1. 1 Type Type system for a programming language = –set of types AND – rules that specify how a typed program is allowed to behave Why? –to generate better code, with less runtime overhead –to avoid runtime errors –to improve expressiveness (see overloading) How? –Type checking Given an operation and an operand of some type, determine whether the operation is allowed on that operand –Type inference Given the type of operands, determine –the meaning of the operation –the type of the operation OR, without variable declarations, infer type from the way the variable is used

2. 2 Issues in typing Does the language have a type system? –Untyped languages (e.g. assembly) have no type system at all When is typing performed? –Static typing: At compile time –Dynamic typing: At runtime –Which is best? How strictly are the rules enforced? –Strongly typed: No exceptions –Weakly typed: With well-defined exceptions Type equivalence & subtyping –When are two types equivalent? What does "equivalent" mean anyway? –When can one type replace another?

3. 3 Components of a type system Built-in types Rules for constructing new types Rules for determining if two types are equivalent Rules for inferring the types of expressions

4. 4 Component: Built-in types These are the basic types. Usually, integer –usual operations: standard arithmetic floating point –usual operations: standard arithmetic character –character set generally ordered lexicographically –usual operations: (lexicographic) comparisons boolean –usual operations: not, and, or, xor

5. 5 Component: type constructors Arrays –array(I,T) denotes the type of an array with elements of type T, and index set I –multidimensional arrays are just arrays where T is also an array –operations: element access, array assignment, products Strings –bitstrings, character strings –operations: concatenation, lexicographic comparison Products –Groups of multiple objects of different types essentially, Cartesian product of types (useful in functions later) Records (structs) –Groups of multiple objects of different types where the elements are given specific names.

6. 6 Component: type constructors Pointers –addresses –operations: arithmetic, dereferencing, referencing –issue: equivalency Function types –A function such as "int add(real, int)" has type real  int  int

7. 7 Component: type equivalence Name equivalence –Two types are equivalent when they have the same name. –Loose vs. strict Structural equivalence –Two types are equivalent when they have the same structure Should records require member name identity to be equivalent? How about Array(int, 1..10) and Array(int, 1..20). Should we consider the bounds or just the element type? How about recursively defined types? Example –C uses structural equivalence for structs and name equivalence for arrays/pointers. Issue –Type equivalence allows type interchangeability If two equivalent types have different meanings, there will be no error message if we interchange them.

8. 8 Component: type equivalence Subtyping –An operation that expects a type T and receives a type T' which is not equivalent to T, may still go ahead if T is a subtype of T'. –But what exactly is a subtype? A subset of the values and superset of the properties? Is a subclass a subtype?

9. 9 Component: type equivalence Type coercion –If x is float, is x=3 acceptable? Disallow Allow and implicitly convert 3 to float. "Allow" but require programmer to explicitly convert 3 to float How do we convert? –Reinterpret bit sequence –Build new object –What should be allowed? float to int ? int to float ? What if both types are equally general? What if multiple coercions are possible? –Consider 3 + "4" in a language that can convert strings to integers.

10. 10 Overloading Same operation name but different effect on different types For an overloaded function f –resolve arguments –look up f in symbol table to get list of all visible "versions" –ignore those with wrong parameter types.

11. 11 A simple type checker We can use an attribute grammar to implement a simple type checker. E  num{E.type = integer} E  E 1 +E 2 {E.type = (E 1.type == integer && E 2.type == integer)? integer : error} E  E 1 [E 2 ]{E.type = (E 1.type == array(int, T) && E 2.type == integer)? T : error} E  *E 1 {E.type = (E 1.type == pointer(T))? T : error} E  E 1 (E 2 ){E.type = (E 1.type == (S  T) && E 2.type == S)? T : error S  if E then S 1 {S.type = (E.type == boolean)? void : error}

12. 12 Inference rules The formal notation for type checking/inference is rules of inference They have the form: If we can prove the hypotheses are all true in the existing environment, then the conclusion is also true. For example: In English: If expressions E1 and E2 both have type int, then E1+E2 also has type int. Environment Hypotheses Environment Conclusions A E1: int A E2: int A E1 + E2 : int

13. 13 Inference rules The inference rules give us templates that describe how to type various expressions. We can use the templates to prove that an expression has a valid type (and find what that type is) The construct parallels the parse tree. Short example: –Type the expression x+a[i] under the assumption that x is an char, a[i] is an array of chars and i is an int. –The environment is A={x:char, a:array(int, char), i:char} A E1: array(T,S), A E2: T A E1[E2] : S [ARRAY] A E1: char, A E2: char A E1 + E2 : char [ADD]

14. 14 Inference rules [ADD] A |– x :char A |– i : int A |– a : array(int,char) A |– a[i] : char A |– x+a[i] : char