A Graph-Rewriting s Perspective of the Beta-Law Dan R. Ghica Koko - - PowerPoint PPT Presentation

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A Graph-Rewriting s Perspective of the Beta-Law Dan R. Ghica Koko - - PowerPoint PPT Presentation

Oct 31, 2022 264 likes •656 views

w o r k p r i n o g r e s A Graph-Rewriting s Perspective of the Beta-Law Dan R. Ghica Koko Muroya Todd Waugh Ambridge (University of Birmingham (University of Birmingham) & RIMS, Kyoto University) S-REPLS 9 (Univ. Sussex),

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Muroya (U. B’ham. & RIMS, Kyoto U.)

A Graph-Rewriting Perspective of the Beta-Law

Dan R. Ghica Todd Waugh Ambridge (University of Birmingham)

S-REPLS 9 (Univ. Sussex), 25 May 2018

Koko Muroya (University of Birmingham & RIMS, Kyoto University) w

  • r

k i n p r

  • g

r e s s

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Muroya (U. B’ham. & RIMS, Kyoto U.)

Equivalence of programs

2

syntactical equation

  • perational

equivalence denotational equality t = u Do t and u denote the same (mathematical)

  • bject?

Given any “closing” context C, do evaluations of C[t] and C[u] yield the same value?

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SLIDE 3

Muroya (U. B’ham. & RIMS, Kyoto U.)

Equivalence of programs

3

syntactical equation

  • perational

equivalence denotational equality

graphically?

t = u Do t and u denote the same (mathematical)

  • bject?

Given any “closing” context C, do evaluations of C[t] and C[u] yield the same value?

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SLIDE 4

Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value equational theory contextual (operational) equivalence [Plotkin ‘75]

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Muroya (U. B’ham. & RIMS, Kyoto U.)

call-by-value equational theory

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contextual (operational) equivalence [Plotkin ‘75]

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Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value equational theory contextual (operational) equivalence [Plotkin ‘75] SECD machine

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Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value equational theory contextual (operational) equivalence soundness [Plotkin ‘75] SECD machine

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SLIDE 8

Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value equational theory contextual (operational) equivalence graph-rewriting machine graphically

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Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value graph-equational theory graphically contextual (operational) equivalence

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Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value graph-equational theory graphically contextual (operational) equivalence all and only values are duplicable

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Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value graph-equational theory graphically contextual (operational) equivalence

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Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value graph-equational theory graphically contextual (operational) equivalence alpha-law: trivial beta-law: refined (cf. explicit substitution)

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Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value graph-equational theory graphically contextual (operational) equivalence alpha-law: trivial beta-law: refined (cf. explicit substitution)

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Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value graph-equational theory graphically contextual (operational) equivalence alpha-law: trivial beta-law: refined (cf. explicit substitution)

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Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value graph-equational theory graphically contextual (operational) equivalence alpha-law: trivial beta-law: refined (cf. explicit substitution)

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Muroya (U. B’ham. & RIMS, Kyoto U.)

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contextual (operational) equivalence graphically call-by-value graph-equational theory alpha-law: trivial beta-law: refined (cf. explicit substitution) SECD machine

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Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value graph-equational theory graphically graph-contextual (operational) equivalence alpha-law: trivial beta-law: refined (cf. explicit substitution) graph-rewriting machine

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Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value graph-equational theory graphically graph-contextual (operational) equivalence alpha-law: trivial beta-law: refined (cf. explicit substitution) dGoI machine

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Muroya (U. B’ham. & RIMS, Kyoto U.)

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SECD machine dGoI machine

  • stack of closures
  • environment
  • control string
  • dump
  • graph
  • evaluation control

(“token”)

  • rewriting flag
  • computation stack
  • box stack
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Muroya (U. B’ham. & RIMS, Kyoto U.)

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SECD machine dGoI machine

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Muroya (U. B’ham. & RIMS, Kyoto U.)

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SECD machine dGoI machine

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Muroya (U. B’ham. & RIMS, Kyoto U.)

dGoI-machine transitions

22

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Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value graph-equational theory graphically graph-contextual (operational) equivalence alpha-law: trivial beta-law: refined (cf. explicit substitution) dGoI machine

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SLIDE 24

Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value graph-equational theory graphically graph-contextual (operational) equivalence alpha-law: trivial beta-law: refined (cf. explicit substitution) dGoI machine soundness

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Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value graph-equational theory graphically graph-contextual (operational) equivalence soundness alpha-law: trivial beta-law: refined (cf. explicit substitution)

  • 1. lift an axiom to a binary relation on

(dGoI-machine) states

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Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value graph-equational theory graphically graph-contextual (operational) equivalence soundness

  • 1. lift an axiom to a binary relation on

(dGoI-machine) states

  • 2. show the binary relation is a

“U-simulation”

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SLIDE 27

Muroya (U. B’ham. & RIMS, Kyoto U.)

27

call-by-value graph-equational theory graphically graph-contextual (operational) equivalence soundness

  • 1. lift an axiom to a binary relation on

(dGoI-machine) states

  • 2. show the binary relation is a

“U-simulation” simulation

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SLIDE 28

Muroya (U. B’ham. & RIMS, Kyoto U.)

28

call-by-value graph-equational theory graphically graph-contextual (operational) equivalence soundness

  • 1. lift an axiom to a binary relation on

(dGoI-machine) states

  • 2. show the binary relation is a

“U-simulation” simulation ...until the difference is reduced

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SLIDE 29

Muroya (U. B’ham. & RIMS, Kyoto U.)

“Until the difference is reduced”

29

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SLIDE 30

Muroya (U. B’ham. & RIMS, Kyoto U.)

30

call-by-value graph-equational theory graphically graph-contextual (operational) equivalence soundness

  • 1. lift an axiom to a binary relation on

(dGoI-machine) states

  • 2. show the binary relation is a

“U-simulation” simulation ...until the difference is reduced

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SLIDE 31

Muroya (U. B’ham. & RIMS, Kyoto U.)

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call-by-value graph-equational theory graphically graph-contextual (operational) equivalence alpha-law: trivial beta-law: refined (cf. explicit substitution) dGoI machine soundness modular proof using U-simulations

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SLIDE 32

Muroya (U. B’ham. & RIMS, Kyoto U.)

Equivalence of programs

32

syntactical equation

  • perational

equivalence denotational equality

graphically?

t = u Do t and u denote the same (mathematical)

  • bject?

Given any “closing” context C, do evaluations of C[t] and C[u] yield the same value?

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SLIDE 33

Muroya (U. B’ham. & RIMS, Kyoto U.)

Equivalence of programs

33

syntactical equation

  • perational

equivalence denotational equality

graphically?

t = u Do t and u denote the same (mathematical)

  • bject?

Given any “closing” context C, do evaluations of C[t] and C[u] yield the same value? modular proof of soundness using U-simulations

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SLIDE 34

Muroya (U. B’ham. & RIMS, Kyoto U.)

so what?

34

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SLIDE 35

Muroya (U. B’ham. & RIMS, Kyoto U.)

Equivalence of programs

35

syntactical equation

  • perational

equivalence

graphically?

t = u Given any “closing” context C, do evaluations of C[t] and C[u] yield the same value? modular proof of soundness using U-simulations

slide-36
SLIDE 36

Muroya (U. B’ham. & RIMS, Kyoto U.)

Equivalence of programs

36

syntactical equation

  • perational

equivalence

graphically?

t = u Given any “closing” context C, do evaluations of C[t] and C[u] yield the same value? modular proof of soundness using U-simulations related proof techniques: logical relations applicative bisimulations envirionmental bisimulations...

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SLIDE 37

Muroya (U. B’ham. & RIMS, Kyoto U.)

Equivalence of programs

37

syntactical equation

  • perational

equivalence

graphically?

t = u Given any “closing” context C, do evaluations of C[t] and C[u] yield the same value? modular proof of soundness using U-simulations semantical criteria of primitive

  • perations (function constants)

to preserve beta-law?

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SLIDE 38

Muroya (U. B’ham. & RIMS, Kyoto U.)

Equivalence of programs

38

syntactical equation

  • perational

equivalence

graphically?

t = u Given any “closing” context C, do evaluations of C[t] and C[u] yield the same value? modular proof of soundness using U-simulations cost-sensitive equivalence?

(cf. [Schmidt-Schauss & Dallmeyer, WPTE ’17]