a graph-rewriting & rims, kyoto university) work in ...kmuroya/talks/srepls9.pdfmuroya (u....
TRANSCRIPT
![Page 1: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/1.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)
A Graph-Rewriting Perspective of the Beta-Law
Dan R. GhicaTodd Waugh Ambridge
(University of Birmingham)
S-REPLS 9 (Univ. Sussex), 25 May 2018
Koko Muroya(University of Birmingham& RIMS, Kyoto University)
work in progress
![Page 2: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/2.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)
Equivalence of programs
2
syntacticalequation
operationalequivalence
denotationalequality
t = u
Do t and u denote the same (mathematical) object?
Given any “closing” context C, do evaluations of C[t] and C[u] yield the same value?
![Page 3: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/3.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)
Equivalence of programs
3
syntacticalequation
operationalequivalence
denotationalequality
graphically?t = u
Do t and u denote the same (mathematical) object?
Given any “closing” context C, do evaluations of C[t] and C[u] yield the same value?
![Page 4: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/4.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)4
call-by-valueequational
theory
contextual(operational)equivalence[Plotkin ‘75]
![Page 5: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/5.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)
call-by-valueequational
theory
5
contextual(operational)equivalence[Plotkin ‘75]
![Page 6: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/6.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)6
call-by-valueequational
theory
contextual(operational)equivalence[Plotkin ‘75]
SECD machine
![Page 7: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/7.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)7
call-by-valueequational
theory
contextual(operational)equivalence
soundness
[Plotkin ‘75]
SECD machine
![Page 8: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/8.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)8
call-by-valueequational
theory
contextual(operational)equivalence
graph-rewritingmachine
graphically
![Page 9: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/9.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)9
call-by-valuegraph-equational
theory graphically
contextual(operational)equivalence
![Page 10: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/10.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)10
call-by-valuegraph-equational
theory graphically
contextual(operational)equivalence
all and only values are duplicable
![Page 11: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/11.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)11
call-by-valuegraph-equational
theory graphically
contextual(operational)equivalence
![Page 12: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/12.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)12
call-by-valuegraph-equational
theory graphically
contextual(operational)equivalence
alpha-law: trivialbeta-law: refined(cf. explicit substitution)
![Page 13: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/13.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)13
call-by-valuegraph-equational
theory graphically
contextual(operational)equivalence
alpha-law: trivialbeta-law: refined(cf. explicit substitution)
![Page 14: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/14.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)14
call-by-valuegraph-equational
theory graphically
contextual(operational)equivalence
alpha-law: trivialbeta-law: refined(cf. explicit substitution)
![Page 15: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/15.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)15
call-by-valuegraph-equational
theory graphically
contextual(operational)equivalence
alpha-law: trivialbeta-law: refined(cf. explicit substitution)
![Page 16: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/16.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)16
contextual(operational)equivalencegraphically
call-by-valuegraph-equational
theory
alpha-law: trivialbeta-law: refined (cf. explicit substitution)
SECD machine
![Page 17: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/17.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)17
call-by-valuegraph-equational
theory graphically
graph-contextual(operational)equivalence
alpha-law: trivialbeta-law: refined (cf. explicit substitution)
graph-rewritingmachine
![Page 18: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/18.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)18
call-by-valuegraph-equational
theory graphically
graph-contextual(operational)equivalence
alpha-law: trivialbeta-law: refined (cf. explicit substitution)
dGoImachine
![Page 19: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/19.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)19
SECD machine dGoI machine
● stack of closures● environment● control string● dump
● graph● evaluation control
(“token”)● rewriting flag● computation stack● box stack
![Page 20: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/20.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)20
SECD machine dGoI machine
![Page 21: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/21.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)21
SECD machine dGoI machine
![Page 22: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/22.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)
dGoI-machine transitions
22
![Page 23: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/23.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)23
call-by-valuegraph-equational
theory graphically
graph-contextual(operational)equivalence
alpha-law: trivialbeta-law: refined (cf. explicit substitution)
dGoImachine
![Page 24: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/24.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)24
call-by-valuegraph-equational
theory graphically
graph-contextual(operational)equivalence
alpha-law: trivialbeta-law: refined (cf. explicit substitution)
dGoImachine
soundness
![Page 25: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/25.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)25
call-by-valuegraph-equational
theory graphically
graph-contextual(operational)equivalence
soundness
alpha-law: trivialbeta-law: refined (cf. explicit substitution)
1. lift an axiom to a binary relation on (dGoI-machine) states
![Page 26: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/26.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)26
call-by-valuegraph-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”
![Page 27: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/27.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)27
call-by-valuegraph-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
![Page 28: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/28.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)28
call-by-valuegraph-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
![Page 29: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/29.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)
“Until the difference is reduced”
29
![Page 30: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/30.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)30
call-by-valuegraph-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
![Page 31: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/31.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)31
call-by-valuegraph-equational
theory graphically
graph-contextual(operational)equivalence
alpha-law: trivialbeta-law: refined (cf. explicit substitution)
dGoImachine
soundness
modular proof using U-simulations
![Page 32: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/32.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)
Equivalence of programs
32
syntacticalequation
operationalequivalence
denotationalequality
graphically?t = u
Do t and u denote the same (mathematical) object?
Given any “closing” context C, do evaluations of C[t] and C[u] yield the same value?
![Page 33: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/33.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)
Equivalence of programs
33
syntacticalequation
operationalequivalence
denotationalequality
graphically?t = u
Do t and u denote the same (mathematical) object?
Given any “closing” context C, do evaluations of C[t] and C[u] yield the same value?
modular proof of soundnessusing U-simulations
![Page 34: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/34.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)
so what?
34
![Page 35: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/35.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)
Equivalence of programs
35
syntacticalequation
operationalequivalencegraphically?t = u
Given any “closing” context C, do evaluations of C[t] and C[u] yield the same value?
modular proof of soundnessusing U-simulations
![Page 36: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/36.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)
Equivalence of programs
36
syntacticalequation
operationalequivalencegraphically?t = u
Given any “closing” context C, do evaluations of C[t] and C[u] yield the same value?
modular proof of soundnessusing U-simulations
related proof techniques:logical relationsapplicative bisimulationsenvirionmental bisimulations...
![Page 37: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/37.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)
Equivalence of programs
37
syntacticalequation
operationalequivalencegraphically?t = u
Given any “closing” context C, do evaluations of C[t] and C[u] yield the same value?
modular proof of soundnessusing U-simulations
semantical criteria of primitive operations (function constants) to preserve beta-law?
![Page 38: A Graph-Rewriting & RIMS, Kyoto University) work in ...kmuroya/talks/srepls9.pdfMuroya (U. B’ham. & RIMS, Kyoto U.) A Graph-Rewriting Perspective of the Beta-Law Dan R. Ghica Todd](https://reader034.vdocuments.site/reader034/viewer/2022042301/5eccd3fc8afc8924536bed82/html5/thumbnails/38.jpg)
Muroya (U. B’ham. & RIMS, Kyoto U.)
Equivalence of programs
38
syntacticalequation
operationalequivalencegraphically?t = u
Given any “closing” context C, do evaluations of C[t] and C[u] yield the same value?
modular proof of soundnessusing U-simulations
cost-sensitive equivalence?(cf. [Schmidt-Schauss & Dallmeyer, WPTE ’17]