On the dynamical behaviour of linear higher-order cellular automata and its decidability
Alberto Dennunzio, Enrico Formenti, Luca Manzoni, Luciano Margara, Antonio E. Porreca
aa r X i v : . [ c s . F L ] F e b On the dynamical behaviour of linear higher-ordercellular automata and its decidability
Alberto Dennunzio a , Enrico Formenti b , Luca Manzoni a , Luciano Margara c , Antonio E. Porreca a,d a Dipartimento di Informatica, Sistemistica e Comunicazione, Università degli Studi di Milano-Bicocca, Viale Sarca 336 /
14, 20126 Milano, Italy b Université Côte d’Azur, CNRS, I3S, France c Department of Computer Science and Engineering, University of Bologna, Cesena Campus, Via Sacchi 3, Cesena, Italy d Aix Marseille Université, Université de Toulon, CNRS, LIS, Marseille, France
Abstract
Higher-order cellular automata (HOCA) are a variant of cellular automata (CA) used in many applications (rang-ing, for instance, from the design of secret sharing schemes to data compression and image processing), and inwhich the global state of the system at time t depends not only on the state at time t −
1, as in the original model,but also on the states at time t −
2, . . . , t − n , where n is the memory size of the HOCA. We provide decidablecharacterizations of two important dynamical properties, namely, sensitivity to the initial conditions and equicon-tinuity, for linear HOCA over the alphabet Z m . Such characterizations extend the ones shown in [ ] for linearCA (LCA) over the alphabet Z nm in the case n =
1. We also prove that linear HOCA of size memory n over Z m form a class that is indistinguishable from a specific subclass of LCA over Z nm . This enables to decide injectivityand surjectivity for linear HOCA of size memory n over Z m using the decidable characterization provided in [ ] and [ ] for injectivity and surjectivity of LCA over Z nm . Finally, we prove an equivalence between LCA over Z nm and an important class of non-uniform CA, another variant of CA used in many applications. Keywords: cellular automata, higher-order cellular automata, linear cellular automata, sensitivity to the initialconditions, decidability, discrete dynamical systems
1. Introduction
Cellular automata (CA) is well-known formal model which has been successfully applied in a wide number offields to simulate complex phenomena involving local, uniform, and synchronous processing (for recent resultsand an up-to date bibliography on CA, see [
12, 13, 1, 14 ] ). More formally, a CA is made of an infinite set ofidentical finite automata arranged over a regular cell grid (usually Z d in dimension d ) and all taking a state froma finite set S called the set of states or the alphabet of the CA. In this paper, we consider one-dimensional CA.A configuration is a snapshot of all states of the automata, i.e., a function c : Z → S . A local rule updates thestate of each automaton on the basis of its current state and the ones of a finite set of neighboring automata. Allautomata are updated synchronously. In the one-dimensional settings, a CA over (the alphabet) S is a structure 〈 S , r , f 〉 where r ∈ N is the radius and f : S r + → S is the local rule which updates, for each i ∈ Z , the state of theautomaton in the position i of the grid Z on the basis of states of the automata in the positions i − r , . . . , i + r . Aconfiguration is an element of S Z and describes the (global) state of the CA. The feature of synchronous updatinginduces the following global rule F : S Z → S Z defined as ∀ c ∈ S Z , ∀ i ∈ Z , F ( c ) i = f ( c i − r , . . . c i + r ) . Email addresses: [email protected] (Alberto Dennunzio), [email protected] (Enrico Formenti), [email protected] (Luca Manzoni), [email protected] (Luciano Margara), [email protected] (Antonio E. Porreca)
Preprint submitted to Elsevier February 20, 2019 s such, the global map F describes the change from any configuration c at any time t ∈ N to the configuration F ( c ) at t + [
17, 24, 9 ] for instance), non-uniform CA relax uniformity ( [
11, 10 ] ), while hormone CA(for instance) relax locality [ ] . However, from the mathematical point of view all those systems, as well as theoriginal model, fall in the same class, namely, the class of autonomous discrete dynamical systems (DDS) andone could also precise memoryless systems. Indeed, the latter compute their next global state just on the basis oftheir current state, while the past ones play no active role. Allowing the original model to take into account paststates leads to a further natural variant which can further extend the application range of the model itself.As a motivating example, consider the following classical network routing problem. Assume to have twopacket sources A and B which are connected to a common router R which in its turn is connected to two receivinghosts O and O as illustrated below. A BRO O If R receives a packet m A from A but none from B , then it sends m A to the output hosts O and O ; if both A and B send a packet, m A and m B , respectively, then m B is enqueued and m A is transmitted to O and O . Of course, themore frequently simultaneous packets from A and B arrive at R , the longer the queue has to be in order to avoidpacket loss. When a whole network is considered, this routing problem can be easily solved using a variant of CAin which the state of each node keeps track of the current state of the router and the past states which representthe received but not yet transmitted packets.As to the possible variants of the original CA model, in [ ] , Toffoli introduced higher-order CA (HOCA),i.e., variants of CA in which the updating of the state of a cell also depends on the past states of the cell itselfand its neighbours. In particular, he showed that any arbitrary reversible linear HOCA can be embedded in areversible linear CA (LCA), where linear means that the local rule is linear. Essentially, the trick consisted inmemorizing past states and recover them later on. Some years later, Le Bruyn and Van Den Bergh explained andgeneralized the Toffoli’s construction and proved that any linear HOCA having the ring S = Z m as alphabet andmemory size n can be simulated by a LCA over the alphabet Z nm (see the precise definition in Section 2) [ ] . Inthis way a practical way to decide injectivity (which is equivalent to reversibility in this setting) and, as we willsee in Section 2, surjectivity of HOCA can be easily derived by the characterization of the these properties forthe corresponding LCA simulating them. Indeed, in [ ] and [ ] , characterizations of injectivity and surjectivityof a LCA over Z nm are provided in terms of properties of the determinant of the matrix associated with it, wherethe determinant turns out to be an other LCA (over Z m and then a LCA simpler than that over Z nm ). Since theproperties of LCA over Z m (i.e., LCA over Z nm with n =
1) have been extensively studied and related decidablecharacterizations have been obtained [
22, 3, 7, 4 ] , one derives the algorithms to decide injectivity and surjectivityfor LCA over Z nm and, then, as we will see in Section 2, also for HOCA over Z m of memory size n , by means of theassociated matrix.Applications of HOCA (in particular the linear ones) cover a wide span of topics, ranging from the design ofsecret sharing schemes [
8, 6 ] to data compression and image processing [ ] . Remark that (linear) HOCA areoften required to exhibit a chaotic behavior in order they can be used in applications, as for instance in thoseabove mentioned. The purpose of the present paper is to study, in the context of linear HOCA, sensitivity to theinitial conditions and equicontinuity, where the former is the well-known basic component and essence of thechaotic behavior of a DDS, while the latter represents a strong form of stability. To do that, we put in evidence thatany linear HOCA of memory size n over Z m is not only simulated by, but also topologically conjugated to a LCAover Z nm defined by a matrix having a specific form. Thus, in order to decide injectivity and surjectivity for linear2OCA of memory size n over Z m , by means of that specific matrix one can use the decidable characterizationprovided in [ ] and [ ] for deciding the same properties for LCA over Z nm . As main result, we prove thatsensitivity to the initial conditions and equicontinuity are decidable properties for linear HOCA of memory size n over Z m (Theorem 14). In particular we provide a decidable characterization of those properties, in terms ofthe matrix associated with a linear HOCA. Remark that if n =
1, starting from our characterizations, one recoverexactly the well known ones of sensitivity and equicontinuity for LCA over Z m . Finally, we prove an equivalencebetween LCA over Z nm and an important class of linear non-uniform cellular automata. This result gives strongmotivations to further study LCA over Z nm in the next future. First of all, non-uniform cellular automata is indeedanother variant of cellular automata which is used in many applications (in particular, the linear ones). Forinstance, as pointed out in [ ] , linear non-uniform cellular automata can be used as subband encoders forcompressing signals and images [ ] . Moreover, little is known for linear non-uniform cellular automata.
2. Higher-Order CA and Linear CA
We begin by reviewing some general notions and introducing notations we will use throughout the paper.A discrete dynamical system (DDS) is a pair ( X , F ) where X is a space equipped with a metric, i.e., a metric space,and F is a transformation on X which is continuous with respect to that metric. The dynamical evolution of aDDS ( X , F ) starting from the initial state x ( ) ∈ X is the sequence { x ( t ) } t ∈ N ⊆ X where x ( t ) = F t ( x ( ) ) for any t ∈ N .When X = S Z for some set finite S , X is usually equipped with the metric d defined as follows ∀ c , c ′ ∈ S Z , d ( c , c ′ ) = n where n = min { i ≥ c i = c ′ i or c ′− i = c ′− i } .Recall that S Z is a compact, totally disconnected and perfect topological space (i.e., S Z is a Cantor space).Any CA 〈 S , r , f 〉 defines the DDS ( S Z , F ) , where F is the CA global rule (which is continuous by Hedlund’sTheorem [ ] ). From now on, for the sake of simplicity, we will sometimes identify a CA with its global rule F or with the DDS ( S Z , F ) .Recall that two DDS ( X , F ) and ( X ′ , F ′ ) are topologically conjugated if there exists a homeomorphism φ : X X ′ such that F ′ ◦ φ = φ ◦ F , while the product of ( X , F ) and ( X ′ , F ′ ) is the DDS ( X × X ′ , F × F ′ ) where F × F ′ is defined as ∀ ( x , x ′ ) ∈ X × X ′ , ( F × F ′ )( x , x ′ ) = ( F ( x ) , F ′ ( x ′ )) and the space X × X ′ is as usual endowedwith the infinite distance. Notation 2.1.
For all i , j ∈ Z with i ≤ j, we write [ i , j ] = { i , i +
1, . . . , j } to denote the interval of integers between iand j. For any n ∈ N and any set Z the set of all n × n matrices with coefficients in Z and the set of Laurent polynomialswith coefficients in Z will be noted by M at ( n , Z ) and Z (cid:2) X , X − (cid:3) , respectively. In the sequel, bold symbols are usedto denote vectors, matrices, and configurations over a set of states which is a vectorial space. Moreover, m will be aninteger bigger than and Z m = {
0, 1, . . . , m − } the ring with the usual sum and product modulo m. For any x ∈ Z n (resp., any matrix M ( X ) ∈ M at (cid:0) n , Z (cid:2) X , X − (cid:3)(cid:1) ), we will denote by [ x ] m ∈ Z nm (resp., [ M ( X )] m ), the vector (resp.,the matrix) in which each component x i of x (resp., every coefficient of each element of M ( X ) ) is taken modulo m.Finally, for any matrix M ( X ) ∈ Z m (cid:2) X , X − (cid:3) and any t ∈ N , the t-th power of M ( X ) will be noted more simply by M t ( X ) instead of ( M ( X )) t . Definition 1 (Higher-Order Cellular Automata). A Higher-Order Cellular Automata (HOCA) is a structure H = 〈 k , S , r , h 〉 where k ∈ N with k ≥ memory size, S is the alphabet , r ∈ N is the radius , and h : S ( r + ) k → S is the local rule . Any HOCA H induces the global rule H : (cid:0) S Z (cid:1) k → (cid:0) S Z (cid:1) k associating any vector e = ( e , . . . , e k ) ∈ (cid:0) S Z (cid:1) k of k configurations of S Z with the vector H ( e ) ∈ (cid:0) S Z (cid:1) k such that H ( e ) j = e j + for each j = k and ∀ i ∈ Z , H ( e ) ki = h e [ i − r , i + r ] e [ i − r , i + r ] ... e k [ i − r , i + r ]
3n this way, H defines the DDS €(cid:0) S Z (cid:1) k , H Š . As for CA, we sometimes identify a HOCA with its global rule or theDDS defined by it. Moreover, we will often refer to a HOCA over S to stress the alphabet over which the HOCAis defined. Remark 2.
It is easy to check that for any HOCA H = 〈 k , S , r , h 〉 there exists a CA (cid:10) S k , r , f (cid:11) which is topologicallyconjugated to H .The study of the dynamical behaviour of HOCA is still at its early stages; a few results are known for theclass of linear HOCA , namely, those HOCA defined by a local rule f which is linear , i.e., S is Z m and there existcoefficients a ji ∈ Z m ( j =
1, . . . , k and i = − r , . . . , r ) such that for any element x = x − r . . . x r x − r . . . x r ... x k − r . . . x kr ∈ Z ( r + ) km , f ( x ) = k X j = r X i = − r a ji x ji m .It is easy to see that linear HOCA are additive, i.e., ∀ c , d ∈ (cid:0) Z Z m (cid:1) k , H ( c + d ) = H ( c ) + H ( d ) where, with the usual abuse of notation, + denotes the natural extension of the sum over Z m to both Z Z m and (cid:0) Z Z m (cid:1) k .In [ ] , a much more convenient representation is introduced for the case of linear HOCA (in dimension d = Definition 3 (Linear Cellular Automata). A Linear Cellular Automaton (LCA) over the alphabet Z nm is a CA L = (cid:10) Z nm , r , f (cid:11) where the local rule f : ( Z nm ) r + → Z nm is defined by 2 r + M − r , . . . , M , . . . , M r ∈ M at ( n , Z m ) as follows: f ( x − r , . . . , x , . . . , x r ) = (cid:2)P ri = − r M i · x i (cid:3) m for any ( x − r , . . . , x , . . . , x r ) ∈ ( Z nm ) r + . Remark 4.
LCA over Z nm have been strongly investigated in the case n = × [
23, 3 ] .We recall that any linear HOCA H can be simulated by a suitable LCA, as shown in [ ] . Precisely, given a linearHOCA H = 〈 k , Z m , r , h 〉 , where h is defined by the coefficients a ji ∈ Z m , the LCA simulating H is L = (cid:10) Z km , r , f (cid:11) with f defined by following matrices M = a a a . . . a k − a k , (1)and, for i ∈ [ − r , r ] with i = M i = a i a i a i . . . a k − i a ki . (2)4 emark 5. We want to put in evidence that a stronger result actually holds (easy proof, important remark):any linear HOCA H is topologically conjugated to the LCA L defined by the matrices in (1) and (2). Clearly,the converse also holds: for any LCA defined by the matrices in (1) and (2) there exists a linear HOCA whichis topologically conjugated to it. In other words, up to a homeomorphism the whole class of linear HOCA isidentical to the subclass of LCA defined by the matrices above introduced. In the sequel, we will call L the matrixpresentation of H .We are now going to show a stronger and useful new fact, namely, that the class of linear HOCA is nothingbut the subclass of LCA represented by a formal power series which is a matrix in Frobenius normal form. Beforeproceeding, let us recall the formal power series (fps) which have been successfully used to study the dynamicalbehaviour of LCA in the case n = [
18, 23, 22, 4 ] . The idea of this formalism is that configurations and globalrules are represented by suitable polynomials and the application of the global rule turns into multiplications ofpolynomials. In the more general case of LCA over Z nm , a configuration c ∈ ( Z nm ) Z can be associated with the fps P c ( X ) = X i ∈ Z c i X i = c ( X ) ... c n ( X ) = P i ∈ Z c i X i ... P i ∈ Z c ni X i ∈ (cid:0) Z m (cid:2) X , X − (cid:3)(cid:1) n ∼ = Z nm (cid:2) X , X − (cid:3) .Then, if F is the global rule of a LCA defined by M − r , . . . , M , . . . , M r , one finds P F ( c ) ( X ) = [ M ( X ) P c ( X )] m where M ( X ) = – r X i = − r M i X − i ™ m is the finite fps , or, the matrix , associated with the LCA F . In this way, for any integer t > F t is M ( X ) t , and then P F t ( c ) ( X ) = [ M ( X ) t P c ( X )] m . Roughly speaking, the action of a LCA over a configuration isgiven by multiplication between elements of M at (cid:0) n , Z m (cid:2) X , X − (cid:3)(cid:1) with elements of (cid:0) Z m (cid:2) X , X − (cid:3)(cid:1) n . Throughoutthis paper, M ( X ) t will refer to [ M ( X ) t ] m .A matrix M ( X ) ∈ M at (cid:0) n , Z (cid:2) X , X − (cid:3)(cid:1) is in Frobenius normal form if M ( X ) = m ( X ) m ( X ) m ( X ) . . . m n − ( X ) m n − ( X ) (3)where each m i ( X ) ∈ Z (cid:2) X , X − (cid:3) From now on, for a given matrix M ( X ) ∈ M at (cid:0) n , Z (cid:2) X , X − (cid:3)(cid:1) in Frobenius normal form, m ( X ) will alwaysmake reference to its n -th row. Definition 6 (Frobenius LCA).
A LCA F over the alphabet Z nm is said to be a Frobenius LCA if the fps M ( X ) ∈ M at (cid:0) n , Z m (cid:2) X , X − (cid:3)(cid:1) associated with F is in Frobenius normal form.It is immediate to see that a LCA is a Frobenius one iff it is defined by the matrices in (1) and (2), i.e., iff it istopologically conjugated to a linear HOCA. This fact together with Remark 5 and Definition 6, allow us to statethe following 5 roposition 7. Up to a homeomorphism, the class of linear HOCA over Z m of memory size n is nothing but the classof Frobenius LCA over Z nm . At this point we want to stress that the action of a Frobenius LCA F , or, equivalently, a linear HOCA, over aconfiguration c ∈ ( Z nm ) Z can be viewed as a bi-infinite array of linear–feedback shift register . Indeed, it holds that P F ( c ) ( X ) = M ( X ) P c ( X ) = c ( X ) ... P ni = m i − ( X ) c i ( X ) ,where the n –th component of the vector P F ( c ) ( X ) is given by the sum of the results of the actions of n one-dimensional LCA over Z m each of them applied on a different component of P c ( X ) , or, in other words, on adifferent element of the memory of the linear HOCA which is topologically conjugated to F . Remark 8.
Actually, in literature a matrix is in Frobenius normal form if either it or its transpose has a formas in (3). Since any matrix in Frobenius normal form is conjugated to its transpose, any Frobenius LCA F istopologically conjugated to a LCA G such that the fps associated with G is the transpose of the fps associated with G . In other words, up to a homeomorphism, such LCA G , linear HOCA, and Frobenius LCA form the same classand, in particular, the action of G on any configuration c ∈ ( Z nm ) Z is such that P G ( c ) ( X ) = M ( X ) T P c ( X ) = m ( X ) c n ( X ) c ( X ) + m ( X ) c n ( X ) ... c n − ( X ) + m n − ( X ) c n ( X ) ,where M ( X ) T is the transpose of the matrix M ( X ) associated to the LCA F which is topologically conjugated to G .From now on, we will focus on Frobenius LCA, i.e., matrix presentations of linear HOCA. Indeed, they allowconvenient algebraic manipulations that are very useful to study formal properties of linear HOCA. For example,in [ ] and [ ] , the authors proved characterizations for injectivity and surjectivity for LCA in terms of the matrix M ( X ) associated to them and which turns out to be decidable by means of the characterization of injectivity andsurjectivity for LCA over Z m shown in [ ] . Proposition 9 ( [
2, 19 ] ). Let €(cid:0) Z nm (cid:1) Z , F Š be a LCA over Z nm and let M ( X ) be the matrix associated with F . Then,F is injective (resp., surjective) if and only if the determinant of M ( X ) is the fps associated with a injective (resp.,surjective) LCA over Z m . We want to stress that, by Remark 5, Definition 6, and Proposition 7, one can use the characterizations fromProposition 9 for deciding injectivity and surjectivity of linear HOCA. Summarizing, the following result holds.
Proposition 10.
Injectivity and surjectivity are decidable properties for HOCA of memory size n over Z m . In this paper we are going to adopt a similar attitude, i.e., we are going to characterise the dynamical behaviourof linear HOCA by the properties of the matrices in their matrix presentation.
3. Dynamical properties
In this paper we are particularly interested to the so-called sensitivity to the initial conditions and equicontinuity .As dynamical properties, they represent the main features of instable and stable DDS, respectively. The former isthe well-known basic component and essence of the chaotic behavior of DDS, while the latter is a strong form ofstability. 6et ( X , F ) be a DDS. The DDS ( X , F ) is sensitive to the initial conditions (or simply sensitive ) if there exists ǫ > x ∈ X and any δ > y ∈ X such that d ( y , x ) < δ and d ( F n ( y ) , F n ( x )) > ǫ for some n ∈ N . Recall that, by Knudsen’s Lemma [ ] , ( X , F ) is sensitive iff ( Y , F ) is sensitive where Y is anydense subset of X which is F -invariant, i.e., F ( Y ) ⊆ Y .In the sequel, we will see that in the context of LCA an alternative way to study sensitivity is via equicontinuitypoints. An element x ∈ X is an equicontinuity point for ( X , F ) if ∀ ǫ > δ > y ∈ X , d ( x , y ) < δ implies that d ( F n ( y ) , F n ( x )) < ǫ for all n ∈ N . The system ( X , F ) is said to be equicontinuous if ∀ ǫ > δ > x , y ∈ X , d ( x , y ) < δ implies that ∀ n ∈ N , d ( F n ( x ) , F n ( y )) < ǫ .Recall that any CA ( S Z , F ) is equicontinuous if and only if there exist two integers q ∈ N and p > F q = F q + p [ ] . Moreover, for the subclass of LCA defined by n = Theorem 11 ( [ ] ). Let ( Z Z m , F ) be a LCA where the local rule f : ( Z m ) r + → Z m is defined by r + coeffiecientsm − r , . . . , m , . . . , m r ∈ Z m . Denote by P the set of prime factors of m. The following statements are equivalent: F is sensitive to the initial conditions; F is not equicontinuous; there exists a prime number p ∈ P which does not divide gcd ( m − r , . . . , m − , m , . . . , m r ) . The dichotomy between sensitivity and equicontinuity still holds for general LCA.
Proposition 12.
Let L = (cid:10) Z nm , r , f (cid:11) be a LCA where the local rule f : ( Z nm ) r + → Z nm is defined by r + matrices M − r , . . . , M , . . . , M r ∈ M at ( n , Z m ) . The following statements are equivalent: F is sensitive to the initial conditions; F is not equicontinuous; (cid:12)(cid:12) { M ( X ) i , i ≥ } (cid:12)(cid:12) = ∞ .Proof . The equivalence between 1. and 2. is a consequence of linearity of F and the Knudsen’s Lemma appliedon the subset of the finite configurations, i.e., those having a state different from the null vector only in a finitenumber of cells. Finally, if condition 3. is false, then F is equicontinuous, that is 2. is false too. ƒ An immediate consequence of Proposition 12 is that any characterization of sensitivity to the initial conditions interms of the matrices defining LCA over Z nm would also provide a characterization of equicontinuity. In the sequel,we are going to show that such a characterization actually exists. First of all, we recall a result that helped in theinvestigation of dynamical properties in the case n = Z nm (immediate generalisation of the result in [
3, 7 ] ).Let (cid:0) ( Z nm ) Z , F (cid:1) be a LCA and let q be any factor of m . We will denote by [ F ] q the map [ F ] q : ( Z nq ) Z → ( Z nq ) Z defined as [ F ] q ( c ) = [ F ( c )] q , for any c ∈ ( Z nq ) Z . Lemma 13 ( [
3, 7 ] ). Consider any LCA (cid:0) ( Z nm ) Z , F (cid:1) with m = pq and gcd ( p , q ) = . It holds that the given LCA istopologically conjugated to € ( Z np ) Z × ( Z nq ) Z , [ F ] p × [ F ] q Š . As a consequence of Lemma 13, if m = p k · · · p k l l is the prime factor decomposition of m , any LCA over Z nm is topologically conjugated to the product of LCAs over Z np kii . Since sensitivity is preserved under topologicalconjugacy for DDS over a compact space and the product of two DDS is sensitive if and only if at least one ofthem is sensitive, we will study sensitivity for Frobenius LCA over Z np k . We will show a decidable characterizationof sensitivity to the initial conditions for Frobenius LCA over Z np k (Lemma 30). Such a decidable characterizationtogether with the previous remarks about the decomposition of m , the topological conjugacy involving any LCAover Z nm and the product of LCAs over Z np kii , and how sensitivity behaves with respect to a topological conjugacyand the product of DDS, immediately lead to state the main result of the paper. Theorem 14.
Sensitivity and Equicontinuity are decidable for Frobenius LCA over Z nm , or, equivalently, for linearHOCA over Z m of memory size n. . Sensitivity of Frobenius LCA over Z np k In order to study sensitivity of Frobenius LCA over Z np k , we introduce two concepts about Laurent polynomials. Definition 15 ( deg + and deg − ). Given any polynomial p ( X ) ∈ Z p k (cid:2) X , X − (cid:3) , the positive (resp., negative ) degreeof p ( X ) , denoted by deg + [ p ( X )] (resp., deg − [ p ( X )] ) is the maximum (resp., minimum) degree among those ofthe monomials having both positive (resp., negative) degree and coefficient which is not multiple of p . If thereis no monomial satisfying both the required conditions, then deg + [ p ( X )] = deg − [ p ( X )] = Example 16.
Consider the Laurent polynomial p ( X ) = X − + X − + + X + X with coefficients in Z . Then, deg + [ p ( X )] = deg − [ p ( X )] = − Definition 17 (Sensitive polynomial).
A polynomial p ( X ) ∈ Z p k (cid:2) X , X − (cid:3) is sensitive if either deg + [ p ( X )] > deg − [ p ( X )] <
0. As a consequence, a Laurent polynomial p ( X ) is not sensitive iff deg + [ p ( X )] = deg − [ p ( X )] = Remark 18.
Consider a matrix M ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) in Frobenius normal form. The characteristic poly-nomial of M ( X ) is then P ( y ) = ( − ) n (cid:0) − m ( X ) − m ( X ) y − · · · − m n − ( X ) y n − + y n (cid:1) . By the Cayley-HamiltonTheorem, one obtains M n ( X ) = m n − ( X ) M ( X ) n − + · · · + m ( X ) M ( X ) + m ( X ) I . (4)We now introduce two further matrices that will allows us to access the information hidden inside M ( X ) . Definition 19 ( U ( X ) , L ( X ) , d + , and d − ). For any matrix M ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) in Frobenius normalform the matrices U ( X ) , L ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) associated with M ( X ) are the matrices in Frobenius normalwhere each component u i ( X ) and l i ( X ) (with i =
0, . . . , n −
1) of the n -th row u ( X ) and l ( X ) of U ( X ) and L ( X ) ,respectively, is defined as follows: u i ( X ) = ¨ monomial of degree deg + [ m i ( X )] inside m i ( X ) if d + i = d + l i ( X ) = ¨ monomial of degree deg − [ m i ( X )] inside m i ( X ) if d − i = d − d + i = deg + [ m i ( X )] n − i , d − i = deg − [ m i ( X )] n − i , d + = max { d + i } , and d − = min { d − i } . Example 20.
Consider the following matrix M ( X ) ∈ M at (cid:0) Z [ X , X − ] (cid:1) in Frobenius normal form M ( X ) = X − + + X + X + X X − + + X X − + X − + X X − + X − + We get d + = d + = , d + = d + = d − = − , d − = − d − = − , d − = −
1. Since d + = d − = −
1, it holds that u ( X ) = X , u ( X ) = u ( X ) = X , u ( X ) = l ( X ) = l ( X ) = X − , l ( X ) = l ( X ) = X − . Therefore, U ( X ) = X X L ( X ) = X − X − Definition 21 ( c M ( X ) and M ( X ) ). For any Laurent polynomial p ( X ) ∈ Z p k (cid:2) X , X − (cid:3) , b p ( X ) and p ( X ) are definedas the Laurent polynomial obtained from p ( X ) by removing all the monomials having coefficients that are multipleof p and p ( X ) = p ( X ) − b p ( X ) , respectively. These definitions extend component-wise to vectors. For any matrix M ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) in Frobenius normal form, c M ( X ) and M ( X ) are defined as the matrix obtainedfrom M ( X ) by replacing its n -th row m ( X ) with c m ( X ) and M ( X ) = M ( X ) − c M ( X ) , respectively.It is clear that any matrix M ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) in Frobenius normal form can be written as M ( X ) = c M ( X ) + p M ′ ( X ) , for some M ′ ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) . Definition 22 (Graph G M ). Let M ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) be any matrix in Frobenius normal form. Thegraph G M = 〈 V M , E M 〉 associated with M ( X ) is such that V M = {
1, . . . , n } and E M = { ( h , k ) ∈ V M | M ( X ) hk = } .Moreover, each edge ( h , k ) ∈ E M is labelled with M ( X ) hk .Clearly, for any matrix M ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) in Frobenius normal form, any natural t >
0, and any pair ( h , k ) of entries, the element M t ( X ) hk is the sum of the weights of all paths of length t starting from h and endingto k , where the weight of a path is the product of the labels of its edges. Example 23.
Consider any matrix M ( X ) ∈ M at (cid:0) Z p k (cid:2) X , X − (cid:3)(cid:1) in Frobenius normal form. The graph G M associated with M ( X ) is represented in Figure 1. It will help to compute M t ( X ) hk . Indeed, M t ( X ) hk is the sum ofthe labels of all paths of length t from vertex k to h (labels along edges of the same path multiply). For examplefor ( h , k ) = (
4, 4 ) one finds M ( X ) = m ( X ) M ( X ) = ( m ( X )) + m ( X ) M ( X ) = m ( X ) + m ( X ) m ( X ) + ( m ( X )) and so on. 1 2 34 m ( X ) m ( X ) m ( X ) m ( X ) Figure 1: Graph G M associated with M ( X ) from Example 23. Lemma 24.
Let p > be a prime number and a , b ≥ , k > be integers such that ≤ a < p k and gcd ( a , p ) = .Then, [ a + pb ] p k = roof . For the sake of argument, assume that [ a + pb ] p k =
0. Thus, a + pb = cp k for some c ≥ a = cp k − pb = p ( cp k − − b ) , that contradicts gcd ( a , p ) = ƒ Lemma 25.
Let p > be a prime number and h , k be two positive integers. Let l , . . . , l h and α , . . . , α h be positiveintegers such that l < l < · · · < l h and for each i =
1, . . . , h both ≤ α i < p k and gcd ( α i , p ) = hold. Considerthe sequence b : Z → Z p k defined for any l ∈ Z asb l = (cid:2) α b l − l + · · · + α h b l − l h (cid:3) p k if l > b = b l = if l < Then, it holds that [ b l ] p = for infinitely many l ∈ N .Proof . Set dz = (cid:12)(cid:12) { l ∈ N : [ b l ] p = } (cid:12)(cid:12) . For the sake of argument, assume that dz is finite. Then, there exists h ∈ N such that (cid:2) b h (cid:3) p = [ b l ] p = l > h . Thus, there exist non negative integers s , s , . . . , s h − such that b l h + h = ps and b l h + h − l i = ps i for each i =
1, . . . , h −
1. Equation (6) can be rewritten with l = l h + h as b l h + h = (cid:2) α b l h + h − l + · · · + α h − b l h + h − l h − + α h b h (cid:3) p k ,which gives ps = (cid:2) α ps + · · · + α h − ps h − + α h b h (cid:3) p k .Thus, there must exist an integer s ≥ p h − X i = α i s i + α h b h = sp k + ps ,or, equivalently, α h b h = p ‚ sp k − + s − h − X i = α i s i Œ ,with ps < p k . If sp k − + s − P h − i = α i s i =
0, we get α h b h = (cid:2) b h (cid:3) p = ≤ α h < p k . Otherwise, p must divide either α h or b h . However, that is impossible sincegcd ( α h , p ) = ( b h , p ) = ƒ For any matrix M ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) in Frobenius normal form, we are now going to study thebehavior of U t ( X ) and L t ( X ) , and, in particular, of their elements U t ( X ) nn and L t ( X ) nn . These will turn out to becrucial in order to establish the sensitivity of the LCA defined by M ( X ) . To make our arguments clearer we preferto start with an example. Example 1
Consider the following matrix M ( X ) ∈ M at (cid:0) Z [ X , X − ] (cid:1) in Frobenius normal form M ( X ) = X − + + X + X X − + + X X − + X X − + ∈ M at (cid:0) Z [ X , X − ] (cid:1) One finds d + = / U ( X ) = X X The graph G U is represented in Figure 2, along with the values U t ( X ) , (cid:2) U t ( X ) (cid:3) , and t d + , for t =
1, . . . , 8. ƒ
10 2 3416 X X U t ( X ) (cid:2) U t ( X ) (cid:3) t d + t = / t = X X t = / t = X X t = / t = X X t = / t = X X Figure 2: The graph G U (on the left), and the values U t ( X ) , (cid:2) U t ( X ) (cid:3) , and td + , for t =
1, . . ., 8 (on the right) from Example 1.
Notation 4.1.
For a sake of simplicity, for any given matrix M ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) in Frobenius normalform, from now on we will denote by u ( t ) ( X ) and l ( t ) ( X ) the elements ( U t ( X )) nn and L t ( X ) nn , respectively. Lemma 26.
Let M ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) be a matrix such that M ( X ) = Ò N ( X ) for some matrix N ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) in Frobenius normal form. For any natural t > , u ( t ) ( X ) (resp., l ( t ) ( X ) ) is either null or amonomial of degree t d + (resp., t d − ).Proof . We show that the statement is true for u ( t ) ( X ) (the proof concerning l ( t ) ( X ) is identical by replacing d + , U ( X ) and related elements with d − , L ( X ) and related elements). For each i ∈ V U , let γ i be the simple cycle of G U from n to n and passing through the edge ( n , i ) . Clearly, γ i is the path n → i → i + → n − → n (with γ n theself-loop n → n ) of length n − i + u i − ( X ) of degree ( n − i + ) d + . We know that u ( t ) ( X ) is the sum of the weights of all cycles of length t starting from n and ending to n in G U if at least one ofsuch cycles exists, 0, otherwise. In the former case, each of these cycles can be decomposed in a certain number s ≥ γ j , . . . , γ sj s of lengths giving sum t , i.e., such that P si = ( n − j i + ) = t . Therefore, ( U t ( X )) nn is a monomial of degree P si = ( n − j i + ) d + = t d + . ƒ Lemma 27.
Let M ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) be any matrix in Frobenius normal form. For every integer t ≥ both the following recurrences hold u ( t ) ( X ) = u n − ( X ) u ( t − ) ( X ) + · · · + u ( X ) u ( t − n + ) ( X ) + u ( X ) u ( t − n ) ( X ) (7) l ( t ) ( X ) = l n − ( X ) l ( t − ) ( X ) + · · · + l ( X ) l ( t − n + ) ( X ) + l ( X ) l ( t − n ) ( X ) (8) with the related initial conditions u ( ) ( X ) = l ( ) ( X ) = u ( l ) ( X ) = l ( l ) ( X ) = for l < . (10) Proof . We show the recurrence involving u ( t ) ( X ) (the proof for l ( t ) ( X ) is identical by replacing U ( X ) and itselements with L ( X ) and its elements). Since U ( X ) is in Frobenius normal form too, by (4), Recurrence (7) holdsfor every t ≥ n . It is clear that u ( ) ( X ) =
1. Furthermore, by the structure of the graph G U and the meaning of U ( X ) nn , Equation (7) is true under the initial conditions (9) and (10) for each t =
1, . . . , n − ƒ Lemma 28.
Let M ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) be a matrix such that M ( X ) = Ò N ( X ) for some matrix N ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) in Frobenius normal form. Let υ ( t ) (resp., λ ( t ) ) be the coefficient of u ( t ) ( X ) (resp., l ( t ) ( X ) ).It holds that gcd [ υ ( t ) , p ] = (resp., gcd [ υ ( t ) , p ] = ), for infinitely many t ∈ N .In particular, if the value d + (resp., d − ) associated with M ( X ) is non null, then for infinitely many t ∈ N both (cid:2) u ( t ) ( X ) (cid:3) p k = and deg ( (cid:2) u ( t ) ( X ) (cid:3) p k ) = (resp., (cid:2) l ( t ) ( X ) (cid:3) p k = and deg ( (cid:2) l ( t ) ( X ) (cid:3) p k ) = ) hold. In other terms,if d + > (resp., d − < ) then |{ u ( t ) ( X ) , t ≥ }| = ∞ (resp., |{ l t ( X ) , t ≥ }| = ∞ ). roof . We show the statements concerning υ ( t ) , U ( X ) , u ( t ) ( X ) , and d + . Replace X by 1 in the matrix U ( X ) . Now,the coefficient υ ( t ) is just the element of position ( n , n ) in the t -th power of the obtained matrix U ( ) . Over U ( ) ,the thesis of Lemma 27 is still valid replacing u ( t ) ( X ) by υ ( t ) . Thus, for every t ∈ N υ ( t ) = u n − ( ) υ ( t − ) + · · · + u ( ) υ ( t − n + ) + u ( ) υ ( t − n ) with initial conditions υ ( ) = υ ( l ) = l < u i ( ) is the coefficient of the monomial u i ( X ) inside U ( X ) . Thus, it follows that [ υ ( t )] p k = [ u n − ( ) υ ( t − ) + · · · + u ( ) υ ( t − n + ) + u ( ) υ ( t − n )] p k By Lemma 25 we obtain that gcd [ υ ( t ) , p ] = [ υ ( t )] p k =
0, too) for infinitely many t ∈ N . In particular, ifthe value d + associated with M ( X ) is non null, then, by the structure of G U and Lemma 26, both (cid:2) u ( t ) ( X ) (cid:3) p k = deg ( (cid:2) u ( t ) ( X ) (cid:3) p k ) = t ∈ N , too. Therefore, |{ u ( t ) ( X ) , t ≥ }| = ∞ . The same proofruns for the statements involving λ ( t ) , L ( X ) , u ( t ) ( X ) , and d − provided that these replace υ ( t ) , U ( X ) , u ( t ) ( X ) ,and d + , respectively. ƒ The following Lemma puts in relation the behavior of u ( t ) ( X ) or l ( t ) ( X ) with that of c M t ( X ) nn , for any matrix M ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) in Frobenius normal form. Lemma 29.
Let M ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) be a matrix in Frobenius normal form. If either |{ u ( t ) ( X ) , t ≥ }| = ∞ or |{ l ( t ) ( X ) , t ≥ }| = ∞ then |{ c M t ( X ) nn , t ≥ }| = ∞ .Proof . Assume that |{ u ( t ) ( X ) , t ≥ }| = ∞ . Since G U is a subgraph of G c M (with different labels), for each integer t from Lemma 28 applied to c M ( X ) , the cycles of length t in G c M with weight containing a monomial of degree t d + are exactly the cycles of length t in G U . Therefore, it follows that |{ c M t ( X ) nn , t ≥ }| = ∞ . The same argumenton G L and involving d − allows to prove the thesis if |{ l ( t ) ( X ) , t ≥ }| = ∞ .We are now able to present and prove the main result of this section. It shows a decidable characterizationof sensitivity for Frobenius LCA over Z np k . Lemma 30.
Let € ( Z np k ) Z , F Š be any Frobenius LCA over Z np k and let ( m ( X ) , . . . , m n − ) be the n-th row of the matrix M ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) in Frobenius normal form associated with F . Then, F is sensitive to the initialconditions if and only if m i ( X ) is sensitive for some i ∈ [ n − ] .Proof . Let us prove the two implications separately.Assume that all m i ( X ) are not sensitive. Then, c M ( X ) ∈ M at (cid:0) n , Z p k (cid:1) , i.e., it does not contain the formal variable X , and M ( X ) = c M ( X ) + p M ′ ( X ) , for some M ′ ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) in Frobenius normal form. Therefore,for any integer t > M t ( X ) is the sum of terms, each of them consisting of a product in which p j appears asfactor, for some natural j depending on t and on the specific term which p j belongs to. Since every elementof M t ( X ) is taken modulo p k , for any natural t > p j appears with j ∈ [ k − ] (we stress that j may depend on t and on the specific term of the sum, but it is always bounded by k ). Therefore, |{ M i ( X ) : i > }| < ∞ and so, by Proposition 12, F is not sensitive to the initial conditions.Conversely, suppose that m i ( X ) is sensitive for some i ∈ [ n − ] and d + > d − < t > M ′ ( X ) ∈ M at (cid:0) n , Z p k (cid:2) X , X − (cid:3)(cid:1) such that M t ( X ) = c M t ( X )+ p M ′ ( X ) . By a combination of Lemmata 28 and 29, we get |{ c M t ( X ) nn , t ≥ }| = ∞ and so, by Lemma 24, |{ M t ( X ) nn , t ≥ }| = ∞ too. Therefore, it follows that |{ M t ( X ) , t ≥ }| = ∞ and, by Proposition 12, we concludethat F is sensitive to the initial conditions. ƒ . Perspectives In this paper we have studied equicontinuity and sensitivity to the initial conditions for linear HOCA over Z m of memory size n , providing decidable characterizations for these properties. Such characterizations extend theones shown in [ ] for linear cellular automata (LCA) over Z nm in the case n = [
8, 6 ] , data compression andimage processing [ ] , where linear HOCA are involved and a chaotic behavior is required. We also proved thatlinear HOCA over Z m of memory size n form a class that is indistinguishable from a subclass of LCA (namely, thesubclass of Frobenius LCA) over Z nm . This enables to decide injectivity and surjectivity for linear HOCA over Z m ofmemory size n by means of the decidable characterizations of injectivity and surjectivity provided in [ ] and [ ] for LCA over Z nm . A natural and pretty interesting research direction consists of investigating other chaotic prop-erties as transitivity and expansivity for linear HOCA. Possible characterizations of the latter properties wouldbe useful in cryptographic applications since they would allow to design schemes that make attacks even moredifficult. Furthermore, all the mentioned dynamical properties, including sensitivity and equicontinuity, couldbe studied for the whole class of LCA over Z nm (more difficult task). Besides leading to a generalization of theshown results, such investigations would indeed help to understand also the behavior of other models used inapplications, as for instance linear non-uniform cellular automata over Z m . Let us explain precisely the meaningof such a sentence.First of all, recall that a non-uniform cellular automaton ( ν -CA) over the alphabet S is a structure (cid:10) S , { f j , r j } j ∈ Z (cid:11) defined by a family of local rules f j : S r j + → S , each of them of its own radius r j . Similarly to CA, the globalrule of a ν -CA is the map F ν : S Z → S Z defined as ∀ c ∈ S Z , ∀ i ∈ Z , F ν ( c ) i = f i ( c i − r , . . . c i + r ) ,and, so, ( S Z , F ν ) is the DDS associated with a given ν -CA. A ν -CA over the alphabet S = Z m is linear if all its localrules are linear. We are going to focus our attention on an interesting class of (linear) ν -CA, namely, the (linear)periodic ones. A periodic ν -CA ( πν -CA) is a ν -CA satisfying the following condition: there exists an integer n > f j = f j mod n for any j ∈ Z . It is clear that a πν -CA of structural period n is defined by the local rules f , . . . , f n − , which, without loss of generality, can be all assumed to have a sameradius r . The following result holds. Proposition 31.
Every linear πν -CA over Z m of structural period n is topologically conjugated to a LCA over Z nm ,and vice versa. In other terms, up to a homeomorphism, the class of linear πν -CA over Z m of structural period n isnothing but the class of LCA over Z nm .Proof . Consider any linear πν -CA of structural period n , radius r , and local rules f , . . . , f n − , where, for each j =
0, . . . , n −
1, the rule f j is defined as follows ∀ ( x − r , . . . , x r ) ∈ Z r + m , f j ( x − r , . . . , x r ) = r X i = − r a j , i x i .Set s = −⌈ r / n ⌉ and let (cid:0) ( Z nm ) Z , F (cid:1) be the LCA over Z nm such that its local rule is defined by the matrices M − s , . . . , M , . . . , M s ∈ M at ( n , Z m ) , where, for each ℓ ∈ [ − s , s ] , M ℓ = a ℓ n a ℓ n + · · · a ℓ n + n − a ℓ n − a ℓ n · · · a ℓ n + n − ... ... ... ... a n − ℓ n − n + a n − ℓ n − n + · · · a n − ℓ n ,with a j , i = i / ∈ [ − r , r ] . One gets that ϕ ◦ F ν = F ◦ ϕ , where ϕ : Z Z m → (cid:0) Z nm (cid:1) Z is the homeomorphismassociating any configuration c ∈ Z Z m with the element ϕ ( c ) ∈ (cid:0) Z nm (cid:1) Z such that for any i ∈ Z and for each j =
0, . . . , n −
1, the ( j + ) –th component of the vector ψ ( c ) i ∈ Z nm is ψ ( c ) j + i = c in + j . Then, the linear πν -CA F ν is topologically conjugated to the LCA F . Similarly, one proves that every LCA over Z π m is topologically conjugatedto a linear πν -CA over Z m of structural period π . ƒ ν -CA, and that linear πν -CA are used in many applications (forinstance, as pointed out in [ ] , they can be used as subband encoders for compressing signals and images [ ] )further motivate the investigation of LCA over Z nm in the next future. Acknowledgements
Enrico Formenti acknowledges the partial support from the project PACA APEX FRI. Alberto Dennunzio andLuca Manzoni were partially supported by Fondo d’Ateneo (FA) 2016 of Università degli Studi di Milano Bic-occa: “Sistemi Complessi e Incerti: teoria ed applicazioni”. Antonio E. Porreca was partially supported by Fondod’Ateneo (FA) 2015 of Università degli Studi di Milano Bicocca: “Complessità computazionale e applicazionicrittografiche di modelli di calcolo bioispirati”.
References [ ] Luigi Acerbi, Alberto Dennunzio, and Enrico Formenti. Shifting and lifting of cellular automata. In S. Barry Cooper, Benedikt Löwe,and Andrea Sorbi, editors,
Computation and Logic in the Real World, Third Conference on Computability in Europe, CiE 2007, Siena, Italy,June 18-23, 2007, Proceedings , volume 4497 of
Lecture Notes in Computer Science , pages 1–10. Springer, 2007. [ ] L. Le Bruyn and M. Van den Bergh. Algebraic properties of linear cellular automata.
Linear algebra and its applications , 157:217–234,1991. [ ] Gianpiero Cattaneo, Alberto Dennunzio, and Luciano Margara. Solution of some conjectures about topological properties of linearcellular automata.
Theor. Comput. Sci. , 325(2):249–271, 2004. [ ] Gianpiero Cattaneo, Enrico Formenti, Giovanni Manzini, and Luciano Margara. Ergodicity, transitivity, and regularity for linear cellularautomata over zm.
Theor. Comput. Sci. , 233(1-2):147–164, 2000. [ ] Julien Cervelle and Grégory Lafitte. On shift-invariant maximal filters and hormonal cellular automata. In
LICS: Logic in ComputerScience , pages 1–10, Reykjavik, Iceland, June 2017. [ ] Zhenchuan Chai, Zhenfu Cao, and Yuan Zhou. Encryption based on reversible second-order cellular automata. In Guihai Chen, Yi Pan,Minyi Guo, and Jian Lu, editors,
Parallel and Distributed Processing and Applications - ISPA 2005 Workshops , pages 350–358, Berlin,Heidelberg, 2005. Springer Berlin Heidelberg. [ ] Michele d’Amico, Giovanni Manzini, and Luciano Margara. On computing the entropy of cellular automata.
Theor. Comput. Sci. ,290(3):1629–1646, 2003. [ ] A. Martín del Rey, J. Pereira Mateus, and G. Rodríguez Sánchez. A secret sharing scheme based on cellular automata.
Applied Mathematicsand Computation , 170(2):1356 – 1364, 2005. [ ] Alberto Dennunzio, Enrico Formenti, Luca Manzoni, and Giancarlo Mauri. m-asynchronous cellular automata: from fairness to quasi-fairness.
Natural Computing , 12(4):561–572, 2013. [ ] Alberto Dennunzio, Enrico Formenti, and Julien Provillard. Non-uniform cellular automata: Classes, dynamics, and decidability.
Infor-mation and Computation , 215:32 – 46, 2012. [ ] Alberto Dennunzio, Enrico Formenti, and Julien Provillard. Three research directions in non-uniform cellular automata.
TheoreticalComputer Science , 559:73 – 90, 2014. Non-uniform Cellular Automata. [ ] Alberto Dennunzio, Enrico Formenti, and Michael Weiss. Multidimensional cellular automata: closing property, quasi-expansivity, and(un)decidability issues.
Theoretical Computer Science , 516:40–59, 2014. [ ] Alberto Dennunzio, Pierre Guillon, and Benoît Masson. Stable dynamics of sand automata. In Giorgio Ausiello, Juhani Karhumäki,Giancarlo Mauri, and C.-H. Luke Ong, editors,
Fifth IFIP International Conference On Theoretical Computer Science - TCS 2008, IFIP20th World Computer Congress, TC 1, Foundations of Computer Science, September 7-10, 2008, Milano, Italy , volume 273 of
IFIP , pages157–169. Springer, 2008. [ ] Alberto Dennunzio, Pietro Di Lena, Enrico Formenti, and Luciano Margara. Periodic orbits and dynamical complexity in cellular au-tomata.
Fundam. Inform. , 126(2-3):183–199, 2013. [ ] Jing Gu and Dianxun Shuai. The faster higher-order cellular automaton for hyper-parallel undistorted data compression.
Journal ofComputer Science and Technology , 15(2):126, Mar 2000. [ ] Gustav Arnold Hedlund. Endomorphisms and automorphisms of the shift dynamical system.
Mathematical Systems Theory , 3:320–375,1969. [ ] T.E. Ingerson and R.L. Buvel. Structure in asynchronous cellular automata.
Physica D: Nonlinear Phenomena , 10(1):59 – 68, 1984. [ ] M. Ito, N. Osato, and Masakazu Nasu. Linear cellular automata over Z m . Journal of Computer and Systems Sciences , 27:125–140, 1983. [ ] Jarkko Kari. Linear cellular automata with multiple state variables. In H. reichel and Sophie Tison, editors,
STACS 2000 , volume 1770of
LNCS , pages 110–121. Springer-Verlag, 2000. [ ] C. Knudsen. Chaos without nonperiodicity.
American Mathematical Monthly , 101:563–565, 1994. [ ] P. K˚urka. Languages, equicontinuity and attractors in cellular automata.
Ergodic Theory & Dynamical Systems , 17:417–433, 1997. [ ] Giovanni Manzini and Luciano Margara. Attractors of linear cellular automata.
J. Comput. Syst. Sci. , 58(3):597–610, 1999. [ ] Giovanni Manzini and Luciano Margara. A complete and efficiently computable topological classification of d-dimensional linear cellularautomata over zm.
Theor. Comput. Sci. , 221(1-2):157–177, 1999. [ ] Birgitt Schönfisch and André c (cid:13) de Roos. Synchronous and asynchronous updating in cellular automata.
Biosystems , 51(3):123 – 143,1999. ] Jerome M. Shapiro. Embedded image coding using zerotrees of wavelet coefficients.
IEEE Trans. Signal Processing , 41(12):3445–3462,1993. [ ] Tommaso Toffoli. Computation and construction universality.
Journal of Computer and Systems Sciences , 15:213–231, 1977., 15:213–231, 1977.