1. Typed Lambda Calculus Adapted from Lectures by Profs Aiken and Necula of Univ. of California at Berkeley Prasad

2. Lecture Outline • Lambda Calculus • Review and examples • Simple types • Type checking • Comparison with OO type systems Prasad

3. Lambda Calculus Review • Grammar: E ::= x variables | E1 E2function application | lx. E function creation • Evaluation: (lx. e) e’®[e’/x]e Prasad

4. Example 1 (lx.x) ly.y ®ly.y (lx.x) ly.y ® [ly.y/x] x = ly.y Prasad

5. Example 2 (lx.ly.x) (lz.z) e ®*lz.z ((lx.ly.x) (lz.z)) e ® ([lz.z/x]ly.x) e = (ly.lz.z) e ® [e/y]lz.z = lz.z Prasad

6. Example 3 (lx. x x) (lx. x x) ® (lx. x x) (lx. x x) ® . . . (lx. x x) (lx. x x) ® [(l x. x x)/x](x x) = (lx. x x) (lx. x x) Prasad

7. Question What programming errors can occur in lambda calculus? Prasad

8. Notes • We can encode anything we like in lambda calculus • Booleans, integers, objects, if-then-else, recursion, . . . • But don’t forget that these are encodings • Akin to programming directly with 0’s and 1’s Prasad

9. Extension • Grammar: E ::= x variables | E1 E2function application | lx. E function creation | 0,1,2,… integers | + addition • Evaluation: (lx. e) e’®[e’/x]e + i j ® k where k = i + j Prasad

10. Digression • There is nothing magic about adding integers and + as constants to the lambda calculus • We could add any data types and operations we like • They can be encoded directly in lambda calculus anyway Prasad

11. Examples (lx. + x 1) (lx. + x x) (lx. + x 1) 3 ® 4 (lx. + x 1) (ly.y) ® ? Prasad

12. What Happens? (lx. + x 1) (ly.y) ® + (ly.y) 1 ® ? Answer: It depends. A runtime error; we can’t add to a function Or no error: If + and 1 are encoded as lambda terms, computation will continue! Prasad

13. Notes • Assume 1, + are encoded as lambda terms • Nothing guarantees the encoding of 1 is used as an integer • E.g., (1 lx.x) • Evaluation doesn’t know what the encodings are supposed to represent Prasad

14. Back to the Future • We need to be able to restrict uses of values to appropriate operations • We need a type system! • Note that we’d like a type system whether or not + (ly.y) 1 causes a runtime error • Catching these errors before program execution is better Prasad

15. Typed Lambda Calculus • Grammar: E ::= x variables | E1 E2function application | lx:t. E function creation | 0,1,2,… integers | + addition • Evaluation: (lx:t. e) e’®[e’/x]e + i j ® k where k = i + j Prasad

16. The Types • We have only two kinds of values • Integers • Functions • Type grammar: t := int | t®t Prasad

17. Examples of Types int int ® int (int ® int) ® int int ® (int ® int) Prasad

18. “has type” “it is provable that” Type environment Type Expression Type Judgments Prasad

19. Examples Prasad

20. More Examples Prasad

21. [Var] Type Rule: Variables A variable has the type assumed in the type environment. Prasad

22. Abstraction A function has type t1®t2 if the function body has type t2 when we assume the argument has type t1. [Abs] Prasad

23. [App] Application Applying a function of type t1®t2 to an argument of type t1gives a result of type t2 Prasad

24. Integers An integer has type int [Int] Prasad

25. Addition Adding two int’s produces an int [Add] Prasad

26. Example 1 Prasad

27. Example 2 Prasad

28. Type Checking • One recursive descent traversal. • Top-down: Add type declarations of formal parameters to environments [Abs] Prasad

29. [App] Type Checking (Cont.) • Bottom-up: Check that types match in applications Prasad

30. Structural Equality • “Types match” means “types are equal” • Equality is recursively defined int = int a ® b = c ® d Û a = c Ù b = d Prasad

31. Notes • In typed lambda calculus: • Types of all variables are declared • Types are structured (e.g., int ® int) • Types are checked for equality • Many typed languages in this class • Nearly all typed languages before 1980 • FORTRAN, ALGOL, Pascal, C • Captures C typing except for casts & coercions Prasad

32. Typed OO Languages • In many typed object-oriented languages • Types of all variables are declared • Types are non-structural (just names) • Declare all types and type relationships • Types are checked for subtyping relationships • What if type declarations are omitted? • A language with type inference Prasad

33. Discussion What about structural types + subtyping? • Area of current research • Currently no consensus on the right way to combine C-like type systems with typical OO-like type systems Prasad