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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
Content
Motivation
Introduction
Asynchronous Cellular Transition Systems
Language Recognizability and Determinization
Content
Motivation
Introduction
Asynchronous Cellular Transition Systems
Language Recognizability and Determinization
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
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
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
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
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
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Recap: Petri Nets
Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}
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}
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
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
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
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
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!?)
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!?)
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!?)
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!?)
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!?)
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!?)
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!?)
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!?)
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!?)
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!?)
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!?)
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!?)
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.
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.
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.
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.
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
Content
Motivation
Introduction
Asynchronous Cellular Transition Systems
Language Recognizability and Determinization
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A Finite Asynchronous Automaton accepting Ln
a c b
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.
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.
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.
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.
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.
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.
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}
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}
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
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}
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
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
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
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
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
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
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
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
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 )
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
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
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
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
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
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
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
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
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
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
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
IndependenceA closer look at the signature
a c b
r1 r2
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 ?
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?
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.
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.
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
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
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
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
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
Content
Motivation
Introduction
Asynchronous Cellular Transition Systems
Language Recognizability and Determinization
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
A particularly simple signature
a
n
b
n
c
1
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
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
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
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
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
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
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
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
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
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
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
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
Motivation Introduction Asynchronous Cellular Transition Systems Language Recognizability and Determinization
Independenceno writing conflicts
a bc
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.
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.
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}
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}
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}
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}
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}
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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.
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.
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.
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.
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
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
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
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
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
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)
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)
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)
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)
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
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
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
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
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
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
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
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
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
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
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
?
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 .
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 .
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 .
Content
Motivation
Introduction
Asynchronous Cellular Transition Systems
Language Recognizability and Determinization
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
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
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
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
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
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.
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.
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.
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.
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.
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.
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.
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.
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}.
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}.
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}.
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}.
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}.
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}.
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}.
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
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
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
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
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.
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.
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.
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.
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
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
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
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
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
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
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
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.