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IR0
interp interp compile / desugar
2 IR1 2
Semantics interp IR 0 2 compile / desugar = interp IR 1 2 - - PowerPoint PPT Presentation
Semantics interp IR 0 2 compile / desugar = interp IR 1 2 - - PowerPoint PPT Presentation
Semantics interp IR 0 2 compile / desugar = interp IR 1 2 Assignment 1 interp Scheme IR 2 church-encode church->nat interp ( (f) ( (x) -calculus (f (f x)))) Formal semantics Axiomatic Semantics Gives axioms for
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Assignment 1
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Scheme IR λ-calculus
interp interp church-encode
2 (λ (f) (λ (x) (f (f x))))
church->nat
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Formal semantics
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Axiomatic Semantics Denotational Semantics Operational Semantics
Gives axioms for constructing sound proofs about programs (typically using Hoare logic). Provides a function that maps language forms into their denotations in a known domain. Provides a step-by-step reduction of the program to a value in terms of program terms or an abstract machine.
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((λ (f) (f (f (λ (x) x)))) (λ (x) x)) ((λ (x) x) ((λ (x) x) (λ (x) x))) ((λ (x) x) (λ (x) x)) (λ (x) x) β β β
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→β
((λ (x) E0) E1) E0[x ← E1]
{
redex
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Capture-avoiding substitution
E0[x ← E1]
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FV(x) = {x} FV((λ (x) E0)) = FV(E0) \ {x} FV((E0 E1)) = FV(E0) ∪ FV(E1)
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x[x ← E] = E y[x ← E] = y where y ≠ x (E0 E1)[x ← E] = (E0[x ← E] E1[x ← E]) (λ (x) E0)[x ← E] = (λ (x) E0) (λ (y) E0)[x ← E] = (λ (y) E0[x ← E])
where y ≠ x and y ∉ FV(E)
β-reduction cannot occur when y ∈ FV(E)
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α - renaming
(λ (x) (λ (y) x)) (λ (a) (λ (b) a))
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→α
(λ (x) E0) (λ (y) E0[x ← y])
α - renaming
=α
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(λ (x) (E0 x)) E0 where x ∉ FV(E0)
η - reduction
→η
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Reduction
(→) = (→β) ∪ (→α) ∪ (→η)
(→*)
reflexive/transitive closure
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Evaluation
E0
*
E1
*
?
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Evaluation to normal form
E0
*
(λ (x) …)
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Evaluation to normal form
E0
*
(λ (x) … (λ (z) ((a …) …)))
function position must be a variable
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Evaluation Strategy
E0
* *
E1 E2
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Evaluation Strategy
((λ (x) ((λ (y) y) x)) (λ (z) z))
→η ((λ (y) y) (λ (z) z)) →β (λ (z) z)
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Evaluation Strategy
((λ (x) ((λ (y) y) x)) (λ (z) z))
→β ((λ (y) y) (λ (z) z)) →β (λ (z) z)
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Evaluation Strategy
((λ (x) ((λ (y) y) x)) (λ (z) z))
→β ((λ (x) x) (λ (z) z)) →β (λ (z) z)
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Confluence
E0
* *
E1 E2
Church-Rosser Theorem
* *
E3
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Applicative evaluation order Normal evaluation order
Always evaluates the innermost leftmost redex first. Always evaluates the outermost leftmost redex first.
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Applicative evaluation order Normal evaluation order
((λ (x) ((λ (y) y) x)) (λ (z) z)) (((λ (x) ((λ (y) y) x)) (λ (z) z)) (λ (w) w))
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Call-by-name semantics Call-by-value semantics
Normal evaluation order, but not under lambdas. Applicative evaluation order, but not under lambdas.
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ℰ ::= (ℰ e)
| (v ℰ) | □
Evaluation contexts
v ::= (λ (x) e) e ::= (λ (x) e) | (e e) | x
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Context and redex
(((λ (x) ((λ (y) y) x)) (λ (z) z)) (λ (w) w))
ℰ = (□ (λ (w) w)) ℰ[(v v)] =
r = ((λ (x) ((λ (y) y) x)) (λ (z) z))
{ {
r
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Context and redex
(((λ (x) ((λ (y) y) x)) (λ (z) z)) (λ (w) w))
ℰ = (□ (λ (w) w)) ℰ[r] =
r = ((λ (x) ((λ (y) y) x)) (λ (z) z))
→β ((λ (y) y) (λ (z) z))
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Put it back together:
ℰ = (□ (λ (w) w))
r = ((λ (x) ((λ (y) y) x)) (λ (z) z))
→β ((λ (y) y) (λ (z) z))
(((λ (y) y) (λ (z) z)) (λ (w) w))
ℰ[r]
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Some exercises
1) (((λ (y) y) (λ (z) z)) (λ (w) w)) 2) ((λ (u) (u u)) (λ (x) (λ (x) x))) 3) (((λ (x) x) (λ (y) y)) ((λ (u) (u u)) (λ (z) (z z))))