Download Lecture Notes for Cellular Automata as Dynamical Systems | MAD 6616 and more Assignments Mathematics in PDF only on Docsity!

April 18, 2005 MAD 6616: ALGEBRAIC AUTOMATA THEORY NOTES # 10 CELLULAR AUTOMATA AS DYNAMICAL SYSTEMS

INSTRUCTOR: NATAˇSA JONOSKA

A dynamical system is a pair (X, f ) where X is a space and f : X → X is a function. A cellular automaton can be viewed as a dynamical system where the space is the set of all bi-infinite sequences AZ^ and the function is the global function Gl. In the following we will concentrate on some introductory aspects of topological dynamics. For the simplicity, we will concentrate our discussion on 1-dim CA. Let Gl be a global CA function defined by (A, 1 , N, l). We can define a metric on the set (space) of configurations Γ = AZ^ defined in the following way:

d(α, α′^ ) =

2 i^ ,^ i^ = min{|j|^ :^ αj^6 =^ α

′ j },^ if^ α^6 =^ α

′ 0 , if α = α′^.

The elements of Γ will be called points.

Proposition 0.1. The function d is a metric, and for this metric AZ^ is compact.

Proof. Exercise. Note that d is symmetric by definition and d(α, α′) = 0 iff α = α′^ also by definition. The triangle inequality remains to be shown. Inorder to show compactness, one has to show that every sequence {αn}n≥ 0 has a convergent subsequence. 

The metric above generalizes to higher dimensions in the following way. Let Nj denote a

box (Moore) neighborhood with radius j and α, α′^ ∈ AZd.

d(α, α′) =

2 i^ ,^ i^ = min{|j|^ :^ α|Nj^6 =^ α

′|N

j },^ if^ α^6 =^ α

′ 0 , if α = α′^

1. Continuity and Reversibility of Cellular Automata

In this first section we will show the well known result of Curtis, Lyndon and Hedlund that a function from AZ^ to AZ^ is continuous and shift commuting iff it is a global function of a cellular automaton. This is an important observation for studying CA as dynamical systems. The propositions below refer to the one-dimensional case, but they can all be extended to higher dimensions by substituting the shift map with the directional shifts σi (see Example 1.3, notes #9). We say that a function f : AZ^ → AZ^ commutes with the shift if f σ = σf where σ is the shift CA σ(α)i = αi+1.

Proposition 1.1. Every 1 dim CA is a continuous function from AZ^ to AZ^ and commutes with the shift.

Proof. Let Gl be a 1 dim CA given with the fourtuple (A, 1 , N, l). Let r = max{|i|, i ∈ N }. Let N ′^ be the 1 dim neighborhood with radius r. Then N ⊆ N ′^ and we can define a local function l′^ : AN^

′ → A such that l′(w) = l(w|N ) where w|N is a restriction of w to N. So Gl = Gl′ and we can assume that N is 1-dim neighborhood with radius r. Let ǫ > 0, and let n be such that 1 2 n^ < ǫ. Choose^ δ^ such that 0^ < δ <^

1 2 n+r^. Let^ α, α

′ (^) ∈ AZ (^) be such that d(α, α′ (^) ) < δ. By the 1

definition of d we have that αi = α′ i for all i = −(n+r),... , 0 ,... , n+r. Then Gl(α)i = Gl(α′)i for all i = −n,... , 0 ,... , n, i.e. d(Gl(α), Gl (α′)) < ǫ. A straight forward check that Glσ = σGl is an exercise. 

Let w = b 1 · · · bn be a word in A∗. The set C(w)i = C(b 1 · · · bn)i = {α ∈ AZ^ | αi · · · αi+n− 1 = b 1 · · · bn} is called a cylinder set around w at index i.

Proposition 1.2. The cylinder sets C(w)i are clopen for all B ∈ An, n ≥ 0 , i ∈ Z. Moreover, the cylinder sets form basis for the metric topology.

Proof. Exercise. To show that a set is open one needs to prove that for every α ∈ C(w)i there is δ > 0 such that id d(α, α′) < δ then α′^ ∈ C(w)i for all α′^ ∈ Γ = AZ. A set is closed if its complement is open (C(w)ci = ∪w 6 =w′,|w|=|w′|C(w′)i). A collection of open sets is a basis for the topology if every open set contains a set from the collection. Note that every open ball contains a cylinder set. 

It follows that cylinder sets are compact, since they are closed subsets of a compact space.

Proposition 1.3. If f : AZ^ → AZ^ is a continuous function that commutes with a shift, then there is a radius r and a local function l defined on 1 dim neighborhood N with radius r such that f = Gl.

Proof. Consider the cylinder sets C(a) 0 , a ∈ A in AZ. They are finite in number, compact and disjoint. Since f is continuous, Ea = f −^1 (C(a) 0 ), a ∈ A are compact and disjoint. In metric spaces, two compact disjoint sets C 1 and C 2 have positive distance, i.e. the d(C, C′) = glb{d(α, α′^ )| α ∈ C 1 , α′^ ∈ C 2 } > 0. Let δ > 0 be such that for every two points α and α′^ if d(α, α′) < δ then there is a ∈ A such that α, α′^ ∈ Ea. One can choose such δ to be δ < d(Ea, Eb) for a, b ∈ A. Let r be such that (^21) r < δ. Then for every pair of points α, α′^ ∈ AZ^ such that αi = α′ i for all i = −r,... , 0 ,... r, there is a ∈ A such that α, α′^ ∈ Ea, i.e. f (α) 0 = f (α′) 0 = a. We define a local function l : A^2 r+1^ → A in the following way. Note that for every w ∈ A^2 r+1^ there is a unique set Ea such that every point α ∈ AZ^ with central (2r + 1)-block equal to w lies in the set Ea. Set l(w) = a. Then Gl = f (since f commutes with the shift f (α)i = σ−if (α) 0 .) 

Let α be a configuration, i.e an element in Γ = AZ. The orbit of α under a global CA map Gl is the sequence α, Gl(α), G^2 l (α),.... For each state a ∈ A the configuration αa is the constant configuration where all cells are in state a. We say that α is ultimately periodic with respect to Gl if the set {α, Gl (α), G^2 l (α),... } is finite. It is periodic with period c with respect to Gl if there is a c > 0 such that Gcl (α) = α. The number p is the least period of α if Gpl (α) = α and Gil (α) 6 = α for all 0 < i < p. The point α is said to be periodic if it is periodic with respect to σ. The limit set Λ(Gl) is the set of all configurations that can occur after arbitrarily many computation steps. Let Si^ denote the set of all configurations obtained in the i-th time step image, Si^ = Gil (AZ). More precisely,

Λ(Gl) = ∩∞ i=0Gil (AZ) = ∩∞ i=0Si By Proposition 2.5 (notes #9), Si+1^ ⊆ Si. A subset Y ⊆ AZ^ is Gl-invariant if Gl(Y ) = Y. It is clear that the limit set Λ(Gl) is Gl-invariant (exercise) and closed and compact subset of AZ^ (exercise). But Λ(Gl) is also a maximal Gl-invariant subset of AZ. Let Y be a Gl-invariant subset of AZ. Then

Y = Gl(Y ) = Gil (Y ) = ∩∞ i=0Gl(Y ) ⊆ Λ(Gl)

Culik-Yu classification Wolfram’s characterization has no precise definitions for may of the notions used and because of that it is very vague. A more precise classification was given by Culik II and Yu. This classification is with respect to finite configurations only. (1) A CA Gl is in class one if there is a state s ∈ A such that l(s,... , s) = s and every s-finite configuration evolves to αs. (s-finite configuration is a configuration that has all but finite number of cells in state s, in other words, s is a quiescent state and s-finite is finite.) (2) A CA Gl is in class two if there is a state s such that every s-finite configuration is ultimately periodic with respect to Gl (3) A CA Gl is in class three if there is a state s ∈ A such that for any pair of s-finite configurations α and β, it is decidable whether there is k such that Gkl (α) = β. (4) Every CA is in class four.

Note that this characterization is such that Clas (1) ⊆ Class (2) ⊆ Class (3) ⊆ Class (4). Moreover, it is undecidable whether a given CA is in class (1) or (2) or (3).

Proposition 1.6. Let Gl be a CA with a quiescent state. It is undecidable whether with Gl all finite configurations evolve to αq.

Proof. (sketch) Consider a Turing Machine M with transition function δ and set of states Q. We assume that M is altered such that whenever it enters a final state it erases all symbols on the ape and leaves the tape blank. Such an alteration is relatively easy to do. There is a cellular automaton CA such that on any finite input which represents the initial configuration of the Turing Machine M, the evolution of the CA simulates step by step the transitions of the Turing Machines. The initial configuration of the CA is the following:

... BBBBq 0 Ba 1 a 2... akBBBBBBB...

The blank symbol B is the quiescent symbol for the CA. For each transition we define the local function

δ(q, a) = (b, q′) ⇒ l(a′, q, a) = q a′, l(q′ a, a, a′′) = b and l(a′, q a′, b) = q′, for anya, a′′′, b δ(q, a) = (L, q′) ⇒ l(a′, q, a) = qa′ , l(b, a′, qa′ ) = q′^ and l(q′, qa′ , a) = a′, for anya′, b δ(q, a) = (R, q′) ⇒ l(a, q, a′) = qa′ , l(qa′ , a′, b) = q′^ and l(qa′ , q′, a) = a′, for any a′, b

If there is an algorithm do determine whether for a given CA all finite configurations evolve to a constant state, we could use that algorithm to determine whether a Turing Machine stops on every input. Since such algorithm doesn’t exist, it is undecidable whether finite configurations evolve to a constant one. 

Proposition 1.7. For every CA Gl : AZ^ → AZ^ consider the complement of Λ(Gl) i.e. Γ
Λ(Gl). Then the language F (Γ − Λ(Gl)) is recursively enumerable.

We l concentrate now on local functions that yield bijective global function. In that case the limit set of Gl is AZ. We first show that every injective CA is necessarily surjective. This holds for arbitrary dimensions of CA, but we will give the proof only for 1 dim CA.

Lemma 1.8. (exercise) The global 1 dim CA function Gl is surjective iff the factor language F (Gl(AZ)) = A∗.

Proof. Follows from the fact that Gl(AZ) is compact and its complement is open. 

Proposition 1.9. If Gl : AZ^ → AZ^ is an injective 1 dim CA, then it is surjective.

Proof. Assume that Gl is not surjective. As we mentioned in the proof of Proposition 1.1, we can assume that the local function is defined on a local neighborhood with radius r. Since Gl is continuous and AZ^ is compact, by Lemma 1.8 there is a finite word w such that w 6 ∈ F (Gl(AZ)). Let k = |w|. We fix a ∈ A and consider the set of configurations Sm = {α| α 1... αmk ∈ Amk^ , and αi = a, i ≤ 0 or i > mk}. Look at Gl(Sm). The cardinality of Sm is |A|mk^ and since w can not appear as a factor in any of the configurations Gl(Sm), the cardinality of Gl(Sm) is at most (|A|k^ − 1)m^ · |A|^2 r^. Let p = |A|k^ and consider

mlim→∞

|Sm| |Gl(Sm)|

= (^) mlim→∞

pm (p − 1)m|A|^2 r^

|A|^2 r^ mlim→∞(^

p p − 1

)m^ = ∞

So there is m such that |Sm| > |Gl(Sm)|, which means that Gl is not injective on Sm, and so it is not injective on AZ. 

For a CA Gl we say that it is reversible, if it is bijective and there is a CA Gl′ such that G− l 1 = Gl′. The above proposition says that every injective CA is also bijective. The following proposition will show that Gl is injective iff it is reversible.

Proposition 1.10. Every bijective CA is reversible.

Proof. Again we will prove this proposition only for the case when Gl is 1 dimensional. Assume that Gl is bijective. Then

Gl(σ(G− l 1 (α))) = σ(Gl(G− l 1 (α))) = σ(α) = Gl(G− l 1 (σ(α)))

Since Gl is injective, G− l 1 commutes with σ. Since Gl : AZ^ → AZ^ is a bijective continuous map from compact to a Hausdorff space, Gl is homeomorphism, i.e. G− l 1 is continuous. By Proposition 1.3, G− l 1 is a global function of a cellular automaton. 

It is not difficult to see that the periodic points in AZ^ are dense (exercise). So it is not surprising that a CA map Gl is injective iff it is injective on the set of periodic points:

Proposition 1.11. For 1 dim CA, the global function Gl is injective iff it is injective on the set of periodic points.

Proof. The ”only if” part is trivial. Assume that Gl is not injective, i.e. there are two configurations α and β such that Gl(α) = Gl(β) = γ. It is sufficient to show that there are two periodic configurations which are mapped to the same configuration. Assume that r is the radius of the neighborhood of l. Let w = sx 1 sx 2 tx 3 t ∈ F (α) and w′^ = uy 1 uy 2 vy 3 v ∈ F (β) be such that x 2 6 = y 2 , |xk| = |yk| (k = 1, 2 , 3) and |s| = |t| = |u| = |v| = 2r. (It follows from the deBruijn graph that such w and w′^ exist, and we leave this fact as an exercise.) Case 1: s 6 = u or t 6 = v. Assume that s 6 = u. We form two periodic configurations α′^ and β′ with α′^ =... sx 1 sx 1 sx 1... and β′^ =... uy 1 uy 1 uy 1.... Clearly, α 6 = β and Gl(α) = Gl(β). Case 2: s = u and t = v. Then we construct two periodic configurations α′^ and β′^ such that α′^ =... sx 2 tsx 2 tsx 2 t... and β′^ =... sy 2 tsy 2 tsy 2 t.... Clearly, α 6 = β (since x 2 6 = y 2 ) and Gl(α) = Gl(β). 

In the early 60’s Moore introduced the concept of Garden of Eden configuration, i.e. a configuration which is not in the image of Gl. This means, once left, the Garden of Eden configuration can never be obtained again. The study of such configurations has been closely related to the study of the injectivity and surjectivity of Gl on finite configurations.

Proposition 2.2. The global CA function Gl is injective iff for every word w ∈ A^2 n

(^2) + there is an edge ew in Gl such that for every path p = e−n 2... e 0... en 2 in Gl such that λ 2 (p) = w we have that e 0 = ew.

Proof. Assume that Gl is not injective, i.e. there are two distinct configurations α and α′^ such that Gl(α) = Gl(α′). Since Gl commutes with the shift, we can assume that α 0 6 = α′ 0. By Proposition 2.2 (notes #9) there are two distinct bi-infinite paths π and π′^ with λ 1 (π) = α and λ 1 (π′) = α′^ such that π 0 6 = π 0 ′. Then for the word w = λ 2 (π−n 2 )... λ 2 (π 0 )... λ 2 (πn 2 ) there are two paths in Gl with output label w and with distinct middle edges. Converse: We can use the same trick as in the proof of Proposition 2.3 (notes # 5). Let w = a−n 2... a 0... an 2 and let p, p′^ be paths in Gl with λ 2 (p) = λ 2 (p′) = w and p 0 6 = p′ 0. We take that p = e−n 2... e 0... en 2 and p′^ = e′−n 2... e′ 0... e′ n 2 with e 0 6 = e′ o. Consider the pairs (t(e 0 ), t(e′ 0 )),... , t(en 2 ), t(e′ n 2 ). Since there are at most n^2 such vertices, there must be i and j with 0 ≤ i < j ≤ n^2 such that (t(ei), t(e′ i)) = (t(ej ), t(e′ j )). Similarly, if we consider the

pairs of vertices (s(e 0 ), s(e′ 0 )),... (s(e−n 2 ), s(e′−n 2 )), there are i′^ and j′^ with −n^2 ≤ j′^ < i′^ ≤ 0 such that s(ei′ ), s(e′ i′ )) = (s(ej′ ), s(e′ j′ )). Let q 1 = ei′... ej′ , let q 2 = ei... ej , let q′ 1 = e′ i′... e′ j′

and q 2 ′ = e′ i... e′ j. Let π =... q 1 q 1 q 1 pq¯ 2 q 2 q 2... and π′^ =... q 1 ′q′ 1 q 1 ′ p¯′q 2 ′q′ 2 q′ 2... such that w′^ = λ 2 (¯p) = λ 2 (¯p′) where ¯p = ei′... e 0... ei and ¯p′^ = e′ i′... e′ 0... e′ i. Then set β = λ 1 (π) and β′^ = λ 1 (π′). By Proposition 2.2 (notes # 9), Gl(β) = Gl(β′) =... xxxwyyy... where x = λ 2 (q 1 ) = λ 2 (q 1 ′) and y = λ 2 (q 2 ) = λ 2 (q 2 ′). Since e 0 6 = e′ 0 by Proposition 2.2 (b) (notes#9), λ 1 (π) 6 = λ 1 (π′) and so Gl is not injective. 

Proposition 2.3. It is decidable whether a global CA function Gl is reversible. Moreover,

if Gl is reversible, then the inverse G− l 1 is defined on local neighborhood of radius R ≤ n^2 = (|A|^2 r^ )^2 = |A|^4 r^ where r is the radius of the local neighborhood of Gl.

Proof. By Propositions 1.9 and 1.10 Gl is reversible iff it is injective. By Proposition 2.2, we have to check whether the de Bruijn graph G′′ l has the property that for every word w with length 2n^2 + 1 each path with label w is uniquely determined by the middle edge. Since there are only finite number of words to check and there are only finite number of paths in G′′ l with length 2n^2 + 1, the decidability follows. If Gl is reversible, then the inverse function G− l 1 = Gl′ is defined on neighborhood with

radius n^2 since we can specify l′^ : A^2 n^2 +1^ → A to be such that for each word w ∈ A^2 n^2 +1, l′(w) = λ 1 (e 0 ) where e 0 is the unique middle edge of a path p = e−n 2... e 0... en 2 in Gl with output label w. Then, by Propositions 1.10, 2.2 Gl′ = G− l 1. 

One can easily check that the above “algorithm” for checking injectivity of CA has expo- nential expansion of the number of steps with respect to the number of vertices n (which also depends exponentially on the size of the alphabet A and the size of the neighborhood N , [see notes # 9]). So one can ask if there is a more efficient algorithm for checking the injectivity of 1 dim CA. The answer is “yes” and we show it in the following.

Proposition 2.4. There is a polynomial time algorithm for deciding whether a global 1 dim CA function Gl is injective.

Proof. We can consider the de Bruijn graph Gl as a non-deterministic sequential machine (NSM) where each edge is labeled with (a, a′) ∈ A × A. We can use the same algorithm for deciding whether Gl is single valued (on bi-infinite sequences) to check whether Gl is injective. Let Gˆ = Gl × Gl be as described in notes # 9 with the following modification: the vertices of Gˆ

are the ordered pairs (v 1 , v 2 ), (v 1 , v 2 ∈ Vl). There is an edge e from (v 1 , v 2 ) to (v′ 1 , v 2 ′) iff there are edges ei from vi to v′ i, (i = 1, 2) such that λ 2 (e 1 ) = λ 2 (e 2 ). We label the new edge e with bits 0 or 1 such that λ(e) = 0 iff λ 1 (e 1 ) = λ 1 (e 2 ). We trim Gˆ such that every vertex in Gˆ has an edge coming in and an edge going out. Then Gl is injective if and only if Gˆ has no edges labeled 1. Assume Gl is not injective. There are two distinct bi-infinite sequences α 1 and α 2 in AZ such that Gl(α 1 ) = Gl(α 2 ). Let k be such that (α 1 )k 6 = (α 2 )k. By Proposition 2.2 (notes

9) (a) and (b), there are two bi-infinite paths in Gl, π 1 and π 2 such that λ 1 (πi) = αi

(i = 1, 2) and (Proposition 2.2 (notes # 9) (c)) λ 2 (π 1 ) = λ 2 (π 2 ). Let π 1 =... , e− 1 , e 0 , e 1 ,... and π 2 =... , e′− 1 , e′ 0 , e′ 1 ,.... Let vi and v′ i be the sources of ei and e′ i respectively, (i ∈ Z).

Since λ 2 (ei) = λ 2 (e′ i) for each (i ∈ Z), each pair (vi, v′ i) is a vertex in the trimmed graph Gˆ.

Since λ 1 (ek ) = (α 1 )k 6 = (α 2 )k = λ 1 (e′ k ) the label of the edge in Gˆ from (vk, v′ k) to (vk+1, v′ k+1) has label 1. Conversely, assume that trimmed Gˆ has an edge with label 1. Since every vertex in Gˆ has an edge coming in and an edge going out, there is a bi-infinite path π =... , e− 1 , e 0 , e 1 ,... in Gˆ such that λ(e 0 ) = 1. Then, by construction of Gˆ, π corresponds to two bi-infinite paths π 1 and π 2 in Gl such that λ 2 (π 1 ) = λ 2 (π 2 ) with α 1 = λ 1 (π 1 ), α 2 = λ 1 (π 2 ) and (α 1 ) 0 6 = (α 2 ) 0. By Proposition 2.2 (notes # 9), Gl(α 1 ) = Gl(α 2 ), and thus Gl is not injective. 

Deciding surjectivity of Gl is very simple.

Corollary 2.5. It is decidable whether a 1 dim CA function Gl is surjective.

Proof. It suffices to check whether L(G′′ l ) = A∗^ and by Proposition 2.3 (notes # 2) this is decidable. 

We have to note that in general G′′ l is not deterministic and checking whether L(G′′ l ) = A∗ may require construction of deterministic FSA recognizing L(G′′ L) which yields an exponential expansion of the number of states. But we have the nice theorem of Richardson (Proposition 1.12) which shows that Gl is surjective iff it is finitely injective. So we have:

Proposition 2.6. There is a polynomial time algorithm for deciding whether a global 1 dim CA function is surjective.

Proof. Again we will use the de Bruijn graph of Gl and we will use the product construction and the previous algorithm. Let Gˆ = Gl × Gl as it was described in Proposition 2.4. Consider the irreducible component C of Gˆ that contains the set of vertices {(v, v)| v ∈ Vl}. Then Gl is finitely injective iff C has an edge with label 1. (In fact that edge must have source or target a vertex (v, v′) with v 6 = v′.) Assume that Gl is not finitely injective. Then there are two configurations α, α′^ such that Gα,α′^ is not injective. This means that there are β, β′^ ∈ [α] (β 6 = β′), such that Gl(β) = Gl(β′). But β and β′^ are same on all but finitely many cells, so the corresponding bi-infinite paths in Gl have infinite number of vertices same (by Proposition 2.2 notes # 9), i.e. starting with some i > 0 all edges with index j > i or j < −i in both paths are equal. So there is a bi-infinite path in C with label 1 on at least one edge. Converse, assume that C has an edge e with label 1. Since the source of the edge s(e) and t(e) are in C, there is a path from (v, v) to s(e) and from t(e) to (v′, v′) for some v, v′^ ∈ Vl. That means that there are bi-infinite paths in Gl such that their input labels differ on only finitely many edges, but they have same output labels. So Gl is not finitely injective.