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 I = { i 1 , i 2 ,..., i k } Concurrent symmetric Programming θ ⊆ I × I irreflexive θ c = I × I \ θ G = � I , θ c � α ⊆ I , θ c ( α ) = { i ∈ I , ∃ j ∈ α | ( i , j ) θ c }

Motivations P i ⊆ I i ∈ { 1 ,..., n } Concurrent Programming P 1 G P P 6 P 8 P 7 P 2 P 5 P 3 P 4

Motivations P i ⊆ I i ∈ { 1 ,..., n } Concurrent Programming P 1 G P Bio- P 6 informatics P 8 P 7 P 2 P 5 P 3 P 4

“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” Lattice Z I = { i 1 , i 2 ,..., i k } States δ : I 2 r + 1 → I Local rule ... -1 0 +1 ...

(Re-)Discoverings

(Re-)Discoverings ... -1 0 +1 ... i 2 i 3 i 1 i 1 i 2 i 2 i 3 i 1 i 3

(Re-)Discoverings ... -1 0 +1 ... i 2 i 3 i 1 i 1 i 2 i 2 i 3 i 1 i 3 0 0 0 0 1 1 1 1 1 ... -1 0 +1 ...

(Re-)Discoverings ... -1 0 +1 ... i 2 i 3 i 1 i 1 i 2 i 2 i 3 i 1 i 3 � �� � 0 · δ ( i 1 , i 2 , i 2 ) 0 0 0 0 1 1 1 1 1 ... -1 0 +1 ...

More (re-)discoverings F ′ : { 0 , 1 } Z × I Z → I Z F ′ ( x , y ) = ( id ( x ) , F ( y )) F ′ ( x , y ) = ( G ( x ) , F ( y ))

So far... CA G shift invariant continuous -CA G continuous ν “P”CA G shift invariant continuous ??? G

So far... CA G shift invariant continuous -CA G continuous ν “P”CA G shift invariant continuous ??? G

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 ... 0 , 0 , 0 , 0 , 0 , 1 , 0 | 0 , 0 , 0 , 0 , 0 , 0 ... t = 1 ... 0 , 0 , 0 , 0 , 0 , 0 , 0 | 0 , 0 , 0 , 0 , 1 , 0 ... t = 0 ... 0 , 0 , 0 , 0 , 0 , 0 , 0 | 1 , 0 , 0 , 0 , 0 , 0 ... t = 2 . . .

Deterministic ACA G = Z N ... 0 , 0 , 0 , 0 , 0 , 1 , 0 | 0 , 0 , 0 , 0 , 0 , 0 ... t = 1 ... 0 , 0 , 0 , 0 , 0 , 0 , 0 | 0 , 0 , 0 , 0 , 1 , 0 ... t = 0 ... 0 , 0 , 0 , 0 , 0 , 0 , 0 | 1 , 0 , 0 , 0 , 0 , 0 ... t = 2 . . .

Global function x t i z 0 i = y i z t i = ( F ′ ) t ( x , y ) i i ) z t − 1 i δ ( z t − 1 i − r ,..., z t − 1 ,..., z t − 1 ( F ′ ) t ( x , y ) i = ( 1 − x t + x t i + r ) i i

Global function x t i z 0 i = y i z t i = ( F ′ ) t ( x , y ) i i ) z t − 1 i δ ( z t − 1 i − r ,..., z t − 1 ,..., z t − 1 ( F ′ ) t ( x , y ) i = ( 1 − x t + x t i + r ) i i

Set properties Definition. F ′ is injective iff Z ∀ x ∈ Z N ∀ y , z ∈ A , F ′ ( x t , y ) � = F ′ ( x t , z ) z � = y ⇒ ∀ t ∈ N

Set properties Definition. F ′ is injective iff Z ∀ x ∈ Z N ∀ y , z ∈ A , F ′ ( x t , y ) � = F ′ ( x t , z ) z � = y ⇒ ∀ t ∈ N

Set properties (2) Definition. F ′ is surjective iff Z ∀ x ∈ Z N ∀ y ∈ A , Z F ′ ( x t , y ) � = F ′ ( x t , z ) ∀ t ∈ ∃ z ∈ A N

Set properties (2) Definition. F ′ is surjective iff Z ∀ x ∈ Z N ∀ y ∈ A , Z F ′ ( x t , y ) � = F ′ ( x t , z ) ∀ t ∈ ∃ z ∈ A N

Set properties (3) Proposition. The following properties are equivalent 1 ) F ′ is injective 2 ) F ′ is surjective is center permutative 3 ) δ

Set properties (3) Proposition. The following properties are equivalent 1 ) F ′ is injective 2 ) F ′ is surjective is center permutative 3 ) δ

Dynamics Proposition. x ∈ Z N If is ultimately periodic, then y , F ′ ( x 1 , y ) , ( F ′ ) 2 ( x 2 , y ) ,..., ( F ′ ) n ( x n , y ) ,... is ultimately periodic.

Dynamics Proposition. x ∈ Z N If is ultimately periodic, then y , F ′ ( x 1 , y ) , ( F ′ ) 2 ( x 2 , y ) ,..., ( F ′ ) n ( x n , y ) ,... is ultimately periodic.

Dynamics (2) Definition. F ′ is sensitive to initial conditions, iff ∃ x ∈ Z N ∃ ε > 0 ∀ y ∈ A Z ∀ δ > 0 ∃ z ∈ B δ ( y ) ∃ t ∈ N such that d (( F ′ ) t ( x , y ) , ( F ′ ) t ( x , z )) > ε

Dynamics (2) Definition. F ′ is sensitive to initial conditions, iff ∃ x ∈ Z N ∃ ε > 0 ∀ y ∈ A Z ∀ δ > 0 ∃ z ∈ B δ ( y ) ∃ t ∈ N such that d (( F ′ ) t ( x , y ) , ( F ′ ) t ( x , z )) > ε

Dynamics (3) Please look at the whiteboard on the right

Dynamics (3) Please look at the whiteboard on the right

Dynamics (4) Definition. F ′ is expansive, iff ∃ ε > 0 ∀ y ∈ A Z ∀ z ∈ A Z ∃ x ∈ Z N ∃ t ∈ N such that d (( F ′ ) t ( x , y ) , ( F ′ ) t ( x , z )) > ε

Dynamics (4) Definition. F ′ is expansive, iff ∃ ε > 0 ∀ y ∈ A Z ∀ z ∈ A Z ∃ x ∈ Z N ∃ t ∈ N such that d (( F ′ ) t ( x , y ) , ( F ′ ) t ( x , z )) > ε

Dynamics (5) Again Please look at the whiteboard on the right

Dynamics (5) Again Please look at the whiteboard on 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. F ′ is transitive if and only if ∃ x ∈ Z N such that 0 ∃ t ∈ N ∀ U , V � = / ( F ′ ) t ( x , U ) ∩ V � = / 0

Dynamics (7) Definition. F ′ is transitive if and only if ∃ x ∈ Z N such that 0 ∃ t ∈ N ∀ U , V � = / ( F ′ ) t ( x , U ) ∩ V � = / 0

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. F ′ has the DPO property if and only if ∃ x ∈ Z N F ′ ( x , · ) has the DPO property. such that F ′ ( x , · ) has the DPO property if and only if its set of periodic points is dense.

Dynamics (10) Definition. F ′ has the DPO property if and only if ∃ x ∈ Z N F ′ ( x , · ) has the DPO property. such that 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. x ∈ Z N F ′ � = id If and has DPO for some x is bounded. then Corollary. x ∈ Z N . Then F ′ ( x , · ) cannot be Devaney Fix chaotic.

Dynamics (12) Lemma. x ∈ Z N F ′ � = id If and has DPO for some x is bounded. then Corollary. x ∈ Z N . Then F ′ ( x , · ) cannot be Devaney Fix 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 ? Topological entropy ? Classification ? Higher dimensions ?

Continuing to conclude Updating schemes

Continuing to conclude Updating schemes More fairness

Continuing to conclude Updating schemes More fairness Structural properties

Continuing to conclude Updating schemes More fairness Structural properties Applications (?)

True conclusions About computability Decidability Tradeoffs

The End. Many thanks for your attention!