CSE-321 Programming Languages ?-Calculus (II)

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CSE-321 Programming Languages?-Calculus (II)
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• POSTECH
• March 27, 2006

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Values and Reductions
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Call-by-name Call-by-value

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Outline

• Abstract syntax of the ?-calculus V
• Operational semantics of the l-calculus V
• Substitutions
• Programming in the ?-calculus

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e' / x e

• Informally"substitute e' for every occurrence of
  x in e."
• Examples

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Easy Cases First
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Two Remaining Cases
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• First (stupid) attempt
• Second attempt
• But wait

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Bound Variables

• "Names of bound variables do not matter."
• Hence
• Because, for a fresh variable y,

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One Remaining Case
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A Naive Attempt

• An anomaly

something for y
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Free Variables

• Those variables that are bound nowhere
• FV(e) set of free variables in e

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Free Variables vs. Bound Variables

• An outside observer does care about free
  variables.
• An outside observer does not care about
  bound variables.
• From the point of view of an outside observer

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Free Variables Remain Free

• From the point of view of an outside observer,a
  free variable remains free until it is
  explicitly replaced.

variable capture
outside observer
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Capture-Avoiding Substitution

• What happens if
• the free variable y is captured and becomes a
  bound variable.
• To an outside observer, it suddenly disappears!

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Substitution Completed
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Capture-Avoiding Substitution in Action

• We have to rename bound variables as necessary.

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?-Conversion

• Renaming bound variables when necessary
• Okay because the names of bound variables do not
  matter.
• Examples

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Formalization of ?-Conversion

• See the course notes!
• It's more interesting than you might think.

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Outline

• Abstract syntax of the ?-calculus V
• Operational semantics of the l-calculus V
• Substitutions V
• Programming in the ?-calculus
• A LOT OF FUN!

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Booleans

• A boolean value
• "Give me two options and I will choose one for
  you!"
• Syntactic sugar

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Examples

• Under the call-by-name strategy,

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Logical Operators
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Natural Numbers

• A natural number n
• has the capability to repeat a given process
  n times.
• "Give me a function f and I will return f
  n f o f ... f o f "

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Church Numerals
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Addition

• Key observation

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Multiplication

• Key observation
• Alternatively

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Others

• Pairs
• Natural numbers
• predecessor DIFFICULT
• subtraction
• exponentiation
• ...
• Lists
• See Course Notes and Pierce Chapter 5 for more
  examples.

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Next Lecture

• Recursion in the ?-calculus
• Read Course Notes!