1. CSE-321 Programming Languages Simply Typed -Calculus
박성우
POSTECH
April 3, 2006

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

3. Simulating Base Types
Booleans
Natural numbers
Fixed point combinator

4. Motivation The -calculus is equally expressive as Turing machines.
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. Huffman Coding in SML
Suppose that you spent 10 hours before getting your program to typecheck.
0am: You start.
10am: Your program compiles with no type errors.
How many more hours did you spend after that?

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 is not 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!