linear cellular automata with boundary conditions

Linear Algebra and its Applications 322 (2001) 193–206www.elsevier.com/locate/laa

Linear cellular automata with boundaryconditions

William Chin∗,1, Barbara Cortzen1, Jerry Goldman2

Department of Mathematical Sciences, DePaul University, Chicago, IL 60614, USA

Received 21 January 2000; accepted 24 July 2000

Submitted by R. Guralnick

Abstract

The main results of the paper concern graphs of linear cellular automata with boundaryconditions. We show that the connected components of such graphs are direct sums of treesand cycles, and we provide a complete characterization of the trees, as well as enumerate thecycles of various lengths. Our work generalizes and clarifies results obtained previously inspecial cases. © 2001 Published by Elsevier Science Inc. All rights reserved.

Keywords: Linear cellular automata; Boundary conditions; State transition diagram

1. Introduction

A linear (or additive) cellular automaton is a system consisting of sites on a lattice,which correspond to cell processor memory having finite field elements as values,and which evolve synchronously over discrete time according to a given local inter-action rule called the state transition function [11,12]. As in [7,12], we consider suchautomata where at most a finite number of the sites of the systems studied containnon-zero values at any instant of time and the set of such sites and site values, aconfiguration of sites and site values of the lattice at timet, is called the state ofthe system at timet. The interaction rule is a function which assigns a site value

∗ Corresponding author.E-mail addresses:[email protected] (W. Chin), [email protected] (B. Cortzen),

[email protected] (J. Goldman).1 Supported by a URC grant at DePaul University.2 Supported by DePaul College of Liberal Arts and Sciences research grant.

0024-3795/01/$ - see front matter( 2001 Published by Elsevier Science Inc. All rights reserved.PII: S 0 0 2 4 - 3 7 9 5 ( 0 0 ) 0 0 2 2 7 - 5

194 W. Chin et al. / Linear Algebra and its Applications 322 (2001) 193–206

at time t + 1 to each site in terms of values at certain of its neighbors at previoustime t. Several authors in the past, following the approach initiated in [7], repre-sented states as elements of a Laurent polynomial ring and most frequently studiedstate transition functions which were linear. For example, see [3,5,6,7]. In this paper,we generalize the notion of a linear cellular automaton by viewing the state transi-tion function abstractly as an endomorphism acting upon a module of states. (For abasic reference on rings and modules, see, for example, [4].) This generalizes the ap-proach of several previous contributors and allows a conceptual understanding of thestructure of cellular automata including those satisfying null and periodic boundaryconditions.

Let Rbe a commutative Artinian ring and letM be a freeR-module of finite rank.The tripleR = (R,M, α) will be called ageneralized linear automatonif R andM are as above andα ∈ EndRM. These are often referred to as “additive” cellularautomata. Here,M is the set of states andα is the state transition function. In theliterature,RandM have been taken as

R = M = F [x1, x2, . . . , xm]I

,

whereF is a finite field, andI is an ideal ofF [x1, x2, . . . , xm] which is thought ofas a set of boundary conditions. WhenI is the ideal generated by somex

ni

i , ni > 0,the boundary conditions are said to benull, and when the generators are of the formx

ni

i − 1, they are said to beperiodic.For example, ifR = M = Z2[x]/(xn − 1) andα is given by left multiplication

by x + x −1, then we get the “rule 90” automaton of Wolfram, extensively studiedby others as well [7]. Hereα(f (x)) = (x + x−1)f (x) for anyf (x) ∈ M. Classical-ly, this is rendered by giving the local transition rulexi(t + 1) = xi−1(t) + xi+1(t),wherexi(t) is the site value at positioni on the one-dimensional lattice at timet.

As another example, consider the two-dimensional cellular automaton given byR = M = Z2[x, y]/(xn − 1, ym − 1), whereα is left multiplication byx + x−1 +y + y−1. Here, the cells lie on a torus and each cell value at timet + 1 is the sum ofthe values of its four orthogonal neighbors at timet.

Following the seminal paper [12], the basic global approach to describing linearautomata was initiated in [7] for certain periodic boundary conditions and some ex-amples and classes of linear transition rules. The state transition graphs were associ-ated with the automata and connected components were computed as being built outof cycles and trees, which corresponded, respectively, to “attractors” and “transients”in the time evolution of the automaton. The connected components were computedand enumerated in a range of specific examples.

More general (higher order) linear cellular automata were studied in [6]. Theyadapted the Laurent polynomial representation of states given by Martin et al. [7]to higher dimensions and a module theoretic setting which allowed the coefficientsof these polynomials to be elements of a finite-dimensional vector space over a fi-nite field. Their transition rules are represented as matrices with Laurent polynomial

W. Chin et al. / Linear Algebra and its Applications 322 (2001) 193–206 195

entries. They fully characterize the transition graphs of automata without boundaryconditions.

In other work, Jen [5] studied linear automata with periodic boundary conditionsand invertible transition rules, utilizing the theory of linear recurrences. In [3], peri-odic boundary conditions were studied using the theory of circulant matrices. Morerecently, in [9,10] Sutner examines the regular languages associated with such au-tomata as well as the complexity of the associated structures. More references con-cerning language-theoretic aspects of linear cellular automata can be found in thesepapers. In [8], the same author investigates linear cellular automata associated todirected graphs, obtaining periodicity and reversibility results in some cases.

In this paper, we define a generalized linear cellular automaton using an arbi-trary endomorphism of the state space as described above, and associate with it itsstate transition graph, a directed graph whose vertices represent states (as configu-rations of sites and values) and whose edges represent the temporal transitions. Weextend the work of earlier authors, who studied periodic boundary conditions, whilecomplementing the boundaryless situation characterized in [6].

We assume in this paper that the boundary conditions are such that the ringR is atleast artinian and, usually, finite-dimensional over a field. The state space is then a fi-nitely generated module, usually finite, overR. Our chief goal is to describe the statetransition graph of the automaton. A principal observation is that the structure of thestate transition graph reduces to the theory of a single linear transformation. As a con-sequence of Fitting’s lemma, the connected components of the graph are products oftrees and cycles. Our work generalizes and clarifies results previouslyobtained in spe-cial casesandspecificexamples,whichwerepredominatelyobtainedviacomputation.Our approach reveals the essential algebraicstructure in thegeneralizedsetting.

Let us outline the contents of this paper. In Section 2, we set up basic definitionsand general results concerning graphs of (generalized) linear cellular automata. Herewe see how the cycles and trees with fixed non-zero in-degree appear. The in-degreeis seen to be the cardinality of the kernel of the transition endomorphism, a factobserved elsewhere in special cases in [3,7], and the height of a tree is its indexof nilpotency. In Section 3, we deal with direct sums, and this yields a picture ofthe connected components of an arbitrary linear cellular automaton as cycles withrooted trees, attached to each node of the cycle. Again, this fact was observed inspecial cases in [3,7]. We briefly characterize balanced trees in Section 4. Finally, inSection 5, we state our results enumerating cycle lengths in terms of the order of thetransition rule. We conclude with an example that was approached computationallyin [7], illustrating our re sults.

2. Definitions and general results

Let R be a commutative ring,M anR-module, and letα ∈ EndRM. We call thetriple R = (R,M, α) a linear cellular automaton(LCA) with state moduleM andtransition ruleα.


ThegraphC(R) of the LCAR = (R,M, α) is the directed graph whose verticesare elements ofM, and whose edges are the ordered pairs(m,m′), with m′ = α(m),for all m ∈ M (symbolized bym 7→ α(m)). Note thatC(R) is simply the functionaldigraph ofα, and it depends only onα andM, not onR.

If α(m) = m′, thenm′ is asuccessorof m, andm is apredecessorof m′. If m ∈ M

has no predecessors, that is, ifm /∈ α(M), thenmis called asource(or, as some authorsprefer, a “garden of Eden”). Callmasinkif α(m) = m, that is, ifC(R)has a loop atm.

A treein thispaper isa rooted tree in theusualsense,except it isadirectedgraphwithall edges oriented towards the root, and, in addition, there is a loop attached to the root.Thus, a tree has a unique sink, namely, the root. Theheightof a tree is the number of di-rected edges along the longest path to the root, not including the loopat the root.

We call a finite directed graph acycle, if it is connected and each vertex has aunique predecessor and a unique successor.

Proposition 2.1. Let R = (R,M, α) be an LCA, where M is a finite, non-zeroR-module.

(i) If α is nilpotent with index of nilpotency k, then the graph ofR is a tree of heightk (with a loop at the root0). Each vertex of the tree which is not a source has|ker(α)| predecessors. The number of non-sources is|M|/|ker(α)|, and0 is theonly sink.

(ii) If α is an automorphism of finite order r, then the graph of(R,M, α) is a unionof cycles and all cycle lengths divide r. There is always at least one cycle oflength1 (namely, 0 with a loop).

(iii) Conversely, if the graph ofR is a tree, thenα is nilpotent, and if it is union ofcycles, thenα is an automorphism.

Proof. (i) The first assertion of (i) is self-evident. Letm ∈ M be a non-source, thatis, m ∈ α(M). If m = α(m1) for somem1 ∈ M, thenα−1(m) = m1 + ker(α), andtherefore there are|ker(α)| predecessors ofm.

The set of all non-sources isα(M) ' M/ker(α), and, obviously,α(m) = m im-plies thatm = 0, which proves the last statement of (i).

(ii) Obviously, each elementm ∈ M has a unique predecessorα−1(m), and aunique successorα(m). Moreover, for everym ∈ M, the size of the orbit ofm,{αi(m)} must divider.

(iii) is self-evident. �

If R = (R,M, α) is an LCA withα nilpotent, we shall call the number|ker(α)|the in-degreeof the treeC(R).

Example 1. Let R = M = Z2[x]/(x3), andα = multiplication byx 2. The graphof R = (R,M, α) is the tree of height 2 and in-degree 4 as shown in Fig. 1. (Hereand subsequently the obvious orientation of the edges will be omitted.)


Fig. 1.

Fig. 2.

Example 2. Let R = M = Z2[x]/(x4 − 1), andα = multiplication byx. Then thegraph ofR = (R,M, α) is the union of three cycles of length 4, one cycle of length2, and two cycles of length 1, as shown in Fig. 2.

3. Direct sums of graphs and LCAs

Definition 1. Let C1 andC2 be two directed graphs with sets of verticesM1 andM2, respectively. We define thedirect sumC1 ⊕ C2 to be the directed graph with theset of verticesM1 × M2, and edges(m1,m2) 7→ (m′

1,m′2) wheneverm1 7→ m′

1 inC1 andm2 7→ m′

2 in C2. What we call here the direct sum is usually referred to asthe synchronous (or direct) product in Automata Theory.

Definition 2. With above notation, directed graphsC1 andC2 areisomorphic(writ-tenC1 ' C2), if there is a 1–1 mappingφ from M1 ontoM2, which preserves thedirected edges.

Definition 3. Let R1 = (R1,M1, α1) andR2 = (R2,M2, α2) be LCAs. ThenM1 ⊕M2 is anR1 ⊕ R2-module, andα1 + α2 is in EndR1⊕R2(M1 ⊕ M2). SoR = (R1 ⊕R2, M1 ⊕ M2, α1 + α2) is an LCA. We callR thedirect sumof the LCAsR1 andR2 and writeR = R1 ⊕ R2.

Proposition 3.1. With above notation, C(R1 ⊕ R2) ' C(R1) ⊕ C(R2).


Proof. The vertex sets of both graphs areM1 × M2. Furthermore, if(m1,m2) 7→(m′

1,m′2) in C(R1 ⊕ R2), then(α1 + α2)(m1,m2) = (m′

1,m′2) in R1 ⊕ R2, that is,

α1(m1) = m′1 andα2(m2) = m′

2. Consequently,(m1,m2) 7→ (m′1,m

′2) in C(R1) ⊕

C(R2). �

Proposition 3.2. Suppose M is an R-module,M = M1 ⊕ M2 (an inner direct sum ofR-submodules), andα an endomorphism of M withα(M1) ⊂ M1 andα(M2) ⊂ M2.ThenC(R,M, α) ' C(R,M1, α|M1) ⊕ C(R,M2, α|M2).

Proof. The same as above withR1 = R2 = R, α1 = α|M1, andα2 = α|M2. (Wheth-er we considerM = M1 ⊕ M2 anR-module or anR ⊕ R-module does not have anyeffect on the graph.) �

Theorem 3.3. LetC1 andC2 be connected components of graphs of LCAs.

(i) SupposeC1 is a cycle of length m andC2 a cycle of length n. ThenC1 ⊕ C2 is aunion of cycles of lengthlcm(m, n).

(ii) SupposeC1 andC2 are graphs of LCAs which are trees. ThenC1 ⊕ C2 is a treewhose height is the maximum of the heights ofC1 andC2. The in-degree ofC isthe product of the in-degrees of the component trees.

(iii) SupposeC1 is an m-cycle andC2 is a graph of an LCA which is a tree. ThenC = C1 ⊕ C2 is an m-cycle with an isomorphic copy of the tree attached to eachof its nodes, which replace the root of the tree.

Proof. (i) Given a cycle of lengthm, c1 7→ c2 7→ · · · 7→ cm 7→ c1, its orbit de-termines the permutationγ = (c1, c2, . . . , cm), with o(γ ) = m. (If α is the under-lying endomorphism of the LCA, thenγ is the restriction ofα to the orbit of thecycle.) SupposeC1 and C2 are graphs which are cycles of lengthsm and n, re-spectively, andγ1, γ2 their associated permutations. Then o(γ1) = m and o(γ2) = n.If (c1, c2) is a vertex ofC1 ⊕ C2, then the connected component ofC1 ⊕ C2 con-taining(c1, c2) is (c1, c2) 7→ (γ1(c1), γ2(c2)) 7→ · · · 7→ (γ l

1(c1), γl2(c2)) = (c1, c2),

wherel = lcm(m, n). In other words,(c1, c2) is a vertex of a cycle whose associatedpermutation acts as a product of disjoint permutations (cycles) of ordersm andn,respectively.

(ii) SupposeC1 andC2 are trees with in-degreesn1 andn2, and rootsr1 andr2,respectively. Letm1 be a vertex inC1 andm2 a vertex inC2. If eitherm1 or m2 isa source, then(m1,m2) is a source inC1 ⊕ C2. If m1 andm2 are not sources, andp1 andp2 are their respective predecessors, then(p1, p2) → (m1,m2) is a direct-ed edge inC1 ⊕ C2. Since there aren1 choices forp1 andn2 choices forp2, weconclude that every non-source inC1 ⊕ C2 hasn1 andn2 predecessors. The root ofC1 ⊕ C2 is (r1, r2), which is also the only vertex with a loop. Since every vertex inCi has a unique successor fori = 1, 2, the same is true ofC1 ⊕ C2, and all its edgesare directed toward the root. Moreover, suppose the height ofC1 is h and the height


of C2 6 h. If v1 → v2 → · · · → vh = r1 is a path of maximum length inC1 (notusing the loop atr1), then(v1, r2) → (v2, r2) → · · · → (vh, r2) = (r1, r2) providesa path with lengthh in C1 ⊕ C2, and, clearly, no longer path exists inC1 ⊕ C2 if onedoes not use the loop at(r1, r2).

(iii) SupposeC1 is the cyclec1 7→ c2 7→ · · · 7→ cm 7→ c1 with the associated per-mutationγ = (c1, c2, . . . , cm) andC2 is a tree rooted atr with in-degreen. Then(c1, r) 7→ (c2, r) 7→ · · · 7→ (cm, r) 7→ (c1, r) is the isomorphic copy of the cycleC1in C1 ⊕ C2. Now each(ci , r) hasn predecessors, namely, the vertices(γ −1(ci), t),wheret’s are predecessors ofr. If t is a vertex of the treeC2 at the levell, then(γ −l (ci), t) is the corresponding vertex of the tree inC1 ⊕ C2 which has the root(ci, r). If t is a source inC1, then(γ −l (ci ), t) is a source inC1 ⊕ C2. Otherwise,thasn predecessors inC2, and(γ −l (ci), t) has the same number of predecessors inC1 ⊕ C2, namely, the set{(γ −l−1(ci), s) | s is a predecessor oft in C2}. Thus, thetree with the root(ci, r) in C1 ⊕ C2 is an isomorphic image of the treeC1. However,the loop at the root ofC2 is missing inC1 ⊕ C2, or rather, it is replaced by a copy ofthe cycleC1. �

The example below illustrates how a graph of an indecomposableR-module de-composes into a direct sum of two graphs, and how changingR to the ringk[α] yieldsa corresponding decomposition of the moduleM.

Example 3. The tree in Example 1 (Fig. 1) is the direct sum of trees which aregraphed in Fig. 3. In Example 1,R = M = Z2[x]/(x3), α = x 2, andM is an inde-composableR-module. However, if we formR′ = Z2[α] = {0, 1, x 2, 1 + x 2}, thenthe graph ofR′ = (R′,M, α) is identical to the graph ofR = (R,M, α), since itdepends only on the action ofα on the module, the base ring being irrelevant. NowasR′-modules,M = M1 ⊕ M2, whereM1 = R′ andM2 = {0, x}. That is,M can bedecomposed into a direct sum ofα-invariantZ2-subspaces.C(R′) is thus the directsum of the graphs of(R′,M1, α|M1) and(R′,M2, α|M2) as shown in Fig. 3.

Example 4. Let R1 = (R,M, α), whereR = M = Z2[x]/(x4) andα is multiplica-tion byx 2. Then the graphC1 of R1 is the tree of height 2 and in-degree 4, in whicheach branch has the same height. LetC2 be any 4-cycle. ThenC1 ⊕ C2 is the graphshown in Fig. 4.

Fig. 3.


Fig. 4.

Theorem 3.4. LetR = (R,M, α), where M is a finite R-module, and R a finite ring.Then theC(R) is a union of direct sums of cycles and identical copies of a tree.

Proof. As a consequence of Fitting’s lemma,M can be decomposed into a directsum of submodulesM1 andM2, such thatα|M1 is nilpotent, andα|M2 is an automor-phism. The graph of(R,M1, α|M1) is a tree, while the graph of(R,M2, α|M2) is aunion of cycles. Thus,C(R) consists of all the cycles from the latter, with the tree ofthe former graph attached to each node of the cycle by the root.�

4. Balanced trees

In this section, we discuss LCAsR = (R,M, α), whereR is a finiteK-algebra,Ma finiteR-module andα a nilpotentR-endomorphism ofM with αk = 0, αk−1 /= 0.As noted in Section 2, the graph ofR is a tree, and if dimK ker(α) = s, then|K|s isthe in-degree of the tree, that is, the number of predecessors of each non-source.

Informally, we say that a tree is balanced, if the height of every branch is the same,that is, if any path leading from any source to 0 is of equal length, sayk. This is equiv-alent to saying that any sourcea /∈ α(M) has the property thatαk−1(a) /= 0. In otherwords, a tree is balanced if ker(αk−1) ⊂ α(M). Since the reverse inclusion alwaysholds, we can now adopt the following formal definition, which concurs with [1].

Definition 4. If R = (R,M, α) is as above (withα nilpotent of indexk), then wecall the graph ofR a balancedtree ifα(M) = ker(αk−1).

Theorem 4.1. Let R = (R,M, α) be as above and denoten = dimK M. Then thefollowing are equivalent:(i) The graph ofR is a balanced tree, that is, α(M) = ker(αk−1).

(ii) ker(α) = αk−1(M).

(iii) αk−i (M) = ker(αi) for all i = 1, . . . , k (with α0(M) denoting M).


Proof. Consider the following chain of submodules ofM and view them asK-sub-spaces ofM:

0 ⊂ αk−1(M) ⊂ αk−2(M) ⊂ · · · ⊂ α(M) ⊂ M

and

0 ⊂ ker(α) ⊂ ker(α2) ⊂ · · · ⊂ ker

(αk−1) ⊂ M.

Obviously we haveαk−i (M) ⊂ ker(αi) for all i. Moreover, as vector spaces,M ' αi(M) ⊕ ker(αi), in particular,M ' α(M) ⊕ ker(α), andM ' αk−1(M) ⊕ker(αk−1).

(i) ⇒ (ii): Suppose the tree is balanced, that is,α(M) = ker(αk−1). Then the lastisomorphism above gives

dimK

(αk−1(M)

)=n − dimK ker(αk−1)

=n − dimK α(M)

=n − (n − s)

=s,

which implies thatαk−1(M) = ker α.(ii) ⇒ (iii): If ker (α) = αk−1(M), then ker(α) ⊂ αi(M) for all i = 1, . . . , k − 1,

and the sequence

0 → ker(α) → αi(M)α→ αi+1(M) → 0

is exact fori = 0, 1, . . . , k − 2. Hence, we have the isomorphism of vector spac-esαi(M) ' ker(α) ⊕ αi+1(M), which implies thatαi(M) ' (ker(α))k−i (that is,αi(M) is isomorphic to the direct sum ofk − i copies of the vector space ker(α))for i = 0, . . . , k − 1. Thus, dimK αi(M) = (k − i)s, in particular, dimK M = n =ks, and dimK ker(αi) = n − (k − i)s = n − ks + is = is. Butαk−i (M) ⊂ ker(αi).Since the dimensions of both arei’s, they must be equal.

(iii) ⇒ (i): Obvious. �

Corollary 4.2. With notation as in Theorem4.1, the graph ofR is a balanced treeif an only ifn = ks, that is, if dimK M = k · dimK ker(α).

Proof. If the graph ofR is a balanced tree, then the proof of Theorem 4.1 impliesthatn = ks. On the other hand, an imbalanced tree would have fewer vertices thana balanced one (assuming the samek ands). Therefore, if a tree has the maximumnumber of elements, namely,|K|ks , it must be balanced.�

Corollary 4.3. LetT1 andT2 be two balanced trees which are graphs of LCAs overthe same field K. ThenT1 ⊕ T2 is balanced if and only ifT1 andT2 have equal height.

Proof. Follows from a dimension count.�


5. Enumeration of cycles

In this section, we assume thatR is a finiteK-algebra, whereK is a finite field,andM is a finiteR-module.

We begin by describing a series of reductions that greatly simplifies the problemof a graph of an arbitrary LCA, using a direct sum decomposition of the graph. Weremark that the graph of(R,M, α) depends only on the action ofα on M, not onthe structure ofR. We can form a subringK[α] of EndRM and viewM as aK[α]-module via the natural extension ofα · m = α(m). It is easy to see that the graphs of(R,M, α) and(K[α],M, α) will be identical.

By the Krull–Schmidt–Azumaya theorem, theK[α]-moduleM is a unique (upto isomorphism) finite direct sum of indecomposable modules. But any such inde-composable module is eitherK[α] itself, or a homomorphic image ofK[α], sinceK[α] is the image of the principal ideal domainK[x] (see, for example, [4]). So thegraph of(K[α],M, α) is uniquely the direct sum of graphs of LCAs of the form(K[α],K[α], α) and(K[α],K[α′], α′), whereα′ is the image ofα under a homo-morphism fromK[α]. However, as we have already observed before, the graph ofthe latter is isomorphic to the graph of(K[α′],K[α′], α′). In other words, our studyof graphs reduces to the case of LCAs of the form(K[α],K[α], α).

Furthermore, by Fitting’s lemma, we can assume thatα is either a unit or thatα isnilpotent. In the former case,K[α] ' K[x]/(f (x)), wheref (x) is a factor ofxn − 1(n being the order ofα), and by [2],f (x) is a power of an irreducible polynomial inK[x]. In the latter case,K[α] ' K[x]/(xk). In either case,K[α] is a local ring.

In view of our results on direct sums of graphs of LCAs, we may therefore restrictourselves to graphs of LCAs of the form(R,M, α), whereR = M = K[α] is a localring.

We now specialize to a fieldK of p elements, for a primep. In what follows, wedescribe the graph of an LCAR = (R,M, α), whereR is a finiteK-algebra,α anR-automorphism ofM, assuming thatR is a local ring andR = M = K[α].

The following three theorems address the cases where the order ofα is p-free, apower ofp, and finally, the mixed case where o(α) = spb, whereb > 1 andp6 | s.

Theorem 5.1. Supposeo(α) = s, wherep6 | s, and f is the order of p modulo s, thatis, the least positive integer d such thatpd ≡ 1(mods). Then the graph ofR consistsof one cycle of length1, and(pf − 1)/s cycles of length s.

Proof. If p does not divide o(α) = s, then by [2, p. 95], the minimal polynomialof α is an irreducible polynomialF(x) ∈ K[x] of degreef = order ofp modulos.Thus,K[α] ' K[x]/(F (x)) is a field extension onK of degreef. Therefore, everynon-zero elementa of K[α] is a unit and lies on a cycle of lengths, namely, on(a, α(a), α2(a), . . . , αs−1(a)). It follows that there is one 0-cycle of length 1, and(pf − 1)/s cycles of lengths. �


Theorem 5.2. Supposeo(α) = pb for some positive integer b. The minimal polyno-mial of α in K[x] is then(x − 1)e for some e withpb−1 < e 6 pb. The graph ofRconsists of p cycles of length1,(ppa − ppa−1

)/pa cycles of lengthpa for 0 < a < b,

and(pe − ppb−1)/pb cycles of maximal lengthpb.

Proof. Sinceα satisfies the polynomialxpb − 1 = (x − 1)pb, the minimal polyno-

mial of α is of the form(x − 1)e, wheree is some integer withpb−1 < e 6 pb.Thus,K[α] ' K[x]/(x − 1)e with α → x. Let Ul = annR(x l − 1) for l | m, that is,for l = pa , wherea 6 b. ThenUl is a subspace ofR consisting of elements that areon cycles of length6 l, that is, on cycles whose length dividesl.

If l = pa , wherepa 6 e, thenUl = (x − 1)e−l = (x − 1)e−pa, which is the sub-

space ofRof dimensionl = pa . (As a vector space,Ul = K[(x − 1)e−pa ] + K[(x −1)e−pa+1] + · · · + K[(x − 1)e−1].) Otherwise, ifl = pb, then annR(x l − 1) = R.

Consequently, ife = pb, then dimK(Ul) = l for all l | pb, and if e < pb,then dimK(Ul) = min(l, e). Thus, we have a filtration of subspacesUl for l =1, p, . . . , pb−1, pb with dimensions 1, p, . . . , pb−1, pe. This shows that the graphof R consists ofp cycles of length 1,(ppa − ppa−1

)/pa cycles of lengthpa for0 < a < b, and(pe − ppb−1

)/pb cycles of maximal lengthpb. �

Example 5. Let p = 2, R = K[x]/(x − 1)5, α = multiplication by x. Theno(α) = 8 and the graph ofR consists of two cycles of length 1, one cycle of length2, three cycles of length 4, and two cycles of length 8.

Theorem 5.3. Leto(α) = m = pbs, where p does not divide s, andb > 1. Then theminimal polynomial ofα over K is of the formF(x)e, whereF(x) is an irreduciblepolynomial over K of degreef = the order of p modulo s, and e is a positive integer.The graph ofR is a union of cycles of lengthsspa for all 1 6 a 6 b, and a singlecycle of length1. Letq = pf . The number of cycles of lengthspa is

qpa − qpa−1

spa

for 1 6 a < b, and the number of cycles of lengthspb is

qe − qpb−1

spb.

Proof. Since o(αpb) = s andp does not divides, the minimal polynomial ofαpb

overK is an irreducible polynomialF(x) ∈ K[x] of degreef = the order ofp mod-ulo s. Thus, we haveF(αpb

) = [F(α)]pb = 0, and the minimal polynomial ofαoverK must beF(x)e for some numbere with pb−1 < e 6 pb, which shows thatR = K[α] ' K[x]/(F (x)e).


As before, letUl = annR(xl − 1) for l | m, that is,Ul is the subspace of elementsthat are on cycles of length dividingl. We first observe thatUl = 0 unlessl = pas,wherea is a non-negative integer6 b. Obviously, the only divisors ofm are ofthe forml = rpa , wherer | s. Moreover, if ann(xl − 1) /= 0, thenF(x) | (xl − 1) =(xr − 1)p

a, and consequentlyF(x) | (xr − 1). However, ifr | s andr /= s, then none

of the rth roots of 1 are primitivesth roots of 1 overK. Since, by [2],F(x) is theproduct of factors of the formx − ω, whereω’s are (certain) primitivesth roots of 1,we conclude thatF(x) does not dividexr − 1, and therefore there are no cycles inRof lengthrpa for r | s, r /= s. In other words, the only possible cycle lengths arespa

and 1.We now proceed to compute the number of cycles of lengthl = spa , with 1 6

a 6 b. We observe that

Ul = annR(xl − 1

) = annR(xs − 1

)pa = (Fe−pa )

.

So we have a chain of subspaces

U1 ⊂ Up0s ⊂ Up1s ⊂ Up2s ⊂ · · · ⊂ Up(b−1)s ⊂ Upbs

with dimensions, respectively,q0 = 1, qp0 = q, qp1, . . . , qpb−1

, qe (where q =pf ). That is, there is a single cycle of length 1,

qpa − qpa−1

spa

cycles of lengthpas if 1 6 a < b, and

qe − qpb−1

spb

cycles of maximal lengthpb. �

Example 6. Let K = Z2, R = M = K[x]/(x12 − 1), and letα be the endomor-phism ofM given by multiplication byx + x −1, that is,α = x + x 11. This is thetype of LCA studied extensively in [7], where the graph of this particular case isgiven (not quite correctly) on p. 221. The graph ofR = (R,M, α) is provided inFig. 7. We show here howM can be decomposed intoK[α]-modules, and how onecan obtain the graph ofR by generating the graphs of the indecomposable directsummands of theK[α]-moduleM.

We immediately get the decomposition

M = K[x]/(x12 − 1

) ' K[x]/(x + 1)4 ⊕ K[x]/(

x2 + x + 1)4

.

If we setM1 = K[x]/(x + 1)4 andM2 = K[x]/(x2 + x + 1)4, thenα|M1 is nilpo-tent, andα|M2 is an automorphism.

Let us now consider the LCA(K[α1],M1, α1), whereα1 = α|M1. Thenα1 =x + x 3, andα2

1 = 0. Now M1 = K[α1] ⊕ xK[α1] as aK[α1]-module. The graphs


Fig. 5.

Fig. 6.

60 copies 6 copies 4 copies

Fig. 7.

of the LCAs(K[α1],K[α1], α1) and(K[α1], xK[α1], α1) are trees of height 2 andin-degree 2, as shown in Fig. 5.

The graph of(K[α1],M1, α1) is their direct sum, that is, a tree of height 2 andin-degree 4 (Fig. 6).

We now proceed to examine the LCA(K[α2],M2, α2), whereα2 = α|M2. In M2we have the relationx 8 = x 4 + 1, so thatα2 = x + x 3 + x 7, α2

2 = x 6, α32 = x 5 +

x 7, andα42 = 1. As aK[α2]-module,M2 = K[α2] ⊕ xK[α2]. Thus, by the Theorem

5.2, the graph of(K[α2],K[α2], α2) consists of two cycles of length 1, one cycle oflength 2, and three cycles of length 4. SincexK[α2] ' K[α2] asK[α2]-modules,the graph of(K[α2], xK[α2], α2) is an isomorphic copy of the former graph, thusconsisting of two cycles of length 1, one cycle of length 2 and three cycles of length 4.

Taking the direct sum of these two isomorphic graphs, we obtain the graph of(K[α2],M2, α2). It consists of four cycles of length 1, six cycles of length 2 (twodirect sums of the 1-cycles and the 2-cycle, two direct sums of the 2-cycle and the1-cycles, and two 2-cycles which arise from the direct sum of the 2-cycle and the2-cycle), and 6+ 6 + 6 · 2 + 9 · 4 = 60 cycles of length 4.


Finally, the graph of the original LCA(K[α],M, α) consists of the 70 cyclesenumerated above, with a copy of the tree in Fig. 6 attached to each node (see Fig. 7).

References

[1] R. Barua, Additive cellular automata and matrices over finite fields, Indian Statistical Institute Tech-nical Report No. 17/91, 1991.

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Phys. 43 (3/4) (1986).[4] T. Hungerford, Algebra. Holt, Rinehart & Winston, New York, 1974.[5] E. Jen, Linear cellular automata and recurring sequences in finite fields, Comm. Math. Phys. 119

(1998) 13–28 (1988).[6] L. LeBruyn, M. Van de Bergh, Algebraic properties of linear cellular automata, Linear Algebra Appl.

157 (1991) 217–234.[7] O. Martin, A. Odlyzko, S. Wolfram, Algebraic properties of cellular automata, Comm. Math. Phys.

93 (1984) 219–258.[8] K. Sutner, Sigma-automata and Chebyshev polynomials, Theoret. Comput. Sci. 230 (1–2) (2000)

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Cellular Automata: A Parallel Model, Kluwer, Dordrecht, 1999.[10] K. Sutner, Linear cellular automata and Fischer automata, Parallel Comput. 23 (11) (1997) 1613–

1634.[11] J. Von Neumann, in: A.W. Burks (Ed.), Theory of Self-reproducing Automata, University of Illinois

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linear cellular automata with boundary conditions

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