Conjugacy Results for Cellular Automata Silvio Capobianco, - - PowerPoint PPT Presentation

Automata 2005, Gda´ nsk 3-5 September 2005

Revised version, 7 September 2005

Conjugacy Results for Cellular Automata

Silvio Capobianco, Universit` a degli Studi di Roma “La Sapienza”, capobian@mat.uniroma1.it

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Introduction A cellular automaton is basically a “short” encoding of a transformation between colorings of the nodes of a regular graph. Transformations having such encodings, in turn, define dynamical systems. This leads to some interesting problems:

• 1. given a dynamical system, determine whether it can be described by a CA;
• 2. in this case, find which classes of CA are fit to describe it.

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References

• S. Capobianco, Structure and invertibility in cellular automata, Tesi di Dottorato
• S. Capobianco, Cellular automata over semi-direct group products: reduction

and invertibility results, submitted for publication

• F

. Fiorenzi, Cellular automata and strongly irreducible shifts of finite type, Theor.

• Comp. Sci. 299 (2003) 477–493
• D. Lind, B. Marcus, An introduction to symbolic dynamics and coding,

Cambridge University Press (1995)

• A. Mach`

ı, F . Mignosi, Garden of Eden configurations for cellular automata on Cayley graphs on groups, SIAM J. Disc. Math. 6 (1993) 44–56

• T. Toffoli, N. Margolus, Invertible cellular automata: A review, Physica D 45

(1990) 229–253

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Evolution of the concept

• von Neumann, 1950s:

uniform local rule on the plane

• Richardson, 1972:

d-dimensional cellular automata

• Hardy, de Pazzis, Pomeau 1976:

lattice gas cellular automata

• Mach`

ı and Mignosi, 1993: cellular automata over groups of polynomial growth

• Lind and Marcus, 1995:

“sliding block codes” between shift subspaces of AZ

• Fiorenzi, 2000:

cellular automata over arbitrary finitely generated groups

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The original form

• Infinite square grid, identifiable with Z2.
• A finite number of states for each point of the grid.
• A finite set {n1, . . . , nk} ⊆ Z2.
• A law of the form (F(c))x = f(cx+n1, . . . , cx+nk).

Hypercubic grid of arbitrary dimension d

⇒ “classical” CA of the form: d, Q, N, f

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Cellular automata as presentations Cellular automata are not dynamical systems per se. Rather, they are “short” ways to encode dynamics which can be very complex. Thus, it is more correct to say that CA are presentations of dynamical systems.

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A special property

A = d, Q, N, f; φ : QZd × Zd → QZd

:

(φ(c, x))y = cx+y

Two properties of φ:

• 1. A’s global evolution function commutes with φ;
• 2. the map:

π : QZd → Q ; π(c) = c0

is continuous and satisfies:

∀c1 = c2 ∈ QZd ∃g ∈ G : π(φ(c1, g)) = π(φ(c2, g)) (X, F) “isomorphic” to the one described by A ⇒ ∃ an action of Zd on X with similar properties.

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A special property (continued) Conjecture 1: (Toffoli, 1980s) Every dynamical system admitting of an action of Zd on its phase space having properties similar to 1 and 2, also admits of a presentation as a d-dimensional cellular automaton. Status: plausible, but missing an important point.

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Terminology Alphabet: 1 < |A| < ∞.

G group, S ⊆ G: S =

• g ∈ G : ∃n ∈ N, s1, . . . , sn ∈ S ∪ S−1 : g = s1 . . . sn
• G = S for |S| < ∞: G finitely generated (f.g.).

Cayley graph of G w.r.t. S:

• V = G,
• E =
• (g, gs), g ∈ G, s ∈ (S ∪ S−1) \ {1G}
• .

Distance of g and h w.r.t. S: minimum length of a path from g to h.

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Examples of Cayley graphs The square grid is the Cayley graph of G = Z2 with respect to S = {(1, 0), (0, 1)}.

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Examples of Cayley graphs (continued) The hexagonal grid is (isomorphic to) the Cayley graph of G = Z2 with respect to S = {(1, 0), (1, 1), (0, 1)}.

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Terminology (continued)

A alphabet, G f.g. group ⇒ c ∈ AG configuration. c ∈ AG : g ∈ G → cg ∈ A AG is given the product topology:

• G finite ⇒ AG discrete;
• G countable ⇒ AG Cantor.

c1, c2 “near” ⇔ “equal on a large disk centered in the origin”.

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Terminology (continued) Action of G on X:

• 1. φ : X × G → X;
• 2. φ(φ(x, g1), g2) = φ(x, g1g2) for all x, g1, g2;
• 3. φ(x, 1G) = x for all x.

φ continuous: x → φ(x, g) continuous ∀g. (σG(c, g))h = cgh natural action of G on AG.

Translations: the maps c → σG(c, g).

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Shift subspaces Origin: Symbolic dynamics.

σ : AZ → AZ the shift map: (σ(c))x = cx+1 X ⊆ AZ: (X, σ) subsystem of (AZ, σ).

Generalization:

X ⊆ AG such that (X, σG(·, g)) subsystem of (AG, σG(·, g)) ∀g ∈ G. AG is called the full shift. σG satisfies properties 1 and 2 when restricted to a shift subspace ⇒ it cannot tell the full shift from an arbitrary shift subspace.

This is precisely the point missed by Conjecture 1.

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Characterization of shift subspaces Pattern: p : E → A, E ⊆ G finite.

p ∈ AE occurs in c ∈ AG: (σG(c, g))|E = p for some g ∈ G. X ⊆ AG. The following are equivalent:

• 1. X is a shift subspace;
• 2. X is closed and translation invariant;
• 3. ∃ a set F of patterns such that:

X = XF = {c ∈ AG : ∀p ∈ F, g ∈ G : p ∈ AE, (σG(c, g))|E = p} X = XF with F finite: X is of finite type.

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UL-definable functions

F : AG → AG UL-definable (i.e., uniformly locally definable) ⇔ ∃ N ⊆ G, |N| < ∞, f : AN → A s.t. (F(c))g = f

• (σG(c, g))|N
• = f
• cgn1, . . . , cgn|N |
• ∀c ∈ AG, g ∈ G

G f.g. ⇒ N can be a disk.

Hedlund’s Theorem:

X ⊆ AG shift subspace. F : X → AG:

• 1. continuous
• 2. σG(F(c), g) = F(σG(c, g)) ∀g ∈ G, c ∈ X

⇒ F is the restriction to X of a UL-definable function.

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The reasons for a broader definition CA are uniform local presentations of global dynamics. Locality is linked to invariance by translation. Uniformity is linked to compactness. Thus: Shift subspaces still possess the requirements for the definition of a dynamics in uniform, local terms.

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Generalized cellular automata

A = X, N, f with:

• X ⊆ AG shift subspace for some G f.g., 1 < |A| < ∞;
• N ⊆ G, |N| < ∞;
• f : AN → A: F(X) ⊆ X, where:

(F(c))g = f

• (σG(c, g)|N )
• G: tessellation group of A.

X: support of A. (X, F): associate dynamical system of A. A is full if X = AG.

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Presenting dynamical systems by generalized CA Topological conjugacy from (X, F) to (X′, F ′): homeomorphism t : X → X′ such that t ◦ F = F ′ ◦ t:

X X X’ X’ F F’ t t

X, N, f presentation of (X′, F ′) ⇔ (X, F) topologically conjugate to (X′, F ′).

Shift subspaces with a countable number of elements exist

⇒ the definition is actually broader.

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A not-so-trivial example of presentation by CA Let A = {0, 1}, G = Z, N = {+1}, f(1 → a) = a. Let σ : {0, 1}Z → {0, 1}Z be the shift map. Then,

• {0, 1}Z, N, f
• is a presentation of
• {0, 1}Z, σ−1

. The topological conjugacy is given by the reversing map:

(r(c))x = c−x

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Disambiguation Lind and Marcus 1995: conjugacy between shift subspaces X, Y ⊆ AZ:

• 1. UL-definable F : AZ → AZ
• 2. F injective over X
• 3. F(X) = Y .

Equivalently: F topological conjugacy between (X, σ) and (Y, σ) (shift conjugacy). The reversing map is not UL-definable, thus it is not a shift conjugacy.

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Discernibility

A alphabet, φ : X × G → X action. X discernible over A by φ ⇔ ∃ π : X → A:

• π is continuous
• ∀x1 = x2 ∈ X ∃g ∈ G : π(φ(x1, g)) = π(φ(x2, g)).

Conjecture 1 can then be restated as such: Commutation with and discernibility by the same group action characterize dynamical systems presentable as CA.

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Theorem 1

(X, F) compact dynamical system. The following are equivalent:

• 1. (X, F) has a presentation as a generalized cellular automaton;
• 2. ∃ A, G f.g., φ : X × G → X continuous s.t.:

(a) F commutes with φ; (b) X is discernible over A by φ. In this case, (X, F) has a presentation as a generalized CA with alphabet A and tessellation group G. In other words: Conjecture 1 is true for generalized CA.

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Discussion Condition 2 only implies that X is homeomorphic to a shift subspace of AG. Reason:

• φ and σG have the same role
• ⇒ neither can tell an arbitrary shift subspace from AG

⇒ Dynamical systems presentable as full CA must possess properties in addition to

those in condition 2.

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Discussion (continued) No special properties of G used in proof of Theorem 1. (Aside from being finitely generated.)

⇒ Further restrictions on the tessellation group can be done, yielding

corresponding variants of Theorem 1. Important if one wants all translations to be global evolution functions of CA, possible iff G is Abelian. (The global map induced by f(g → a) = a obeys (F(c))h = chg, which in general is not cgh unless g is central.)

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Discussion (continued) Generalized CA are “more powerful” than canonical CA at presenting dynamical systems. Other side of the coin: they are finite presentations ⇔ their support has one. Two cases when this is possible:

• 1. X is of finite type

(finite presentation: F s.t. |F| < ∞, X = XF)

• 2. X = F(Y ) for Y of finite type, F UL-definable

(finite presentation: F for Y plus f for F ) The latter example generalizes the sofic shifts of AZ.

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Lattice gases Special kinds of cellular automata, obeying a two-step discipline:

• 1. collision;
• 2. propagation.

Well suited to describe invertible microscopical dynamics. A presentation as a lattice gas is also a presentation as a cellular automaton.

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A presentation problem Does the existence of a presentation as a cellular automaton imply the existence of a presentation as a lattice gas? Conjecture 2: (Toffoli and Margolus, 1990) Cellular automata and lattice gases are presentations of the same class of dynamical systems. A special form of Conjecture 2 is known to be true for invertible dynamics over Zd. (Durand-Lose, 2001)

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Generalized lattice gases Introduction of f.g. groups and shift subspaces, as with cellular automata. Association of collision maps to pointwise transformations f : AN → AN :

Cf : (AN )G → (AN )G ; (Cf(c))g = f(cg)

and propagation maps to symmetric neighborhood indexes:

PN : (AN )G → (AN )G ; ((PN (c))g)i = (cgi−1)i

(Going in direction i, means coming from direction i−1)

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Generalized lattice gases (continued) A generalized lattice gas will be given by:

• 1. a finite symmetric N ⊆ G;
• 2. a shift subspace X ⊆ (AN )G;
• 3. a map f : AN → AN

such that (PN ◦ Cf)(X) ⊆ X. An alternative definition asks (Cf ◦ PN )(X) ⊆ X: e.g., the CAM8 cellular automata machine. This is equivalent (the PN ’s are homeomorphisms).

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Theorem 2 Any generalized CA X, N, f is topologically conjugate to a generalized LG

N ′, X′, f ′ where:

• 1. N ′ = N ∪ N −1;
• 2. X′ =
• c ∈ (AN ′)G : (cg)i = (cg)j∀i, j ∈ N, g ∈ G; g → (cg)i ∈ X
• ;
• 3. (f ′(α))i = f(α|N )

This transformation turns shifts of finite type into shifts of finite type, and sofic shifts into sofic shifts. In other words: Conjecture 2 is true for generalized LG and CA.

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Discussion A well known and widely used procedure has good properties: not too surprising. The same procedure fits well into a different context: more noteworthy. The same procedure preserves properties it was not designed to: very interesting. Other side of the coin:

• nly graph properties of lattice taken into account.

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Group extensions

H and K groups.

Extension of H by K: a group G such that ∃K′ G : K′ ∼

= K, G/K′ ∼ = H.

Equivalently: a group G such that the sequence:

1 → K → G → H → 1

is exact. Observe that G is f.g. if H and K are.

Z2 is an extension of Z by Z. S3 is an extension of Z2 by Z3.

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Semi-direct products

τ : H → Aut(K) group homomorphism.

Semi-direct product of H and K by τ: the group with support H × K and product:

(h1, k1)(h2, k2) = (h1h2, τh2(k1)k2)

Semi-direct products correspond to split exact sequences.

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Groups with finite parts

1 → K → G → H → 1 exact. A CA with tessellation group G. H or K finite ⇒ seems natural to “embed” it into the alphabet.

Possible use: switching from a presentation as a CA on a group to one as a CA on a subgroup or a quotient. Question: can this preserve the dynamics presented by A?

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Example Before:

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Example (continued) After?

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Theorem 3

G = H ⋉τ K semi-direct product.

• 1. K finite ⇒ every CA over G is conjugate to a CA over H.
• 2. H finite ⇒ every CA over G is conjugate to a CA over K.
• 3. The transformations of points 1 and 2 turn shifts of finite type into shifts of finite

type, and sofic shifts into sofic shifts.

• 4. The transformations of points 1 and 2 are computable if the word problem for H

and K is decidable. In other words: The answer to the question is positive if the short exact sequence is split.

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Discussion The finite part of the semi-direct product is easily thought of as a “curled up” dimension.

⇒ In cellular automata, “curled up” dimensions can be perceived as state rather

than as directions. In particular, full cellular automata on Abelian groups can always be seen as “classical” cellular automata. Reason: the structure theorem for f.g. Abelian groups.

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Discussion (continued) Theorem 3 can be used to extend decidability results. Example: the invertibility problem for full cellular automata is decidable on any group

• f the form Z ⋉τ K or K ⋉τ Z with K finite.

Consider the infinite dihedral group:

D∞ =

• a, b | a2 = 1, ab = b−1a
• Observe that D∞ ∼

= Z2 ⋉τ Z with τ1(x) = −x. ⇒ It is decidable if a cellular automaton of the form

• AD∞, N, f
• is invertible.

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Future research

• 1. Add conditions on φ so that Conjecture 1 becomes true for full CA.
• 2. Prove or disprove Conjecture 2 for “standard” lattice gases.
• 3. Extend Theorem 3 to other classes of group extensions.

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Thank you for your attention!

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