Post on 21-Dec-2015
Compilation 2007Compilation 2007
Static Type CheckingStatic Type Checking
Michael I. Schwartzbach
BRICS, University of Aarhus
2Static Type Checking
The Type CheckerThe Type Checker
The type checker has several tasks:• determine the types of all expressions• check that values and variables are used correctly• resolve certain ambiguities by transformations
Some languages have no type checker• no compile-time guarantees• runtime tagging and checking become necessary
3Static Type Checking
Types Describe Possible ValuesTypes Describe Possible Values
The Joos types are:• void (the empty set of values)• byte• short • int • char• boolean• C (objects of class C or any subclass)• null (the polymorphic null constant)• σ[] (for any type σ)
4Static Type Checking
Type AnnotationsType Annotations
Type annotations:
int x;
Foo y;
specify invariants about the runtime behavior:• x will always contain an integer value• y will always contain null or an object of class Foo or
any subclass
Type annotations are not very expressive viewed as invariants
5Static Type Checking
A program is type correct if the type annotations are valid invariants
Type correctness is trivially undecidable:
int x = 0;
TM(j);
x = false;
The program is type correct iff the j'th Turing machine does not halt on empty input
Type CorrectnessType Correctness
6Static Type Checking
Static Type CorrectnessStatic Type Correctness
A program is statically type correct if it satisfies some type rules
The rules are chosen to be:• simple to understand• efficient to decide• conservative with respect to type correctness
Type systems are not canonical They are designed like the rest of the language
7Static Type Checking
SlackSlack
Static type systems are necessarily flawed:
There is always slack: programs that are unfairly rejected by the type checker
statically type correcttype correct
slack
8Static Type Checking
Examples of SlackExamples of Slack
Trivial example:
x = 87;
if (false) x = true;
Useful example:
Map m = new Hashmap();
m.put("bar","foo");
String x = m.get("bar");
Java 1.5 is designed to pick up some slack
9Static Type Checking
Specifying Type RulesSpecifying Type Rules
Prose:
"The argument of the sqrt function must be of type int, the result is of type double"
Logical rules:
C |- x: int
C |- sqrt(x): double
10Static Type Checking
The Shortcomings of ProseThe Shortcomings of Prose
The first expression must be of type boolean, or a compile-time error occurs. The conditional operator
may be used to choose between second and third operands of numeric type, or second and third
operands of type boolean, or second and third operands that are each of either reference type or the null
type. All other cases result in a compile-time error. Note that it is not permitted for either the second or
the third operand expression to be an invocation of a void method. In fact, it is not permitted for a
conditional expression to appear in any context where an invocation of a void method could appear. The
type of a conditional expression is determined as follows: If the second and third operands have the
same type (which may be the null type), then that is the type of the conditional expression. Otherwise, if
the second and third operands have numeric type, then there are several cases:If one of the operands is
of type byte and the other is of type short, then the type of the conditional expression is short. If one of
the operands is of type T where T is byte, short, or char, and the other operand is a constant expression
of type int whose value is representable in type T, then the type of the conditional expression is T.
Otherwise, binary numeric promotion is applied to the operand types, and the type of the conditional
expression is the promoted type of the second and third operands. Note that binary numeric promotion
performs value set conversion. If one of the second and third operands is of the null type and the type of
the other is a reference type, then the type of the conditional expression is that reference type. If the
second and third operands are of different reference types, then it must be possible to convert one of the
types to the other type (call this latter type T) by assignment conversion; the type of the conditional
expression is T. It is a compile-time error if neither type is assignment compatible with the other type.
11Static Type Checking
Three Kinds of Type RulesThree Kinds of Type Rules
Declarations:
this variable is declared to have this type Propagations:
if the argument is of this type, then the result is of that type
Restrictions:
the argument must be of this type or that type
The logical notation handles all three uniformly
12Static Type Checking
Judging ExpressionsJudging Expressions
The judgment:
C, L, X, σ |- E: τ
means that the expression E is statically type correct and has type τ in a context where:• C is the current class• L is the current local environment• X is the current set of exceptions• σ is the return type of the current method
13Static Type Checking
Judging StatementsJudging Statements
The judgment:
C, L, X, σ |- S
means that the statement S is statically type correct in a context where:• C is the current class• L is the current local environment• X is the current set of exceptions• σ is the return type of the current method
14Static Type Checking
The class environment is implicitly available The notation D≤C means that D is a subclass of C We can query a class C:
• C |- extends D (super class)• C |- σ f (fields)• C |- static σ f (static fields)
• C |- (σ1,..., σk) X (constructors)
• C |- σ m (σ1,..., σk) X (methods)
• C |- static σ m (σ1,..., σk) X (static methods)
Class EnvironmentsClass Environments
return type
name argument typesthrown exceptions
15Static Type Checking
Assignment CompatibilityAssignment Compatibility
The relation σ := τ means that values of type τ may be assigned to variables of type σ:σ := σ short := byte σ :- σ
int := byte int := short σ[] :- τ[], if σ :- τ
int := char C := null java.lang.Cloneable :- σ[]
σ[] := null Object := σ[] java.io.Serializable :- σ[]
java.lang.Cloneable := σ[] Object :- σ[]
java.io.Serializable := σ[] C :- D, if D ≤ C
σ[] := τ[], if σ :- τ
C := D, if D ≤ C
It is reflexive, transitive, and anti-symmetric
16Static Type Checking
Assignment CompatibilityAssignment Compatibility
The relation σ := τ means that values of type τ may be assigned to variables of type σ:σ := σ short := byte σ :- σ
int := byte int := short σ[] :- τ[], if σ :- τ
int := char C := null java.lang.Cloneable :- σ[]
σ[] := null Object := σ[] java.io.Serializable :- σ[]
java.lang.Cloneable := σ[] C :- D, if D ≤ C
java.io.Serializable := σ[]
σ[] := τ[], if σ :- τ
C := D, if D ≤ C
It is reflexive, transitive, and anti-symmetric
Turbo-Pascal:
reflexive transitive anti-symmetric symmetric
17Static Type Checking
StatementsStatements
C,L,X,σ |- S1 C,L,X,σ |- S2
C, L, X, σ |- S1 S2
C, L[n→ τ], X, σ |- S
C, L, X, σ |- { τ n; S }
C, L, X, σ |- E: τ
C, L, X, σ |- E;
18Static Type Checking
Control StatementsControl Statements
C,L,X,σ |- E: boolean C,L,X,σ |- S
C, L, X, σ |- if (E) S
C,L,X,σ |- E: boolean C,L,X,σ |- Si
C, L, X, σ |- if (E) S1 else S2
C,L,X,σ |- E: boolean C,L,X,σ |- S
C, L, X, σ |- while (E) S
19Static Type Checking
Return StatementsReturn Statements
σ = void
C, L, X, σ |- return
C, L, X, σ |- E: τ σ := τ
C, L, X, σ |- return E
20Static Type Checking
Throw StatementThrow Statement
C,L,X, σ |- E: τ throwok({τ},X)
C, L, X, σ |- throw E
throwok(A,B) ≡ αA: Error:= α
RuntimeException:= α βB: β:=α
21Static Type Checking
LocalsLocals
L(n) = τ
C, L, X, σ |- n: τ
22Static Type Checking
FieldsFields
C,L,X,σ |- this: C
D |- static τ f
C,L,X,σ |- D.f: τ
C,L,X,σ |- E: D D |- τ f
C, L, X, σ |- E.f: τ
23Static Type Checking
AssignmentsAssignments
C,L,X,σ |- E: τ2 L(n)=τ1 τ1:=τ2
C, L, X, σ |- n = E: τ1
C,L,X,σ |- E: τ2 D |- static τ1 f τ1:=τ2
C, L, X, σ |- D.f = E: τ1
C,L,X,σ |- E2: τ2 C,L,X,σ |- E1: D D |- τ1 f τ1:=τ2
C, L, X, σ |- E1.f = E2: τ1
24Static Type Checking
ArraysArrays
C,L,X,σ |- E1: τ1[] C,L,X,σ |- E2: τ2 num(τ2)
C, L, X, σ |- E1[E2]: τ1
C, L, X, σ |- E1: τ1[] C, L, X, σ |- E2: τ2
C, L, X, σ |- E3: τ3 num(τ2) τ1:= τ3
C, L, X, σ |- E1[E2]= E3: τ1
num(σ) ≡ σ{ byte, short, int, char }
25Static Type Checking
Array OperationsArray Operations
C,L,X,σ |- E: τ[]
C,L,X,σ |- E.length: int
C,L,X,σ |- E: τ[]
C,L,X,σ |- E.clone(): Object
C,L,X,σ |- E: τ2 num(τ2)
C,L,X,σ |- new τ1[E]: τ1[]
26Static Type Checking
OperatorsOperators
C,L,X,σ |- E: boolean
C,L,X,σ |- !E: boolean
C,L,X,σ |- E1: τ1 C,L,X,σ |- E2: τ2 num(τ1) num(τ2)
C, L, X, σ |- E1*E2: int
27Static Type Checking
LiteralsLiterals
C,L,X,σ |- true: boolean
C,L,X,σ |- "abc": String
C,L,X,σ |- null: null C,L,X,σ |- '@': char
C,L,X,σ |- 42: int
28Static Type Checking
PlusPlus
C,L,X,σ |- E1: τ1 C,L,X,σ |- E2: τ2 num(τ1) num(τ2)
C, L, X, σ |- E1+E2: int
C,L,X,σ |- E1: String C,L,X,σ |- E2: τ2 τ2≠void
C, L, X, σ |- E1+E2: String
C,L,X,σ |- E1: τ1 C,L,X,σ |- E2: String τ1≠void
C, L, X, σ |- E1+E2: String
29Static Type Checking
EqualityEquality
C,L,X,σ |- E1: τ1 C,L,X,σ |- E2: τ2
(num(τ1)num(τ2)) τ1:=τ2 τ2:=τ1
C, L, X, σ |- E1==E2: boolean
30Static Type Checking
Casts and InstanceofCasts and Instanceof
C,L,X,σ |- E: τ1 (num(τ1)num(τ2)) τ1:=τ2 τ2:=τ1
C, L, X, σ |- (τ2)E: τ2
C,L,X,σ |- E: τ τ:=D D:=τ
C, L, X, σ |- E instanceof D: boolean
31Static Type Checking
Why Restrict Casts?Why Restrict Casts?
τ1
τ2
τ2
τ1
τ1τ2
Alway succeeds, but downcasting is useful for selecting different methods
τ2 := τ1
Really useful, the object may or may not by a subclass of τ2
τ1 := τ2
Always fails, don't bother to execute
τ1 :≠ τ2 τ1 :≠ τ2
32Static Type Checking
Method InvocationMethod Invocation
C,L,X,σ |- E: D C,L,X,σ |- Ei: τi
D |- τ m(τ1,...,τk) Y
C, L, X, σ |- E.m(E1,...,Ek): τ
C,L,X,σ |- Ei: τi
D |- static τ m(τ1,...,τk) Y
C, L, X, σ |- D.m(E1,...,Ek): τ
33Static Type Checking
Constructor InvocationConstructor Invocation
C,L,X,σ |- Ei: τi
D |- (τ1,...,τk) Y throwok(Y,X)
C, L, X, σ |- new D(E1,...,Ek): D
C,L,X,σ |- Ei: τi C |- extends D
D |- (τ1,...,τk) Y throwok(Y,X)
C, L, X, σ |- super(E1,...,Ek)
34Static Type Checking
Closest Match Method InvocationClosest Match Method Invocation
C,L,X,σ |- E: D C,L,X,σ |- Ei: σi
D |- τ m(τ1,...,τk) Y τi:= σi
D |- γ m(γ1,..., γk) Z : (i: γi:= σi) (i: γi:= τi)
C, L, X, σ |- E.m(E1,...,Ek): τ
• This rule does not describe access modifiers
35Static Type Checking
Checking ExceptionsChecking Exceptions
C, L, X, σ |- E: D C, L, X, σ |- E.m(E1,...,Ek): τ
throwok({Yi| D |- τ m(τ1,...,τk) Yi }, X)
YZ = YZ Z Y
YZ = { yY | throwok({y},Z) }
36Static Type Checking
Judging Methods and ConstructorsJudging Methods and Constructors
C, [ai→ σi], { Xi }, σ |- S Throwable := Xi
C |- σ m(σ1 a1,...,σk ak)throws X1,...,Xn { S }
C, [ai→ σi], { Xi }, void |- S Throwable := Xi
C |- C(σ1 a1,...,σk ak)throws X1,...,Xn { S }
37Static Type Checking
Different Kinds of Type RulesDifferent Kinds of Type Rules
Axioms:
Predicates: τ1:=τ2 τ1:=τ2
Inferences:
C,L,X,σ |- this: C
C,L,X,σ |- E: D D |- τ f
C, L, X, σ |- E.f: τ
38Static Type Checking
Type ProofsType Proofs
A type proof is a tree in which:• nodes are inferences• leaves are axioms or true predicates
A judgment is valid
it is the root of some type proof
39Static Type Checking
A Type ProofA Type Proof
[x→A,y→B](x)=A
C,[x→A,y→B],X,σ |- x:A ABBA [x→A,y→B](y)=B C,[x→A,y→B],X,σ |- (B)x: B B:=B
C,[x→A,y→B],X, σ |- y=(B)x: B
C,[x→A,y→B],X, σ |- y=(B)x;
C,[x→A],X, σ |- {B y; y=(B)x;}
C,[],X,σ |- {A x; {B y; y=(B)x;}}
Equal to a trace of a successful run of the type checker
(Assuming that B is a subclass of A)
40Static Type Checking
Static Type CorrectnessStatic Type Correctness
A program is statically type correct
all method and constructor judgments are valid
A type system is sound
static type correctness implies type correctness
41Static Type Checking
Java's Type System is UnsoundJava's Type System is Unsound
The following is a valid judgement:
C, [b→B,x→A[],y→B[],c→C], X, σ |- x=y; x[0]=c; b=y[0];
where BA, CA, and B and C are incomparable Afterward, b contains an object of class C But its declared type only allows B objects
42Static Type Checking
GreedGreed
Java's type system is too greedy:
Runtime type checks catch the unsound cases
statically type correcttype correct
unsound
43Static Type Checking
Fixed Assignment CompatibilityFixed Assignment Compatibility
By design, the rule for assignment of arrays has been chosen to be convenient but unsound
It should have looked like:σ := σ short := byte σ :- σ
int := byte int := short σ[] :- τ[], if σ :- τ
int := char C := null java.lang.Cloneable :- σ[]
σ[] := null Object := σ[] java.io.Serializable :- σ[]
java.lang.Cloneable := σ[] C :- D, if D ≤ C
java.io.Serializable := σ[]
σ[] := τ[], if σ :- τ
C := D, if D ≤ C
But that would then introduce annoying slack...
44Static Type Checking
Complexity of Inference SystemsComplexity of Inference Systems
Deciding valid judgments of an inference system depends on its rules and axioms and may be:• undecidable• (hyper)exponential time (backtracking)• linear time
The Java type system is designed to be simple A type checker performs a single traversal of the
parse tree in linear time
45Static Type Checking
Type Checking in JavaType Checking in Java
Type checking is performed bottom-up The rule to apply is uniquely determined by the
types of subexpressions and the syntax
The type proof also specifies how to transform the program with respect to:• address comparisons• implicit string conversions• concat(...) code
46Static Type Checking
Implicit CoercionsImplicit Coercions
Suppose we add a coercion type rule:
corresponding to the JVM bytecode i2f
Type proofs are now no longer unique!
C,L,X,σ |- E: int
C,L,X,σ |- E: float
47Static Type Checking
Ambiguous Type ProofsAmbiguous Type Proofs
L(i)=int L(j)=int
C,L,X,σ |-i: int C,L,X,σ |- j: int
C,L,X,σ |-i: float C,L,X,σ |- j: float L(f)=float
C,L,X,σ |- i+j: float C,L,X,σ |- f: float
C,L,X,σ |- i+j==f: boolean
L(i)=int L(j)=int
C,L,X,σ |-i: int C,L,X,σ |- j: int
C,L,X,σ |-i+j: int L(f)=float
C,L,X,σ |- i+j: float C,L,X,σ |- f: float
C,L,X,σ |- i+j==f: boolean
48Static Type Checking
Semantic CoherenceSemantic Coherence
Different type proofs generate different transformed programs
The type system is semantically coherent if they always have the same semantics
The previous hypothetical type system is not:• i = 2147400000• j = 83647• f = 2.14748365E9
Java chooses the proof with coercions as near the root as possible