julian jarecki -...

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

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

Overview

Recap: Petri Nets

Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}

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}

Recap: Petri Nets

Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}

. . . b

a a b a b b c

Recap: Petri Nets

Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}

. . . b

c1 2 n n + 1

a a b a b b c

Recap: Petri Nets

Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}

. . . b

a a b a b b c

Exactly one final marking:

p 7→ 0

Recap: Petri Nets

Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}

. . . b

a a b a b b c

Exactly one final marking:

p 7→ 0

Recap: Petri Nets

Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}

. . . b

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!?)

Recap: Petri Nets

Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}

. . . b

a a b a b b c

• Net is 1-safe.

(orly!?)

Recap: Petri Nets

Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}

. . . b

a a b a b b c

• Net is 1-safe.

(orly!?)

Recap: Petri Nets

Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}

a a b a b b c

• Net is 1-safe.

(orly!?)

Recap: Petri Nets

Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}

a a b a b b c

• Net is 1-safe.

• The Net is deterministic. (orly!?)

Recap: Petri Nets

Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}

a b a b b c

• Net is 1-safe.

(orly!?)

Recap: Petri Nets

Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}

b a b b c

• Net is 1-safe.

(orly!?)

Recap: Petri Nets

Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}

ca a b

a b b c

• Net is 1-safe.

(orly!?)

Recap: Petri Nets

Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}

ca a b a

• Net is 1-safe.

(orly!?)

Recap: Petri Nets

Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}

ca a b a b

• Net is 1-safe.

(orly!?)

Recap: Petri Nets

Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}

ca a b a b b

• Net is 1-safe.

(orly!?)

Recap: Petri Nets

Ln = {wc | w ∈ {a, b}∗ ∧ |w |a = |w |b = n}

ca a b a b b c

• Net is 1-safe.

(orly!?)

Recap: Petri Nets

Can Petri Nets make it better ?

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.

Recap: Petri Nets

computations on numbers of tokens to be expressed, which makesthe formalism Turing complete.1

1http://en.wikipedia.org/wiki/Petri net

Content

Motivation

Introduction

A Finite Asynchronous Automaton accepting Ln

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.

A Signature:

σ = (Σ,R, E)

A Signature:

σ = (Σ,R, E)

A Signature:

σ = (Σ,R, E)

A Signature:

σ = (Σ,R, E)

A Signature:

σ = (Σ,R, E)

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}

τ = (Σ,R, E ,X ,∆)

• X = N≤n ∪ {s}

• ∆ = {δa, δb, δc}

a c bn

τ = (Σ,R, E ,X ,∆)

• X = N≤n ∪ {s}

• ∆ = {δa, δb, δc}

Global states:S = XR

τ = (Σ,R, E ,X ,∆)

• X = N≤n ∪ {s}• ∆ = {δa, δb, δc}

Local transitions:δa ⊆ X Ea × X aE

δcr1 r2 r1

δar1 r1

i i − 1

δcr1 r2 r1

δar1 r1

i i − 1

δcr1 r2 r1

δar1 r1

i i − 1

δcr1 r2 r1

0 0 sδa

i i − 1

δcr1 r2 r1

0 0 sδa

i i − 1

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ n

δcr1 r2 r1

0 0 sδa

i i − 1δb

i i − 1

δcr1 r2 r1

0 0 sδa

i i − 1δb

i i − 1

Finite Asynchronous Automaton (FAA):

A = ( Σ,R, E ,X ,∆︸︷︷︸fat-system

, I ,F )

a c bn

δcr1 r2 r1

0 0 sδa

i i − 1δb

i i − 1

A = (Σ,R, E ,X ,∆, I ,F )

I = {fI}fI (r1) = n

fI (r2) = n

a c bs

δcr1 r2 r1

0 0 sδa

i i − 1δb

i i − 1

A = (Σ,R, E ,X ,∆, I ,F )

F = {fF}fF (r1) = s

fF (r2) = 0

A Finite Asynchronous Automaton accepting L3Example Run for aababbc ∈ L3

a a b a b b c

δcr1 r2 r1

a a b a b b c

δcr1 r2 r1

a a b a b b c

δcr1 r2 r1

a a b a b b c

δcr1 r2 r1

a a b a b b c

δcr1 r2 r1

a a b a b b c

δcr1 r2 r1

a a b a b b c

δcr1 r2 r1

a a b a b b c

δcr1 r2 r1

IndependenceA closer look at the signature

When are two Actions a, b ∈ Σ independent ?

What does independent mean?

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.

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.

Cσ = E2 ∩ Σ2 a→©→ b

Wσ = E × E−1 ∩ Σ2 a→©← b

Dσ = Cσ ∪ C−1σ ∪Wσ

Iσ = Σ2 \ Dσ

Cσ = E2 ∩ Σ2 a→©→ b

Wσ = E × E−1 ∩ Σ2 a→©← b

Iσ = Σ2 \ Dσ

Cσ = E2 ∩ Σ2 a→©→ b

Wσ = E × E−1 ∩ Σ2 a→©← b

Whiteboard Example

Iσ = Σ2 \ Dσ

Cσ = E2 ∩ Σ2 a→©→ b

Wσ = E × E−1 ∩ Σ2 a→©← b

Iσ = Σ2 \ Dσ

Cσ = E2 ∩ Σ2 a→©→ b

Wσ = E × E−1 ∩ Σ2 a→©← b

Iσ = Σ2 \ Dσ

Content

Motivation

Introduction

A particularly simple signature

signature of a fat-system:

σ = (Σ,R, E)

Where E ⊆ R × Σ ∪ Σ× R

New Type of signature:

σ = (Σ,C )

Where C ⊆ Σ2

Finite Asynchronous Cellular Transition System (fact-system):

τ = (Σ,C ,X ,∆)

Where∆ = {δa | a ∈ Σ}

And for a ∈ Σδa ⊆ XCa × X

τ = (Σ,C ,X ,∆)

δaa a

i i − 1δb

i i − 1

δaa a

i i − 1δb

i i − 1

δaa a

i i − 1

0 < i ≤ n

δbb b

i i − 1

δaa a

i i − 1δb

i i − 1δc

δaa a

i i − 1δb

i i − 1δc

Independenceno writing conflicts

When are two Actions a, b ∈ Σ dependent ?

σ = (Σ,C )

Cσ = C (a→©→ b)

Wσ = idΣ (a→©← b)

Iσ = Σ2 \ Dσ

{(a, b) | a, b not adjacent}

σ = (Σ,C )

Cσ = C (a→©→ b)

Iσ = Σ2 \ Dσ

σ = (Σ,C )

Cσ = C (a→©→ b)

Iσ = Σ2 \ Dσ

σ = (Σ,C )

Cσ = C (a→©→ b)

Iσ = Σ2 \ Dσ

σ = (Σ,C )

Cσ = C (a→©→ b)

Iσ = Σ2 \ Dσ {(a, b) | a, b not adjacent}

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 .

SimulationsNDT → fact

τC = (Σ,C ,X ,∆) s.t.:

• C = Σ2

• X = {0, . . . , k − 1}• ∆ =

⋃a∈Σ {δa}

(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 .

τC = (Σ,C ,X ,∆) s.t.:

• C = Σ2

• X = {0, . . . , k − 1}• ∆ =

⋃a∈Σ {δa}

(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 .

τC = (Σ,C ,X ,∆) s.t.:

• C = Σ2

• X = {0, . . . , k − 1}• ∆ =

⋃a∈Σ {δa}

(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 .

τC = (Σ,C ,X ,∆) s.t.:

• C = Σ2

• X = {0, . . . , k − 1}• ∆ =

⋃a∈Σ {δa}

(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 .

τC = (Σ,C ,X ,∆) s.t.:

• C = Σ2

• X = {0, . . . , k − 1}• ∆ =

⋃a∈Σ {δa}

(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 .

τC = (Σ,C ,X ,∆) s.t.:

• C = Σ2

• X = {0, . . . , k − 1}

• ∆ =⋃

a∈Σ {δa}

(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 .

τC = (Σ,C ,X ,∆) s.t.:

• C = Σ2

• X = {0, . . . , k − 1}• ∆ =

⋃a∈Σ {δa}

(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 .

τC = (Σ,C ,X ,∆) s.t.:

• C = Σ2

• X = {0, . . . , k − 1}• ∆ =

⋃a∈Σ {δa}

(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 .

τC = (Σ,C ,X ,∆) s.t.:

• C = Σ2

• X = {0, . . . , k − 1}• ∆ =

⋃a∈Σ {δa}

(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 .

τC = (Σ,C ,X ,∆) s.t.:

• C = Σ2

• X = {0, . . . , k − 1}• ∆ =

⋃a∈Σ {δa}

(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 .

τC = (Σ,C ,X ,∆) s.t.:

• C = Σ2

• X = {0, . . . , k − 1}• ∆ =

⋃a∈Σ {δa}

(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 .

τC = (Σ,C ,X ,∆) s.t.:

• C = Σ2

• X = {0, . . . , k − 1}• ∆ =

⋃a∈Σ {δa}

(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 .

τC = (Σ,C ,X ,∆) s.t.:

• C = Σ2

• X = {0, . . . , k − 1}• ∆ =

⋃a∈Σ {δa}

(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 .

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.

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

Fat signatureσ = (Σ,R, E)

C = E2 ∩ Σ2Fact signatureσ′ = (Σ,C )

σ induces C

If there are no writing-conflicts

(x , y , z)

Writing-conflict

This writing conflict isω-redundant.

Writing-conflict

Add a time-stamp

(xb), (tb)

(xc), (tc)

(xa), (ta)

Add a time-stamp

(xb), (tb)

(xc), (tc)

(xa), (ta)

Add a time-stamp

(xb), (tb)

(xc), (tc)

(xa), (ta)

Add a time-stamp

(xb), (tb)

(xc), (tc)

(xa), (ta)

Simulationsfat → fact: Example

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

Σ = Σfat

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

C = E2fat ∩ Σ2

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

C = E2fat ∩ Σ2

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

C = E2fat ∩ Σ2

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

C = E2fat ∩ Σ2

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

C = E2fat ∩ Σ2

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

C = E2fat ∩ Σ2

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

X =⋃a∈Σ

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

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

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

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

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

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

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

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

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

ta r1 tc

i 0 * 0 i − 1 0

i 1 * 1 i − 1 1

* 0 i 1 i − 1 1

* 1 i 0 i − 1 0

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

i 0 * 0 i − 1 0

i 1 * 1 i − 1 1

* 0 i 1 i − 1 1

* 1 i 0 i − 1 0

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

r1 ta r1

i 0 * 0 i − 1 0

i 1 * 1 i − 1 1

* 0 i 1 i − 1 1

* 1 i 0 i − 1 0

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

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

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

a (0) c

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

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

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

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

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

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

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

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

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

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

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

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

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

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

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

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

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

a c b3

δcr1 r2 r1

0 0 sδa

i i − 1

0 < i ≤ 3

δbr2 r2

i i − 1

X =⋃

a∈Σ Xa

δa ⊆∏α∈Ca

Xα × Xa

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

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

Languages

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

Languages

A = (Σ,R, E︸︷︷︸σ

,X ,∆, I ,F )

L(A) = {w ∈ Σ∗ | ∃s0 ∈ I , sf ∈ F : sf ∈ ∆(s0,w)}

Languages

A = (Σ,R, E︸︷︷︸σ

,X ,∆, I ,F )

L(A) = {w ∈ Σ∗ | ∃s0 ∈ I , sf ∈ F : sf ∈ ∆(s0,w)}

Languages

A = (Σ,R, E︸︷︷︸σ

,X ,∆, I ,F )

L(A) = {w ∈ Σ∗ | ∃s0 ∈ I , sf ∈ F : sf ∈ ∆(s0,w)}

Languages

A = (Σ,R, E︸︷︷︸σ

,X ,∆, I ,F )

L(A) = {w ∈ Σ∗ | ∃s0 ∈ I , sf ∈ F : sf ∈ ∆(s0,w)}

Languages

• 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.

Languages

ThenLdσ1

= Ldσ2and Lnσ1

= Lnσ2

Languages

ThenLdσ1

= Ldσ2and Lnσ1

= Lnσ2

Languages

ThenLdσ1

= Ldσ2and Lnσ1

= Lnσ2

Languages

ThenLdσ1

= Ldσ2and Lnσ1

= Lnσ2

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.

Languages

Proof.boring

Languages

Proof.boring

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}.

Languages

Σ1 ⊆ Σ2 and C1 = C2 ∩ Σ21

proof idea.

0|Σ2\Σ1.

Pick F1 = {s|Σ1| s ∈ F2}.

Languages

Σ1 ⊆ Σ2 and C1 = C2 ∩ Σ21

Then for each language L ⊆ Σ∗1:

L ∈ Lnσ1⇔ L ∈ Lnσ2

proof idea.

0|Σ2\Σ1.

Pick F1 = {s|Σ1| s ∈ F2}.

Languages

Σ1 ⊆ Σ2 and C1 = C2 ∩ Σ21

proof idea.

0|Σ2\Σ1.

Pick F1 = {s|Σ1| s ∈ F2}.

Languages

Σ1 ⊆ Σ2 and C1 = C2 ∩ Σ21

Then for each language L ⊆ Σ∗1: L ∈ Ldσ1⇔ L ∈ Ldσ2

proof idea.

0|Σ2\Σ1.

Pick F1 = {s|Σ1| s ∈ F2}.

Languages

Σ1 ⊆ Σ2 and C1 = C2 ∩ Σ21

Then for each language L ⊆ Σ∗1: L ∈ Luσ1⇔ L ∈ Luσ2

proof idea.

0|Σ2\Σ1.

Pick F1 = {s|Σ1| s ∈ F2}.

Languages

Σ1 ⊆ Σ2 and C1 = C2 ∩ Σ21

Then for each language L ⊆ Σ∗1: L ∈ Luσ1⇔ L ∈ Luσ2

proof idea.

0|Σ2\Σ1.

Pick F1 = {s|Σ1| s ∈ F2}.

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

Languages

Ldσ1⊆ Ldσ2

and Lnσ1⊆ Lnσ2

(s, x) ∈ δ2a ∈ ∆2 iff (s|C1a, x) ∈ δ1

a ∈ ∆1

Languages

Ldσ1⊆ Ldσ2

and Lnσ1⊆ Lnσ2

(s, x) ∈ δ2a ∈ ∆2 iff (s|C1a, x) ∈ δ1

a ∈ ∆1

Languages

Ldσ1⊆ Ldσ2

and Lnσ1⊆ Lnσ2

(s, x) ∈ δ2a ∈ ∆2 iff (s|C1a, x) ∈ δ1

a ∈ ∆1

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.

Languages

Ldσ1⊆ Ldσ2

iff C1 ⊆ C2

⇒: blurp.

Languages

Ldσ1⊆ Ldσ2

iff C1 ⊆ C2

⇒: blurp.

Languages

Ldσ1⊆ Ldσ2

iff C1 ⊆ C2

⇒: blurp.

Determinization

Theorem (Zielonka’s Theorem)

Let σ = (Σ,C ). If C is symmetric and reflexive then

Ldσ = Lnσ

Determinization

Theorem (Zielonka’s Theorem)

Let σ = (Σ,C ). If C is symmetric and reflexive then

Ldσ = Lnσ

Complexity

• indirect proof construction

• triple-exponential blow-up in size

• direct determinization shown in [KMS94]

• ”only” double-exponential

← essentially optimal

Complexity

• ”only” double-exponential ← essentially optimal

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.

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