1. 1 CS 2104 Type checking / Polymorphism Reading: Chapters 3.1, 4.3, 4.4.1 Dr. Abhik Roychoudhury Adapted from Goh Aik Hui’s lecture notes.

2. 2 Overview  Types  Motivation for typed languages  Issues in Type Checking  How to type check?  How to cater for Polymorphism  Type Equivalence  When to type check?  Strong and Weak typed languages

3. 3 1. Motivation for typed languages  Untyped Languages: perform any operation on any data.  Example: Assembly movi 5 r0 // Move integer 5 (2’s complement) to r0 addf 3.6 r0 // Treat bit representation in r0 as a // Floating point representation and add 3.6 // to it. Result? You can be sure that r0 does not contain 8.6!  (+) Flexibility : “I can do anything I want to and you can’t stop me”  (–) Ease of Error Checking. (programs are prone to errors, especially huge ones). “I am human, my brain is limited, I can’t remember and monitor everything.”

4. 4 1. Motivation for typed languages Typed Languages:  A type represents a set of values. Programs / procedures / operators are functions from an input type to an output type.  Type Checking is the activity of ensuring that the operands / arguments of an operator / procedure are of compatible type  This is done by using a set of rules for associating a type with every expression in the language. (These rules are known as the type system).  A type error results when an operator is applied to an operand of inappropriate/incompatible type.  Output of a type system:  There are type-errors (wrt type system) => Program is NOT type-safe.  There are no type-errors (wrt type system) => Program is type-safe.

5. 5 1. Motivation for typed languages TC says that there are type errors TC says that there are no type errors Program really has errors Program really does not have errors Usually truePossible TC errs on the conservative side ????? Program MAY still have errors 1.It may still have type errors due to unsafe features of a language. This is due to bad type system design. 2.It may have logic errors. This serves to show that type errors is but one of the many errors you encounter.

6. 6 1. Motivation for typed languages TC says that there are type errors Program really has errors Program really does not have errors Usually truePossible TC errs on the conservative side Typed Languages (+) Error Detection (+) Program documentation (–) Loss of Flexibility (but it’s ok, I don’t lose much freedom anyway since I don’t usually program in that way in the first place. I gain more than what I lose).

7. 7 2. Issues in Type Checking  How to type-check?  How to cater for polymorphism?  What is your definition of “compatible type”?  When to perform type checking?  Is your language strongly or weakly typed?

8. 8 2.1 How to type-check? Definition:  Type statements are of the form: : meaning that an expression ‘is-of-the- type’ (the ‘:’ symbol).  Examples:  3 : int  3+4 : int  3.14 : real  “abc” : String  while (x < 5) {x++;} : Stmt

9. 9 2.1 How to type-check? Definition:  Type rules are of the form: e 1 : t 1 e 2 : t 2 … e n : t n f e 1 e 2 …e n : t (rule name) where each e i : t i is a type statement, n  0. The rule is interpreted as “IF e 1 is of type t 1 and … and e n is of type t n THEN f e 1 e 2 …e n is of type t.” This is similar to the rules we saw when we studied semantics.

10. 10 2.1 How to type-check? Examples of type rules:  Rule for constants: E 1 : int E 2 : int E 1 + E 2 : int (+)  Rule for addition: 1 : int2 : int3 : int E 1 : int E 2 : int E 1 == E 2 : bool (==)  Rule for boolean comparison:

11. 11 2.1 How to type-check? Examples of type rules: x : T E : T x := E; : Stmt (:=)  Rule for if-statment:  Rule for assignment statement: E 1 : Bool S 1 : Stmt S 2 : Stmt if (E 1 ) {S 1 } else {S 2 } : Stmt (if)

12. 12 2.1 How to type-check?  Given the program: int x; … x := x+1; … …And Given the rules: 1 : int2 : int3 : int E 1 : int E 2 : int E 1 + E 2 : int (+) E 1 : int E 2 : int E 1 == E 2 : bool (==) x : T E : T x := E; : Stmt (:=) E 1 : Bool S 1 : Stmt S 2 : Stmt if (E 1 ) {S 1 } else {S 2 } : Stmt (if) A program/expression is type- safe if we can construct a derivation tree to give a type for that program/expression. x:=x+1; : Stmt x : int x+1 : int (:=) x : int 1 : int (+)

13. 13 2.1 How to type-check?  Given the program: int x; float y; … if (x == 3) { y := x; } else { x := x+1; } … …And Given the rules: 1 : int2 : int3 : int E 1 : int E 2 : int E 1 + E 2 : int (+) E 1 : int E 2 : int E 1 == E 2 : bool (==) x : T E : T x := E; : Stmt (:=) E 1 : Bool S 1 : Stmt S 2 : Stmt if (E 1 ) {S 1 } else {S 2 } : Stmt (if) A program/expression is type- safe if we can construct a derivation tree to give a type for that program/expression. if (x==3) {y:=x;} else {x:=x+1;} : Stmt x==3 : Bool y:=x; : Stmt x:=x+1; : Stmt (if) x : int 3 : int (==) ??? (:=) x : int x+1 : int (:=) x : int 1 : int (+) Follow the rules! Try to build tree. Cannot build tree => Not type safe

14. 14 Issues in Type Checking  How to type-check?  How to cater for polymorphism?  What is your definition of “compatible type”?  When to perform type checking?  Is your language strongly or weakly typed?

15. 15 2.2 How to cater for Polymorphism  Polymorphism = poly (many) + morph (form)  Polymorphism is the ability of a data object to that can take on or assume many different forms.  Polymorphism can be categorised into 2 types  Ad-hoc Polymorphism ( discussed now)  Universal Polymorphism  Parametric (discussed with Functional Programming)  Inclusion (discussed later in the lecture)

16. 16 2.2 How to cater for Polymorphism CoercionOverloadingParametricInclusion Polymorphism Ad-HocUniversal Ad-Hoc polymorphism is obtained when a function works, or appears to work on several different types (which may not exhibit a common structure) and may behave in unrelated ways for each type. Universal polymorphism is obtained when a function works uniformly on a range of types; these types normally exhibit some common structure. Cardelli and Wegner’s classification (1985)

17. 17 2.2 Polymorphism – Coercion COERCION A coercion is a operation that converts the type of an expression to another type. It is done automatically by the language compiler. (If the programmer manually forces a type conversion, it’s called casting) E : int E : float (Int-Float Coercion) int x; float y;... y := x;...

18. 18 2.2 Polymorphism – Coercion Example of the use of COERCION int x; float y; … if (x == 3) { y := x; } else { x := x+1; } … 1 : int2 : int3 : int E 1 : int E 2 : int E 1 + E 2 : int (+) E 1 : int E 2 : int E 1 == E 2 : bool (==) x : T E : T x := E; : Stmt (:=) E 1 : Bool S 1 : Stmt S 2 : Stmt if (E 1 ) {S 1 } else {S 2 } : Stmt (if) if (x==3) {y:=x;} else {x:=x+1;} : Stmt x==3 : Bool y:=x; : Stmt x:=x+1; : Stmt (if) x : int 3 : int (==) y : float x : float (:=) x : int x+1 : int (:=) x : int 1 : int (+) E : int E : float (Int-Float Coercion) Add in new rule… x : int (Coercion)

19. 19 2.2 Polymorphism – Coercion Coercion WideningNarrowing Widening coercion converts a value to a type that can include (at least approximations of) all of the values of the original type. Widening is safe most of the time. It can be unsafe in certain cases. Narrowing coercion converts a value to a type that cannot store (even approximations of) all of the values of the original type. Narrowing is unsafe. Information is lost during conversion of type. int float <- Widening Narrowing ->

20. 20 2.2 Polymorphism – Coercion Coercions (+) Increase flexibility in programming  Example: float x,y,z; int a,b,c;  If I have no coercions, and I intend to add y and a and store in x, then writing… x = y + ((float) a); …is too much of a hassle.  Therefore coercion can be useful.

21. 21 2.2 Polymorphism – Coercion Coercions (–) Decrease Reliability (error detection)  Example: float x,y,z; int a,b,c;  If I have coercions and I intend to add x and y and store in z, but I accidentally write… z = x + a; …then my error will go undetected because the compiler will simply coerce the a to a float.  Therefore coercion can lead to problems.

22. 22 2.2 Polymorphism – Overloading OVERLOADING An overloaded operation has different meanings, and different types, in different contexts. E 1 : int E 2 : int E 1 + E 2 : int (+-int) E 1 : float E 2 : float E 1 + E 2 : float (+-float)

23. 23 2.2 Polymorphism – Overloading 1 : int Example of the use of Overloading int x,y,z; float a,b,c; … if (x == 3) { x := y + z; } else { a := b + c; } … 2 : int3 : int E 1 : int E 2 : int E 1 + E 2 : int (+) E 1 : int E 2 : int E 1 == E 2 : bool (==) x : T E : T x := E; : Stmt (:=) E 1 : Bool S 1 : Stmt S 2 : Stmt if (E 1 ) {S 1 } else {S 2 } : Stmt (if) if (x==3) {x:=y+z;} else {a:=b+c;} : Stmt x==3 : Bool x:=y+z; : Stmt a:=b+c; : Stmt (if) x : int 3 : int (==) b:float c:float (+ -float) a : float b+c : float (:=) x : int y+z : int (:=) y:int z:int (+) Add in new rule… E 1 : float E 2 : float E 1 + E 2 : float (+-float)

24. 24 2.2 Polymorphism – Overloading Overloading (+) Increase flexibility in programming  Examples are when user wants to use an operator to express similar ideas.  Example: int a,b,c; int p[10], q[10], r[10]; int x[10][10], y[10][10], z[10][10]; a = b * c; // integer multiplication p = a * q; // Scalar multiplication x = y * z; // Matrix multiplication  Therefore overloading is good.

25. 25 2.2 Polymorphism – Overloading Overloading (–) Decrease Reliability (error detection)  Examples are when user intends to use the operator in one context, but accidentally uses it in another.  Example  In many languages, the minus sign is overloaded to both unary and binary uses. x = z–y and x = -y will both compile. What if I intend to do the first, but accidentally leave out the ‘z’?  Similarly, in C, we can have a situation when x = z&y and x = &y will both compile. Is overloading good?

26. 26 2.2 Polymorphism – Overloading  Even for common operations, overloading may not be good.  Example int sum, count; float average;... average = sum / count; Since sum and count are integers, integer division is performed first before result is coerced to float. That’s why Pascal has div for integer division and / for floating point division. Overloading (–) Decrease Reliability (error detection)

27. 27 2.2 How to cater for Polymorphism CoercionOverloadingParametricInclusion Polymorphism Ad-HocUniversal Ad-Hoc polymorphism is obtained when a function works, or appears to work on several different types (which may not exhibit a common structure) and may behave in unrelated ways for each type. Universal polymorphism is obtained when a function works uniformly on a range of types; these types normally exhibit some common structure. Cardelli and Wegner’s classification (1985)

28. 28 2.2 Inclusion Polymorphism  Q: Is the subclass regarded as a subtype of the parent class?  Yes – Inclusion Polymorphism (Sub-typing) class A {…} class B extends A {…} Note that B  A (Inclusion) A a = new B(); A a = new A(); Polymorphism  

29. 29 2.2 Inclusion Polymorphism  Q: Is the subclass regarded as a subtype of the parent class?  Yes – Inclusion Polymorphism (Sub-typing)  Some people call it the IS-A relationship between parent and derived class.  “class Table extends Furniture”  Table IS-A Furniture.  Table  Furniture

30. 30 2.2 Inclusion Polymorphism  Variables are polymorphic – since they can refer to the declared class and to subclasses too.  Requirement (Do you know why?):  Subclass must INHERIT EVERYTHING from the base class.  Subclass must NOT MODIFY ACCESS CONTROL of the base class methods/data.  That’s why C++ Inclusion Polymorphism definition adds a ‘public’ to the derived class since a private derived class modifies access control of base class methods/data.

31. 31 Issues in Type Checking  How to type-check?  How to cater for polymorphism?  What is your definition of “compatible type”?  When to perform type checking?  Is your language strongly or weakly typed?

32. 32 2.3 Type Equivalence type// type definitions Q = array [1..10] of integer; S = array [1..10] of integer; T = S; var// variable declarations a : Q; b : S; c : T; d : array [1..10] of integer; begin a := b; // Is this allowed? // Meaning to say “Is a and b // the same type?” a := c; // Is this allowed? a := d; // Is this allowed? b := c; // Is this allowed? end. type// type definitions Queue = array [1..10] of integer; Stack = array [1..10] of integer; Tree = Stack; var// variable declarations a : Queue; b : Stack; c : Tree; d : array [1..10] of integer; begin a := b; // Is this allowed? // Meaning to say “Is a and b // the same type?” a := c; // Is this allowed? a := d; // Is this allowed? b := c; // Is this allowed? end. If you had said “yes” to most of it, chances are that you are adopting structural equivalence. If you had said “no” most of the time, then it is likely you are adopting name equivalence.

33. 33 2.3 Type Equivalence Difference between type names and anonymous type names.  The type of a variable is either described through:  A type name: (1) those names defined using a type definition command. (eg. ‘type’ for Pascal, ‘typedef’ for C.), or… (2) the primitive numeric types (eg. int, float)  Or directly through a type constructor (eg. array-of, record-of, pointer-to). In this case, the variable has an anonymous type name.

34. 34 2.3 Type Equivalence type// type definitions Q = array [1..10] of integer; S = array [1..10] of integer; T = S; var// variable declarations a : Q; b : S; c : T; d : array [1..10] of integer; begin a := b; // Is this allowed? // Meaning to say “Is a and b // the same type?” a := c; // Is this allowed? a := d; // Is this allowed? b := c; // Is this allowed? end.  Example Q,S,T are type names d has a type, but d does not have a type name.

35. 35 2.3 Type Equivalence  When are two types equivalent (  )? Rule 1: For any type name T, T  T. Rule 2: If C is a type constructor and T 1  T 2, then CT 1  CT 2. Rule 3: If it is declared that type name = T, then name  T. Rule 4 (Symmetry): If T 1  T 2,then T 2  T 1. Rule 5 (Transitivity): If T 1  T 2 and T 2  T 3, then T 1  T 3.  What rules do you want to use?

36. 36 2.3 Type Equivalence  When are two types equivalent (  )? Rule 1: For any type name T, T  T. Rule 2: If C is a type constructor and T 1  T 2, then CT 1  CT 2. Rule 3: If it is declared that type name = T, then name  T. Rule 4 (Symmetry): If T 1  T 2,then T 2  T 1. Rule 5 (Transitivity): If T 1  T 2 and T 2  T 3, then T 1  T 3.  Structural Equivalence will use all the rules to check for type equivalence.

37. 37 2.3 Type Equivalence  When are two types equivalent (  )? Rule 1: For any type name T, T  T. Rule 2: If C is a type constructor and T 1  T 2, then CT 1  CT 2. Rule 3: If it is declared that type name = T, then name  T. Rule 4 (Symmetry): If T 1  T 2,then T 2  T 1. Rule 5 (Transitivity): If T 1  T 2 and T 2  T 3, then T 1  T 3.  (Pure) Name Equivalence will use only the first rule. Unless the two variables have the same type name, they will be treated as different type    

38. 38 2.3 Type Equivalence  When are two types equivalent (  )? Rule 1: For any type name T, T  T. Rule 2: If C is a type constructor and T 1  T 2, then CT 1  CT 2. Rule 3: If it is declared that type name = T, then name  T. Rule 4 (Symmetry): If T 1  T 2,then T 2  T 1. Rule 5 (Transitivity): If T 1  T 2 and T 2  T 3, then T 1  T 3.  Declarative Equivalence will leave out the second rule. 

39. 39 2.3 Type Equivalence type// type definitions Q = array [1..10] of integer; S = array [1..10] of integer; T = S; var// variable declarations a,x : Q; b : S; c : T; d : array [1..10] of integer; e : array [1..10] of integer; begin a := x; // Is this allowed? // Meaning to say “Is a and b // the same type?” a := b; // Is this allowed? a := c; // Is this allowed? a := d; // Is this allowed? b := c; // Is this allowed? d := e; // Is this allowed? end.  Example yes SE yes no NE yes no yes no DE R1: For any type name T, T  T. R2: If C is a type constructor and T 1  T 2, then CT 1  CT 2. R3: If it is declared that type name = T, then name  T. R4 (Symmetry): If T 1  T 2,then T 2  T 1. R5 (Transitivity): If T 1  T 2 and T 2  T 3, then T 1  T 3.

40. 40 2.3 Type Equivalence Name Equivalence  Easy to implement checking, since we need only compare the name.  Very restrictive, inflexible. type idxtype = 1..100; var count : integer; index : idxtype; Structure Equivalence  Harder to implement since entire structures must be compared. Other issues to consider: eg. arrays with same sizes but different subscripts – are they the same type? (similar for records and enumerations)  More flexible, yet the flexibility can be bad too. type celsius = real; fahrenheit = real; var x : celsius; y : fahrenheit;...x := y; // Allowed?

41. 41 2.3 Type Equivalence  Different Languages adopt different rules. And the rules may change for one language (people can change their minds too!)  Pascal  Before 1982 – unknown.  ISO1982 – Declarative Equivalence.  ISO1990 – Structural Eqivalence.  C : Structural Equivalence, except for structs and unions, for which C uses declarative equivalence. If the two structs are in different files, then C goes back to structural equivalence.  C++ : Name Equivalence  Haskell/SML : Structural Equivalence.

42. 42 Issues in Type Checking  How to type-check?  How to cater for polymorphism?  What is your definition of “compatible type”?  When to perform type checking?  Is your language strongly or weakly typed?

43. 43 2.4 When to perform Type Checking? Compile-Time (Static Type Binding) In theory, you can choose to type check at compile time or run-time. In practice, languages try to do it as much statically as possible. Eg. SML, Pascal Run-Time (Dynamic Type Binding) No choice but to do dynamic type checking. Eg. JavaScript, APL When is the variable bound to the type? When can I type check?

44. 44 2.4 When to perform Type Checking?  Static Type Checking – done at compile time.  (+) Done only once  (+) Earlier detection of errors  (–) Less Program Flexibility (Fewer shortcuts and tricks)

45. 45 2.4 When to perform Type Checking?  Dynamic Type Checking – done at run time.  (–) Done many times  (–) Late detection of errors  (–) More memory needed, since we need to maintain type information of all the current values in their respective memory cells.  (–) Slows down overall execution time, since extra code is inserted into the program to detect type error.  (+) Program Flexibility (Allows you to ‘hack’ dirty code.)

46. 46 Issues in Type Checking  How to type-check?  How to cater for polymorphism?  What is your definition of “compatible type”?  When to perform type checking?  Is your language strongly or weakly typed?

47. 47 2.5 Strong Type Systems  A programming language is defined to be strongly typed if type errors are always detected STATICALLY.  A language with a strong-type system only allows type- safe programs to be successfully compiled into executables. (Otherwise, language is said to have a weak type system).  Programs of strong-type systems are guaranteed to be executed without type-error. (The only error left to contend with is logic error).

48. 48 2.5 Strong Type Systems Fortran Ada Modula-3 C, C++ Java Pascal SML Haskell Strongly Typed? LanguageWhy? NoAllows variable of one type to refer to value of another type through EQUIVALENCE keyword. NoLibrary function UNCHECKED_CONVERSION suspends type checking. NoSame as Ada through use of keyword LOOPHOLE No1. Forced conversion of type through type casting 2. Union Types can compromise type safety NoType Casting AlmostVariant Records can compromise type safety Yes All variables have STATIC TYPE BINDING.

49. 49 2.5 Weak-Type Systems: Variant Recs  Variant Records in C (via union keyword) compromises Type Safety... typedef union { int X; float Y; char Z[4];} B;... B P;  Variant part all have overlapping (same) L-value!!!  Problems can occur. What happens to the code below? P.X = 142; printf(“%O\n”, P.Z[3])  All 3 data objects have same L-value and occupy same storage. No enforcement of type checking.  Poor language and type system design

50. 50 2.5 Weak-Type Systems: Variant Recs  Variant Records in Pascal tries to overcome C’s deficiency. They have a tagged union type. type whichtype = (inttype, realtype); type uniontype = record case V : whichtype of inttype : (X: integer); realtype: (Y: real); end  But the compiler usually doesn’t check the consistency between the variant and the tag. So we can ‘subvert’ the tagged field: var P: uniontype P.V = inttype; P.X = 142; P.V = realtype; // type safety compromised