Classification of Elementary Cellular Automata Up to Topological Conjugacy

. Topological conjugacy is the natural notion of isomorphism in topological dynamics. It can be used as a very ﬁne grained classiﬁcation scheme for cellular automata. In this article, we investigate diﬀerent invariants for topological conjugacy in order to distinguish between non-conjugate systems. In particular we show how to compute the cardinality of the set of points with minimal period n for one-dimensional CA. Applying these methods to the 256 elementary one-dimensional CA, we show that up to topological conjugacy there are exactly 8 3 of them.


Introduction
One-dimensional cellular automata can be topologically characterized as the continuous σ-commuting endomorphisms of the space A Z . Topological dynamics is therefore a natural framework to study their dynamics and has shown to be rather fruitful [6].
Topological dynamics in our sense is the study of compact metrizable space X together with a continuous map F : X → X. The classical notion of isomorphism in this setting is that of a topological conjugacy. Two topological dynamical systems F : X → X and G : Y → Y are called conjugate, if there is a homeomorphism ϕ : X → Y such that ϕ • F = G • ϕ [7]. It is easily seen that this defines an equivalence relation on topological dynamical systems. A natural problem now is to classify a certain class of such systems up to conjugacy.
This problem received a lot of attention for the case of subshifts of finite type. While there has been substantial progress and some powerful invariants have been found, there still remain many questions, ranging from the question if conjugacy is decidable for SFTs, to the question of deciding conjugacy for two concretely given edge shifts [2].
The corresponding problem of classifying CA up to topological conjugacy has up to now seen very little activity, although many classification schemes for CA have been proposed (see [8] for a survey). As a starting point we will classify the elementary one-dimensional cellular automata, mainly using the cardinality of the set of points with minimal period n, the Cantor-Bendixson derivative of the periodic points and various ad-hoc arguments.
of these conjugacies. The next theorem will show, that these are in a sense the only general methods to get a conjugate CA from another. Theorem 1. Let ϕ : A Z → A Z be a homeomorphism. Then the following are equivalent.
By setting F = σ, we see that ϕ −1 • σ • ϕ is a CA. Now Ryan's theorem [9] tells us that the center of the group H A ∩ CA A consists only of powers of the shift, i.e. if an invertible CA commutes with all other invertible CA, it must be a power of the shift. Hence This first of all implies that k = 0. Now take any point y ∈ Per 1 (σ k ). Then (σ • ϕ)(y) = (ϕ • σ k )(y) = ϕ(y). Hence ϕ(y) ∈ Per 1 (σ) and therefore ϕ defines an injective mapping from Per 1 (σ k ) into Per 1 (σ). Having a look at the cardinalities we see that |A| |k| = | Per 1 (σ k )| ≤ | Per 1 (σ)| = |A|, implying k = ±1. In the case of k = 1 we are done. In the other case In the light of Theorem 1, we call a conjugacy ϕ ∈ CA ∪ CA • τ a strong conjugacy. In Section 7 we will see conjugate cellular automata, that are not strongly conjugate.

Periodic Points and the Cantor-Bendixson Derivative
Consider two conjugate cellular automata F and G := ϕ • F • ϕ −1 with ϕ ∈ H A . The first invariant of topological conjugacy normally considered is the number of periodic points, . While for shifts have only finitely many periodic points of a given period, this is in general not true any more for cellular automata.
To deal with this, we use standard cardinal arithmetic in order to extend the addition on N to C := N ∪ { 0 , 1 } by defining This turns C into a commutative monoid. The justification for this definition is given by the fact, that for A 1 , . . . , A pairwise disjoint sets with Notice however, that for two disjoint sets A, B with |A|, |B| ∈ C it is no longer possible to recover the cardinality of B from the knowledge of |A| and |A ∪ B|.
While these are already nice invariants they do not use the fact that ϕ is continuous at all but only its bijectivity. However, two spaces with cardinality 1 might look rather different from a topological point of view. We therefore look at the set of all limit points D(Per n (F )) of Per n (F ) defined as follows.
Definition 2. Let B ⊆ A Z . The set of limit points of B, also called its Cantor-Bendixson derivative, is defined by It is well known and easy to proof that ϕ(D(B)) = D(ϕ(B)) for any homeomorphism ϕ : A Z → A Z . For a subshift (X, σ) and a subset B ⊆ X we can characterize the set of limit points as follows. A configuration (x i ) i∈Z is a limit point of B if for all k ∈ N the word x −k,...,k can be extended to a configuration in B different from X. We will use this characterization at the end of Section 5 to compute D(Per n (F )). Now we fix n ∈ N and a cellular automaton F : A Z → A Z with radius r ≥ 1 and local rule f : A 2r+1 → A, and try to determine quantities | P er n (F )| and |D(Per n (F ))|.
We define the De Bruijn graph D = (V, E, t, h) by (see Sec. 6 for notation) to 1011011.
A direct calculation (see Fig. 1 for an illustration) shows, that F (x) = x iff ∈ i∈Z p(ψ(x) i ). Now we take the subgraph of D containing only those edges e with n ∈ p(e) and then remove all edges not contained in any infinite path and call the result G. By this construction Ψ (Per n (F )) = Path(G) =: Per n (G) and Ψ ( P er n (F )) = {γ ∈ Path(G) ; i∈Z p(γ i ) = {n}} =: P er n (G). See Fig. 2 for an example.

Computing the Invariants
In this section we show how to compute P er n (G) and D(Per n (G)). Let S G be the set of strongly connected components of G, that is the maximal strongly connected subgraphs of G. Define the strong component digraph S G (see [1]) of G as the acyclic digraph with vertex set S G , edge set E(S G ) := {(s 1 , s 2 ) ; ∃e ∈ E(G) : t(e) ∈ s 1 and h(e) ∈ s 2 } and tail resp. head being the first resp. second entry of the edge. For each vertex i ∈ V (G) there is a unique component s(i) ∈ S G such that i ∈ V (s(i)). Each bi-infinite path (γ i ) i∈Z in G induces a unique finite vertex-path s(γ) = (s(γ) 1 , . . . , s(γ) ) in S G (since S G is a finite acyclic digraph, it contains only finite paths) such that Thus s(γ) is the path in S G traversed by the vertices on γ.
For components s 1 , . . . , s k ∈ S G we define Path(s 1 , . . . , s k ) as the set of all bi-infinite paths in G that traverse the components s 1 , ..., s k in that order, i.e.
Path(s 1 , . . . , s k ) = {γ ∈ Path(G) ; s(γ) = (s 1 , . . . , s k )} We now annotate the vertices and edges of S G by three functions defined as follows (remember that the vertices of S G are subgraphs of G).  With these annotations, we can calculate the cardinality of Path(s 1 , . . . , s k ) as follows: Together with the following theorem this gives an algorithm for computing | P er n (F )| = | P er n (G)|.
Theorem 3. Let m be the length of the longest vertex path in S G and let M k be the set of all vertex paths (s 1 , . . . , Proof. We first show that a vertex path (s 1 , . . . , On the other hand let (s 1 , . . . , s k ) ∈ M k . There are edges e 1 , . . . , be the set of all bi-infinite paths containing all of the edges in E(s 1 ) ∪ · · · ∪ E(s k ) ∪ {e 1 , . . . , e k−1 } and no other edges. Then L ⊆ Path(s 1 , . . . , s k ) and for γ ∈ L we have Hence γ ∈ P er n (G) and ∅ = L ⊆ Path(s 1 , . . . , s k ) ∩ P er(G).
The set L contains 1 elements iff one of the components s 1 , . . . , s k is not a directed circle or a single vertex. If this is not the case and there are at least two components, then |L| = 0 . If k = 1 and s 1 is a directed circle or a single vertex, then L = Path(s 1 , . . . , s k ). Therefore by (1) Determining the derived set of Per n (G) is simpler. By the definition of the topology on E(G) Z we have that Path(s 1 , . . . , s k ) = ∅ is either contained in D(Per n (G)) or its complement D(Per n (G)) c . The first case happens if and only if at least one of the following conditions is met

Data for the 25Elementary CA
Armed with the algorithm to compute the number of minimally p-periodic points of a CA F we can now set forth and apply this to the classification of the 256 elementary CA, the CA with alphabet {0, 1} and radius 1. We enumerate them according to their Wolfram code [10], so W k is the CA with Wolfram code k.
There remains one issue. All periodic points of F lie in its eventual image ω(F ) := t∈N F t (A Z ). If two CA are conjugate when restricted to their eventual image but differ in their transient behaviour, we have no possibility to detect this up to now. As a very simple invariant capturing some transient behaviour we therefore check We already know from Section 3, that we can always get an conjugate elementary CA by conjugation with the homeomorphisms of {0, 1} Z induced by Each equivalence class of CA up to conjugation with these two homeomorphisms contains at most four elements (it contains less if e.g. F = υF υ −1 ). It is well known that 88 of these equivalence classes remain [8]. We represent each of them by the member with the smallest Wolfram code. For each equivalence class we compute the invariants and group them by this data. The results are shown in Table 1.

The Special Cases
We still have 10 classes of elementary cellular automata left, that we could not distinguish with the invariants considered up to now. We start with the nontrivially conjugate CA.
The following pairs of cellular automata are conjugate by These (together with their conjugates with respect to υ) are exactly the leftand right-permutive elementary CA. Therefore by a result of Kurka and Nasu [5] they are conjugate to the one-sided full shift with alphabet {1, . . . , 4} and in particular they are conjugate to each other.
We will show on a case by case basis, that all CA in the remaining classes are pairwise non-conjugate. For this we use two new invariants, again only using the bijectivity of the conjugation ϕ. Let Fix k (F ) be the set of all fixed points of F with k preimages, that is, It is straightforward to see, that |F −1 (Per 1 (F ))| and | Fix k (F )| both remain invariant under conjugation.
For each CA F with local rule f : {0, 1} 3 → {0, 1} the De Bruijn graph for n = 1 with edges annotated by f is shown. A edge is drawn thickly if f (x −1 x 0 x 1 ) = x 0 , therefore the edge shift of the subgraph defined by the thick edges is Ψ (Per 1 (F )) = Per 1 (G).  We have that Hence |W −1 6 (Per 1 (W 6 ))| = 1 . On the other hand and thus |W −1 134 (Per 1 (W 134 ))| = 0 . Therefore W 134 and W 6 are not conjugate. Rules 78 and 140 From Fig. 6 we derive that and each occurrence of 01010 resp. 10110 might be replaced by 01110 resp. 10010 in fixed points of W 78 without changing the image. As a last invariant we look at the possible cardinalities of the preimage of a point and define PF(F ) := {|F −1 (x)| ; x ∈ A Z } ⊆ C. Let Fib be the set of Fibonacci numbers, defined by a 1 = 1, a 2 = 2, a k+2 = a k+1 + a k for k ∈ N. We will show that PF(W 200 ) = PF(W 12 ) In the case of W 200 the ambiguity in forming the preimage comes from blocks of the form 110 k 11, see Fig. 9b. Since isolated 1s are erased by W 200 , the number of preimages of ∞ 1.0 k 1 ∞ equals the number of words of length k − 2 containing no two consecutive 1s, which equals a k−1 ∈ Fib. If more then one block of the form 110 k 11 occurs, one can independently put isolated 1s in these blocks without changing the image, hence the number of the preimages is the product of those for the single blocks. The same is true for W 76 but here we look at blocks terminated by 11 on each side and containing only isolated 1s, e.g. 11001001010001011. We can replace 010 k 10 by 01 k+2 0 without changing the image. But since we can not do this for adjacent occurrences of 010 k 10, again the number of preimages of ∞ 10w01 ∞ with w containing isolated 1s is a . On the other hand so 0 ∈ PF(W 12 ). But 0 ∈ PF(W 4 ), since any point having infinitely many preimages wrt. W 4 must contain infinitely many occurrences of blocks of the form 10 k 1 with k ≥ 2 or start resp. end in ∞ 0 resp. 0 ∞ , thus already having uncountably many predecessors. Consequently W 12 is not conjugate to any of W 4 , W 76 and W 200 . This leaves us with these three cellular automata. Next we look at W −1 4 (x) for x = ∞ (01).000000(10) ∞ ).
Each element of this set has to coincide with x everywhere except for the underlined block of four consecutive zeros. In this block we only have to ensure that no isolated 1s occur. So we have to determine the number of 0, 1 blocks of length 4 where ones only occur in blocks of length at least two. Therefore there can be only either zero or one block of ones, of length from 2 to 4. This gives 1 + 3 + 2 + 1 = 7 possibilities. But 7 is not a product of Fibonacci numbers, hence W 4 is not conjugate to either W 76 or W 200 . Finally we differentiate between these two CA. Notice that Fix 3 (W 200 ) consists of all configuration in Per 1 (W 200 ) containing the block 11000011 but no other block of zeros of length greater then two. Hence the closure of Fix 3 (W 200 ) is contained in Fix 3 (W 200 )∪Fix 1 (W 200 ). On the other hand we have ( ∞ 0.10 ∞ ) ∈ Fix 3 (W 76 ), hence there is (x n ) n∈N in Fix 3 (W 76 ) with x n → ∞ 0 ∞ ∈ Fix 2 (W 76 ). With that we have finally shown that W 200 and W 76 are not topologically conjugate.
Notice however, that | Fix k (W 76 )| = | Fix k (W 200 )| for all k ∈ C. Therefore W 76 and W 200 are conjugate when {0, 1} Z is endowed with the discrete topology.

Conclusion
We showed that there are exactly 83 equivalence classes of topologically conjugate elementary CA. Among them we saw examples of pairs of CA that are (a) conjugate, but not strongly conjugate, e.g. W 170 = σ and W 15 = σ • ν, (b) not conjugate, but conjugate if one neglects the topology, e.g. W 200 and W 76 , (c) not conjugate, but conjugate when restricted to their eventual image, e.g. W 4 and W 12 .
Our main tool in differentiating non-conjugate CA was the number of minimally n-periodic points. In higher dimensions this is in general not computable, as already being able to decide if | Per 1 (F )| = 0 is equivalent to deciding the tiling problem. Therefore it would be interesting how far one can get in deciding conjugacy of higher-dimensional CA with small radius and alphabet size.
A cellular automaton is nilpotent, iff restricted to its eventual image it is conjugate to the dynamical system whose state space consists of a single point. This implies that all nilpotent CA are conjugate when restricted to their eventual image. Nilpotency is undecidable already in dimension one [4]. Hence it is undecidable if two CA are topological conjugacy when restricted to their eventual image. But this does not immediately imply that topological conjugacy is undecidable. Therefore we finish with the following conjecture.

Conjecture 4.
Topological conjugacy of one-dimensional cellular automata is undecidable.