SLIDE 1
Objectives Evaluation Order How Far to Evaluate For Us
Evaluation Order
- Dr. Mattox Beckman
Objectives Evaluation Order How Far to Evaluate For Us
Evaluation Order Dr. Mattox Beckman University of Illinois at - - PowerPoint PPT Presentation
Evaluation Order Dr. Mattox Beckman University of Illinois at - - PowerPoint PPT Presentation
Objectives Evaluation Order How Far to Evaluate For Us Evaluation Order Dr. Mattox Beckman University of Illinois at Urbana-Champaign Department of Computer Science Objectives Evaluation Order How Far to Evaluate For Us Objectives
SLIDE 2
SLIDE 3
Objectives Evaluation Order How Far to Evaluate For Us
Things We Didn’t Mention Last Time ...
◮ If there is more than one β-reduction, which one do you do fjrst? ◮ Do you always have to do all β-reductions, or should some be left alone?
SLIDE 4
Objectives Evaluation Order How Far to Evaluate For Us
Applicative Order
◮ Applicative order is like call-by-value in many programming languages. ◮ Start with the left-most outer-most application. ◮ Evaluate the argument before doing the β-reduction.
(λxf.fx)(λz.z)((λq.q)(λr.r))
SLIDE 5
Objectives Evaluation Order How Far to Evaluate For Us
Applicative Order
◮ Applicative order is like call-by-value in many programming languages. ◮ Start with the left-most outer-most application. ◮ Evaluate the argument before doing the β-reduction.
(λxf.fx)(λz.z)((λq.q)(λr.r)) ⇒ (λf.fλz.z)((λq.q)(λr.r))
SLIDE 6
Objectives Evaluation Order How Far to Evaluate For Us
Applicative Order
◮ Applicative order is like call-by-value in many programming languages. ◮ Start with the left-most outer-most application. ◮ Evaluate the argument before doing the β-reduction.
(λxf.fx)(λz.z)((λq.q)(λr.r)) ⇒ (λf.fλz.z)((λq.q)(λr.r)) ⇒ (λf.fλz.z)(λr.r)
SLIDE 7
Objectives Evaluation Order How Far to Evaluate For Us
Applicative Order
◮ Applicative order is like call-by-value in many programming languages. ◮ Start with the left-most outer-most application. ◮ Evaluate the argument before doing the β-reduction.
(λxf.fx)(λz.z)((λq.q)(λr.r)) ⇒ (λf.fλz.z)((λq.q)(λr.r)) ⇒ (λf.fλz.z)(λr.r) ⇒ (λr.r)(λz.z) ⇒ (λz.z)
SLIDE 8
Objectives Evaluation Order How Far to Evaluate For Us
Normal Order
◮ Normal order is like call-by-name, which is what the C preprocessor uses. ◮ Again, start with the left-most outer-most application. ◮ But this time do the β-reduction right away.
(λxf.fx)(λz.z)((λq.q)(λr.r))
SLIDE 9
Objectives Evaluation Order How Far to Evaluate For Us
Normal Order
◮ Normal order is like call-by-name, which is what the C preprocessor uses. ◮ Again, start with the left-most outer-most application. ◮ But this time do the β-reduction right away.
(λxf.fx)(λz.z)((λq.q)(λr.r)) ⇒ (λf.fλz.z)((λq.q)(λr.r))
SLIDE 10
Objectives Evaluation Order How Far to Evaluate For Us
Normal Order
◮ Normal order is like call-by-name, which is what the C preprocessor uses. ◮ Again, start with the left-most outer-most application. ◮ But this time do the β-reduction right away.
(λxf.fx)(λz.z)((λq.q)(λr.r)) ⇒ (λf.fλz.z)((λq.q)(λr.r)) ⇒ ((λq.q)(λr.r))λz.z
SLIDE 11
Objectives Evaluation Order How Far to Evaluate For Us
Normal Order
◮ Normal order is like call-by-name, which is what the C preprocessor uses. ◮ Again, start with the left-most outer-most application. ◮ But this time do the β-reduction right away.
(λxf.fx)(λz.z)((λq.q)(λr.r)) ⇒ (λf.fλz.z)((λq.q)(λr.r)) ⇒ ((λq.q)(λr.r))λz.z ⇒ (λr.r)λz.z
SLIDE 12
Objectives Evaluation Order How Far to Evaluate For Us
Normal Order
◮ Normal order is like call-by-name, which is what the C preprocessor uses. ◮ Again, start with the left-most outer-most application. ◮ But this time do the β-reduction right away.
(λxf.fx)(λz.z)((λq.q)(λr.r)) ⇒ (λf.fλz.z)((λq.q)(λr.r)) ⇒ ((λq.q)(λr.r))λz.z ⇒ (λr.r)λz.z ⇒ λz.z
SLIDE 13
Objectives Evaluation Order How Far to Evaluate For Us
Interesting Effects
◮ Applicative order often has fewer reductions.
E.g., (λxf.fxxxx)((λa.a)(λb.b)) Normal order can win sometimes. E.g., xf fffff a a b b If it terminates, applicative order will yield the same result as normal order. E.g., xf fffff a aa b bb
SLIDE 14
Objectives Evaluation Order How Far to Evaluate For Us
Interesting Effects
◮ Applicative order often has fewer reductions.
E.g., (λxf.fxxxx)((λa.a)(λb.b))
◮ Normal order can win sometimes.
E.g., (λxf.fffff)((λa.a)(λb.b)) If it terminates, applicative order will yield the same result as normal order. E.g., xf fffff a aa b bb
SLIDE 15
Objectives Evaluation Order How Far to Evaluate For Us
Interesting Effects
◮ Applicative order often has fewer reductions.
E.g., (λxf.fxxxx)((λa.a)(λb.b))
◮ Normal order can win sometimes.
E.g., (λxf.fffff)((λa.a)(λb.b))
◮ If it terminates, applicative order will yield the same result as normal order.
E.g., (λxf.fffff)((λa.aa)(λb.bb))
SLIDE 16
Objectives Evaluation Order How Far to Evaluate For Us
When Can We Stop?
◮ Consider this function defjnition. ◮ When do you expect the (\z.z) y function call to occur?
1 foo x y = 2
x + (\z . z) y
SLIDE 17
Objectives Evaluation Order How Far to Evaluate For Us
Weak Head Normal Form
◮ If the “head node” (root node of the syntax tree) is a lambda, then everything inside is the
body of the function.
◮ This is weak head normal form. ◮ This form more closely resembles what “real programming languages” do.
λa @ λx x λy y λa.(λx.x)(λy.y)
SLIDE 18
Objectives Evaluation Order How Far to Evaluate For Us
Normal Form
◮ In normal form, once the outermost node is a lambda, you descend into the body and
continue there.
◮ You get maximally reduced expressions: “normalized” ◮ It’s possible to have α-capture though.
E.g., λy.(λxy.x)y λa λy y λa.λy.y
SLIDE 19
Objectives Evaluation Order How Far to Evaluate For Us