On deterministic ACA Enrico Formenti University of Nice-Sophia - - PowerPoint PPT Presentation

On deterministic ACA

Enrico Formenti

University of Nice-Sophia Antipolis,France.

Something that is not synchronous is asynchronous!

The definition

Something that is not synchronous is asynchronous!

Motivations

Concurrent Programming

Motivations

Concurrent Programming

I = {i1,i2,...,ik} θ ⊆ I ×I

symmetric irreflexive

θc = I ×I \θ

G = I,θc

α ⊆ I,θc(α) = {i ∈ I,∃j ∈ α|(i, j)θc}

Motivations

Concurrent Programming

Pi ⊆ I i ∈ {1,...,n} GP

P1 P2 P3 P4 P5 P6 P7 P8

Motivations

Concurrent Programming

Pi ⊆ I i ∈ {1,...,n} GP

P1 P2 P3 P4 P5 P6 P7 P8

Bio- informatics

“Generalizing”

Finite vs Infinite Finite Graph vs Lattice Non-uniform vs Uniform Drop dependency graph

“Generalizing”

Finite vs Infinite Finite Graph vs Lattice Non-uniform vs Uniform Drop dependency graph

“Generalizing”

Z

Lattice

I = {i1,i2,...,ik}

States

δ : I2r+1 → I

Local rule

• 1

+1 ... ...

(Re-)Discoverings

(Re-)Discoverings

i3 i3 i2 i2 i1 i1 i1 i2 i3

• 1

+1 ... ...

(Re-)Discoverings

i3 i3 i2 i2 i1 i1 i1 i2 i3

• 1

+1 ... ...

• 1

+1 ... ...

1 1 1 1 1

(Re-)Discoverings

i3 i3 i2 i2 i1 i1 i1 i2 i3

• 1

+1 ... ...

• 1

+1 ... ...

1 1 1 1 1

• 0·δ(i1,i2,i2)

More (re-)discoverings

F′: {0,1}Z×IZ → IZ F′(x,y) = (id(x),F(y)) F′(x,y) = (G(x),F(y))

So far...

G

shift invariant continuous

CA

G

continuous

• CA

ν

G

???

G

shift invariant continuous

“P”CA

So far...

G

shift invariant continuous

CA

G

continuous

• CA

ν

G

???

G

shift invariant continuous

“P”CA

Time for pictures

ECA 54 (54,90)

Time for pictures

ECA 54 (54,90)

More pictures

ECA 54 (54,18)

More pictures

ECA 54 (54,18)

Even more pictures

ECA 54 (54,id)

Even more pictures

ECA 54 (54,id)

Deterministic ACA

G = Z N t = 0 t = 1 t = 2

. . .

...0,0,0,0,0,1,0|0,0,0,0,0,0... ...0,0,0,0,0,0,0|0,0,0,0,1,0... ...0,0,0,0,0,0,0|1,0,0,0,0,0...

Deterministic ACA

G = Z N t = 0 t = 1 t = 2

. . .

...0,0,0,0,0,1,0|0,0,0,0,0,0... ...0,0,0,0,0,0,0|0,0,0,0,1,0... ...0,0,0,0,0,0,0|1,0,0,0,0,0...

Global function

xt

i

z0

i = yi

zt

i = (F′)t(x,y)i

(F′)t(x,y)i = (1−xt

i)zt−1 i

+xt

iδ(zt−1 i−r,...,zt−1 i

,...,zt−1

i+r)

Global function

xt

i

z0

i = yi

zt

i = (F′)t(x,y)i

(F′)t(x,y)i = (1−xt

i)zt−1 i

+xt

iδ(zt−1 i−r,...,zt−1 i

,...,zt−1

i+r)

Set properties

,

Definition.

F′ is injective iff ∀y,z ∈ A

Z ∀x ∈ Z N

N

∀t ∈ z = y ⇒ F′(xt,y) = F′(xt,z)

Set properties

,

Definition.

F′ is injective iff ∀y,z ∈ A

Z ∀x ∈ Z N

N

∀t ∈ z = y ⇒ F′(xt,y) = F′(xt,z)

Set properties (2)

,

Definition.

F′ is surjective iff ∀y ∈ A

Z

∃z ∈ A

Z

∀x ∈ Z N

N

∀t ∈ F′(xt,y) = F′(xt,z)

Set properties (2)

,

Definition.

F′ is surjective iff ∀y ∈ A

Z

∃z ∈ A

Z

∀x ∈ Z N

N

∀t ∈ F′(xt,y) = F′(xt,z)

Set properties (3)

Proposition.

The following properties are equivalent

1) F′ is injective 2) F′ is surjective

3) δ

is center permutative

Set properties (3)

Proposition.

The following properties are equivalent

1) F′ is injective 2) F′ is surjective

3) δ

is center permutative

Dynamics

Proposition.

If is ultimately periodic, then is ultimately periodic.

x ∈Z N

y,F′(x1,y),(F′)2(x2,y),...,(F′)n(xn,y),...

Dynamics

Proposition.

If is ultimately periodic, then is ultimately periodic.

x ∈Z N

y,F′(x1,y),(F′)2(x2,y),...,(F′)n(xn,y),...

Dynamics (2)

Definition.

is sensitive to initial conditions, iff such that

∃ε > 0∀y ∈ AZ

∃t ∈N

F′

d((F′)t(x,y),(F′)t(x,z)) > ε

∀δ > 0∃z ∈ Bδ(y)

∃x ∈ Z N

Dynamics (2)

Definition.

is sensitive to initial conditions, iff such that

∃ε > 0∀y ∈ AZ

∃t ∈N

F′

d((F′)t(x,y),(F′)t(x,z)) > ε

∀δ > 0∃z ∈ Bδ(y)

∃x ∈ Z N

Dynamics (3)

Please look at the whiteboard

• n the right

Dynamics (3)

Please look at the whiteboard

• n the right

Dynamics (4)

Definition.

is expansive, iff such that

F′

∃ε > 0∀y ∈ AZ

∃t ∈N

d((F′)t(x,y),(F′)t(x,z)) > ε

∀z ∈ AZ

∃x ∈ Z N

Dynamics (4)

Definition.

is expansive, iff such that

F′

∃ε > 0∀y ∈ AZ

∃t ∈N

d((F′)t(x,y),(F′)t(x,z)) > ε

∀z ∈ AZ

∃x ∈ Z N

Dynamics (5)

Again Please look at the whiteboard

• n the right

Dynamics (5)

Again Please look at the whiteboard

• n the right

Dynamics (6)

Proposition.

Leftmost or rightmost permutative ACA are sensitive to initial conditions.

Proposition.

Leftmost and rightmost permutative ACA are expansive.

Dynamics (6)

Proposition.

Leftmost or rightmost permutative ACA are sensitive to initial conditions.

Proposition.

Leftmost and rightmost permutative ACA are expansive.

Dynamics (7)

Definition.

such that is transitive if and only if

F′ ∀U,V = / 0 ∃t ∈N

(F′)t(x,U)∩V = /

∃x ∈ Z N

Dynamics (7)

Definition.

such that is transitive if and only if

F′ ∀U,V = / 0 ∃t ∈N

(F′)t(x,U)∩V = /

∃x ∈ Z N

Dynamics (8)

Look at the whiteboard

Dynamics (8)

Look at the whiteboard

Dynamics (9)

Proposition.

Leftmost or rightmost permutative ACA are transitive. (Recall that they are also sensitive)

Dynamics (9)

Proposition.

Leftmost or rightmost permutative ACA are transitive. (Recall that they are also sensitive)

Dynamics (10)

Definition.

such that

F′ has the DPO property if and only if

∃x ∈ Z N

F′(x,·) has the DPO property. F′(x,·) has the DPO property if and only if

its set of periodic points is dense.

Dynamics (10)

Definition.

such that

F′ has the DPO property if and only if

∃x ∈ Z N

F′(x,·) has the DPO property. F′(x,·) has the DPO property if and only if

its set of periodic points is dense.

Dynamics (11)

Proposition.

Deterministic ACA have DPO iff they are surjective.

Dynamics (11)

Proposition.

Deterministic ACA have DPO iff they are surjective.

Dynamics (12)

Lemma.

If and has DPO for some then

F′= id

x ∈ Z N

x is bounded. Corollary.

x ∈ Z N

Fix . Then F′(x,·) cannot be Devaney chaotic.

Dynamics (12)

Lemma.

If and has DPO for some then

F′= id

x ∈ Z N

x is bounded. Corollary.

x ∈ Z N

Fix . Then F′(x,·) cannot be Devaney chaotic.

Beginning to conclude

Deterministic ACA are interesting! What about

Beginning to conclude

Deterministic ACA are interesting! What about Nilpotency ?

Beginning to conclude

Deterministic ACA are interesting! What about Nilpotency ? Topological entropy ?

Beginning to conclude

Deterministic ACA are interesting! What about Nilpotency ? Topological entropy ? Classification ?

Beginning to conclude

Deterministic ACA are interesting! What about Nilpotency ? Higher dimensions ? Topological entropy ? Classification ?

Continuing to conclude

Updating schemes

Continuing to conclude

More fairness

Updating schemes

Continuing to conclude

More fairness

Updating schemes

Structural properties

Continuing to conclude

More fairness

Updating schemes

Applications (?) Structural properties

True conclusions

About computability

Decidability Tradeoffs

The End.

Many thanks for your attention!