CSE-321 Programming Languages

-Calculus (II)

POSTECH

March 26, 2007

박성우

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Values and Reductions

redex = reducible expression

: -reduction

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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 -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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• Names of bound variables do not matter.

• Hence

– for a fresh variable y,

Bound Variables

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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• Variables that are bound nowhere

FV(e) = set of free variables in e

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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.

outside observer

? variable capture

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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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• We have to rename bound variables as necessary.

Capture-Avoiding Substitution in Action

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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 -calculus V• Substitutions V• Programming in the -calculus

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• A boolean value– "Give me two options and I will choose one for

you!"

• Syntactic sugar

Booleans

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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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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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Now we only need to find a fixed point of:

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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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Magic Revealed• Fixed point combinator / Y combinator

(call-by-value)

• fix F returns a fixed point of F.

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Now we only need to find a fixed point of:

Now we only need to compute:

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Answer