CS412/413

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CS412/413

• Introduction to
• Compilers and Translators
• Spring 99
• Lecture 10 More Type Checking

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Administration

• Homework 2 due Friday

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Review

• During type checking, associate types from a type
  system with expressions and variables in program
• Types primitive types, type constructors,
  pseudo-types (e.g. void, error)
• Type constructors
• Arrays array(T)
• Records/structs id1 T1, id2 T2,
• Functions T1?T2 T1?T2 ?... ? Tn
  ? Tr

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Review

• Formally symbol table is map from identifiers to
  types, referred to as type-checking environment
• A id1 T1, id2 T2,
• Type checking is attempt to make type judgments
• A E T
• means
• in the environment A, expression E has type T

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Review

• Type inference rules allow precise, compact
  expression of type-checking process
• Semantic analysis corresponds to working
  backwards through inference rules
• A E bool
• A ! E bool

!
bool?
E
What if cant get desired type judgment?
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Function-checking rule

• Notation A F means F is type-safe
• Create an environment A containing all the
  argument variables
• Check the body E in this environment
• Type of E must match declared return type of
  function
• A ? ..., ai Ti, ... E T
• A id ( ..., ai Ti, ... ) T E

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Example

• fact(x int) int
• if (x0) 1 else x fact(x - 1)
• A fact(x int) int
• A ? x int if (x0) ... else ... int
• A ? x int x0 bool
• A ? x int x T1 (T1 int)
• A ? x int 0 int
• A ? x int 1 T2 (T2 int)
• A ? x int x fact(x-1) int

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Remainder of proof

• A ? x int x fact(x-1) int
• A ? x int x int
• A ? x int fact(x-1) int
• A ? x int x-1 int
• A ? x int x int
• A ? x int 1 int
• A ? x int fact int?int

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Sequence

• Rule A sequence of statements is type-safe if
  the first statement is type-safe, and the
  remaining are type-safe too
• A S1 T1
• A S2 S3 ... Sn T
• A S1 S2 ... Sn T
• What about variable declarations ?

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Declarations

• A id T1 E T1
• A ? id T1 S2 ... Sn T
• A id T1 E S2 ... Sn Tn
• This formally describes the type-checking code
  from two lectures ago!

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Implementation

• class Block Stmt stmts
• Type typeCheck(SymTab s) Type t
• for (int i 0 i lt stmts.length i)
• t stmtsi.typeCheck(s)
• if (stmtsi instanceof Decl)
• s s.add(Decl.id, Decl.typeExpr.evaluate())
  
• return t

A S1 T1 S1 not a decl. A S2 S3... Sn
T A S1 S2 ... Sn T
A id T1 E T1 A ? id T1
S2 ... Sn T A id T1 E S2 ... Sn
T
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Completing static semantics

• Rest of static semantics can be written in this
  style
• Provides complete recipe (inductively) for how to
  show a program type-safe
• Induction
• have rules for atoms A true bool
• for every valid syntactic construct in language,
  have a rule showing how to prove it type-safe
• Therefore, have rules for checking all
  syntactically valid programs for type safety

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Implementing Rules

• Start from goal judgments for each function
• A id ( ..., ai Ti, ... ) T E
• Work backward applying inference rules to
  sub-trees of abstract syntax trees
• Same recursive traversal described earlier
• Need efficient way to create new environments ,
  e.g. for
• A ? id T1 S2 ... Sn T

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Implementing Types

• How do we represent types during type checking?
  Need a data structure.
• Types need operations
• creation from type expressions in the program
• interpretType(AST_node n, SymTab s)
• ability to be checked for equivalence
• application of type constructors

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Type Class

• abstract class Type
• boolean equals(Type t)
• // type equivalence
• void unparse(OutputStream o)
• // for debugging

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Primitive Types

• class PrimitiveType extends Type
• string name
• boolean equals(Type t)
• return this t
• void unparse(OutputStream o)
• o.print(name)

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Type Constructor Array

• class ArrayType extends Type
• Type T // represents the type arrayT
• Array(Type t) T t
• boolean equals(Type t)
• if (t instanceof ArrayType
• ((ArrayType)t).T.equals(T))
• return true
• else return false

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Type Equivalence

• Structural equivalence vs. name equivalence
• Record types in C
• struct Point int x,y
• struct Range int low, int high
• struct Point2 int x, y
• struct Point3 int y, x
• Name equivalence all different since created by
  different constructor invocations
• Structural equivalence depends on structure
• possibilities all same, PointPoint2, Point3

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Type Equivalence, Formally

• Add a new kind of judgment
• A T1 T2
• Meaning types T1 and T2 are equivalent in
  environment A
• Axioms A int int, A bool bool
• Arrays
• A T1 T2
• A array(T1) array(T2)

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Record Equivalence

• Name equivalence record type has a name
  generated (or assigned) at declaration
• n1 n2
• A n1id1 T1 n2id1 T1
• Structural equivalence field names types must
  match
• m n idi idi A Ti Ti
• A id1T1...idnTn id1T1 idm Tm

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Interpreting Types

• Iota
• type id array type
• Suppose type TypeExpr
• class TypeId extends TypeExpr
• class ArrayTypeExpr extends TypeExpr

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Interpreting Types

• abstract class TypeExpr
• Type interpret(SymTab A)
• class TypeId string name
• Type interpret(SymTab A)
• if (name.equals(int)) return Int
• if (name.equals(string)) return Str
• ...

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Interpreting Arrays

• class ArrayTypeExpr extends TypeExpr
• TypeExpr te
• Type interpret(SymTab A)
• return new ArrayType(te.interpret())

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More language features

• Recursive definitions forward decls
• Returns and exceptions
• Subtyping
• Classes
• Inheritance
• Parametric polymorphism

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Handling Recursion

• A fact(x int) int
• if (x0) 1 else x fact(x - 1)
• To type-check this, A must contain entry fact
  int ? int
• Java, Iota functions (methods, classes) may be
  mutually recursive.
• f(x int) int if (xlt1) 1 else g(f(x-1))
• g(x int) int if (xlt1) 2 else f(g(x-1))

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Handling Recursion

• Need environment containing at least
• f int ? int, g int ? int
• when checking both f and g
• Two-step approach
• Scan top level of AST picking up all function
  signatures and creating an environment binding
  all global identifiers
• Type-check each function individually using this
  global environment

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Non-local control flow

• How to check non-local control transfers?
• return, throw, break, continue
• Fits into existing framework for type checking
  (mostly)
• Question what is the type of a return statement?
• A return E ?

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How to check return?

• A E T
• A return E void
• A return statement has no value, so its type is
  void
• But how do make sure the return type of the
  current function is T ?

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• Answer put it in the symbol table
• Add entry return int when we start checking
  the function, look up this entry when we hit a
  return statement.

A A ? ..., ai Ti, ... ? return T A
? ..., ai Ti, ... E T A id (
..., ai Ti, ... ) T E
A E T return T ? A A return E
void
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Summary

• Type systems and static semantics
• Type-checking functions
• How bindings are added to environment
• local variables
• global definitions (possibly recursive)
• Representing, interpreting types
• Type equivalence
• return statement