CSE-321 Programming Languages Fixed Point Combinator

1. CSE-321 Programming LanguagesFixed Point Combinator 박성우 POSTECH March 29, 2006

2. : -reduction redex = reducible expression Values and Reductions

3. Substitution Completed avoid variable captures

4. Outline • Abstract syntax of the -calculus V • Operational semantics of the l-calculus V • Substitutions V • Programming in the -calculus V • Fixed point combinator • non-termination in the -calculus • recursive functions in the -calculus

5. Non-termination in -calculus

6. Recursive Functions in -calculus • Plan of attack • Assume a recursive function constructfun f x. e • Rewrite it as an expression in the -calculusi.e., show that it is syntactic sugar.

7. Example: Factorial • Plan of attack • Assume a recursive function construct • Rewrite it as an expression in the -calculusi.e., show that it is syntactic sugar.

8. fac to FAC

9. FAC produces a new function from f

10. new! KEY Observation on FAC

11. FAC improves the input function f :computes fac(0), fac(1), ..., fac(k) :computes fac(0), fac(1), ..., fac(k), fac(k + 1)

12. Special Case no matter what

13. Invariant:

14. FAC fac

15. Key Equation

16. What is f ?

18. (Earlier Slide)

21. Now we only need to solve:

22. Fixed Point • Fixed point of function f V such that V = f (V) • Ex. f (x) = 2 - x fixed point of f = 11 = f (1)

23. Now we only need to solve: Now we only need to find a fixed point of:

24. Magic Revealed • Fixed point combinator / Y combinator (call-by-value) • fix F returns a fixed point of F.

25. Magic Explained

26. Now we only need to solve: Now we only need to find a fixed point of: Now we only need to compute:

27. Answer

28. If you are a masochist, • See the course notes to learn how to derive the fixed point combinator! • There is a really crazy idea!