![Page 1: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/1.jpg)
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Kleisli Simulation for Real-Weighted Automata and Its Algorithm
卜部 夏木 #(東京大学理学部情報科学科4年)
![Page 2: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/2.jpg)
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Overview● Simulation#● Kleisli Simulation#
1. Definition#2. Searching Kleisli Simulation for Probabilistic System#
● Contribution#1. Implementation#2. Increasing the Chance of Simulation#
● Experimental Results and Comparison
![Page 3: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/3.jpg)
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Overview● Simulation#● Kleisli Simulation#
1. Definition#2. Searching Kleisli Simulation for Probabilistic System#
● Contribution#1. Implementation#2. Increasing the Chance of Simulation#
● Experimental Results and Comparison
![Page 4: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/4.jpg)
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目標● State-based system の trace inclusion を調べる
● Trace inclusion = Trace Semantics の間の包含関係● Trace semantics = Linear-time behavior
![Page 5: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/5.jpg)
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目標● State-based system の trace inclusion を調べる
● 例:非決定性オートマトンの場合● Trace inclusion = Trace Semantics の間の包含関係● Trace semantics = Linear-time behavior
![Page 6: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/6.jpg)
Spec
Impl
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目標● State-based system の trace inclusion を調べる
● 例:非決定性オートマトンの場合● Trace inclusion = Trace Semantics の間の包含関係● Trace semantics = Linear-time behavior
![Page 7: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/7.jpg)
Spec
Impl
�4
目標● State-based system の trace inclusion を調べる
● 例:非決定性オートマトンの場合● Trace inclusion = Trace Semantics の間の包含関係● Trace semantics = Linear-time behavior
![Page 8: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/8.jpg)
Spec
Impl
�4
目標● State-based system の trace inclusion を調べる
● 例:非決定性オートマトンの場合● Trace inclusion = Trace Semantics の間の包含関係● Trace semantics = Linear-time behavior
![Page 9: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/9.jpg)
Spec
Impl
�4
目標● State-based system の trace inclusion を調べる
● 例:非決定性オートマトンの場合● Trace inclusion = Trace Semantics の間の包含関係● Trace semantics = Linear-time behavior
![Page 10: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/10.jpg)
● State-based system の trace inclusion を調べる
● Trace inclusion = Trace Semantics の間の包含関係● Trace semantics = Linear-time behavior
目標
![Page 11: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/11.jpg)
Imple
Spec
● State-based system の trace inclusion を調べる
● Trace inclusion = Trace Semantics の間の包含関係● Trace semantics = Linear-time behavior
目標
![Page 12: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/12.jpg)
Imple
Spec
● State-based system の trace inclusion を調べる
● Trace inclusion = Trace Semantics の間の包含関係● Trace semantics = Linear-time behavior
目標
![Page 13: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/13.jpg)
Imple
Spec
● State-based system の trace inclusion を調べる
● Trace inclusion = Trace Semantics の間の包含関係● Trace semantics = Linear-time behavior
目標
![Page 14: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/14.jpg)
Imple
Spec
● State-based system の trace inclusion を調べる
● Trace inclusion = Trace Semantics の間の包含関係● Trace semantics = Linear-time behavior
目標
![Page 15: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/15.jpg)
Imple
Spec
● State-based system の trace inclusion を調べる
● Trace inclusion = Trace Semantics の間の包含関係● Trace semantics = Linear-time behavior
目標
![Page 16: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/16.jpg)
Imple
Spec
● State-based system の trace inclusion を調べる
● Trace inclusion = Trace Semantics の間の包含関係● Trace semantics = Linear-time behavior
目標
![Page 17: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/17.jpg)
Imple
Spec
● State-based system の trace inclusion を調べる
● Trace inclusion = Trace Semantics の間の包含関係● Trace semantics = Linear-time behavior
目標
![Page 18: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/18.jpg)
Imple
Spec
● State-based system の trace inclusion を調べる
● 例:非決定性オートマトンの場合● Trace inclusion = Trace Semantics の間の包含関係● Trace semantics = Linear-time behavior
目標
![Page 19: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/19.jpg)
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目標● State-based system の trace inclusion を調べる#● Trace semantics は普通は無限#
#
#
#
![Page 20: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/20.jpg)
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目標● State-based system の trace inclusion を調べる#● Trace semantics は普通は無限#
#
#
#
自動で調べることは難しい
![Page 21: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/21.jpg)
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● 全体の trace inclusion を調べる代わりに#
stepwise な trace inclusion を調べる
Simulation = Stepwise Trace Inclusion
![Page 22: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/22.jpg)
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● 全体の trace inclusion を調べる代わりに#
stepwise な trace inclusion を調べる
Simulation = Stepwise Trace Inclusion
A B
AからBへのsimulationが存在
Simulation
R ✓ A ⇥ B s.t.
xRy, x
a�! x
0 ) 9y0.y
a�! y
0, x
0Ry
0
![Page 23: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/23.jpg)
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健全性
A から B への simulation が存在
![Page 24: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/24.jpg)
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健全性
A から B への simulation が存在
A の trace は B の trace に含まれる (trace inclusion)
![Page 25: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/25.jpg)
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健全性
健全性
A から B への simulation が存在
A の trace は B の trace に含まれる (trace inclusion)
![Page 26: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/26.jpg)
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健全性
健全性
A から B への simulation が存在
A の trace は B の trace に含まれる (trace inclusion)
![Page 27: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/27.jpg)
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健全性
健全性完全性は必ずしも成り立たない
A から B への simulation が存在
A の trace は B の trace に含まれる (trace inclusion)
![Page 28: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/28.jpg)
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Overview● Simulation#● Kleisli Simulation#
1. Definition#2. Searching Kleisli Simulation for Probabilistic System#
● Contribution#1. Implementation#2. Increasing the Chance of Simulation#
● Experimental Results and Comparison
![Page 29: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/29.jpg)
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Kleisli 圏による表現• Sets 上の Kleisli 圏でシステムを表現
1s�!p X
c�!p FX K`(T )in
s : 1!p XX
c : X!p FX
・・・状態集合・・・始状態・・・遷移
F : Sets ! Sets
F : K`(T ) ! K`(T )
![Page 30: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/30.jpg)
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Kleisli 圏による表現
• モナドTごとに様々なシステム例:
→ 非決定性オートマトン
→ 確率的オートマトン
Powerset monad
Subdistribution monad
![Page 31: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/31.jpg)
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Trace Semantics via Coinduction (Hasuo, Jacobs, and Sokolova, 2006)
● Kleisli 圏を用いたtrace semantics の一般的な定義#● 様々な遷移系に trace semantics を定義できる
Final Coalgebra
![Page 32: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/32.jpg)
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Trace Semantics via Coinduction (Hasuo, Jacobs, and Sokolova, 2006)
● Kleisli 圏を用いたtrace semantics の一般的な定義#● 様々な遷移系に trace semantics を定義できる
Final Coalgebra
![Page 33: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/33.jpg)
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Trace Semantics via Coinduction (Hasuo, Jacobs, and Sokolova, 2006)
● Kleisli 圏を用いたtrace semantics の一般的な定義#● 様々な遷移系に trace semantics を定義できる
Final Coalgebra
![Page 34: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/34.jpg)
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Trace Inclusion via Kleisli Category
• の各 homset に順序 (Cppo) を入れるK`(T )Trace inclusion が定義できる
vtr
![Page 35: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/35.jpg)
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Trace Inclusion via Kleisli Category
• の各 homset に順序 (Cppo) を入れるK`(T )Trace inclusion が定義できる
vtrdef,
1
![Page 36: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/36.jpg)
�13
Trace Inclusion via Kleisli Category
• の各 homset に順序 (Cppo) を入れるK`(T )Trace inclusion が定義できる
vtrdef,
1
![Page 37: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/37.jpg)
�13
Trace Inclusion via Kleisli Category
• の各 homset に順序 (Cppo) を入れるK`(T )Trace inclusion が定義できる
vtrdef,
1
![Page 38: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/38.jpg)
�13
Trace Inclusion via Kleisli Category
• の各 homset に順序 (Cppo) を入れるK`(T )Trace inclusion が定義できる
vtrdef,
1
![Page 39: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/39.jpg)
�14
Kleisli Simulation (Hasuo, 2006)
● Kleisli 圏を用いたsimulation の一般的な定義#● 様々な遷移系に simulation (Kleisli simulation)を定義できる
BackwardForwardFX
v
FYF (f)oo
X
c
OO
v
Y
d
OO
foo
1
s
OOt
OO
FX
v
F (b) // FY
X
c
OO
v
b // Y
d
OO
1
s
OOt
OO
![Page 40: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/40.jpg)
�15
Kleisli Simulation● Forward Simulation
9f.
FX
v
FYF (f)oo
X
c
OO
v
Y
d
OO
foo
1
s
OOt
OOvFdef,
![Page 41: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/41.jpg)
�16
Kleisli Simulation● Backward Simulation
9b.
FX
v
F (b) // FY
X
c
OO
v
b // Y
d
OO
1
s
OOt
OOvBdef,
![Page 42: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/42.jpg)
�17
Kleisli Simulation の健全性(Hasuo, 2006)
9f.
![Page 43: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/43.jpg)
�17
Kleisli Simulation の健全性(Hasuo, 2006)
9f.
![Page 44: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/44.jpg)
�17
Kleisli Simulation の健全性(Hasuo, 2006)
9f.
9b.
![Page 45: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/45.jpg)
�18
Overview● Simulation#● Kleisli Simulation#
1. Definition#2. Searching Kleisli Simulation for Probabilistic System#
● Contribution#1. Implementation#2. Increasing the Chance of Simulation#
● Experimental Results and Comparison
![Page 46: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/46.jpg)
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での Kleisli Simulation の探索• の射は行列で表現できるK`(D)
f : X!p Y
K`(D)
![Page 47: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/47.jpg)
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での Kleisli Simulation の探索• の射は行列で表現できるK`(D)
f : X ! DY
K`(D)
![Page 48: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/48.jpg)
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での Kleisli Simulation の探索• の射は行列で表現できるK`(D)
f : X ! DY M
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での Kleisli Simulation の探索• の射は行列で表現できるK`(D)
f : X ! DY M
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• Kleisli Simulation の探索は行列の探索に帰着
K`(D)
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f : X ! DY M
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• Kleisli Simulation の探索は行列の探索に帰着9f.
K`(D)
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での Kleisli Simulation の探索• の射は行列で表現できるK`(D)
f : X ! DY M
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,9Mf .
MfMc MdMFf
Ms MtMf
9f.
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● 確率的オートマトンの trace inclusion は#
決定不能 (Blondel & Canterni, 2003)
(Hasuo, 2006)
確率的オートマトンのKleisli Simulation
![Page 53: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/53.jpg)
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● 確率的オートマトンの trace inclusion は#
決定不能 (Blondel & Canterni, 2003)
(Hasuo, 2006)
確率的オートマトンのKleisli Simulation
![Page 54: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/54.jpg)
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Overview● Simulation#● Kleisli Simulation#
1. Definition#2. Searching Kleisli Simulation for Probabilistic System#
● Contribution#1. Implementation#2. Increasing the Chance of Simulation#
● Experimental Results and Comparison
![Page 55: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/55.jpg)
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Contributions
1. 線形不等式を解くことで確率的オートマトンの #
Kleisli simulation を調べるプログラムを実装#
![Page 56: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/56.jpg)
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Contributions
2. Kleisli simulation を存在しやすくする工夫#
- Subdistribution Monad → Multiset Monad# - オートマトンの変形
(Hasuo, 2006)
![Page 57: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/57.jpg)
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Contributions
2. Kleisli simulation を存在しやすくする工夫#
- Subdistribution Monad → Multiset Monad# - オートマトンの変形
(Hasuo, 2006)
![Page 58: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/58.jpg)
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Subdistribution Monad → Multiset Monad
• モナドSubdistribution Monad
DX = {f : X ! [0, 1] |X
x
f(x) 1}
![Page 59: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/59.jpg)
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Subdistribution Monad → Multiset Monad
• モナドSubdistribution Monad
DX = {f : X ! [0, 1] |X
x
f(x) 1}
Multiset MonadMX = {f : X ! [0,1] | supp(f) is countable}
![Page 60: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/60.jpg)
�24
Subdistribution Monad → Multiset Monad
確率的オートマトン
• モナドSubdistribution Monad
DX = {f : X ! [0, 1] |X
x
f(x) 1}
Multiset MonadMX = {f : X ! [0,1] | supp(f) is countable}
![Page 61: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/61.jpg)
�24
Subdistribution Monad → Multiset Monad
確率的オートマトン
実数重み付きオートマトン
• モナドSubdistribution Monad
DX = {f : X ! [0, 1] |X
x
f(x) 1}
Multiset MonadMX = {f : X ! [0,1] | supp(f) is countable}
![Page 62: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/62.jpg)
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確率 → 実数重み● 新しく Kleisli simulation が見つかることがある
1
1
1
2
![Page 63: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/63.jpg)
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オートマトンの変形
● オートマトンを変形し 、# Kleisli simulation を存在しやすく
![Page 64: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/64.jpg)
4種の変形Split Backward Merge Backward
Split Forward Merge Forward
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変形の性質Split Backward Merge Backward
![Page 66: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/66.jpg)
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変形の性質Split Backward Merge Backward
Forward Simulationの左側#Backward Simulationの右側} に「適切」な変形
A vF B A vB B
![Page 67: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/67.jpg)
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変形の性質
Split Forward Merge Forward
![Page 68: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/68.jpg)
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変形の性質
Split Forward Merge Forward
Forward Simulationの右側#Backward Simulationの左側} に「適切」な変形
A vF B A vB B
![Page 69: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/69.jpg)
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変形の性質
● 適切な変形は既存の Kleisli simulation を壊さない
● 適切な変形は Kleisli simulation の健全性を保つA0 vF B0
A0 vB B0A vtr B
A0 vF B0
A0 vB B0
A vF B
A vB B
)
))
)
![Page 70: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/70.jpg)
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変形の性質
● これ以上変形できないオートマトンの間の# Kleisli simulation は完全
![Page 71: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/71.jpg)
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変形の性質
● これ以上変形できないオートマトンの間の# Kleisli simulation は完全
● 普通は無限回変形できる
![Page 72: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/72.jpg)
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変形の性質● 確率的オートマトンのtrace inclusion は半決定不能#● 何度変形してもKleisli simulationが# 存在するようにならないことがある
(Hasuo, 2006)
![Page 73: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/73.jpg)
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変形の性質● 確率的オートマトンのtrace inclusion は半決定不能#● 何度変形してもKleisli simulationが# 存在するようにならないことがある
(Hasuo, 2006)
![Page 74: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/74.jpg)
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Overview● Simulation#● Kleisli Simulation#
1. Definition#2. Searching Kleisli Simulation for Probabilistic System#
● Contribution#1. Implementation#2. Increasing the Chance of Simulation#
● Experimental Results and Comparison
![Page 75: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/75.jpg)
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実装● Kleisli simulation を探すプログラム#
- 線形計画問題ソルバを用いて線形不等式を解く#- GLPK (GNU Linear Programmin Kit) © Andrew Makhorin#- 単体法#
● オートマトンを変形するプログラム
![Page 76: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/76.jpg)
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実験結果● Grade protocol の正しさの検証#● オートマトンの等価性を調べることで検証できる# (Kiefer et al. 2011)
parm protocol specific type time(sec) space ex
G S st tr st tr al dir f/b user real (GB)
2 8 578 1522 130 642 11 p ! s fwd 1.77 2.00 1.21 E
bwd 2.03 2.20 1.21 N
s ! p fwd 1.94 2.08 1.14 N
bwd 1.72 1.88 1.22 E
2 10 1102 2982 202 1202 13 p ! s fwd 9.42 9.95 4.05 E
bwd 11.51 12.00 4.08 N
s ! p fwd 10.99 11.45 3.82 N
bwd 9.25 9.81 4.09 E
2 12 1874 5162 290 2018 15 p ! s fwd 38.60 40.16 11.51 E
bwd 49.03 50.38 11.60 N
s ! p fwd 47.99 49.40 10.83 N
bwd 38.34 39.78 11.63 E
3 8 1923 7107 243 2163 20 p ! s fwd 44.43 62.04 12.26 E
bwd 56.92 79.31 12.59 N
s ! p fwd 55.48 57.12 12.27 N
bwd 44.11 67.60 12.64 E
4 6 1636 7468 196 1924 23 p ! s fwd 30.28 31.64 10.39 E
bwd 38.79 39.97 10.49 N
s ! p fwd 37.86 39.10 9.79 N
bwd 29.94 31.25 10.49 E
![Page 77: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/77.jpg)
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実験結果● Grade protocol の正しさの検証#● オートマトンの等価性を調べることで検証できる# (Kiefer et al. 2011)
parm protocol specific type time(sec) space ex
G S st tr st tr al dir f/b user real (GB)
2 8 578 1522 130 642 11 p ! s fwd 1.77 2.00 1.21 E
bwd 2.03 2.20 1.21 N
s ! p fwd 1.94 2.08 1.14 N
bwd 1.72 1.88 1.22 E
2 10 1102 2982 202 1202 13 p ! s fwd 9.42 9.95 4.05 E
bwd 11.51 12.00 4.08 N
s ! p fwd 10.99 11.45 3.82 N
bwd 9.25 9.81 4.09 E
2 12 1874 5162 290 2018 15 p ! s fwd 38.60 40.16 11.51 E
bwd 49.03 50.38 11.60 N
s ! p fwd 47.99 49.40 10.83 N
bwd 38.34 39.78 11.63 E
3 8 1923 7107 243 2163 20 p ! s fwd 44.43 62.04 12.26 E
bwd 56.92 79.31 12.59 N
s ! p fwd 55.48 57.12 12.27 N
bwd 44.11 67.60 12.64 E
4 6 1636 7468 196 1924 23 p ! s fwd 30.28 31.64 10.39 E
bwd 38.79 39.97 10.49 N
s ! p fwd 37.86 39.10 9.79 N
bwd 29.94 31.25 10.49 E
● Space consuming
![Page 78: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/78.jpg)
�35
実験結果● Grade protocol の正しさの検証#● オートマトンの等価性を調べることで検証できる# (Kiefer et al. 2011)
parm protocol specific type time(sec) space ex
G S st tr st tr al dir f/b user real (GB)
2 8 578 1522 130 642 11 p ! s fwd 1.77 2.00 1.21 E
bwd 2.03 2.20 1.21 N
s ! p fwd 1.94 2.08 1.14 N
bwd 1.72 1.88 1.22 E
2 10 1102 2982 202 1202 13 p ! s fwd 9.42 9.95 4.05 E
bwd 11.51 12.00 4.08 N
s ! p fwd 10.99 11.45 3.82 N
bwd 9.25 9.81 4.09 E
2 12 1874 5162 290 2018 15 p ! s fwd 38.60 40.16 11.51 E
bwd 49.03 50.38 11.60 N
s ! p fwd 47.99 49.40 10.83 N
bwd 38.34 39.78 11.63 E
3 8 1923 7107 243 2163 20 p ! s fwd 44.43 62.04 12.26 E
bwd 56.92 79.31 12.59 N
s ! p fwd 55.48 57.12 12.27 N
bwd 44.11 67.60 12.64 E
4 6 1636 7468 196 1924 23 p ! s fwd 30.28 31.64 10.39 E
bwd 38.79 39.97 10.49 N
s ! p fwd 37.86 39.10 9.79 N
bwd 29.94 31.25 10.49 E
● Space consuming
• Kieferらの実装より遅い# - Trace inclusionは# trace equivalenceより# 難しい
![Page 79: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/79.jpg)
�36
実験結果● Crowds protocol の probable innocence の検証#● Trace inclusion を調べることで検証できる (Hasuo, Kawabe, Sakurada, 2010)#
● 変形が必要になる場合が発生→変形・探索を繰り返すparm orig inno time(sec) space itern c pf st tr st tr al f/b user real (GB)
5 1 910
7 44 7 56 18 fwd 52.48 52.56 0.01 2Lbwd 0.01 0.06 0.01 2L
7 1 34
9 88 9 118 26 fwd 0.15 0.21 0.03 2Lbwd 0.02 0.07 0.01 2L
10 2 45
12 224 12 280 54 fwd 802.47 803.64 0.35 2Lbwd 0.05 0.12 0.03 2L
20 6 45
22 1514 22 1696 238 fwd TObwd 1.32 1.51 0.78 2L
30 6 45
32 4732 32 5112 550 fwd SFbwd 11.84 12.82 5.99 2L
![Page 80: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/80.jpg)
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実験結果● Crowds protocol の probable innocence の検証#● Trace inclusion を調べることで検証できる (Hasuo, Kawabe, Sakurada, 2010)#
● 変形が必要になる場合が発生→変形・探索を繰り返すparm orig inno time(sec) space itern c pf st tr st tr al f/b user real (GB)
5 1 910
7 44 7 56 18 fwd 52.48 52.56 0.01 2Lbwd 0.01 0.06 0.01 2L
7 1 34
9 88 9 118 26 fwd 0.15 0.21 0.03 2Lbwd 0.02 0.07 0.01 2L
10 2 45
12 224 12 280 54 fwd 802.47 803.64 0.35 2Lbwd 0.05 0.12 0.03 2L
20 6 45
22 1514 22 1696 238 fwd TObwd 1.32 1.51 0.78 2L
30 6 45
32 4732 32 5112 550 fwd SFbwd 11.84 12.82 5.99 2L
![Page 81: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/81.jpg)
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実験結果● Crowds protocol の probable innocence の検証#● Trace inclusion を調べることで検証できる (Hasuo, Kawabe, Sakurada, 2010)#
● 変形が必要になる場合が発生→変形・探索を繰り返すparm orig inno time(sec) space itern c pf st tr st tr al f/b user real (GB)
5 1 910
7 44 7 56 18 fwd 52.48 52.56 0.01 2Lbwd 0.01 0.06 0.01 2L
7 1 34
9 88 9 118 26 fwd 0.15 0.21 0.03 2Lbwd 0.02 0.07 0.01 2L
10 2 45
12 224 12 280 54 fwd 802.47 803.64 0.35 2Lbwd 0.05 0.12 0.03 2L
20 6 45
22 1514 22 1696 238 fwd TObwd 1.32 1.51 0.78 2L
30 6 45
32 4732 32 5112 550 fwd SFbwd 11.84 12.82 5.99 2L
![Page 82: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/82.jpg)
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実験結果● Crowds protocol の probable innocence の検証#● Trace inclusion を調べることで検証できる (Hasuo, Kawabe, Sakurada, 2010)#
● 変形が必要になる場合が発生→変形・探索を繰り返す
● Backward simulation is faster
parm orig inno time(sec) space itern c pf st tr st tr al f/b user real (GB)
5 1 910
7 44 7 56 18 fwd 52.48 52.56 0.01 2Lbwd 0.01 0.06 0.01 2L
7 1 34
9 88 9 118 26 fwd 0.15 0.21 0.03 2Lbwd 0.02 0.07 0.01 2L
10 2 45
12 224 12 280 54 fwd 802.47 803.64 0.35 2Lbwd 0.05 0.12 0.03 2L
20 6 45
22 1514 22 1696 238 fwd TObwd 1.32 1.51 0.78 2L
30 6 45
32 4732 32 5112 550 fwd SFbwd 11.84 12.82 5.99 2L
![Page 83: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/83.jpg)
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他のSimulationとの比較
(Hasuo, 2006) (Hughes & Jacobs, 2004)
![Page 84: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/84.jpg)
�37
他のSimulationとの比較
(Hasuo, 2006) (Hughes & Jacobs, 2004)
(Hasuo, 2010)
![Page 85: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/85.jpg)
�37
他のSimulationとの比較
(Hasuo, 2006) (Hughes & Jacobs, 2004)
(Hasuo, 2010)
![Page 86: Kleisli Simulation for Real-Weighted Automata and Its ...group-mmm.org/~nurabe/slides/cscat_slide.pdf1 Kleisli Simulation for Real-Weighted Automata and Its Algorithm 卜部 夏木](https://reader035.vdocuments.site/reader035/viewer/2022070217/611f00f3264a940f3d47ea3f/html5/thumbnails/86.jpg)
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まとめ● Kleisli simulationを探すプログラムを実装#● Simulation を存在しやすくする工夫
● オートマトンの変形を圏論的に#● 他のシステムに対するKleisli simulationの実装
Future Work