Church call-by-name (x.e) e x e{x:=e x - - PowerPoint PPT Presentation

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Church call-by-name (x.e) e x e{x:=e x - - PowerPoint PPT Presentation

Mar 10, 2023 684 likes •1.59k views

T HE C ALL-BY-NEED L AMBDA C ALCULUS, R EVISITED Stephen Chang and Matthias Felleisen Northeastern University 26/3/2012 Church call-by-name (x.e) e x e{x:=e x } ()

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THE CALL-BY-NEED LAMBDA CALCULUS, REVISITED

Stephen Chang and Matthias Felleisen Northeastern University

26/3/2012

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Church

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λ

call-by-name

(λx.e) ex → e{x:=ex} (β)

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λ

call-by-name

E[(λx.e) ex] → E[e{x:=ex}] (β)

"leftmost-outermost"

E @ λx e ex

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λv

call-by-value

E[(λx.e) vx] → E[e{x:=vx}] (βv)

"leftmost-outermost"

E @ λx e vx

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call-by-need

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call-by-need 1) Evaluate argument only when needed.

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call-by-need 1) Evaluate argument only when needed. 2) Evaluate argument at most once.

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λneed

? ?

call-by-need 1) Evaluate argument only when needed. 2) Evaluate argument at most once.

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SLIDE 15

λneed

λneed-af

Ariola/Felleisen '94,'95,'97

?

=

?

=

λneed-mow

Maraist/Odersky/Wadler '94,'95,'98

call-by-need 1) Evaluate argument only when needed. 2) Evaluate argument at most once.

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SLIDE 16

λneed

λneed-af

Ariola/Felleisen '94,'95,'97

  • λneed-mow

Maraist/Odersky/Wadler '94,'95,'98

call-by-need 1) Evaluate argument only when needed. 2) Evaluate argument at most once.

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SLIDE 17

λneed

λneed-af

Ariola/Felleisen '94,'95,'97

  • λneed-mow

Maraist/Odersky/Wadler '94,'95,'98

call-by-need

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SLIDE 18

λneed

λneed-af

Ariola/Felleisen '94,'95,'97

  • λneed-mow

Maraist/Odersky/Wadler '94,'95,'98

call-by-need

x

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SLIDE 19

λneed

λneed-af

Ariola/Felleisen '94,'95,'97

  • λneed-mow

Maraist/Odersky/Wadler '94,'95,'98

call-by-need

x λx

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SLIDE 20

λneed

λneed-af

Ariola/Felleisen '94,'95,'97

  • λneed-mow

Maraist/Odersky/Wadler '94,'95,'98

call-by-need

x λx ex

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SLIDE 21

λneed

λneed-af

Ariola/Felleisen '94,'95,'97

  • λneed-mow

Maraist/Odersky/Wadler '94,'95,'98

call-by-need

x λx ex → → @ λx x ex (reshuffle)

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SLIDE 22

λneed

λneed-af

Ariola/Felleisen '94,'95,'97

  • λneed-mow

Maraist/Odersky/Wadler '94,'95,'98

call-by-need

@ λx x vx → @ λx vx vx (like β)

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  • ur λneed

call-by-need

x λx ex

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  • ur λneed

call-by-need

x λx ex

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  • ur λneed

call-by-need "demand path"

x λx ex

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  • ur λneed

call-by-need "demand path"

x λx ex (βneed)

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OLD λneed: OPERATIONAL OVERVIEW 1) Find the next demanded variable. 2) Find its corresponding argument and evaluate it. 3) Substitute evaluated argument for demanded variable.

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OLD λneed: OPERATIONAL OVERVIEW 1) Find the next demanded variable. 2) Find its corresponding argument and evaluate it. 3) Substitute evaluated argument for demanded variable.

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OLD λneed: DEMAND CONTEXTS

D = [ ] | D e

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OLD λneed: DEMAND CONTEXTS

D = [ ] | D e

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OLD λneed: DEMAND CONTEXTS

D = [ ] | D e | (λx.D) e

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OLD λneed: DEMAND CONTEXTS

D = [ ] | D e | (λx.D) e

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OLD λneed: DEMAND CONTEXTS

D = [ ] | D e | (λx.D) e

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OLD λneed: DEMAND CONTEXTS

D = [ ] | D e | (λx.D) e

binding structure

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OLD λneed: DEMAND CONTEXTS

D = [ ] | D e | B[D]

binding structure

B = [ ] | (λx.B) e

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OLD λneed: BINDING STRUCTURE

B = [ ] | (λx.B) e

(λx.(λy.(λz. ...) ez) ey) ex

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OLD λneed: BINDING STRUCTURE

B = [ ] | (λx.B) e

(λx.(λy.(λz. ...) ez) ey) ex

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OLD λneed: BINDING STRUCTURE

B = [ ] | (λx.B) e

(λx.(λy.(λz. ...) ez) ey) ex

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OLD λneed: BINDING STRUCTURE

B = [ ] | (λx.B) e

(λx.(λy.(λz. ...) ez) ey) ex @ λx @ λy @ λz ... ez ey ex

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OLD λneed: RESHUFFLING OF BINDINGS

B = [ ] | (λx.B) e

(((λx.λy.λz. ...) ex) ey) ez @ @ @ λx λy λz ... ex ey ez

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OLD λneed: RESHUFFLING OF BINDINGS

B = [ ] | (λx.B) e

(((λx.λy.λz. ...) ex) ey) ez @ @ @ λx λy λz ... ex ey ez

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OLD λneed: RESHUFFLING OF BINDINGS

B = [ ] | (λx.B) e

(((λx.λy.λz. ...) ex) ey) ez @ @ @ λx λy λz ... ex ey ez

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OLD λneed: RESHUFFLING OF BINDINGS

B = [ ] | (λx.B) e

((λx.(λy.λz. ...) ey) ex) ez @ @ λx @ λy λz ... ey ex ez

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OLD λneed: RESHUFFLING OF BINDINGS

B = [ ] | (λx.B) e

((λx.(λy.λz. ...) ey) ex) ez @ @ λx @ λy λz ... ey ex ez

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OLD λneed: RESHUFFLING OF BINDINGS

B = [ ] | (λx.B) e

(λx.((λy.λz. ...) ey) ez) ex @ λx @ @ λy λz ... ey ez ex

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OLD λneed: RESHUFFLING OF BINDINGS

B = [ ] | (λx.B) e

(λx.((λy.λz. ...) ey) ez) ex @ λx @ @ λy λz ... ey ez ex

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OLD λneed: RESHUFFLING OF BINDINGS

B = [ ] | (λx.B) e

(λx.(λy.(λz. ...) ez) ey) ex @ λx @ λy @ λz ... ez ey ex

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PROBLEMS WITH OLD CALL-BY-NEED CALCULUS

1) Reshuffling rules.

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OLD λneed: OPERATIONAL OVERVIEW 1) Find the next demanded variable. 2) Find its corresponding argument and evaluate it. 3) Substitute evaluated argument for demanded variable.

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OLD λneed: OPERATIONAL OVERVIEW 1) Find the next demanded variable. 2) Find its corresponding argument and evaluate it. 3) Substitute evaluated argument for demanded variable.

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OLD λneed: OPERATIONAL OVERVIEW 1) Find the next demanded variable. 2) Find its corresponding argument and evaluate it. 3) Substitute evaluated argument for demanded variable.

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OLD λneed: DEREFERENCING

(λx.(λy.(λz. ...) ez) ey) ex @ λx @ λy @ λz ... ez ey ex

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OLD λneed: DEREFERENCING

(λx.(λy.(λz. y ) ez) ey) ex @ λx @ λy @ λz y ez ey ex

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OLD λneed: DEREFERENCING

(λx.(λy.(λz. y ) ez) vy) ex @ λx @ λy @ λz y ez vy ex

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OLD λneed: DEREFERENCING

(λx.(λy.(λz. y ) ez) vy) ex @ λx @ λy @ λz y ez vy ex

(λy.D[y]) v → (λy.D[v]) v (deref)

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OLD λneed: DEREFERENCING

(λx.(λy.(λz. vy ) ez) vy) ex @ λx @ λy @ λz vy ez vy ex

(λy.D[y]) v → (λy.D[v]) v (deref)

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PROBLEMS WITH OLD CALL-BY-NEED CALCULUS

1) Reshuffling rules. 2) Arguments and applications never go away.

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NEW λneed: HANDLING ARBITRARY BINDING STRUCTURE

@ λx @ λy @ λz ... ez ey ex

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NEW λneed: HANDLING ARBITRARY BINDING STRUCTURE

@ λx @ λy @ λz ... ez ey ex B = [ ] | (λx.B) e

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NEW λneed: HANDLING ARBITRARY BINDING STRUCTURE

@ @ @ λx λy λz ... ex ey ez A = [ ] | ???

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NEW λneed: HANDLING ARBITRARY BINDING STRUCTURE

@ λx @ @ λy λz ... ey ez ex A = [ ] | ???

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NEW λneed: HANDLING ARBITRARY BINDING STRUCTURE

@ @ λx @ λy λz ... ey ex ez A = [ ] | ???

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NEW λneed: HANDLING ARBITRARY BINDING STRUCTURE

@ @ λx @ λy λz ... ey ex ez A = [ ] | (λx.A) e

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NEW λneed: HANDLING ARBITRARY BINDING STRUCTURE

@ @ λx @ λy λz ... ey ex ez A = [ ] | (λx.A) e

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NEW λneed: HANDLING ARBITRARY BINDING STRUCTURE

@ @ λx @ λy λz ... ey ex ez A = [ ] | (λx.A) e

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NEW λneed: HANDLING ARBITRARY BINDING STRUCTURE

@ @ λx @ λy λz ... ey ex ez A = [ ] | A[(λx.A)] e

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NEW λneed: HANDLING ARBITRARY BINDING STRUCTURE

@ @ λx @ λy λz ... ey ex ez A = [ ] | A[(λx.A)] e D = [ ] | D e | A[D]

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PROBLEMS WITH OLD CALL-BY-NEED CALCULUS

1) Reshuffling rules. 2) Arguments and applications never go away.

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NEW λneed: SPLITTING CONTEXTS

@ @ @ λx λy λz ... ex ey ez

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NEW λneed: SPLITTING CONTEXTS

@ @ @ λx λy λz ... ex ey ez A[ ...]

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NEW λneed: SPLITTING CONTEXTS

@ @ @ λx λy λz D[y] ex ey ez A[D[y]]

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NEW λneed: SPLITTING CONTEXTS

@ @ @ λx λy λz D[y] ex ey ez A[D[y]]

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NEW λneed: SPLITTING CONTEXTS

@ @ @ λx λy λz D[y] ex ey ez ... (λy...D[y])...ey...

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NEW λneed: SPLITTING CONTEXTS

@ @ @ λx λy λz D[y] ex ey ez ... (λy...D[y])...ey...

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NEW λneed: SPLITTING CONTEXTS

@ @ @ λx λy λz D[y] ex ey ez ...A[λy...D[y] ] ey ...

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NEW λneed: SPLITTING CONTEXTS

@ @ @ λx λy λz D[y] ex ey ez ...A[λy...D[y] ] ey ...

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NEW λneed: SPLITTING CONTEXTS

@ @ @ λx λy λz D[y] ex ey ez ...A[λy. [D[y]]] ey ... = [ ] | A[λx. ]

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NEW λneed: SPLITTING CONTEXTS

@ @ @ λx λy λz D[y] ex ey ez ...A[λy. [D[y]]] ey ... = [ ] | A[λx. ]

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NEW λneed: SPLITTING CONTEXTS

@ @ @ λx λy λz D[y] ex ey ez [A[λy. [D[y]]] ey] = [ ] | A[λx. ] = [ ] | A[ ] e

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NEW λneed: SPLITTING CONTEXTS

@ @ @ λx λy λz D[y] ex vy ez [A[λy. [D[y]]] vy] = [ ] | A[λx. ] = [ ] | A[ ] e

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NEW λneed: βneed

@ @ @ λx λy λz D[y] ex vy ez [A[λy. [D[y]]] vy]

→

[A[ [D[y]]{y:=vy}]] (βneed)

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NEW λneed: βneed

@ @ @ λx λy λz D[y] ex vy ez → @ @ λx λz D[y]{y:=vy} ex ez [A[λy. [D[y]]] vy]

→

[A[ [D[y]]{y:=vy}]] (βneed)

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PROBLEMS WITH PREVIOUS CALL-BY-NEED CALCULUS

1) Reassociation rules. 2) Function calls not resolved.

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NEW λneed: EVALUATING ARGUMENTS

D = [ ] | D e | A[D]

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NEW λneed: EVALUATING ARGUMENTS

D = [ ] | D e | A[D] | [A[λy. [D[y]]] D]

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OTHER INTERESTING THINGS IN THE PAPER . . . Correspondence to Launchbury's (1993) machine semantics.

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OTHER INTERESTING THINGS IN THE PAPER . . . Correspondence to Launchbury's (1993) machine semantics. Confluence, Standardization properties.

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OTHER INTERESTING THINGS IN THE PAPER . . . Correspondence to Launchbury's (1993) machine semantics. Confluence, Standardization properties. Soundness with respect to observational equivalence.

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Thanks!