CSE321 Programming Languages Fixed Point Combinator POSTECH March

CSE-321 Programming Languages Fixed Point Combinator 박성우 POSTECH March 29, 2006

Values and Reductions : -reduction redex = reducible expression 2

Substitution Completed avoid variable captures 3

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

Non-termination in -calculus 5

Recursive Functions in -calculus • Plan of attack 1. Assume a recursive function construct fun f x. e 2. Rewrite it as an expression in the -calculus i. e. , show that it is syntactic sugar. 6

Example: Factorial • Plan of attack 1. Assume a recursive function construct 2. Rewrite it as an expression in the -calculus i. e. , show that it is syntactic sugar. 7

fac to FAC 8

FAC produces a new function from f 9

KEY Observation on FAC new! 10

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

Special Case no matter what 12

Invariant: 13

FAC fac 14

Key Equation 15

What is f ? 16

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(Earlier Slide) 18

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Now we only need to solve: 21

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

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

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

Magic Explained 25

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

Answer 27

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