CSE-321 Programming Languages

Fixed Point Combinator

POSTECH

March 29, 2006

박성우

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Values and Reductions

redex = reducible expression

: -reduction

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Substitution Completed

avoid variable captures

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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

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Non-termination in -calculus

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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 -calculusi.e., show that it is syntactic sugar.

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• Plan of attack

1. Assume a recursive function construct

2. Rewrite it as an expression in the -calculusi.e., show that it is syntactic sugar.

Example: Factorial

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fac to FAC

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FAC produces a new function from f

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KEY Observation on FAC

new!

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FAC improves the input function f

:computes fac(0), fac(1), ..., fac(k)

:computes fac(0), fac(1), ..., fac(k), fac(k + 1)

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Special Case

no matter what

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Invariant:

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FAC fac

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Key Equation

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What is f ?

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(Earlier Slide)

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Now we only need to solve:

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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)

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Now we only need to solve:

Now we only need to find a fixed point of:

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Magic Revealed• Fixed point combinator / Y combinator

(call-by-value)

• fix F returns a fixed point of F.

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Magic Explained

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Now we only need to solve:

Now we only need to find a fixed point of:

Now we only need to compute:

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Answer

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If you are a masochist,• See the course notes to learn

how to derive the fixed point combinator!– There is a really crazy idea!