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Fixed Point Illustrations

Fixed Point Illustrations. Simple Examples. f : int -> int f(n) = 5 n = 5 is a unique fixed point. f(n) = n^2 – 2 n = 2 and n = -1 are both fixed points. f(n) = n This has infinite fixed points. Fixed point computing function (fix f) = f (fix f) Declarative specification

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Fixed Point Illustrations

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  1. Fixed Point Illustrations

  2. Simple Examples • f : int -> int • f(n) = 5 • n = 5 is a unique fixed point. • f(n) = n^2 – 2 • n = 2 and n = -1 are both fixed points. • f(n) = n • This has infinite fixed points.

  3. Fixed point computing function (fix f) = f (fix f) • Declarative specification • In certain formalizations of mathematics, such as the lambda calculus and combinatorial calculus, every function has a fixed point. • In Denotational Semantics, the meaning associated with a recursive definition is given in terms of least fixed point. E.g., while-loop, etc • From a more practical point of view, this fixed point approach allows the definition of anonymous recursive functions. Somewhat surprisingly, the fixed-point function can be defined with non-recursive lambda abstractions.

  4. Fixed Point Combinator • A fixed point combinator Y is a function such that f(Y(f)) = Y(f) for all functions f. • Well-known fixed point combinators • Discovered by Haskell B. Curry • Y = λf.(λx.(f (x x)) λx.(f (x x))) • Discovered by Alan Turing • Θ = (λx.λy.(y (x x y)) λx.λy.(y (x x y)))

  5. Factorial Function • fac :: Int -> Int • fac n = if (n<1) 1 (n * fac (n-1)) • Now the factorial (using a fixed point combinator) looks like: • factorial = Y (lfac ln -> if (n<1) 1 (n * fac (n-1)))

  6. F = \f.\x.if (x = 0) then 1 else (f (x-1)) • Now we run factorial(3) as follows... • ((Y F) 3) => F (Y F) 3 • (note we now have F followed by 2 arguments, the first is the function copy and the second is the number 3). • F (Y F) 3 => if (3 = 0) then 1 else ((Y F) (3-1)) • Where Y F is used to copy F once again • if (3 = 0) then 1 else 3 * ((Y F) (3-1)) • if (3 = 0) then 1 else 3 * F (Y F) 2 • if (3 = 0) then 1 else 3 * (if (2 = 0) then 1 else 2 * (Y F) 1) • if (3 = 0) then 1 else 3 * (if (2 = 0) then 1 else 2 * F (Y F) 1) • if (3 = 0) then 1 else 3 * (if (2 = 0) then 1 else 2 * (if (1 = 0) then 1 else 1 * (Y F) 0)) • …

  7. … • if (3 = 0) then 1 else 3 * (if (2 = 0) then 1 else 2 * (if (1 = 0) then else 1 * (if (0 = 0) then 1 else 0 * (Y F) -1) )) • Which we can now simply evaluate... • if (3 = 0) then 1 else 3 * (if (2 = 0) then 1 else 2 * (if (1 = 0) then 1 else 1 * (1) )) • if (3 = 0) then 1 else 3 * (if (2 = 0) then 1 else 2 * 1) • if (3 = 0) then 1 else 3 * 2 • 6

  8. SML code • fun fix f = f (fix f); • (* val fix = fn : ('a -> 'a) -> 'a *) • fix (fn n => 5); • ( fix ( fn f => (fn n => if n = 0 then 1 else n * f (n-1)) ) ) 5; • (* • Infinite loop due to "call by value" parameter passing mechanism • GC #0.0.0.0.1.8: (0 ms) • GC #0.0.0.1.2.11: (156 ms) • *)

  9. Haskell Code • -- :load "I:\tkprasad\cs776\test.hs" • fix f = f (fix f) • -- fix (\n -> 5) • -- Main> fix (\n -> 5) • -- 5 :: Integer • -- Main> ( fix ( \f -> (\n -> if n == 0 then 1 else n * f (n-1)) ) ) 3 • -- 6 :: Integer • -- Main> ( fix ( \f -> (\n -> if n == 0 then 1 else n * f (n-1)) ) ) 5 • -- 120 :: Integer • -- Main> ( fix ( \f -> (\n -> if n == 0 then 1 else n * f (n-1)) ) ) 6 • -- 720 :: Integer • -- Main> fix (\n -> n*n - 2) • -- ERROR - C stack overflow

  10. Scheme Code • (define (Y f) • ( (lambda (x) (f f)) (lambda (x) (f f)) ) ) • (define (fact-nr f) • (lambda (n) • (if (zero? n) 1 (* n ((f f) (- n 1)))) )) • ((Y fact-nr) 3) • 6 • ((Y fact-nr) 5) • 120

  11. References for Combinatory Logic • To Kill a Mocking Bird • by Harper Lee • To Mock a Mocking Bird • by Raymond Smullyan

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