Programming Language Technology DAT151/DIT231 Andreas Abel

Type Checking

Introduction

Purposes of types

Where types make a difference

... divide(... x, ... y) { return x / y; }

printDouble (divide (5, 3));

Judgements and rules

Rule format:

J₁ ... Jₙ
--------- C
    J

Logical interpretation: Judgement J holds if C and all of J₁ ... Jₙ hold.

Algorithmic interpretation: To check J, check J₁ ... Jₙ (in this order) and C (in a suitable place).

Typing of arithmetic expressions

Judgement version 1: "expression e has type t"

e : t

Concrete or abstract syntax

Rules may be written using concrete syntax:

e₁ : int    e₂ : int
--------------------
e₁ / e₂ : int

e₁ : double    e₂ : double
--------------------------
e₁ / e₂ : double

Generic version, using side condition:

e₁ : t    e₂ : t
---------------- t ∈ {int, double}
e₁ / e₂ : t

Using abstract syntax:

TInt.    Type ::= "int";  
TDouble. Type ::= "double";  

EInt.     Exp6 ::= Integer;  
EDouble.  Exp6 ::= Double;  
EDiv.     Exp2 ::= Exp3 "/" Exp2;  
-------------        -------------------
EInt i : TInt        EDouble d : TDouble

e₁ : t    e₂ : t
---------------- t ∈ {TInt, TDouble}
EDiv e₁ e₂ : t

Type checking and type inference

Checking: Given e and t, check whether e : t.
Inference: Given e, compute t such that e : t.

check (Exp e, Type t): Bool
infer (Exp e) : Maybe Type

Example implementation:

check (EInt i, t):  
    t == TInt

check (EDiv e₁ e₂, t):  
    check (e₁, t) && check (e₂, t) && (t == TInt || t == TDouble)

infer (EInt i):  
    return TInt

infer (EDiv e₁ e₂):  
    t ← infer (e₁)
    check (e₂, t)
    return t

Coercion and subtyping

int values can be coerced to double values.
int is a subtype of double, written int ≤ double.
Subtyping t₁ ≤ t₂ is a preorder, i.e., a reflexive-transitive relation.

e : t₁
------ t₁ ≤ t₂
e : t₂

Subtyping should be coherent, up-casting via an intermediate type should not make a difference:

short int i;  
double x = (double)((int)i);  
double y = (double)i;

In practice, it often does. E.g. with coercions to string:

int ≤ string             1 → "1"
int ≤ double ≤ string    1 → 1.0 → "1.0"

Quiz: What is the value of this expression?

1 + 2 + "hello" + 1 + 2

This should better be a type error! The value should not depend on the associativity of "+".

Elaboration in the type checker

Type checker can insert coercion ECoerce when subtyping int < double was applied.

internal ECoerce.  Exp ::= "(double)" Exp;

Checking/inference: also return elaborated expression.

check (Exp e, Type t): Maybe Exp
infer (Exp e) : Maybe (Exp, Type)

Example implementation:

check (EInt i, t):  
    if t == TInt then return (EInt i)
    else fail "expected int"

check (EDiv e₁ e₂, t):  
    assert (t ∈ {TInt, TDouble})
    e₁' ← check (e₁, t)
    e₂' ← check (e₂, t)
    return (EDiv e₁' e₂')

infer (EInt i):  
    return (EInt i, TInt)

infer (EDiv e₁ e₂):  
    (e₁', t₁) ← infer (e₁)
    (e₂', t₂) ← infer (e₂)
    t ← max t₁ t₂
    e₁'' = coerce (e₁', t₁, t)
    e2'' = coerce (e₂', t₂, t)
    return (EDiv e₁'' e₂'', t)

coerce (e, t₁, t₂):  
    if t₁ == TInt and t₂ == TDouble then ECoerce(e)
    else e

Boolean expressions

Comparison operators return a boolean:

e₁ : t    e₂ : t
---------------- t ∈ {int, double}
e₁ < e₂ : bool

e₁ : t    e₂ : t
---------------- t ∈ {bool, int, double}
e₁ == e₂ : bool

Statements

Statements all have type void so we can omit it.
Judgements version 1:

⊢ s            Statement s is well-typed
⊢ s₁...sₙ      Statement sequence s₁...sₙ is well-typed

NB: ⊢ = "turnstile" (With TeX input mode: \vdash)

Rules for conditional statements:

e : bool    ⊢ s
---------------
⊢ while (e) s

e : bool    ⊢ s₁    ⊢ s₂
------------------------
⊢ if (e) s₁ else s₂

Statement sequences:

       ⊢ s₀    ⊢ s₁...sₙ
---    -----------------
⊢ ε    ⊢ s₀ s₁...sₙ

Expressions as statements

e : t
-----
⊢ e;

Return statements

A return statement needs to have a return expression of the correct type.

int main () {
  ...  
  if (...) return 5;  
  else return 1.0;    // error
}

Judgements version 1.5:

⊢ᵗ s            Statement s is well-typed, may return t
⊢ᵗ s₁...sₙ      Statement sequence s₁...sₙ is well-typed, may return t

Rule:

e : t
-----------
⊢ᵗ return e;

Checking functions

DFun.  Def ::= Type Id "(" ")" "{" [Stm] "}"

Checking function definitions (version 1):

    checkStm (Type t, Stm s)

    checkStms (Type t, [Stm] ss):  
       for (Stm s ∈ ss):  
           checkStm(t, s)

    checkDef (DFun t x ss):  
      checkStms (t, ss)

Variables and blocks

Typing environments (aka contexts) assign types to variables.

Example:

  int f (double x) {
    int i = 2 ;  
    int j = 5 ;  
    // int i;         // illegal, i already declared in block
    // int x;         // illegal, x already declared in block
    {
      int    x = 0;   // legal, shadows function parameter.  
      double i = 3 ;  // shadows the previous variable i
      {
                   // Environment: (x:double,i:int,j:int).(x:int,i:double).()
      }
      i = i + j;   // the inner i becomes 8.0
      int k;  
    }
    i++ ;          // the outer i becomes 3
    return i + j;  
  }

Environments Γ are structured as stack of blocks.
Each block Δ is a (partial and finite) map from identifier to type.

Example: in the innermost block, the typing environment is a stack of three blocks

  1. (x:double,i:int,j:int): function parameters and top local variables
  2. (x:int,i:double): variables declared in the first inner block
  3. (): variables declared in the second inner block (none)

Declaration statements like int i, j; declare a new block component Δ = (i:int, j:int).
Notation:

E.g. (x:t),(x:...) is a clash.

Judgements version 2:

Rules for declarations and blocks.

------------------------------------- no xᵢ ∈ Δ, and xᵢ ≠ xⱼ when i ≠ j
Γ.Δ ⊢ᵗ⁰ t x₁,...,xₙ; ⇒ (x₁:t,...xₙ:t)

Γ.(Δ,x:t) ⊢ e : t
------------------------ x ∉ Δ
Γ.Δ ⊢ᵗ⁰ t x = e; ⇒ (x:t)

Γ.() ⊢ᵗ⁰ ss ⇒ Δ
-----------------
Γ ⊢ᵗ⁰ { ss } ⇒ ()

Valid but pointless example for initialization statement:

int main () {
  int i = i;  
}

t x = e is sugar for t x; x = e;.

Rules for sequences:

-----------
Γ ⊢ᵗ⁰ ε ⇒ ()

Γ ⊢ᵗ⁰ s ⇒ Δ₁    Γ,Δ₁ ⊢ᵗ⁰ ss ⇒ Δ₂
--------------------------------
Γ ⊢ᵗ⁰ s ss ⇒ Δ₁,Δ₂

Rules for conditional statements:
Branches need to be in new scope.

Γ ⊢ᵗ⁰ e : bool    Γ. ⊢ᵗ⁰ s₁ ⇒ Δ₁    Γ. ⊢ᵗ⁰ s₂ ⇒ Δ₂
-------------------------------------------------
Γ ⊢ᵗ⁰ if (e) s₁ else e₂ ⇒ ()

Γ ⊢ᵗ⁰ e : bool    Γ. ⊢ᵗ⁰ s ⇒ Δ
-----------------------------
Γ ⊢ᵗ⁰ while (e) s ⇒ ()

Example:

    int main () {
      if (condition) int i = 1; else int j = 2;  
      return i;  
    }

Same as if (condition) { int i = 1; } else { int j = 2; }.

Functions

When calling a function, we need to provide arguments of the correct type.

Global environment Σ (for function signature) maps function names to function types.

   bool foo (int x, double y) { ... }

Type of foo is bool(int,double) also written as (int,double)→bool.

Judgements version 3:

Rule for function application:

Σ;Γ ⊢ e₁ : t₁ ... Σ;Γ ⊢ eₙ : tₙ
------------------------------- Σ(f) = t(t₁,...,tₙ)
Σ;Γ ⊢ f(e₁,...,eₙ) : t

Rule for function definition:

Σ; (x₁:t₁,...,xₙ:tₙ) ⊢ᵗ ss ⇒ Δ
---------------------------------
Σ ⊢ t f (t₁ x₁, ... tₙ xₙ) { ss }

The correctness of the signature, Σ(f) = t(t₁,...,tₙ), can be assumed in the last rule.

Programs

A program is a list of functions. Checking a program:

The elaborating type checker will compute a new body ss' for each function definition, resulting in an elaborated program.

The result of type checking is a map from function names to their types and elaborated definitions.

More type annotations

Elaboration:

Type-annotating checker: annotate each subexpression with its type.