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Lecture # 20

Lecture # 20. Type Systems. Type Systems. A type system defines a set of types and rules to assign types to programming language constructs Informal type system rules, for example “ if both operands of addition are of type integer, then the result is of type integer ”

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Lecture # 20

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  1. Lecture # 20 Type Systems

  2. Type Systems • A type system defines a set of types and rules to assign types to programming language constructs • Informal type system rules, for example “if both operands of addition are of type integer, then the result is of type integer” • Formal type system rules: Post system

  3. Type Rules in Post System Notation Type judgmentse : where e is an expression and  is a type Environment maps objects v to types :(v) =  (v) =   v :   (v) =   e :   v := e : void   e1 : integer e2 : integer     e1+e2 : integer  

  4. Type System Example Environment  is a set of name, type pairs, for example:  = { x,integer, y,integer, z,char, 1,integer, 2,integer } From  and rules we can check types:type checking = theorem proving The proof that x := y + 2 is typed correctly: (y) = integer (2) = integer y : integer 2 : integer     y + 2 : integer (x) = integer   x := y + 2 : void  

  5. A Simple Language Example E true  false  literal  num  id  EandE  E+E  E[E]  E^ P D;SD D;D  id : TT boolean  char  integer  array[ num ] ofT  ^TS id :=E  ifEthenS  whileEdoS  S;S Pointer to T Pascal-like pointer dereference operator

  6. Simple Language Example: Declarations D id : T { addtype(id.entry, T.type) }T boolean { T.type := boolean }T char { T.type := char }T integer { T.type := integer }T array[ num ] ofT1 { T.type := array(1..num.val, T1.type) }T ^T1 { T.type := pointer(T1) Parametric types:type constructor

  7. Simple Language Example: Checking Statements (v) =   e :   v := e : void   S id :=E { S.type := ifid.type = E.type thenvoidelsetype_error } Note: the type of id is determined by scope’s environment:id.type = lookup(id.entry)

  8. Simple Language Example: Checking Statements (cont’d) e : boolean  s :     if e then s :   S ifEthenS1 { S.type := ifE.type = booleanthenS1.typeelsetype_error }

  9. Simple Language Example: Statements (cont’d)  e: boolean  s :     while e do s :   S whileEdoS1 { S.type := ifE.type = booleanthenS1.typeelsetype_error }

  10. Simple Language Example: Checking Statements (cont’d)  s1 : void  s2 : void   s1; s2: void   S S1;S2 { S.type := ifS1.type = void and S2.type = void then void elsetype_error }

  11. Simple Language Example: Checking Expressions (v) =   v :   E true { E.type = boolean }E false { E.type = boolean }E literal { E.type = char }E num { E.type = integer } E id { E.type = lookup(id.entry) }…

  12. Simple Language Example: Checking Expressions (cont’d) e1 : integer e2 : integer      e1+e2 : integer  E E1+E2 { E.type := ifE1.type = integerandE2.type = integerthenintegerelsetype_error }

  13. Simple Language Example: Checking Expressions (cont’d) e1 : boolean e2 : boolean     e1ande2 : boolean   E E1andE2{ E.type := ifE1.type = booleanandE2.type = booleanthenbooleanelsetype_error }

  14. Simple Language Example: Checking Expressions (cont’d) e2 : integer  e1 : array(s, )     e1[e2] :   E E1[E2] { E.type := ifE1.type = array(s, t) andE2.type = integerthentelsetype_error }

  15. Type Conversion and Coercion • Type conversion is explicit, for example using type casts • Type coercion is implicitly performed by the compiler to generate code that converts types of values at runtime (typically to narrow or widen a type) • Both require a type system to check and infer types from (sub)expressions

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