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Finite Asynchronous Automata
Julian Jarecki
Department of Computer Science,SWT, Freiburg, Germany
http://swt.informatik.uni-freiburg.de
18. Juni 2012
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Content
Motivation
Introduction
Asynchronous Cellular Transition Systems
Language Recognizability and Determinization
![Page 3: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/3.jpg)
Content
Motivation
Introduction
Asynchronous Cellular Transition Systems
Language Recognizability and Determinization
![Page 4: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/4.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Overview
• introduced by Wies law Zielonka [Zie87]
• motivated by Trace Theory [Maz77]
• concurrent systems
• similar to Petri Nets [DR95]
• restriction to a finite statespace
![Page 5: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/5.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Overview
• introduced by Wies law Zielonka [Zie87]
• motivated by Trace Theory [Maz77]
• concurrent systems
• similar to Petri Nets [DR95]
• restriction to a finite statespace
![Page 6: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/6.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Overview
• introduced by Wies law Zielonka [Zie87]
• motivated by Trace Theory [Maz77]
• concurrent systems
• similar to Petri Nets [DR95]
• restriction to a finite statespace
![Page 7: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/7.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Overview
• introduced by Wies law Zielonka [Zie87]
• motivated by Trace Theory [Maz77]
• concurrent systems
• similar to Petri Nets [DR95]
• restriction to a finite statespace
![Page 8: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/8.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Overview
• introduced by Wies law Zielonka [Zie87]
• motivated by Trace Theory [Maz77]
• concurrent systems
• similar to Petri Nets [DR95]
• restriction to a finite statespace
![Page 9: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/9.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
![Page 10: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/10.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
• L1 = {abc, bac}• L2 = {aabbc, ababc, baabc, babac, bbaac}• L3 = {aaabbbc, aababbc, . . . , bbbaaac}
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
a a
b
. . .
. . . b
c
a a b a b b c
![Page 12: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/12.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
a a
b
. . .
. . . b
c1 2 n n + 1
a a b a b b c
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
a a
b
. . .
. . . b
c
a a b a b b c
Exactly one final marking:
p 7→ 0
![Page 14: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/14.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
a a
b
. . .
. . . b
c
a a b a b b c
Exactly one final marking:
p 7→ 0
![Page 15: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/15.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
a a
b
. . .
. . . b
c
a a b a b b c
• Ln is finite, thus regular.
• Net has linear size (O(n2) for NFA)
• Net is 1-safe.
• The Net is deterministic.
(orly!?)
![Page 16: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/16.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
a a
b
. . .
. . . b
c
a a b a b b c
• Ln is finite, thus regular.
• Net has linear size (O(n2) for NFA)
• Net is 1-safe.
• The Net is deterministic.
(orly!?)
![Page 17: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/17.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
a a
b
. . .
. . . b
c
a a b a b b c
• Ln is finite, thus regular.
• Net has linear size (O(n2) for NFA)
• Net is 1-safe.
• The Net is deterministic.
(orly!?)
![Page 18: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/18.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
a a a
b b b
c
a a b a b b c
• Ln is finite, thus regular.
• Net has linear size (O(n2) for NFA)
• Net is 1-safe.
• The Net is deterministic.
(orly!?)
![Page 19: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/19.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
a a a
b b b
c
a a b a b b c
• Ln is finite, thus regular.
• Net has linear size (O(n2) for NFA)
• Net is 1-safe.
• The Net is deterministic. (orly!?)
![Page 20: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/20.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
a a a
b b b
ca
a b a b b c
• Ln is finite, thus regular.
• Net has linear size (O(n2) for NFA)
• Net is 1-safe.
• The Net is deterministic.
(orly!?)
![Page 21: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/21.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
a a a
b b b
ca a
b a b b c
• Ln is finite, thus regular.
• Net has linear size (O(n2) for NFA)
• Net is 1-safe.
• The Net is deterministic.
(orly!?)
![Page 22: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/22.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
a a a
b b b
ca a b
a b b c
• Ln is finite, thus regular.
• Net has linear size (O(n2) for NFA)
• Net is 1-safe.
• The Net is deterministic.
(orly!?)
![Page 23: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/23.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
a a a
b b b
ca a b a
b b c
• Ln is finite, thus regular.
• Net has linear size (O(n2) for NFA)
• Net is 1-safe.
• The Net is deterministic.
(orly!?)
![Page 24: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/24.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
a a a
b b b
ca a b a b
b c
• Ln is finite, thus regular.
• Net has linear size (O(n2) for NFA)
• Net is 1-safe.
• The Net is deterministic.
(orly!?)
![Page 25: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/25.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
a a a
b b b
ca a b a b b
c
• Ln is finite, thus regular.
• Net has linear size (O(n2) for NFA)
• Net is 1-safe.
• The Net is deterministic.
(orly!?)
![Page 26: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/26.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
a a a
b b b
ca a b a b b c
• Ln is finite, thus regular.
• Net has linear size (O(n2) for NFA)
• Net is 1-safe.
• The Net is deterministic.
(orly!?)
![Page 27: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/27.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Can Petri Nets make it better ?
n n
a b
An inhibitor arc imposes the precondition that the transition mayonly fire when the place is empty; this allows arbitrary
computations on numbers of tokens to be expressed, which makesthe formalism Turing complete.
![Page 28: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/28.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Can Petri Nets make it better ?
n n
a b
An inhibitor arc imposes the precondition that the transition mayonly fire when the place is empty; this allows arbitrary
computations on numbers of tokens to be expressed, which makesthe formalism Turing complete.
![Page 29: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/29.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Can Petri Nets make it better ?
n n
a b
c
An inhibitor arc imposes the precondition that the transition mayonly fire when the place is empty; this allows arbitrary
computations on numbers of tokens to be expressed, which makesthe formalism Turing complete.
![Page 30: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/30.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Can Petri Nets make it better ?
n n
a b
c
An inhibitor arc imposes the precondition that the transition mayonly fire when the place is empty; this allows arbitrary
computations on numbers of tokens to be expressed, which makesthe formalism Turing complete.
![Page 31: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/31.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Can Petri Nets make it better ?
n n
a bc
An inhibitor arc imposes the precondition that the transition mayonly fire when the place is empty; this allows arbitrary
computations on numbers of tokens to be expressed, which makesthe formalism Turing complete.1
1http://en.wikipedia.org/wiki/Petri net
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Content
Motivation
Introduction
Asynchronous Cellular Transition Systems
Language Recognizability and Determinization
![Page 33: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/33.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
![Page 34: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/34.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
A Signature:
σ = (Σ,R, E)
• Σ is an Alphabet (the set of Actions)
• R is the Set of Registers
• E ⊆ R × Σ ∪ Σ× R is the connection relation.
![Page 35: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/35.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
A Signature:
σ = (Σ,R, E)
• Σ is an Alphabet (the set of Actions)
• R is the Set of Registers
• E ⊆ R × Σ ∪ Σ× R is the connection relation.
![Page 36: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/36.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
r1 r2
A Signature:
σ = (Σ,R, E)
• Σ is an Alphabet (the set of Actions)
• R is the Set of Registers
• E ⊆ R × Σ ∪ Σ× R is the connection relation.
![Page 37: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/37.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
r1 r2
A Signature:
σ = (Σ,R, E)
• Σ is an Alphabet (the set of Actions)
• R is the Set of Registers
• E ⊆ R × Σ ∪ Σ× R is the connection relation.
![Page 38: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/38.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
r1 r2
A Signature:
σ = (Σ,R, E)
• Σ is an Alphabet (the set of Actions)
• R is the Set of Registers
• E ⊆ R × Σ ∪ Σ× R is the connection relation.
![Page 39: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/39.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
r1 r2
A Signature:
σ = (Σ,R, E)
• Σ is an Alphabet (the set of Actions)
• R is the Set of Registers
• E ⊆ R × Σ ∪ Σ× R is the connection relation.
![Page 40: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/40.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
r1 r2
Finite Asynchronous Transition System (fat-system):
τ = ( Σ,R, E︸ ︷︷ ︸σ
,X ,∆)
• X is the a finite set of values
• ∆ = {δa | a ∈ Σ} is a familyof transition Relations
• X = N≤n ∪ {s}• ∆ = {δa, δb, δc}
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
r1 r2
Finite Asynchronous Transition System (fat-system):
τ = (Σ,R, E ,X ,∆)
• X is the a finite set of values
• ∆ = {δa | a ∈ Σ} is a familyof transition Relations
• X = N≤n ∪ {s}
• ∆ = {δa, δb, δc}
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c bn
r1
1
r2
Finite Asynchronous Transition System (fat-system):
τ = (Σ,R, E ,X ,∆)
• X is the a finite set of values
• ∆ = {δa | a ∈ Σ} is a familyof transition Relations
• X = N≤n ∪ {s}
• ∆ = {δa, δb, δc}
Global states:S = XR
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
r1 r2
Finite Asynchronous Transition System (fat-system):
τ = (Σ,R, E ,X ,∆)
• X is the a finite set of values
• ∆ = {δa | a ∈ Σ} is a familyof transition Relations
• X = N≤n ∪ {s}• ∆ = {δa, δb, δc}
![Page 44: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/44.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
r1 r2
Local transitions:δa ⊆ X Ea × X aE
![Page 45: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/45.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
r1 r2
δcr1 r2 r1
0 0 s
δar1 r1
i i − 1
Local transitions:δa ⊆ X Ea × X aE
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
r1 r2
δcr1 r2 r1
0 0 s
δar1 r1
i i − 1
Local transitions:δa ⊆ X Ea × X aE
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
r1 r2
δcr1 r2 r1
0 0 s
δar1 r1
i i − 1
Local transitions:δa ⊆ X Ea × X aE
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
r1 r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
Local transitions:δa ⊆ X Ea × X aE
![Page 49: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/49.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
r1 r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
Local transitions:δa ⊆ X Ea × X aE
![Page 50: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/50.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
r1 r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ n
Local transitions:δa ⊆ X Ea × X aE
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
r1 r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1δb
r2 r2
i i − 1
Local transitions:δa ⊆ X Ea × X aE
![Page 52: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/52.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
r1 r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1δb
r2 r2
i i − 1
Finite Asynchronous Automaton (FAA):
A = ( Σ,R, E ,X ,∆︸ ︷︷ ︸fat-system
, I ,F )
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c bn
r1
n
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1δb
r2 r2
i i − 1
Finite Asynchronous Automaton (FAA):
A = (Σ,R, E ,X ,∆, I ,F )
I = {fI}fI (r1) = n
fI (r2) = n
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c bs
r1
0
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1δb
r2 r2
i i − 1
Finite Asynchronous Automaton (FAA):
A = (Σ,R, E ,X ,∆, I ,F )
F = {fF}fF (r1) = s
fF (r2) = 0
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting L3Example Run for aababbc ∈ L3
a c b
a a b a b b c
3
r1
3
r2
δa
r1 r1
3 2
2 1
1 0
δb
r2 r2
3 2
2 1
1 0
δcr1 r2 r1
0 0 s
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting L3Example Run for aababbc ∈ L3
a c b
a a b a b b c
2
r1
3
r2
δa
r1 r1
3 2
2 1
1 0
δb
r2 r2
3 2
2 1
1 0
δcr1 r2 r1
0 0 s
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting L3Example Run for aababbc ∈ L3
a c b
a a b a b b c
1
r1
3
r2
δa
r1 r1
3 2
2 1
1 0
δb
r2 r2
3 2
2 1
1 0
δcr1 r2 r1
0 0 s
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting L3Example Run for aababbc ∈ L3
a c b
a a b a b b c
1
r1
2
r2
δa
r1 r1
3 2
2 1
1 0
δb
r2 r2
3 2
2 1
1 0
δcr1 r2 r1
0 0 s
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting L3Example Run for aababbc ∈ L3
a c b
a a b a b b c
0
r1
2
r2
δa
r1 r1
3 2
2 1
1 0
δb
r2 r2
3 2
2 1
1 0
δcr1 r2 r1
0 0 s
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting L3Example Run for aababbc ∈ L3
a c b
a a b a b b c
0
r1
1
r2
δa
r1 r1
3 2
2 1
1 0
δb
r2 r2
3 2
2 1
1 0
δcr1 r2 r1
0 0 s
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting L3Example Run for aababbc ∈ L3
a c b
a a b a b b c
0
r1
0
r2
δa
r1 r1
3 2
2 1
1 0
δb
r2 r2
3 2
2 1
1 0
δcr1 r2 r1
0 0 s
![Page 62: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/62.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting L3Example Run for aababbc ∈ L3
a c b
a a b a b b c
s
r1
0
r2
δa
r1 r1
3 2
2 1
1 0
δb
r2 r2
3 2
2 1
1 0
δcr1 r2 r1
0 0 s
![Page 63: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/63.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
IndependenceA closer look at the signature
a c b
r1 r2
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
IndependenceA closer look at the signature
a c b
r1 r2
When are two Actions a, b ∈ Σ independent ?
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
IndependenceA closer look at the signature
a c b
r1 r2
When are two Actions a, b ∈ Σ independent ?
What does independent mean?
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
IndependenceA closer look at the signature
a c b
r1 r2
When are two Actions a, b ∈ Σ independent ?
In a signature σ two Actions a, b are called independent,if σ’s structure doesn’t allow a fat-system to distinguish ab and ba.
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
IndependenceA closer look at the signature
a c b
r1 r2
When are two Actions a, b ∈ Σ independent ?
In a signature σ two Actions a, b are called independent,if for all fat-systems with σ a and b can be executed in any order
without altering the result of the computation.
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
IndependenceA closer look at the signature
a c b
r1 r2
Cσ = E2 ∩ Σ2 a→©→ b
Wσ = E × E−1 ∩ Σ2 a→©← b
Dσ = Cσ ∪ C−1σ ∪Wσ
Iσ = Σ2 \ Dσ
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
IndependenceA closer look at the signature
a c b
r1 r2
Cσ = E2 ∩ Σ2 a→©→ b
Wσ = E × E−1 ∩ Σ2 a→©← b
Dσ = Cσ ∪ C−1σ ∪Wσ
Iσ = Σ2 \ Dσ
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
IndependenceA closer look at the signature
a c b
r1 r2
Cσ = E2 ∩ Σ2 a→©→ b
Wσ = E × E−1 ∩ Σ2 a→©← b
Whiteboard Example
Dσ = Cσ ∪ C−1σ ∪Wσ
Iσ = Σ2 \ Dσ
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
IndependenceA closer look at the signature
a c b
r1 r2
Cσ = E2 ∩ Σ2 a→©→ b
Wσ = E × E−1 ∩ Σ2 a→©← b
Dσ = Cσ ∪ C−1σ ∪Wσ
Iσ = Σ2 \ Dσ
![Page 72: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/72.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
IndependenceA closer look at the signature
a c b
r1 r2
Cσ = E2 ∩ Σ2 a→©→ b
Wσ = E × E−1 ∩ Σ2 a→©← b
Dσ = Cσ ∪ C−1σ ∪Wσ
Iσ = Σ2 \ Dσ
![Page 73: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/73.jpg)
Content
Motivation
Introduction
Asynchronous Cellular Transition Systems
Language Recognizability and Determinization
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A particularly simple signature
a
n
b
n
c
1
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A particularly simple signature
a
n
b
n
c
1
signature of a fat-system:
σ = (Σ,R, E)
Where E ⊆ R × Σ ∪ Σ× R
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A particularly simple signature
a
n
b
n
c
1
New Type of signature:
σ = (Σ,C )
Where C ⊆ Σ2
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A particularly simple signature
a
n
b
n
c
1
Finite Asynchronous Cellular Transition System (fact-system):
τ = (Σ,C ,X ,∆)
Where∆ = {δa | a ∈ Σ}
And for a ∈ Σδa ⊆ XCa × X
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A particularly simple signature
a
n
b
n
c
1
Finite Asynchronous Cellular Transition System (fact-system):
τ = (Σ,C ,X ,∆)
Where∆ = {δa | a ∈ Σ}
And for a ∈ Σδa ⊆ XCa × X
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A particularly simple signature
a
n
b
n
c
1
Finite Asynchronous Cellular Transition System (fact-system):
τ = (Σ,C ,X ,∆)
Where∆ = {δa | a ∈ Σ}
And for a ∈ Σδa ⊆ XCa × X
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Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A particularly simple signature
an
bn
c
1
Finite Asynchronous Cellular Transition System (fact-system):
τ = (Σ,C ,X ,∆)
Where∆ = {δa | a ∈ Σ}
And for a ∈ Σδa ⊆ XCa × X
![Page 81: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/81.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A particularly simple signature
an
bn
c
1
Finite Asynchronous Cellular Transition System (fact-system):
τ = (Σ,C ,X ,∆)
Where∆ = {δa | a ∈ Σ}
And for a ∈ Σδa ⊆ XCa × X
![Page 82: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/82.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A particularly simple signature
an
bn
c
1
δaa a
i i − 1δb
b b
i i − 1
![Page 83: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/83.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A particularly simple signature
an
bn
c
1
δaa a
i i − 1δb
b b
i i − 1
![Page 84: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/84.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A particularly simple signature
an
bn
c
1
δaa a
i i − 1
0 < i ≤ n
δbb b
i i − 1
![Page 85: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/85.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A particularly simple signature
an
bn
c
1
δaa a
i i − 1δb
b b
i i − 1δc
a b c
0 0 0
![Page 86: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/86.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A particularly simple signature
an
bn
c
1
δaa a
i i − 1δb
b b
i i − 1δc
a b c
0 0 0
![Page 87: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/87.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Independenceno writing conflicts
a bc
![Page 88: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/88.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Independenceno writing conflicts
a bc
When are two Actions a, b ∈ Σ dependent ?
In a signature σ two Actions a, b are called independent,if σ’s structure doesn’t allow a fat-system to distinguish ab and ba.
![Page 89: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/89.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Independenceno writing conflicts
a bc
When are two Actions a, b ∈ Σ dependent ?
In a signature σ two Actions a, b are called independent,if σ’s structure doesn’t allow a fat-system to distinguish ab and ba.
![Page 90: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/90.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Independenceno writing conflicts
a bc
σ = (Σ,C )
Cσ = C (a→©→ b)
Wσ = idΣ (a→©← b)
Dσ = Cσ ∪ C−1σ ∪Wσ
Iσ = Σ2 \ Dσ
{(a, b) | a, b not adjacent}
![Page 91: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/91.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Independenceno writing conflicts
a bc
σ = (Σ,C )
Cσ = C (a→©→ b)
Wσ = idΣ (a→©← b)
Dσ = Cσ ∪ C−1σ ∪Wσ
Iσ = Σ2 \ Dσ
{(a, b) | a, b not adjacent}
![Page 92: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/92.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Independenceno writing conflicts
a bc
σ = (Σ,C )
Cσ = C (a→©→ b)
Wσ = idΣ (a→©← b)
Dσ = Cσ ∪ C−1σ ∪Wσ
Iσ = Σ2 \ Dσ
{(a, b) | a, b not adjacent}
![Page 93: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/93.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Independenceno writing conflicts
a bc
σ = (Σ,C )
Cσ = C (a→©→ b)
Wσ = idΣ (a→©← b)
Dσ = Cσ ∪ C−1σ ∪Wσ
Iσ = Σ2 \ Dσ
{(a, b) | a, b not adjacent}
![Page 94: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/94.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Independenceno writing conflicts
a bc
σ = (Σ,C )
Cσ = C (a→©→ b)
Wσ = idΣ (a→©← b)
Dσ = Cσ ∪ C−1σ ∪Wσ
Iσ = Σ2 \ Dσ {(a, b) | a, b not adjacent}
![Page 95: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/95.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulations
Are fact-systems regular?
Proof.Let τ = (Σ,Q, δ) be a regular finite transition system.WLOG let Q = {q0, . . . , qk−1}.
τC = (Σ,C ,X ,∆) s.t.:
• C = Σ2
• X = {0, . . . , k − 1}• ∆ =
⋃a∈Σ {δa}
let a ∈ Σ, β ∈ X Σ and x ∈ X .
(β, x) ∈ δa
⇔ ∃s, s ′ :
(qs , a, qs′) ∈ δ
∧∑b∈Σ
β(b) mod k = s
∧s − β(a) + x mod k = s ′
τC simulates τ .
![Page 96: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/96.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
SimulationsNDT → fact
Proof.Let τ = (Σ,Q, δ) be a regular finite transition system.WLOG let Q = {q0, . . . , qk−1}.
τC = (Σ,C ,X ,∆) s.t.:
• C = Σ2
• X = {0, . . . , k − 1}• ∆ =
⋃a∈Σ {δa}
let a ∈ Σ, β ∈ X Σ and x ∈ X .
(β, x) ∈ δa
⇔ ∃s, s ′ :
(qs , a, qs′) ∈ δ
∧∑b∈Σ
β(b) mod k = s
∧s − β(a) + x mod k = s ′
τC simulates τ .
![Page 97: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/97.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
SimulationsNDT → fact
Proof.Let τ = (Σ,Q, δ) be a regular finite transition system.WLOG let Q = {q0, . . . , qk−1}.
τC = (Σ,C ,X ,∆) s.t.:
• C = Σ2
• X = {0, . . . , k − 1}• ∆ =
⋃a∈Σ {δa}
let a ∈ Σ, β ∈ X Σ and x ∈ X .
(β, x) ∈ δa
⇔ ∃s, s ′ :
(qs , a, qs′) ∈ δ
∧∑b∈Σ
β(b) mod k = s
∧s − β(a) + x mod k = s ′
τC simulates τ .
![Page 98: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/98.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
SimulationsNDT → fact
Proof.Let τ = (Σ,Q, δ) be a regular finite transition system.WLOG let Q = {q0, . . . , qk−1}.
τC = (Σ,C ,X ,∆) s.t.:
• C = Σ2
• X = {0, . . . , k − 1}• ∆ =
⋃a∈Σ {δa}
let a ∈ Σ, β ∈ X Σ and x ∈ X .
(β, x) ∈ δa
⇔ ∃s, s ′ :
(qs , a, qs′) ∈ δ
∧∑b∈Σ
β(b) mod k = s
∧s − β(a) + x mod k = s ′
τC simulates τ .
![Page 99: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/99.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
SimulationsNDT → fact
Proof.Let τ = (Σ,Q, δ) be a regular finite transition system.WLOG let Q = {q0, . . . , qk−1}.
τC = (Σ,C ,X ,∆) s.t.:
• C = Σ2
• X = {0, . . . , k − 1}• ∆ =
⋃a∈Σ {δa}
let a ∈ Σ, β ∈ X Σ and x ∈ X .
(β, x) ∈ δa
⇔ ∃s, s ′ :
(qs , a, qs′) ∈ δ
∧∑b∈Σ
β(b) mod k = s
∧s − β(a) + x mod k = s ′
τC simulates τ .
![Page 100: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/100.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
SimulationsNDT → fact
Proof.Let τ = (Σ,Q, δ) be a regular finite transition system.WLOG let Q = {q0, . . . , qk−1}.
τC = (Σ,C ,X ,∆) s.t.:
• C = Σ2
• X = {0, . . . , k − 1}• ∆ =
⋃a∈Σ {δa}
let a ∈ Σ, β ∈ X Σ and x ∈ X .
(β, x) ∈ δa
⇔ ∃s, s ′ :
(qs , a, qs′) ∈ δ
∧∑b∈Σ
β(b) mod k = s
∧s − β(a) + x mod k = s ′
τC simulates τ .
![Page 101: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/101.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
SimulationsNDT → fact
Proof.Let τ = (Σ,Q, δ) be a regular finite transition system.WLOG let Q = {q0, . . . , qk−1}.
τC = (Σ,C ,X ,∆) s.t.:
• C = Σ2
• X = {0, . . . , k − 1}
• ∆ =⋃
a∈Σ {δa}
let a ∈ Σ, β ∈ X Σ and x ∈ X .
(β, x) ∈ δa
⇔ ∃s, s ′ :
(qs , a, qs′) ∈ δ
∧∑b∈Σ
β(b) mod k = s
∧s − β(a) + x mod k = s ′
τC simulates τ .
![Page 102: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/102.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
SimulationsNDT → fact
Proof.Let τ = (Σ,Q, δ) be a regular finite transition system.WLOG let Q = {q0, . . . , qk−1}.
τC = (Σ,C ,X ,∆) s.t.:
• C = Σ2
• X = {0, . . . , k − 1}• ∆ =
⋃a∈Σ {δa}
let a ∈ Σ, β ∈ X Σ and x ∈ X .
(β, x) ∈ δa
⇔ ∃s, s ′ :
(qs , a, qs′) ∈ δ
∧∑b∈Σ
β(b) mod k = s
∧s − β(a) + x mod k = s ′
τC simulates τ .
![Page 103: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/103.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
SimulationsNDT → fact
Proof.Let τ = (Σ,Q, δ) be a regular finite transition system.WLOG let Q = {q0, . . . , qk−1}.
τC = (Σ,C ,X ,∆) s.t.:
• C = Σ2
• X = {0, . . . , k − 1}• ∆ =
⋃a∈Σ {δa}
let a ∈ Σ, β ∈ X Σ and x ∈ X .
(β, x) ∈ δa
⇔ ∃s, s ′ :
(qs , a, qs′) ∈ δ
∧∑b∈Σ
β(b) mod k = s
∧s − β(a) + x mod k = s ′
τC simulates τ .
![Page 104: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/104.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
SimulationsNDT → fact
Proof.Let τ = (Σ,Q, δ) be a regular finite transition system.WLOG let Q = {q0, . . . , qk−1}.
τC = (Σ,C ,X ,∆) s.t.:
• C = Σ2
• X = {0, . . . , k − 1}• ∆ =
⋃a∈Σ {δa}
let a ∈ Σ, β ∈ X Σ and x ∈ X .
(β, x) ∈ δa ⇔ ∃s, s ′ :
(qs , a, qs′) ∈ δ
∧∑b∈Σ
β(b) mod k = s
∧s − β(a) + x mod k = s ′
τC simulates τ .
![Page 105: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/105.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
SimulationsNDT → fact
Proof.Let τ = (Σ,Q, δ) be a regular finite transition system.WLOG let Q = {q0, . . . , qk−1}.
τC = (Σ,C ,X ,∆) s.t.:
• C = Σ2
• X = {0, . . . , k − 1}• ∆ =
⋃a∈Σ {δa}
let a ∈ Σ, β ∈ X Σ and x ∈ X .
(β, x) ∈ δa ⇔ ∃s, s ′ :
(qs , a, qs′) ∈ δ
∧∑b∈Σ
β(b) mod k = s
∧s − β(a) + x mod k = s ′
τC simulates τ .
![Page 106: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/106.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
SimulationsNDT → fact
Proof.Let τ = (Σ,Q, δ) be a regular finite transition system.WLOG let Q = {q0, . . . , qk−1}.
τC = (Σ,C ,X ,∆) s.t.:
• C = Σ2
• X = {0, . . . , k − 1}• ∆ =
⋃a∈Σ {δa}
let a ∈ Σ, β ∈ X Σ and x ∈ X .
(β, x) ∈ δa ⇔ ∃s, s ′ :
(qs , a, qs′) ∈ δ
∧∑b∈Σ
β(b) mod k = s
∧s − β(a) + x mod k = s ′
τC simulates τ .
![Page 107: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/107.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
SimulationsNDT → fact
Proof.Let τ = (Σ,Q, δ) be a regular finite transition system.WLOG let Q = {q0, . . . , qk−1}.
τC = (Σ,C ,X ,∆) s.t.:
• C = Σ2
• X = {0, . . . , k − 1}• ∆ =
⋃a∈Σ {δa}
let a ∈ Σ, β ∈ X Σ and x ∈ X .
(β, x) ∈ δa ⇔ ∃s, s ′ :
(qs , a, qs′) ∈ δ
∧∑b∈Σ
β(b) mod k = s
∧s − β(a) + x mod k = s ′
τC simulates τ .
![Page 108: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/108.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
SimulationsNDT → fact
Proof.Let τ = (Σ,Q, δ) be a regular finite transition system.WLOG let Q = {q0, . . . , qk−1}.
τC = (Σ,C ,X ,∆) s.t.:
• C = Σ2
• X = {0, . . . , k − 1}• ∆ =
⋃a∈Σ {δa}
let a ∈ Σ, β ∈ X Σ and x ∈ X .
(β, x) ∈ δa ⇔ ∃s, s ′ :
(qs , a, qs′) ∈ δ
∧∑b∈Σ
β(b) mod k = s
∧s − β(a) + x mod k = s ′
τC simulates τ .
![Page 109: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/109.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Can a finite asynchronous cellular automaton (faca) simulate a faa?
– both accept regular languages.
What would be an interesting simulation?
– One that maintains independence.
![Page 110: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/110.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Can a finite asynchronous cellular automaton (faca) simulate a faa?
– both accept regular languages.
What would be an interesting simulation?
– One that maintains independence.
![Page 111: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/111.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Can a finite asynchronous cellular automaton (faca) simulate a faa?
– both accept regular languages.
What would be an interesting simulation?
– One that maintains independence.
![Page 112: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/112.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Can a finite asynchronous cellular automaton (faca) simulate a faa?
– both accept regular languages.
What would be an interesting simulation?
– One that maintains independence.
![Page 113: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/113.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
How can we assure that all independence is maintained?
Try not to interfere with the signature:
Fat signatureσ = (Σ,R, E) C = E2 ∩ Σ2
Fact signatureσ′ = (Σ,C )
σ induces C
![Page 114: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/114.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
How can we assure that all independence is maintained?
Try not to interfere with the signature:
Fat signatureσ = (Σ,R, E)
C = E2 ∩ Σ2Fact signatureσ′ = (Σ,C )
σ induces C
![Page 115: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/115.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
How can we assure that all independence is maintained?
Try not to interfere with the signature:
Fat signatureσ = (Σ,R, E) C = E2 ∩ Σ2
Fact signatureσ′ = (Σ,C )
σ induces C
![Page 116: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/116.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
How can we assure that all independence is maintained?
Try not to interfere with the signature:
Fat signatureσ = (Σ,R, E) C = E2 ∩ Σ2
Fact signatureσ′ = (Σ,C )
σ induces C
![Page 117: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/117.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
How can we assure that all independence is maintained?
Try not to interfere with the signature:
Fat signatureσ = (Σ,R, E) C = E2 ∩ Σ2
Fact signatureσ′ = (Σ,C )
σ induces C
![Page 118: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/118.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
If there are no writing-conflicts
a
x
r1
y
r2
z
r3
a
(x , y , z)
![Page 119: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/119.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
If there are no writing-conflicts
a
x
r1
y
r2
z
r3
a
(x , y , z)
![Page 120: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/120.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
If there are no writing-conflicts
a
x
r1
y
r2
z
r3
a
(x , y , z)
![Page 121: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/121.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
If there are no writing-conflicts
a
x
r1
y
r2
z
r3
b
a
(x , y , z)
![Page 122: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/122.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
If there are no writing-conflicts
a
x
r1
y
r2
z
r3
b
a
(x , y , z)
b
![Page 123: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/123.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Writing-conflict
a
b
c
This writing conflict isω-redundant.
b
c
a
d
![Page 124: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/124.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Writing-conflict
a
b
c
r
This writing conflict isω-redundant.
b
c
a
d
![Page 125: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/125.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Writing-conflict
a
b
c
r
This writing conflict isω-redundant.
b
c
a
d
![Page 126: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/126.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Writing-conflict
a
b
c
r
This writing conflict isω-redundant.
b
c
a
d
![Page 127: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/127.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Writing-conflict
a
b
c
xb
rxb
This writing conflict isω-redundant.
b
(xb)
c
a
d
![Page 128: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/128.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Writing-conflict
a
b
c
xc
rxb
xc
This writing conflict isω-redundant.
b
(xb)
c
(xc)
a
d
![Page 129: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/129.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Writing-conflict
a
b
c
xa
rxb
xc
xa
This writing conflict isω-redundant.
b
(xb)
c
(xc)
a
(xa)
d
![Page 130: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/130.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Writing-conflict
a
b
c
xa
rxb
xc
xa
This writing conflict isω-redundant.
db
(xb)
c
(xc)
a
(xa)
d
![Page 131: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/131.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Writing-conflict
a
b
c
xa
rxb
xc
xa
This writing conflict isω-redundant.
db
(xb)
c
(xc)
a
(xa)
d
![Page 132: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/132.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Writing-conflict
a
b
c
xa
rxb
xc
xa
This writing conflict isω-redundant.
db
(xb)
c
(xc)
a
(xa)
d
?
![Page 133: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/133.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Add a time-stamp
a
b
c
xa
rxb
xc
xa
This writing conflict isω-redundant.
db
(xb), (tb)
c
(xc), (tc)
a
(xa), (ta)
d
![Page 134: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/134.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Add a time-stamp
a
b
c
xa
rxb
xc
xa
This writing conflict isω-redundant.
db
(xb), (tb)
c
(xc), (tc)
a
(xa), (ta)
d
![Page 135: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/135.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Add a time-stamp
a
b
c
xa
rxb
xc
xa
This writing conflict isω-redundant.
db
(xb), (tb)
c
(xc), (tc)
a
(xa), (ta)
d
![Page 136: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/136.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact
Add a time-stamp
a
b
c
xa
rxb
xc
xa
This writing conflict isω-redundant.
db
(xb), (tb)
c
(xc), (tc)
a
(xa), (ta)
d
![Page 137: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/137.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b
3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
![Page 138: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/138.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
![Page 139: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/139.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
![Page 140: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/140.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
Σ = Σfat
![Page 141: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/141.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
C = E2fat ∩ Σ2
![Page 142: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/142.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
C = E2fat ∩ Σ2
![Page 143: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/143.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
C = E2fat ∩ Σ2
![Page 144: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/144.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
C = E2fat ∩ Σ2
![Page 145: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/145.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
C = E2fat ∩ Σ2
![Page 146: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/146.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
C = E2fat ∩ Σ2
![Page 147: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/147.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
X =⋃a∈Σ
Xa
![Page 148: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/148.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a
(0)
c
(1)
a
(0)
r1 ta r1 tc r1 ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 149: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/149.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a
(0)
c
(1)
a
(0)
r1 ta r1 tc r1 ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 150: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/150.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a
(0)
c
(1)
a
(0)
r1 ta r1 tc r1 ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 151: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/151.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a
(0)
c
(1)
a
(0)
r1 ta r1 tc r1 ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 152: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/152.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a
(0)
c
(1)
a
(0)
r1
ta r1 tc
r1
ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 153: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/153.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a
(0)
c
(1)
a
(0)
r1 ta
r1 tc
r1 ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 154: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/154.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a
(0)
c
(1)
a
(0)
r1 ta r1
tc
r1 ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 155: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/155.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a
(0)
c
(1)
a
(0)
r1 ta r1 tc r1 ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 156: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/156.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a (0) c
(1)
a (0)
r1 ta r1 tc r1 ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 157: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/157.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a (0) c (1) a (0)
r1 ta r1 tc r1 ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 158: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/158.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a (0) c (1) a (0)
r1 ta r1 tc r1 ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 159: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/159.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a (0) c (1) a (0)
r1 ta r1 tc r1 ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 160: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/160.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a (0) c (1) a (0)
r1 ta r1 tc r1 ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 161: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/161.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a (0) c (1) a (0)
r1 ta r1 tc r1 ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 162: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/162.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a (0) c (1) a (0)
r1 ta r1 tc r1 ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 163: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/163.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a (0) c (1) a (0)
r1 ta r1 tc r1 ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 164: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/164.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a (0) c (1) a (0)
r1 ta r1 tc r1 ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 165: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/165.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfat → fact: Example
a c b3
r1
3
r2
δcr1 r2 r1
0 0 sδa
r1 r1
i i − 1
0 < i ≤ 3
δbr2 r2
i i − 1
a c b
X =⋃
a∈Σ Xa
δa ⊆∏α∈Ca
Xα × Xa
δa:
a (0) c (1) a (0)
r1 ta r1 tc r1 ta
i 0 * 0 i − 1 0
i 1 * 1 i − 1 1
* 0 i 1 i − 1 1
* 1 i 0 i − 1 0
![Page 166: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/166.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfact → fat
Does the other direction work too?
Let τ = (Σ,C ,X ,∆) be a fact-system and σ an ω-redundantsignature that induces C .
Then there exists a fat-system τ ′ = (Σ,R, E ,X ′,∆′) over σcovering τ .
![Page 167: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/167.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfact → fat
Does the other direction work too?
Let τ = (Σ,C ,X ,∆) be a fact-system and σ an ω-redundantsignature that induces C .
Then there exists a fat-system τ ′ = (Σ,R, E ,X ′,∆′) over σcovering τ .
![Page 168: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/168.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Simulationsfact → fat
Does the other direction work too?
Let τ = (Σ,C ,X ,∆) be a fact-system and σ an ω-redundantsignature that induces C .
Then there exists a fat-system τ ′ = (Σ,R, E ,X ′,∆′) over σcovering τ .
![Page 169: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/169.jpg)
Content
Motivation
Introduction
Asynchronous Cellular Transition Systems
Language Recognizability and Determinization
![Page 170: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/170.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Finite Asynchronous Automaton (FAA):
A = (Σ,R, E︸ ︷︷ ︸σ
,X ,∆, I ,F )
L(A) = {w ∈ Σ∗ | ∃s0 ∈ I , sf ∈ F : sf ∈ ∆(s0,w)}
Ldσ := class of languages recognized by deterministic faa over thesignature σ
Lnσ := class of languages recognized by non-deterministic faa overthe signature σ
Defined analogously for FACA
![Page 171: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/171.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Finite Asynchronous Automaton (FAA):
A = (Σ,R, E︸ ︷︷ ︸σ
,X ,∆, I ,F )
L(A) = {w ∈ Σ∗ | ∃s0 ∈ I , sf ∈ F : sf ∈ ∆(s0,w)}
Ldσ := class of languages recognized by deterministic faa over thesignature σ
Lnσ := class of languages recognized by non-deterministic faa overthe signature σ
Defined analogously for FACA
![Page 172: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/172.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Finite Asynchronous Automaton (FAA):
A = (Σ,R, E︸ ︷︷ ︸σ
,X ,∆, I ,F )
L(A) = {w ∈ Σ∗ | ∃s0 ∈ I , sf ∈ F : sf ∈ ∆(s0,w)}
Ldσ := class of languages recognized by deterministic faa over thesignature σ
Lnσ := class of languages recognized by non-deterministic faa overthe signature σ
Defined analogously for FACA
![Page 173: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/173.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Finite Asynchronous Automaton (FAA):
A = (Σ,R, E︸ ︷︷ ︸σ
,X ,∆, I ,F )
L(A) = {w ∈ Σ∗ | ∃s0 ∈ I , sf ∈ F : sf ∈ ∆(s0,w)}
Ldσ := class of languages recognized by deterministic faa over thesignature σ
Lnσ := class of languages recognized by non-deterministic faa overthe signature σ
Defined analogously for FACA
![Page 174: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/174.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Finite Asynchronous Automaton (FAA):
A = (Σ,R, E︸ ︷︷ ︸σ
,X ,∆, I ,F )
L(A) = {w ∈ Σ∗ | ∃s0 ∈ I , sf ∈ F : sf ∈ ∆(s0,w)}
Ldσ := class of languages recognized by deterministic faa over thesignature σ
Lnσ := class of languages recognized by non-deterministic faa overthe signature σ
Defined analogously for FACA
![Page 175: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/175.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Let
• σ1 = (Σ,R, E) be an ω-redundant signature of a fat-system
• σ2 = (Σ,C ) be the induced signature of a fact-system(C = E2 ∩ Σ2)
ThenLdσ1
= Ldσ2and Lnσ1
= Lnσ2
Proof.Simulation fat↔fact plus a little magic regarding initial/final statesand languages.
In the sequel we restrain our attention to languages recognized byfaca.
![Page 176: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/176.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Let
• σ1 = (Σ,R, E) be an ω-redundant signature of a fat-system
• σ2 = (Σ,C ) be the induced signature of a fact-system(C = E2 ∩ Σ2)
ThenLdσ1
= Ldσ2and Lnσ1
= Lnσ2
Proof.Simulation fat↔fact plus a little magic regarding initial/final statesand languages.
In the sequel we restrain our attention to languages recognized byfaca.
![Page 177: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/177.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Let
• σ1 = (Σ,R, E) be an ω-redundant signature of a fat-system
• σ2 = (Σ,C ) be the induced signature of a fact-system(C = E2 ∩ Σ2)
ThenLdσ1
= Ldσ2and Lnσ1
= Lnσ2
Proof.Simulation fat↔fact plus a little magic regarding initial/final statesand languages.
In the sequel we restrain our attention to languages recognized byfaca.
![Page 178: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/178.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Let
• σ1 = (Σ,R, E) be an ω-redundant signature of a fat-system
• σ2 = (Σ,C ) be the induced signature of a fact-system(C = E2 ∩ Σ2)
ThenLdσ1
= Ldσ2and Lnσ1
= Lnσ2
Proof.Simulation fat↔fact plus a little magic regarding initial/final statesand languages.
In the sequel we restrain our attention to languages recognized byfaca.
![Page 179: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/179.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Let
• σ1 = (Σ,R, E) be an ω-redundant signature of a fat-system
• σ2 = (Σ,C ) be the induced signature of a fact-system(C = E2 ∩ Σ2)
ThenLdσ1
= Ldσ2and Lnσ1
= Lnσ2
Proof.Simulation fat↔fact plus a little magic regarding initial/final statesand languages.
In the sequel we restrain our attention to languages recognized byfaca.
![Page 180: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/180.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
For each faca A = (Σ,C ,X ,∆, I ,F ) there exists a faca A′ overthe same signature σ = (Σ,C ) with only one initial state s.t.L(A) = L(A′).
Proof.boring
In the sequel wlog we assume that there is only one initial state.
![Page 181: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/181.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
For each faca A = (Σ,C ,X ,∆, I ,F ) there exists a faca A′ overthe same signature σ = (Σ,C ) with only one initial state s.t.L(A) = L(A′).
Proof.boring
In the sequel wlog we assume that there is only one initial state.
![Page 182: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/182.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
For each faca A = (Σ,C ,X ,∆, I ,F ) there exists a faca A′ overthe same signature σ = (Σ,C ) with only one initial state s.t.L(A) = L(A′).
Proof.boring
In the sequel wlog we assume that there is only one initial state.
![Page 183: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/183.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Let σ1 = (Σ1,C1) and σ2 = (Σ2,C2) be two signatures, such that
Σ1 ⊆ Σ2 and C1 = C2 ∩ Σ21
Then for each language L ⊆ Σ∗1: L ∈ Lnσ1⇔ L ∈ Lnσ2
proof idea.
⇒: Let A1 be a faca over σ1. Construct A2 over σ2 such that itworks like A1 for Actions in Σ1. For Actions in Σ2 \Σ1 it goes intoa state such that a final state cannot be reached anymore.
⇐: Let A2 be a faca over σ2 with initial state s20 , final states F2.
Construct A1 over σ1 with the same values X that works like A2
for Actions in Σ1. Let s be a final state, then s|Σ2\Σ1= s2
0|Σ2\Σ1.
Pick F1 = {s|Σ1| s ∈ F2}.
![Page 184: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/184.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Let σ1 = (Σ1,C1) and σ2 = (Σ2,C2) be two signatures, such that
Σ1 ⊆ Σ2 and C1 = C2 ∩ Σ21
Then for each language L ⊆ Σ∗1: L ∈ Lnσ1⇔ L ∈ Lnσ2
proof idea.
⇒: Let A1 be a faca over σ1. Construct A2 over σ2 such that itworks like A1 for Actions in Σ1. For Actions in Σ2 \Σ1 it goes intoa state such that a final state cannot be reached anymore.
⇐: Let A2 be a faca over σ2 with initial state s20 , final states F2.
Construct A1 over σ1 with the same values X that works like A2
for Actions in Σ1. Let s be a final state, then s|Σ2\Σ1= s2
0|Σ2\Σ1.
Pick F1 = {s|Σ1| s ∈ F2}.
![Page 185: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/185.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Let σ1 = (Σ1,C1) and σ2 = (Σ2,C2) be two signatures, such that
Σ1 ⊆ Σ2 and C1 = C2 ∩ Σ21
Then for each language L ⊆ Σ∗1:
L ∈ Lnσ1⇔ L ∈ Lnσ2
proof idea.
⇒: Let A1 be a faca over σ1. Construct A2 over σ2 such that itworks like A1 for Actions in Σ1. For Actions in Σ2 \Σ1 it goes intoa state such that a final state cannot be reached anymore.
⇐: Let A2 be a faca over σ2 with initial state s20 , final states F2.
Construct A1 over σ1 with the same values X that works like A2
for Actions in Σ1. Let s be a final state, then s|Σ2\Σ1= s2
0|Σ2\Σ1.
Pick F1 = {s|Σ1| s ∈ F2}.
![Page 186: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/186.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Let σ1 = (Σ1,C1) and σ2 = (Σ2,C2) be two signatures, such that
Σ1 ⊆ Σ2 and C1 = C2 ∩ Σ21
Then for each language L ⊆ Σ∗1: L ∈ Lnσ1⇔ L ∈ Lnσ2
proof idea.
⇒: Let A1 be a faca over σ1. Construct A2 over σ2 such that itworks like A1 for Actions in Σ1. For Actions in Σ2 \Σ1 it goes intoa state such that a final state cannot be reached anymore.
⇐: Let A2 be a faca over σ2 with initial state s20 , final states F2.
Construct A1 over σ1 with the same values X that works like A2
for Actions in Σ1. Let s be a final state, then s|Σ2\Σ1= s2
0|Σ2\Σ1.
Pick F1 = {s|Σ1| s ∈ F2}.
![Page 187: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/187.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Let σ1 = (Σ1,C1) and σ2 = (Σ2,C2) be two signatures, such that
Σ1 ⊆ Σ2 and C1 = C2 ∩ Σ21
Then for each language L ⊆ Σ∗1: L ∈ Ldσ1⇔ L ∈ Ldσ2
proof idea.
⇒: Let A1 be a faca over σ1. Construct A2 over σ2 such that itworks like A1 for Actions in Σ1. For Actions in Σ2 \Σ1 it goes intoa state such that a final state cannot be reached anymore.
⇐: Let A2 be a faca over σ2 with initial state s20 , final states F2.
Construct A1 over σ1 with the same values X that works like A2
for Actions in Σ1. Let s be a final state, then s|Σ2\Σ1= s2
0|Σ2\Σ1.
Pick F1 = {s|Σ1| s ∈ F2}.
![Page 188: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/188.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Let σ1 = (Σ1,C1) and σ2 = (Σ2,C2) be two signatures, such that
Σ1 ⊆ Σ2 and C1 = C2 ∩ Σ21
Then for each language L ⊆ Σ∗1: L ∈ Luσ1⇔ L ∈ Luσ2
proof idea.
⇒: Let A1 be a faca over σ1. Construct A2 over σ2 such that itworks like A1 for Actions in Σ1. For Actions in Σ2 \Σ1 it goes intoa state such that a final state cannot be reached anymore.
⇐: Let A2 be a faca over σ2 with initial state s20 , final states F2.
Construct A1 over σ1 with the same values X that works like A2
for Actions in Σ1. Let s be a final state, then s|Σ2\Σ1= s2
0|Σ2\Σ1.
Pick F1 = {s|Σ1| s ∈ F2}.
![Page 189: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/189.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Let σ1 = (Σ1,C1) and σ2 = (Σ2,C2) be two signatures, such that
Σ1 ⊆ Σ2 and C1 = C2 ∩ Σ21
Then for each language L ⊆ Σ∗1: L ∈ Luσ1⇔ L ∈ Luσ2
proof idea.
⇒: Let A1 be a faca over σ1. Construct A2 over σ2 such that itworks like A1 for Actions in Σ1. For Actions in Σ2 \Σ1 it goes intoa state such that a final state cannot be reached anymore.
⇐: Let A2 be a faca over σ2 with initial state s20 , final states F2.
Construct A1 over σ1 with the same values X that works like A2
for Actions in Σ1. Let s be a final state, then s|Σ2\Σ1= s2
0|Σ2\Σ1.
Pick F1 = {s|Σ1| s ∈ F2}.
![Page 190: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/190.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Lemma: Let C1 ⊆ C2 ⊆ Σ2 and σi = (Σ,Ci ), i = 1, 2. Then
Ldσ1⊆ Ldσ2
and Lnσ1⊆ Lnσ2
Proof idea.Given A1 faca over σ1, construct A2 over σ2 s.t. L(A1) = L(A2):
Let a ∈ Σ, s ∈ XC2a, x ∈ X then
(s, x) ∈ δ2a ∈ ∆2 iff (s|C1a, x) ∈ δ1
a ∈ ∆1
![Page 191: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/191.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Lemma: Let C1 ⊆ C2 ⊆ Σ2 and σi = (Σ,Ci ), i = 1, 2. Then
Ldσ1⊆ Ldσ2
and Lnσ1⊆ Lnσ2
Proof idea.Given A1 faca over σ1, construct A2 over σ2 s.t. L(A1) = L(A2):
Let a ∈ Σ, s ∈ XC2a, x ∈ X then
(s, x) ∈ δ2a ∈ ∆2 iff (s|C1a, x) ∈ δ1
a ∈ ∆1
![Page 192: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/192.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Lemma: Let C1 ⊆ C2 ⊆ Σ2 and σi = (Σ,Ci ), i = 1, 2. Then
Ldσ1⊆ Ldσ2
and Lnσ1⊆ Lnσ2
Proof idea.Given A1 faca over σ1, construct A2 over σ2 s.t. L(A1) = L(A2):
Let a ∈ Σ, s ∈ XC2a, x ∈ X then
(s, x) ∈ δ2a ∈ ∆2 iff (s|C1a, x) ∈ δ1
a ∈ ∆1
![Page 193: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/193.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Lemma: Let C1 ⊆ C2 ⊆ Σ2 and σi = (Σ,Ci ), i = 1, 2. Then
Ldσ1⊆ Ldσ2
and Lnσ1⊆ Lnσ2
Proof idea.Given A1 faca over σ1, construct A2 over σ2 s.t. L(A1) = L(A2):
Let a ∈ Σ, s ∈ XC2a, x ∈ X then
(s, x) ∈ δ2a ∈ ∆2 iff (s|C1a, x) ∈ δ1
a ∈ ∆1
![Page 194: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/194.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Theorem (Hierarchy Theorem)
C1,C2 ⊆ Σ2 and σi = (Σ,Ci ), i = 1, 2. Then
Ldσ1⊆ Ldσ2
iff C1 ⊆ C2
Proof.⇐: given by the Lemma.
⇒: blurp.
![Page 195: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/195.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Theorem (Hierarchy Theorem)
C1,C2 ⊆ Σ2 and σi = (Σ,Ci ), i = 1, 2. Then
Ldσ1⊆ Ldσ2
iff C1 ⊆ C2
Proof.⇐: given by the Lemma.
⇒: blurp.
![Page 196: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/196.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Theorem (Hierarchy Theorem)
C1,C2 ⊆ Σ2 and σi = (Σ,Ci ), i = 1, 2. Then
Ldσ1⊆ Ldσ2
iff C1 ⊆ C2
Proof.⇐: given by the Lemma.
⇒: blurp.
![Page 197: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/197.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Languages
Theorem (Hierarchy Theorem)
C1,C2 ⊆ Σ2 and σi = (Σ,Ci ), i = 1, 2. Then
Ldσ1⊆ Ldσ2
iff C1 ⊆ C2
Proof.⇐: given by the Lemma.
⇒: blurp.
![Page 198: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/198.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Determinization
Theorem (Zielonka’s Theorem)
Let σ = (Σ,C ). If C is symmetric and reflexive then
Ldσ = Lnσ
![Page 199: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/199.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Determinization
Theorem (Zielonka’s Theorem)
Let σ = (Σ,C ). If C is symmetric and reflexive then
Ldσ = Lnσ
![Page 200: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/200.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Complexity
• indirect proof construction
• triple-exponential blow-up in size
• direct determinization shown in [KMS94]
• ”only” double-exponential
← essentially optimal
![Page 201: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/201.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Complexity
• indirect proof construction
• triple-exponential blow-up in size
• direct determinization shown in [KMS94]
• ”only” double-exponential
← essentially optimal
![Page 202: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/202.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Complexity
• indirect proof construction
• triple-exponential blow-up in size
• direct determinization shown in [KMS94]
• ”only” double-exponential
← essentially optimal
![Page 203: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/203.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Complexity
• indirect proof construction
• triple-exponential blow-up in size
• direct determinization shown in [KMS94]
• ”only” double-exponential
← essentially optimal
![Page 204: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/204.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Complexity
• indirect proof construction
• triple-exponential blow-up in size
• direct determinization shown in [KMS94]
• ”only” double-exponential ← essentially optimal
![Page 205: Julian Jarecki - uni-freiburg.deswt.informatik.uni-freiburg.de/teaching/SS2012/AutomataTheory/... · Content Motivation Introduction Asynchronous Cellular Transition Systems Language](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1aa30ff766eb15cf295138/html5/thumbnails/205.jpg)
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Sources
V. Diekert and G. Rozenberg.The Book of Traces.World Scientific, 1995.
Nils Klarlund, Madhavan Mukund, and Milind A. Sohoni.Determinizing asynchronous automata.In Proceedings of the 21st International Colloquium onAutomata, Languages and Programming, ICALP ’94, pages130–141, London, UK, UK, 1994. Springer-Verlag.
Antoni Mazurkiewicz.Concurrent program schemes and their interpretations.Technical report, DAIMI Aarhus University, 1977.Report PB 78.
Wieslaw Zielonka.Notes on finite asynchronous automata.ITA, 21(2):99–135, 1987.