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Understanding Fixed-Point Combinators in Lambda Calculus

This presentation discusses fixed-point combinators within the framework of lambda calculus, focusing on $beta$-reduction, the definition of reducible expressions, and the concept of values and substitutions without variable captures. The abstract syntax, operational semantics, and recursive functions in lambda calculus are explored, with emphasis on the Y combinator and its applications, including non-termination scenarios. Illustrative examples, like the factorial function, clarify these concepts, showcasing the power of fixed points in recursive function definitions.

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Understanding Fixed-Point Combinators in Lambda Calculus

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  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 ?

  17. (Earlier Slide)

  18. Now we only need to solve:

  19. 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)

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

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

  22. Magic Explained

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

  24. Answer

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

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