2.5 int * int int 5 / 2.5 (ML-

error) (coercion) real * real real 2.0

( int + int ) / real int / real real


Expression Grammar: Type Inference Example

E -> E + E | E * E | x | y | i | j

• Arithmetic Evaluationx, y in {…, -1.1, …, 2.3, …}i, j in {…, -1,0,1,…}

+, * : “infinite table”

• Type Inference

x, y : real i, j : int


+,* int realint int realreal real real

int intreal real

Values can be abstracted as type names and arithmetic operations can be abstracted as operations on these type names.


if true then “5” else 0.5

• Not type correct, but runs fine.

if true then 1.0/0.0 else 3.5

• Type correct, but causes run-time error.

Type correctness is neither necessary nor sufficient for programs to run successfully.


Assigning types to expressions• Uniquely determined

fn s => s ^ “.\n”; val it = fn : string -> string

• Over-constrained (type error without coercion) (2.5 + 2)

• Under-constrained – Overloadingfn x => fn y => x + y; (* resolvable *)fn record => #name(record); (* error *)

– Polymorphism fn x => 1 ;

val it = fn : 'a -> int


Type Signatures

• fun rdivc x y = x / y rdivc : real -> real -> real• fun rdivu (x,y) = x / y rdivu : real * real -> real•fun plusi x y = x + y plusi : int -> int -> int• fun plusr (x:real,y) = x + y plusr : real * real -> real


Polymorphic Types

• Semantics of operations on data structures such as stacks, queues, lists, and tables, are independent of the component type.

• Polymorphic type system provides a natural representation of generic data structures without sacrificing type safety.

• Polymorphism fun I x = x; I 5; I “x”; for all types I:


Composition

val h = f o g; fun comp f g = let fun h x = f (g x) in h; “Scope vs Lifetime : Closure”

comp:

– Generality– Equality constraints


map-functionfun map f [] = [] | map f (x::xs) = f x :: map f xs

map listlist

map (fn x => 0) [1, 2, 3]map (fn x => x::[]) [“a”, “b”]

• list patterns; term matching. • definition by cases; ordering of rules


Conventions

• Function application is left-associative.

f g h = ( ( f g ) h )• -> is right-associative. int->real->bool = int->(real->bool)• :: is right-associative. a::b::c::[] = a::(b::(c::[]))

• Function application binds stronger than ::. f x :: xs = ( f x ) :: xs


Polymorphic Type System


Goals• Allow expression of “for all types T”

fun I x = x

I : ’a -> ’a

• Allow expression of type-equality constraints

fun fst (x,y) = x

fst : ’a *’b -> ’a

• Support notion of instance of a type I 5

I : int -> int


Polymorphic Type System

• Type constants int, bool, …

• Type variables ’a, …, ’’a, …

• Type constructors->, *, …

• Principal type of an expression is the most general type, which ML system infers.

Type Checking RuleType Checking Rule

An expression of a type can legally appear in all contexts where an instance of that type can appear.


Signature of Equality = : ’a * ’a -> bool ?• Equality (“= ”) is not computable for all

types. E.g., function values.

• So types (’a,’b, …) are partitioned into • types that support equality (’’a, ’’b, …)• types that do not support equality.

= : ’’a * ’’a -> bool• int, string, etc. are equality types. They

are closed under cartesian product, but not under function space constructor.


Type Inference

I (3,5) (3,5) (int*int)->(int*int) (int*int)

Principal type of I is the generalization of

all possible types of its uses.

Curry2 : (’a * ’b -> ’c) ->

(’a -> ’b -> ’c)

Uncurry2 : (’a -> ’b -> ’c) ->

(’a * ’b -> ’c)


Subtle Points

• The type of a use of a function must be an instance of the principal type inferred in the definition.

• The type of a value is fixed. So, multiple occurrences of a symbol denoting the same value (such as several uses of a formal parameter in a function body) must have identical type.


Systematic Type Derivation

fun c f g x = f(g(x))• Step1: Assign most general types to left-

hand-side arguments and the (rhs) result. f : t1 g : t2 x : t3 f(g(x)) : t4 Thus, type of c is: c: t1 -> t2 -> t3 -> t4


• Step 2: Analyze and propagate type constraints– Application RuleApplication Rule If f x : t then x : t1 and f : t1 -> t, for some new t1.– Equality RuleEquality Rule If both x:t and x:t1 can be deduced for the value of a variable x, then t = t1.– Function RuleFunction Rule (t->u)=(t1->u1) iff (t=t1) /\ (u=u1)


f(g x): t4

AR

(g x) : t5

f : t5 -> t4

AR

x : t6

g: t6 -> t5

ER

t1 = t5 -> t4

t2 = t6 -> t5

t3 = t6

• Step 3: Overall deduced type

c : (t5 -> t4) ->

(t6 -> t5) ->

(t6 -> t4)

(unary function

composition)


Examplefun f x y = fst x + fst ygiven op + : int ->int -> int fst : ’a * ’b -> ’aStep 1: x: t1 y : t2 (fst_1 x + fst_2 y) : t3 The two instantiations of fst need

not have the same type. fst_1 : u1 * u2 -> u1 fst_2 : v1 * v2 -> v1

f: t1 -> t2 -> t3


• Step 2: Applying rule for +

(fst_1 x) : int

(fst_2 y) : int

t3 = int• Step 3: fst_1 : int*u2 -> int

fst_2 : int*v2 -> int

t1 = int * u2

t2 = int * v2

f: int * ’a -> int * ’b -> int


Example (fixed point)

fun fix f = f (fix f)

1. Assume f : ’a -> ’b

2. From (fix f = f (…)) infer

fix : (’a -> ’b) -> ’b

3. From ( ... = f (fix f)) infer

fix : (’a -> ’b) -> ’a

fix : ’a -> ’a -> ’a


Recursive Definition (curried function)

fun f x y = f (f x) 0;

f: (int -> ’a) -> int -> ’a

fun f x y = f (f x) (f x y);

f: (’a -> ’a) -> ’a -> ’a

fun f f = f; (* identity function *) (* names of formals and functions come from disjoint namespaces *)

fun f x y = f x; (* illegal *)

fun f g = g f; (* illegal *)


Example (ill-typed definition)fun selfApply f = f f

1. f_1:t1 (f_2 f_3):t2

selfApply : t1 -> t2

2. f_3 : t3 f_2 : t3 -> t2

3. t1 = t3 = t3 -> t2

(unsatisfiable)

(cf. val selfApply = I I)


Problematic Case

fn x => (1, “a”);val it = fn : 'a -> int * string

fn f => (f 1, f “a”);

Type error“Least upper bound” of int and string does

not exist, and the expression cannot be typed as ((’a->’b) ->’b*’b) because the following expression is not type correct.

( (fn f => (f 1, f “a”)) (Real.Math.sqrt) );


(cont’d)fn (x,y) => (size x, length y);val it = fn : string * 'a list -> int * int

fn z => (size z, length z);

Type Error

string and list cannot be unified to obtain ’a -> (int * int).

Or else, it will conflict with type instantiation rule, compromising type safety.


Equivalence?

(let val V = E in F end) =/= ((fn V => F) E)

Even though both these expressions have the same behavior, the lambda abstraction is more restricted because it is type checked independently of the context of its use. In particular, the polymorphic type variable introduced in an abstraction must be treated differently from that introduced in a local definition.

Type Checking and Type Inference

Documents

type errors

type names

type safety

type correctness

component type

type signaturesfun rdivc

x y rdivc

x y plusi