CSE-321%20Programming%20Languages%20Fixed%20Point%20Combinator

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CSE-321 Programming LanguagesFixed Point
Combinator
???

• POSTECH
• March 29, 2006

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Values and Reductions
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Substitution Completed
avoid variable captures
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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

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Non-termination in ?-calculus
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Recursive Functions in ?-calculus

• Plan of attack
• Assume a recursive function construct fun f
  x. e
• Rewrite it as an expression in the
  ?-calculusi.e., show that it is syntactic sugar.

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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.

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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
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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!