CSE-321 Programming Languages Simply Typed  -Calculus

1. CSE-321 Programming LanguagesSimply Typed -Calculus 박성우 POSTECH March 28, 2007

2. -Calculus • Abstract syntax • Operational semantics (call-by-value)

3. Simulating Base Types • Booleans • Natural numbers • Fixed point combinator

4. Motivation • The -calculus is expressive. • booleans, integers, lists, recursive functions, ... • But it is highly inefficient to program in the -calculus. • Why not just use instead of ? • So we introduce a type system!

5. Simply Typed -Calculus • An extension of the untyped-calculus with types • Assumes a fixed set of base types • E.g. • base type • primitive constructs • A subset of Standard ML

6. Outline • Introduction V • The simply typed -calculus • Abstract syntax • Operational semantics • Type system • Type safety

7. Abstract Syntax

8. What if there is no base type? No interesting expression!

9. Outline • Introduction V • The simply typed -calculus • Abstract syntax V • Operational semantics • Type system • Type safety

10. Simply Untyped -Calculus

11. Reduction Rules for Booleans

12. Capture-Avoiding Substitutions • Completely analogous

13. Free Variables • Completely analogous

14. Outline • Introduction V • The simply typed -calculus • Abstract syntax V • Operational semantics V • Type system • Type safety

15. What is the type of ? Answer:

16. How to find the type of • Assume that the type of x is A. • okay • Find the type of x. • A • Build a function type • A ! A Need to make assumptions on types of variables!

17. Type System • Typing context • Typing judgment

18. Typing Rules --- Top-down

19. Typing Rules --- Bottom-up

20. Typing Rules for Booleans

21. Typing Derivation

22. Typing Derivation

23. Outline • Introduction V • The simply typed -calculus V • Abstract syntax V • Operational semantics V • Type system V • Type safety

24. Unsafe Operations in C • Adding two pointers • Subtracting an integer from a string • which is okay, but likely to be unintended • Null-pointer dereferencing • Argh... segmentation fault! • Using an integer as a destination address in a function call • ...

25. Assignment 3+ • Count the number of hours that you spend before getting your Matrix program to typecheck. • Example: • 0am: You start. • 10am: Your program compiles with no type errors. • Then count the number of extra hours that you spend before completing Assignment 3.

26. Type Safety • Slogan"well-typed expressions never go wrong" • Two theorems • Type preservation:"A well-typed expression reduces to another expression of the same type." • Progress:"A well-typed expression does not get stuck: either it is a value or reduces to another expression."

27. Type Preservation + Progress • A well-typed expression e: • If it is a value, we are finished. • If it is not, • It reduces to another e' [Progress] • e' has the same type as e. [Type preservation]

28. Type Safety • Type preservation • Progress

29. Proof of Type Safety • Use the rule induction. • In the next lecture!