Type Checking

1. Type Checking Textbook: Types and Programming Languages Benjamin Pierce

2. Plan • Motivation (Chapter 1) • Untyped Arithmetic Expressions (Chapter 3) • Typed Arithmetic Expressions (Chapter 8) • Untyped Lambda Calculus (Chapter 5) • Typed Lambda Calculus (Chapter 9) • Extensions

3. Specification of software behavior Reasoning about logical formulas The annotation burden The dynamic nature of software Automatically infer safe invariants from the code Concentrate on very simple properties with lightweight annotations Challenges of Program Verification

4. June 4, 1996 The European Ariane5 rocket explodes 40s into its maiden flight due to a software bug.

5. 1997 Mars Rover Loses Contact 1999 Mars Climate Orbiter is Lost 1999 Mars polar Lander is lost 2004 Mar rover freezes

6. August 2005 As a Malaysia Airlines jetliner cruised from Perth, Australia, to Kuala Lumpur, Malaysia, one evening last August, it suddenly took on a mind of its own and zoomed 3,000 feet upward. The captain disconnected the autopilot and pointed the Boeing 777's nose down to avoid stalling, but was jerked into a steep dive. He throttled back sharply on both engines, trying to slow the plane. Instead, the jet raced into another climb. The crew eventually regained control and manually flew their 177 passengers safely back to Australia. Investigators quickly discovered the reason for the plane's roller-coaster ride 38,000 feet above the Indian Ocean. A defective software program had provided incorrect data about the aircraft's speed and acceleration, confusing flight computers. August 2005

7. Type Systems • Atractable syntactic method for proving absence of certain program behaviors by classifying phrases according to the kinds they compute • Examples • Whenever f is called, its argument must be integer • The arguments of f are not aliased • The types of dimensions must match • …

8. What is a type • A denotation of set of values • Int • Bool • … • A set of legal operations

9. Static Type Checking • Performed at compile-time • Conservative (sound but incomplete) • if <complex test> then 5 else <type error> • Usually limited to simple properties • Prevents runtime errors • Enforce modularity • Protects user-defined abstractions • Allows tractable analysis • Properties beyond scope (usually) • Array out of bound • Division by zero • Non null reference

10. Error detection • Early error detection • Logical errors • Interface errors • Dimension analysis • Effectiveness also depends on the programmer • Can be used for code maintenance

11. Abstraction • Types define interface between different software components • Enforces disciplined programming • Ease software integration

12. Documentation • Types are useful for reading programs • Can be used by language tools

13. Language Safety • A safe programming language protects its own abstraction • Can be achieved by type safety

14. Statically vs. Dynamically Checked Languages

15. Eiffel, 1989 Cook, W.R. (1989) - A Proposal for Making Eiffel Type-Safe, in Proceedings of ECOOP'89. S. Cook (ed.), pp. 57-70. Cambridge University Press. Betrand Meyer, on unsoundness of Eiffel: “Eiffel users universally report that they almost never run into such problems in real software development.”

16. Ten years later: Java

17. Efficiency • Compilers can used types to optimize computations • Pointer scope (Titanium) • Region inference

18. Language Design • Design the programming language with the type system • But types incur some notational overhead • Implicit vs. explicit types • The annotation overhead

19. Untyped Arithmetic Expressions Chapter 3

20. Untyped Arithmetic Expressions t ::= terms true constant true false constant false if t then t else t conditional 0 constant zero succ t successor pred t predecessor izzero t zero test if false then 0 else 1 1 iszero (pred (succ 0)) true

21. Untyped Arithmetic Expressions t ::= terms true constant true false constant false if t then t else t conditional 0 constant zero succ t successor pred t predecessor izzero t zero test succ true type error if 0 then 0 else 0 type error

22. t1 t’1 (E-IF) if t1 then t1 else t2 if t’1 then t1 else t2 SOS for Booleans t1t’1 t ::= terms true constant true false constant false if t then t else t conditional v ::= values true false if true then t1 else t2 t1 (E-IFTRUE) if false then t1 else t2 t2 (E-IFFALSE)

23. SOS for Numbers t1 t’1 (E-SUCC) succ t1  succ t’1 t1 t’1 (E-PRED) pred t1 pred t’1 t1 t’1 (E-ISZERO) iszero t1  iszero t’1 t1t’1 t ::= terms 0 constant zero succ t successor pred t predecessor iszero t zero test v ::= values true true value false false value nv numeric values nv ::= numeric values 0 zero value succ nv successor value pred 0  0 (E-PREDZERO) pred (succ nv1)  nv1 (E-PREDSUCC) iszero 0  true (E-ISZEROZERO) iszero (succ nv1)  false (E-ISZEROSUCC)

24. Typed Arithmetic Expressions Chapter 8

25. Stuck Computations • The goal of the type system is to ensure at compile-time that no stuck ever occurs at runtime • Safety (soundness) • Progress: A well-typed term t never gets stuck • Either it has value or there exists t’ such that t t’ • Preservation (subject reduction) • If well type term takes a step in evaluation, then the resulting term is also well typed

26. t1 : Bool t2 : T t3 : T (T-IF) if t1 then t2 else t3 : T Type Rules for Boolean t : T T ::= types Bool type of Boolean true : Bool (T-TRUE) false : Bool (T-FALSE)

27. t1 : Nat t1 : Nat t1 : Nat T-SUCC T-PRED ISZERO succ(t1) : Nat pred(t1) : Nat iszero(t1) : Bool Type Rules for Numbers t : T T ::= types Nat type of Natural numbers 0 : Nat (T-ZERO)

28. The Typing Relation • Formally the typing relation is the smallest binary relation between terms and types • in terms of inclusion • A term t is typable (well typed) if there exists some type T such that t : T

29. Inversion of the typing relation • true : R  R = Bool • false : R  R = Bool • if t1 then t2 else t3 : R  t1: Bool, t2 : R, t3: R • 0 : R  R = Nat • succ t1 : R  R = Nat and t1 : Nat • pred t1 : R  R = Nat and t1 : Nat • iszero t1 : R  R = Bool and t1 : Nat

30. Uniqueness of Types • Each term t has at most one type • If t is typable then • its type is unique • There is a unique type derivation tree for t

31. Safety • Canonical Forms: • If v is a value of type Boolean then v =true or v=false • If v is a value of type Nat then v belongs to nv • Progress: If t is well defined then either t is a value or for some t’: t  t’ • Preservation: if t : T and t  t’ then t’ : T nv ::= numeric values 0 zero value succ nv successor value

32. Untyped Lambda Calculus Chapter 5

33. Untyped Lambda Calculus t ::= terms x variable  x. t abstraction t t application ( x. x) ( x. x) ( x. x) • Syntactic Conventions • Applications associates to left • The body of abstraction extends as far as possible

34. Scope • An occurrence of x is free in a term t if it is not in the body on an abstraction x. t • otherwise it is bound • x is a binder • FV: t  P(Var) is the set free variables of t • FV(x) = {x} • FV( x. t) = FV(t) – {x} • FV (t1 t2) = FV(t1) FV(t2) • Terms w/o free variables are combinators • Example:  x. x

35. Operational Semantics ( x. t12) t2  [x t2] t12 (-reduction) [xs]x =s [x s]y=y if y  x [x s ] (y. t1) = y. [x s ] t1 if y  x and yFV(s) [x s ] (t1 t2) = ([x s ] t1) ([x s ] t2) ( x. x) y  y ( x. x ( x. x) ) (u r) u r ( x. x) ( x (y. x y)) (y z)  ( x (w. x w)) (y z)  w. y z w

36. Operational Semantics ( x. t12) t2  [x t2] t12 (-reduction) [xs]x =s [x s]y=y if y  x [x s ] (y. t1) = y. [x s ] t1 if y  x and yFV(s) [x s ] (t1 t2) = ([x s ] t1) ([x s ] t2) • Evaluation orders: • Full beta reduction • Normal order • Call by name • Call by value )x. x)((x. x)(z. (x. x) z))

37. Programming in the Lambda Calculus • Turing complete • Multiple arguments • (x, y). s = x. y.s • Church Booleans • tru = t. f. t • fls =t. f. f • test =l. m. n. l m n • and = b. c. b c fls • Pairs • pair = f. b. s. b f s • fst = p. p tru • snd = p. p fls • Church Numerals • c0 = f. z. z • c1 =f. z. s z • c2 = f. z. s (s z) • c3 = f. z. s (s (s z)) • Divergence • omega= (x. x x) (x. x x)

38. Summary Lambda Calculus • Powerful • Useful to illustrate ideas • But can be counterintuitive • Usually extended with useful syntactic sugars

39. Simple Typed Lambda Calculus Chapter 9

40. Simple Typed Lambda Calculus t ::= terms x variable  x: T. t abstraction t t application T::= types  T  T types of functions

41. t1 t’1 t2 t’2 (E-APP1) (E-APP2) t1 t2 t’1 t2 t1 t2 t1 t’2 SOS for Simple Typed Lambda Calculus t ::= terms x variable  x: T. t abstraction t t application t1 t2 v::= values  x: T. t abstraction values ( x: T11. t12) t2  [x t2] t12 (E-APPABS) T::= types  T  T types of functions

42. x : T  t1 : Bool t2 : T t3 : T (T-VAR)  x : T if t1 then t2 else t3 : T , x : T1 t2 : T2 (T-ABS)  x : T1. t2 : T2 : T1 T2  t2 : T11  t1 : T11T12 (T-APP)  t1 t2 : T12 Type Rules  t : T t ::= terms x variable  x: T. t abstraction T::= types  T  T types of functions ::= context   empty context , x : T term variable binding

43.   true : Bool (T-TRUE) x : T  t1 : Bool t2 : T t3 : T (T-VAR)   false : Bool (T-FALSE)  x : T if t1 then t2 else t3 : T , x : T1 t2 : T2 t1 : Bool t2 : T t3 : T (T-ABS)  x : T1. t2 : T2 : T1 T2 (T-IF)  if t1 then t2 else t3 : T  t2 : T11  t1 : T11T12 (T-APP)  t1 t2 : T12  t : T t ::= terms x variable  x: T. t abstraction t t application true constant true false constant false if t then t else t conditional T::= types Bool Boolean type  T  T types of functions ::= context   empty context , x : T term variable binding

44. Examples • )x:Bool. x ) true • if true then )x:Bool. x) else ) x:Bool. x) • if true then )x:Bool. x) else ) x:Bool. y:Bool. x)

45. The Typing Relation • Formally the typing relation is the smallest ternary relation on contexts, terms and types • in terms of inclusion • A term t is typable in a given context  (well typed) if there exists some type T such that t : T

46. Inversion of the typing relation •   x : R  x: R  •   x : T1. t2 : R  R = T1 R2 for some R2 with  t2 : R2 •   t1 t2 : R  there exists T11 such that  t1 : T11 R and  t2 : T11 •   true : R  R = Bool •   false : R  R = Bool •   if t1 then t2 else t3 : R   t1: Bool,   t2 : R,   t3: R

47. Uniqueness of Types • Each term t has at most one type in any given context • If t is typable then • its type is unique • There is a unique type derivation tree for t

48. Safety • Canonical Forms: • If v is a value of type Boolean then v =true or v=false • If v is a value of type T1 T2 then v =x. T1. t2 • Progress: If t is closed and well defined (t : T for some T) then either t is a value of for some t’: t  t’ • Permutation: if   t : T and  is a permutation of  then   t : T • Weakening: if   t : T and x  dom() then , x : S  t : T • Preservation of types under substitution: • If , x : S  t : T and   s : S then  [x s] t : T • Preservation: if  t : T and t  t’ then  t’ : T

49. Implicit vs. Explicit Types • Do we have to spell out the type of every argument? • Implicit type systems allow the programmers to omit the types as long as the resulting type is unique • Reduces the annotation burden • A type inference algorithm infers a unique type or issues an error

50. Other Issues • The Curry-Howard Correspondence • Erasure of typability • Curry-Stlyle vs. Church style of languages definition