An Easy to Check Characterization of Positive Expansivity for Additive Cellular Automata Over a Finite Abelian Group

Additive cellular automata over a finite abelian group are a wide class of cellular automata (CA) that are able to exhibit most of the complex behaviors of general CA and they are often exploited for designing applications in different practical contexts. We provide an easy to check algebraic characterization of positive expansivity for Additive Cellular Automata over a finite abelian group. We stress that positive expansivity is an important property that defines a condition of strong chaos for CA and, for this reason, an easy to check characterization of positive expansivity turns out to be crucial for designing proper applications based on Additive CA and where a condition of strong chaos is required. First of all, in the paper an easy to check algebraic characterization of positive expansivity is provided for the non trivial subclass of Linear Cellular Automata over the alphabet $(\mathbb {Z}/m \mathbb {Z})^{n}$ . Then, we show how it can be exploited to decide positive expansivity for the whole class of Additive Cellular Automata over a finite abelian group.


I. INTRODUCTION
Cellular automata (CA) are well-known formal models that find application in several disciplines and their different subdomains.This is essentially due to three reasons: the huge variety of distinct CA dynamical behaviors; the emergence of complex behaviors from simple local interactions; the ease of their implementation (even at a hardware level).
In practical applications one needs to know if the CA used for modelling a certain system exhibits some specific property.However, this can be a severe issue.Indeed, a strong result by Jarkko Kari [6] states that all non-trivial dynamical behaviors are undecidable.From this seminal result, a long sequence followed.
Luckily, the undecidability issue can be tackled by imposing some constraints on the model.In the specific case of The associate editor coordinating the review of this manuscript and approving it for publication was Ludovico Minati .this paper, the alphabet and the global updating map are constrained to be a finite abelian group and an additive function, respectively, giving rise to Additive CA over a finite abelian group or, briefly, Additive CA (see [1], [2], [12], e.g., for studies regarding linear and group CA).We stress that such requirements do not prevent Additive CA at all from being successfully used for practical purposes.On the contrary, since Additive CA are able to exhibit most of the complex behaviors of general CA, they are often exploited for designing many applications (see, for instance, [11], [13]).Moreover, Additive CA are more expressive and they give rise to much more complex dynamics than the already investigated subclass of Linear Cellular Automata over the alphabet (Z/mZ) n with n = 1.
Among the dynamical properties, positive expansivity received and still receives a significant attention by researchers since it is a stronger form of sensitive dependence on the initial conditions, the latter being the essence of a chaotic behavior.Moreover, in CA settings positive expansivity plays an important role since it is just a condition of strong chaos, since positively expansive CA are chaotic besides exhibiting sensitive dependence on the initial conditions.While the related (un)decidability issue is still an open problem for general CA, positive expansivity turns out to be decidable as far as Additive CA are concerned.Indeed, this decidability result follows from a combination of two known facts: i) positive expansivity is decidable for CA with sofic traces [5]; ii) the traces of any Additive CA are subshifts of finite type (and, hence, they are sofic), a consequence of the classical result by Kitchens and Schmidt that all group subshifts are of finite type [7].However, the decidability result is not useful for practical purposes at all, i.e., the corresponding algorithm is impractical.Thus, besides being of theoretical interest, an easy to check characterization of positive expansivity for Additive CA would turn out to be crucial for designing proper applications based on such CA and in situations where, as often happens, a condition of strong chaos is required.
In this paper we provide an easy to check characterization of positive expansivity for Additive CA over a finite abelian group.First of all, an easy to check algebraic characterization of positive expansivity is provided for the non trivial subclass of Linear Cellular Automata over the alphabet (Z/mZ) n , where m is any natural greater than 1.Namely, for any Linear CA over (Z/mZ) n such a characterization is expressed as an easy to check condition on the degrees of the coefficients (that are Laurent polynomials) of the characteristic polynomial of the matrix associated with the Linear CA.Then, we show how it can be exploited to decide positive expansivity for the whole class of Additive Cellular Automata over a finite abelian group.
The main and more difficult part of this work consists of the proof of an easy to check algebraic characterization of positive expansivity for Linear Cellular Automata over (Z/pZ) n , where p is any prime number (Theorem 1).To reach that result 1) first of all, we provide an easy to check algebraic characterization of positively expansive Linear Cellular Automata over (Z/pZ) n with associated matrix that is (the traspose of one) in a rational canonical form consisting of only one block; this is the heart of our work; 2) then, we prove that such an algebraic characterization turns out to hold also for positively expansive Linear Cellular Automata over (Z/pZ) n with associated matrix that is in a rational canonical form possibly consisting of more than one block; 3) finally, we prove that such an algebraic characterization turns out to hold also for all positively expansive Linear Cellular Automata over (Z/pZ) n .
Afterwards, the easy to check algebraic characterization of positive expansivity is extended first from Linear Cellular Automata over (Z/pZ) n to Linear Cellular Automata over where k is any non zero natural (Theorem 2).If the prime factor decomposition of m is assumed to be known, that characterization immediately extends to Linear Cellular Automata over (Z/mZ) n (Corollary 1).However, since the prime factor decomposition is a well-known difficult task (i.e., no algorithm has been published yet that can factor any natural in polynomial time), it should be avoided, especially for practical purposes.Actually, we have gone one step further: in Section III-B we show how the easy to check characterization of positive expansivity for Linear Cellular Automata over (Z/pZ) n can be exploited to decide positive expansivity for Linear Cellular Automata over (Z/mZ) n in an efficient way (i.e., without decomposing m into its prime factors and by only making use of gcd operations) and, at the end, for whole class of Additive Cellular Automata over a finite abelian group (Theorem 3).The paper is structured as follows.Next section introduces all the necessary background and notions.Section III contains our results.Section IV is entirely devoted to the proof of Theorem 1.In the last section we draw our conclusion.

II. BASIC NOTIONS
Let K be any commutative ring and let A ∈ K n×n be an n × nmatrix over K.We denote by A T the transpose matrix of A and by χ A the characteristic polynomial det (tI n − A) ∈ K [t] of A, where I n always stands for the n × n identity matrix (over whatever ring we are considering).Furthermore, K[X , X −1 ] and K[[X , X −1 ]] denote the set of Laurent polynomials and series, respectively, with coefficients in K.In particular, whenever K = Z/mZ for some natural m > 1, we will write L m and S m instead of Z/mZ[X , X −1 ] and Z/mZ[[X , X −1 ]], respectively.
Let K = Z/mZ for some natural m > 1 and let q be a natural with q < m.If P is any polynomial from K[t] (resp., a Laurent polynomial from L m ) (resp., a matrix from (L m ) n×n ), P mod q denotes the polynomial (resp., the Laurent polynomial) (resp., the matrix) obtained by P by taking all its coefficients modulo q.
Let be a finite set (also called alphabet).A CA configuration (or, briefly, a configuration) is any function from Z to .Given a configuration c ∈ Z and any integer i ∈ Z, the value of c in position i is denoted by c i .The set Z , called configuration space, is as usual equipped with the standard Tychonoff distance d defined as Whenever the term linear is involved the alphabet is K n , where K = Z/mZ for some natural m > 1.Clearly, in that case both K n and (K n ) Z become K-modules in the obvious (i.e., entrywise) way.On the other hand, whenever the term additive is involved the alphabet is a finite abelian group G and the configuration space turns G Z turns out to be an abelian group, too, where the group operation of G Z is the componentwise extension of the group operation of G, both of them will be denoted by +.
A one-dimensional CA (or, briefly, a CA) over is a pair ( Z , F), where F : Z → Z is the uniformly continuous transformation (called global rule) defined as ∀c ∈ Z , ∀i ∈ Z, F(c) i = f (c i−r , . . ., c i+r ), for some fixed natural number r ∈ N (called radius) and some fixed function f : 2r+1 → (called local rule of radius r).In the sequel, when no misunderstanding is possible, we will sometimes identify any CA with its global rule.
We recall that a CA ( Z , F) is positively expansive if for some constant ε > 0 it holds that for any pair of distinct configurations c, c ′ ∈ Z there exists a natural number ℓ such that d(F ℓ (c), F ℓ (c ′ )) ≥ ε.We stress that CA positive expansivity is a condition of strong chaos.Indeed, on a hand, positive expansivity is a stronger condition than sensitive dependence on the initial conditions, the latter being the essence of the chaos notion.On the other hand, any positively expansive CA is also topologically transitive and, at the same time, it has dense periodic orbits.Therefore, any positively expansive CA is chaotic according to the Devaney definition of chaos (see [4], for the definitions of chaos, sensitive dependence on the initial conditions, topological transitivity, and denseness of dense periodic orbits).Finally, we recall that if a CA F is positively expansive then F is surjective.

A. LINEAR AND ADDITIVE CA
Let K = Z/mZ for some natural m > 1 and let n ∈ N with n ≥ 1.Let G be a finite abelian group.
A local rule f : (K n ) 2r+1 → K n of radius r is said to be linear if it is defined by 2r + 1 matrices A −r , . . ., A r ∈ K n×n as follows: ∀(x −r , . . ., x r ) ∈ (K n ) 2r+1 , f (x −r , . . ., x r ) = r i=−r A i • x i .A one-dimensional linear CA (LCA) over K n is a CA F based on a linear local rule.
The Laurent polynomial (or matrix) is said to be the the matrix associated with F.
We now recall the notion of Additive CA, a wider class than LCA.An Additive CA over G is a CA (G Z , F) where the global rule F : G Z → G Z is an endomorphism of G Z .We stress that the local rule f : G 2r+1 → G of an Additive CA of radius r over a finite abelian group G can be written as ∀(x −r , . . ., x r ) ∈ G 2r+1 , f (x −r , . . ., x r ) = r i=−r f i (x i ), where the functions f i are endomorphisms of G.Moreover, as a consequence of the application of the fundamental theorem of finite abelian groups to Additive CA (see [3], for details), without loss of generality we can assume that Let Ĝ = (Z/p k 1 Z) n and let ψ : G → Ĝ be the map defined as ∀h ∈ G, ∀i = 1, . . ., n, ψ(h) i = h i p k 1 −k i , where, for a sake of clarity, we stress that h i denotes the i-th component of h, while p k 1 −k i is just the (k 1 − k i )-th power of p.Let : G Z → ĜZ be the componentwise extension of ψ, i.e., the function defined as ∀c ∈ G Z , ∀j ∈ Z, (c) j = ψ(c j ).The function turns out to be continuous and injective.We recall that for any Additive CA over G an LCA over (Z/p k 1 Z) n associated with it can be defined as follows.With a further abuse of notation, in the sequel we will write p −m with m ∈ N even if this quantity might not exist in Z/p k Z.However, we will use it only when it multiplies p m ′ for some integer m ′ > m.In such a way p m ′ −m is well-defined in Z/p k Z and we will note it as product p −m • p m ′ .Let (G Z , F) be any Additive CA and let f : G 2r+1 → G be its local rule defined, by 2r + 1 endomorphisms f −r , . . ., f r of G.For each z ∈ {−r, . . ., r}, let A z = (a (z) i,j ) 1≤i≤n, 1≤j≤n ∈ (Z/p k 1 Z) n×n be the matrix such that ∀i, j ∈ {1, . . ., n}, a , where L is defined by A −r , . . ., A r or, equivalently, by A = r z=−r A z X −z ∈ Z/p k 1 Z X , X −1 n×n .We stress that the following diagram commutes i.e., L • = • F. Therefore, ( ĜZ , L) is said to be the LCA associated with (G Z , F) via the embedding .In general, (G Z , F) is not topologically conjugated (i.e., homeomorphic) to ( ĜZ , L) but (G Z , F) is a subsystem of ( ĜZ , L) and the latter condition alone is not enough in general to lift dynamical properties from a one system to the other one.Despite this obstacle, in the sequel we will succeed in doing such a lifting, as far as positive expansivity is concerned.

III. RESULTS
The following two notions are fundamental throughout this paper.
Definition 1 (Positive and Negative Degree): The positive (resp., negative) degree of any given polynomial α ∈ L m with α ̸ = 0, denoted by deg + (α) (resp., deg − (α)), is the maximum (resp, minimum) value among the degrees of the monomials of α.Such notions extend to any element υ ̸ = 0 of S n m when υ is considered as a formal power series with coefficients in (Z/mZ) n instead of a vector of n elements from S m and with the additional defining clause that deg maximum (resp., minimum) does not exist.Furthermore, the previous notions are extended to both α = 0 and υ = 0 as follows: Example 1: The following are the values of the positive and negative degree of some polynomials: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
from L m [t].We say that π (t) is expansive if both the following two conditions are satisfied: +t n is expansive then it must necessarily hold that α 0 ̸ = 0.
Lemma 1: For any three polynomials π, ρ, τ ∈ L p [t] such that π = ρ • τ it holds that π is expansive if and only if ρ and τ are both expansive.
We prove now that if π is expansive then ρ and τ are both expansive.Assume that the consequent is not true.We deal with the two following cases: (1) ρ is expansive but τ is not (by symmetry, we do not consider the situation in which τ is expansive but ρ is not); (2) neither ρ nor τ are expansive.
(1) Suppose that ρ is expansive but τ is not.If γ 0 = 0 then it trivially follows that π is not expansive.
Otherwise, let min be the minimum index such that Since ρ is expansive, β 0 γ min is the (only) addend of maximum degree in the sum from Equation (1) considered for i = min.Thus, deg . In a symmetric way, one also gets that deg − (α i ) ≤ deg − (α 0 ) for some i ∈ {1, . . ., n − 1}.(2) Suppose that neither ρ nor τ are expansive.If β 0 = 0 ∨ γ 0 = 0 then it trivially follows that π is not expansive.Otherwise, let max 1 and max 2 be the maximum indexes such that deg + (β max 1 ) = max 0≤i 1 <n 1 {deg + (β i 2 ) : , respectively.Consider now Equation ( 1) for i = max 1 + max 2 .Take any pair of indexes i 1 , i 2 of the sum such that i 1 ̸ = max 1 , i 2 ̸ = max 2 , β i 1 ̸ = 0, and γ i 2 ̸ = 0.If i 1 < max 1 (and then i 2 > max 2 ), resp., if i 1 > max 1 (and then i 2 < max 2 ), we get that ) and in a symmetric way, one also gets that deg Therefore, in both cases it follows that π is not expansive and this concludes the proof.□ In the sequel, we will often make use of the following Definition 3: Let K ⊆ S n m .For any s ∈ Z we define the following sets: Definition 3 along with the notion of positively expansive CA when reformulated for LCA and the compactness of the configuration space immediately allow stating the following Lemma 2: Let F be any LCA over (Z/mZ) n and let A ∈ L n×n m be the matrix associated with F, where m and n are any two naturals with m > 1 and n > 1.The LCA F is not positively expansive if and only if there exists an integer s > 0 such that at least one of the following two conditions holds It is an immediate consequence of the fact that F is open and, hence, it is both left and right closing (see [8]).
□ We now state the result that is the heart of our work, i.e., providing an easy to check characterization of positive expansivity for LCA over (Z/pZ) n .Since its proof is very long, we place it in Section IV.
Theorem 1: Let F be any LCA over (Z/pZ) n where p and n are any two naturals such that p is prime and n > 1.The LCA F is positively expansive if and only if the matrix associated with F is expansive.
The following Theorem extends the characterization result provided by Theorem 1 from LCA over (Z/pZ) n to LCA over (Z/p k Z) n .
Theorem 2: Let G be any LCA over (Z/p k Z) n and let B ∈ L n×n p k be the matrix associated with G, where p, k, n are any three naturals such that p is prime, k > 1, and n > 1.The LCA G is positively expansive if and only if the LCA F over (Z/pZ) n having A as associated matrix is too, where A = (B mod p) ∈ L n×n p .Equivalently, G is positively expansive if and only if the matrix B mod p is expansive.
Proof: We start to prove that if G is positively expansive then F is too.Let us suppose that F is not positively expansive and the first condition from Lemma 2 holds, i.e., there exists υ ∈ Left(S n p , 0) with υ ̸ = 0 such that A ℓ υ ∈ Left * (S n p , s) for every natural ℓ > 0, where s > 0 is some integer constant depending on F (the proof is symmetric if one supposes that the second condition holds).Set ω = p k−1 υ.Clearly, ω ∈ Left(S n p k , 0) and ω ̸ = 0. Since B can be written as A + pN for some matrix N ∈ L n×n p k−1 , it holds that for every natural ℓ > 0 where all the sums and products of coefficients of the Laurent polynomials/series inside the previous equalities are now meant in Z p k .Hence, although A ∈ L n×n p and υ ∈ S n p , in general A ℓ υ now belongs to S n p k instead of S n p and, as a consequence, we can not immediately conclude that B ℓ ω = p k−1 A ℓ υ ∈ Left * (S n p k , s) immediately follows just from the hypothesis as it is, i.e., A ℓ υ ∈ Left * (S n p , s) when A ℓ υ is considered as an element of S n p .However, since A ℓ υ ∈ S n p k can be written as (A ℓ υ) mod p + pω ′ for some ω ′ ∈ S n p k−1 and, by hypothesis, (A ℓ υ) mod p ∈ Left * (S n p , s), we get that for every natural ℓ > 0. Therefore, by Lemma 2, G is not positively expansive.
We now prove that if F is positively expansive then G is too.Let us suppose that G is not positively expansive and the first condition from Lemma 2 holds, i.e., there exists ω ∈ Left(S n p k , 0) with ω ̸ = 0 such that B ℓ ω ∈ Left * (S n p k , s) for every natural ℓ > 0, where s > 0 is some integer constant depending on G (again, the proof is symmetric if one supposes that the second condition holds).We deal with the following two mutually exclusive cases.
If there is no ω ′ such that ω = pω ′ , then set υ = ω mod p.Clearly, υ ̸ = 0 and υ ∈ Left(S n p , h) for some integer h ≤ 0.Moreover, it holds that A ℓ υ = (B mod p) ℓ (ω mod p) = (B ℓ ω) mod p ∈ Left * (S n p , s) for every natural ℓ > 0. Otherwise, let j ∈ {1, . . ., k − 1} and ω ′ ∈ S n p k be such that ω = p j ω ′ , where j is the largest natural such that all the coefficients of the n Laurent series forming ω are multiple of p j .Set υ = ω ′ mod p.Clearly, υ ̸ = 0 and υ ∈ Left(S n p , h) for some integer h ≤ 0. Since ∈ Left * (S n p k , s) happens, but, in both situations it must hold that (B ℓ ω ′ ) mod p ∈ Left * (S n p , s).Therefore, it follows that A ℓ υ = (B mod p) ℓ (ω ′ mod p) = (B ℓ ω ′ ) mod p ∈ Left * (S n p , s) for every natural ℓ > 0. In both cases, the first condition of Lemma 2 is satisfied as far as F is concerned.Thus, F is not positively expansive and this concludes the proof that G is positively expansive if and only if F is positively expansive.By Theorem 1, it follows that G is positively expansive if and only if the matrix B mod p is expansive.
□ At this point, we are able to extend the characterization result regarding positive expansivity to the whole class of LCA over (Z/mZ) n where m is any natural with m > 1.
Corollary 1: Any LCA F over (Z/mZ) n is positively expansive if and only if A mod p j is expansive for every j ∈ {1, . . ., l}, where A is the matrix associated with F and p 1 , . . ., p l are all the primes appearing in the prime factor decomposition of m (i.e., m

is the prime factor decomposition of m).
Proof: For each j ∈ {1, . . ., l}, let F j be the LCA over (Z/(p j ) k j Z) n having (A mod (p j ) k j ) ∈ L n×n (p j ) Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

B. BYPASSING THE PRIME FACTOR DECOMPOSITION OF m IN THE CHARACTERIZATION OF POSITIVE EXPANSIVITY FOR LCA OVER (Z/mZ) n
The characterization of positively expansive LCA over (Z/mZ) n provided by Corollary 1 requires that the prime factor decomposition of m is known.We now illustrate how deciding positive expansivity without decomposing m into prime factors and by only making use of gcd operations.Let m = p k 1 1 • • • p k l l be the prime factor decomposition of m and let A be the matrix associated with a given LCA F over (Z/mZ) n .By Corollary 1, F is positively expansive if and only if both the following conditions are satisfied: (C1) for every j ∈ {1, . . ., l} it holds that deg + (α 0 mod p j ) > 0 and deg − (α 0 mod p j ) < 0; (C2) for every i ∈ {1, . . ., n − 1} it holds that (C2a) for every j ∈ {1, . . ., l}, deg + (α 0 mod p j ) > deg + (α i mod p j ) and (C2b) for every j ∈ {1, . . ., l}, deg − (α 0 mod p j ) < deg − (α i mod p j ), where It is not difficult to see that, by the characterization of positively expansive LCA over Z/mZ provided in [9], condition (C1) can be rewritten in an equivalent form as follows: where A −r , . . ., A r ∈ Z/mZ define α 0 .Therefore, (C1) is an efficiently testable condition.
We are now going to provide a method that checks condition (C2) without decomposing m in prime factors and under the assumption that condition (C1) is satisfied.To proceed, for any a ∈ Z/mZ define otherwise, where P a = {j ∈ {1, . . ., l} : gcd(a, p j ) = 1}.We emphasize that y a is the greatest divisor of m having as prime factors all (and only) those prime factors of m that are not prime factors of a.Moreover, y a can be computed without knowing p 1 , . . ., p l and k 1 , . . ., k l , i.e., without the need of decomposing m into its prime factors.Indeed, consider the elements of the sequence m ℓ recursively defined by m ℓ+1 = m ℓ / gcd(m ℓ , a), where m 0 = m.Clearly, there exists ℓ * such that m ℓ * +1 = m ℓ * and it holds that y a = m ℓ * +1 = m ℓ * .

C. AN EASY TO CHECK CHARACTERIZATION OF POSITIVE EXPANSIVITY FOR ADDITIVE CA OVER A FINITE ABELIAN GROUP
Finally, we lift the characterization result regarding positive expansivity for LCA over (Z/mZ) n to the whole class Additive CA over any finite abelian group.We stress that although the characterization result is stated for Additive CA over G = Z/p k 1 Z × . . .× Z/p k n Z, this is not a restriction.Indeed, any finite abelian group is isomorphic to a direct sum of a certain number of its subgroups (with pairwise coprime cardinality), each of them being as such a G, and an Additive CA over a finite abelian group splits into the direct sum of Additive CA over those subgroups.Hence, the former CA turns out to be positively expansive if and only if all the CA components of that sum are positively expansive.
Theorem 3: Let F : G Z → G Z be any Additive CA over a finite abelian group G, where G = Z/p k 1 Z × . . .× Z/p k n Z for some prime p and some non zero naturals k 1 , . . ., k n with k 1 ≥ k 2 ≥ . . .≥ k n .Let L be the LCA over Ĝ associated with F via the embedding , where Ĝ = (Z/p k 1 Z) n .It holds that F is positively expansive if and only if L is positively expansive if and only if A is expansive, where A is the matrix associated with L.
Proof: We are going to show that F is positively expansive if and only if L is too.By Theorem 2, this is enough to conclude that the statement is true.
We start to prove that if L is positively expansive then F is too.Let us suppose that F is not positively expansive. □

IV. PROOF OF THEOREM 1
We now recall some useful notions and known fact from abstract algebra.In the sequel, the standard acronyms PID and UFD stand for principal ideal domain and unique factorization domain, respectively.Let P be PID and let A ∈ P n×n .The elementary divisors, or invariants, or invariant factors associated with A are the elements a 1 , . . ., a n ∈ P defined as follows: ∀i ∈ {1, . . ., n}, a i = i (A)/ i−1 (A), where i (A) is the greatest common divisor of the i-minors of A and 0 (A) = 1.
The companion matrix of a monic polynomial π (t where each C π i is the companion matrix of some monic polynomial π i of non null degree and π i divides π j for i ≤ j.
It is well-known that if A ∈ F n×n is any matrix where F is a field all the following facts hold: -A is similar to a unique matrix in rational canonical form and this latter is called the rational canonical form of A; -the monic polynomials π 1 , . . ., π s defining the blocks of the rational canonical form of A are the nonconstant invariant factors of tI − A; -χ A (t) = s i=1 π i (t), where π s is the minimal polynomial of A; -there exist v 1 , . . ., v s ∈ F n such that is a basis of F n with respect to which A becomes in rational canonical form C, i.e., A = P −1 CP, where d i = deg(π i ) and P is the matrix having the elements of that basis as columns.
We now report the following known result which will be very useful in the sequel (see [10, Proposition 1], for instance).
Lemma 4: Let U be a UFD and let F U be the field of fractions of U .For any monic polynomials π The following is an important consequence of Lemma 4.
Lemma 5: Let U be a UFD and let F U be the field of fractions of U .Let A = U n×n and let π 1 , . . ., π s ∈ F U [t] be the invariant factors of tI − A when A is considered as an element of F n×n U .Then, for every i ∈ {1, . . ., s} it holds that Proof: Since π 1 , . . ., π s , χ A are all monic and χ , by a repeated application of Lemma 4, we get that every □ We now deal with the algebraic structures of our interest, namely, the PID L p and the UFD L p [t].Clearly, L p is also an UFD, but, since L p is not a field, L p [t] is not a PID, while F p [t] is, where F p is the field of fraction of L p .Therefore, we can not refer to invariant factors when the involved set is L p [t], while we can as far as F p [t] is concerned.
Lemma 6: For any matrix A ∈ L n×n p with det(A) ̸ = 0 there exist two matrices Q, C ∈ L n×n p such that det(Q) ̸ = 0, C is in rational canonical form, QA = CQ, and χ A = χ C .
Proof: Choose arbitrarily a matrix A ∈ L n×n p .Cleary, it holds that A ∈ F n×n p , where F p is the field of fractions of L p .Hence, there exist v 1 , . . ., v s ∈ F n such that, A = P −1 CP, C ∈ F n×n p is the matrix in canonical rational form, the blocks of which are defined by the invariant factors π 1 , . . ., π s ∈ F p [t] of tI − A, and P is the matrix having the elements of the basis {v 1 , Av 1 , . . ., p , as desired.Furthermore, by Lemma 5, it follows that C ∈ L n×n p , too.Moreover, we get that Therefore, QA = CQ and this concludes the proof.□ Lemma 7: For any A, B, Q ∈ L n×n p such that det(Q) ̸ = 0 and AQ = QB it holds that the LCA F over (Z/pZ) n having A as associated matrix is positively expansive if and only if the LCA G over (Z/pZ) n having B as associated matrix is, too.
Proof: First of all, it is easy to see that A ℓ Q = QB ℓ for every natural ℓ > 0.Moreover, det(A) = 0 if and only if det(B) = 0. So, the thesis turns out to be trivially true if det(A) = det(B) = 0. Therefore, in the sequel of the proof, we will assume that det(A) ̸ = 0 and det(B) ̸ = 0, i.e., both F and G are surjective.
We now start to prove that if F is positively expansive then G is too.Suppose that G is not positively expansive and the first condition from Lemma 2 holds, i.e., there exists υ ∈ Left(S n p , 0) with υ ̸ = 0 such that B ℓ υ ∈ Left * (S n p , s) for every natural ℓ > 0, where s > 0 is some integer constant depending on G (the proof is symmetric if one supposes that the second condition holds).Since Q is the matrix associated with a surjective LCA over (Z/pZ) n condition C 3 of Lemma 3 is satisfied as far as Q is concerned.Hence, setting ω = Qυ, for some natural constant c > 0 depending on Q and some integer h with −c ≤ h ≤ c, it holds that A ℓ ω = QB ℓ υ ∈ Left * (S n p , h) for every natural ℓ > 0. Clearly, ω ̸ = 0 since det(Q) ̸ = 0.In addition, it holds that ω ∈ Left(S n p , h ′ ) for some integer h ′ .Thus, it follows that F is not positively expansive.
We now prove that if G is positively expansive then F is too.Assume that F is not positively expansive and the first condition from Lemma 2 holds, i.e., there exists υ ∈ Left(S n p , 0) with υ ̸ = 0 such that A ℓ υ ∈ Left * (S n p , s) for every natural ℓ > 0, where s > 0 is some integer constant depending on G (again, the proof is symmetric if one supposes that the second condition holds).Since Q is the matrix associated with a surjective LCA over (Z/pZ) n , condition C 1 of Lemma 3 is satisfied as far as Q is concerned.Thus, for some natural constant c > 0 depending on Q and some integer h with −c ≤ h ≤ c, there exists ω ∈ Left(S n p , h) such that Qω = υ.Clearly, ω ̸ = 0 since υ ̸ = 0. Furthermore, QB ℓ ω = A ℓ Qω = A ℓ υ ∈ Left * (S n p , s) for every natural ℓ > 0. Therefore, there exists an integer constant s ′ > s > 0 such that B ℓ ω ∈ Left * (S n p , s ′ ) for every natural ℓ > 0. So, by Lemma 2, G is not positively expansive and this concludes the proof.
call pick any natural J > 0 such that deg + (υ j ) ≤ deg + (υ J ) for every natural j with 0 < j < J .It holds that for any pick J there exists a pick J ′ ∈ {J + 1, . . ., J + n − min + 1}.In particular, for any pick the number of positions within which there is a further pick does not depend on the values of the initial elements υ 1 , . . ., υ n of the sequence.Proof: Clearly, the set of picks is non empty since 1, . . ., n are picks.Let J be any pick.If there exists j ∈ {J + 1, . . ., J + n − min} such that deg + (υ j ) ≥ deg + (υ J ), necessarily there must be a further pick inside the integer interval {J + 1, . . ., J + n − min}.Otherwise, it holds that deg + (υ j ) < deg + (υ J ) for every j ∈ {J + 1, . . ., J + n − min} and, since it follows that the natural J + n − min + 1 turns out to be a pick.
□ The following result is the heart of our work.It provides a decidable characterization of positively expansive LCA over (Z/pZ) n with associated matrix such that its transpose is in a rational canonical form consisting of only one block.
Lemma 9: Let A ∈ L n×n p be the matrix such that its transpose A T is the companion matrix of any monic polynomial where p and n are any two naturals such that p is prime and n > 1, and let F be the LCA over (Z/pZ) + t n is expansive.Since F is surjective if and only if −α 0 ̸ = 0, the thesis turns out to be trivially true if α 0 = 0. Hence, in the sequel of the proof we can assume that α 0 ̸ = 0.
Proof: Clearly, A is as in (2).It holds that A T Q = QA, where Since det(Q) ̸ = 0, the thesis directly follows from Lemma 7. □ We now prove that the decidable characterization of positive expansivity provided by Lemma 9 also holds also in a more general situation, namely, for LCA over (Z/pZ) n with associated matrix that is in a rational canonical form possibly consisting of more than one block.
Lemma We provided an easy to check algebraic characterization of positive expansivity for Additive CA over a finite abelian group.Besides having a theoretical value, this characterization turns out to be useful for designing proper applications based on these CA and where a condition of strong chaos is required.Providing (efficient) algorithms that, as far as such CA are concerned, decide other meaningful dynamical properties such as strong transitivity or compute some useful quantities as topological entropy is an important step for further research in this domain.This would also allow one to build even more robust methods based on such CA in applications.Another important research direction consists in considering the multidimensional setting.Besides having a theoretical value, providing algorithms that decide dynamical properties for multidimensional Additive CA will be certainly useful in many applications involving multidimensional data.

0 .m
of degree d inside α i , with a little abuse of notation, let us denote by y i,d the quantity y a (i) d .Clearly, for every j ∈ P a (i) d it holds that a (i) d mod p j ̸ = 0. Fix now i ∈ {1, . . ., n − 1}.The following procedure to be repeated for every i ∈ {1, . . ., n − 1} tests condition (C2a), i.e., as far as α i is concerned, it checks whether deg + (α 0 mod p j ) > deg + (α i mod p j ) for every j ∈ {1, . . ., l}.The procedure consists of the following steps: (S1) Let a (0) d 0 X d 0 be the monomial of maximum degree inside α 0 with y 0,d 0 ̸ = 1.If condition (C1) is satisfied then d 0 > 0. We stress that a (0) d 0 mod p j ̸ = 0 for every j ∈ P a (0) of degree d ≥ d 0 inside α i (while jump to step (S3.2) if deg + (α i ) < d 0 ).For each of such monomials compute y i,d and g d = gcd(y 0,d 0 , y i,d ).(S3.1)If g d > 1 for some d ≥ d 0 , it means that there exists j ∈ P a (0) d 0 ∩ P a (i) d ̸ = ∅ such that deg + (α 0 mod p j ) ≤ deg + (α i mod p j ) and, hence, (C2a) is not satisfied.(S3.2) Otherwise, since a (0) d 0 mod p j ̸ = 0 and a (i) d mod p j = 0 for every j ∈ P a (0) d 0 and every d ≥ d 0 , the inequality inside condition (C2a) holds for every j ∈ P a (0) d To check if it also holds for every j / ∈ P a (0) by m/y 0,d 0 , and, referring to this new value of m, if m ̸ = 1, restart from step (S1) with (α 0 − a (0) d 0 X d 0 ) mod m and α 1 mod m in place of α 0 and α 1 , respectively, inside the new background Z/mZ.If, on the contrary, m = 1, it means that condition (C2a) is satisfied.
s)On the contrary, the LCA F is positively expansive if and only if there exists a natural number l > 0 such that for any d ∈ Z and any υ ∈ Left(S n m , d) it holds that A ℓ υ ∈ Left(S n m , d + 1) for some ℓ ≤ l and, symmetrically, for any d ∈ Z and any υ ∈ Right(S n m , d) it holds that A ℓ υ ∈ Right(S n m , d − 1) for some ℓ ≤ l.Lemma 3: Let F be any surjective LCA over (Z/mZ) n and let A ∈ L n×n m be the matrix associated with F, where m and n are any two naturals with m > 1 and n > 1.Then there exists an integer constant c ≥ 0 (that only depends on A) such that all the following conditions C 1 , C 2 , C 3 , and C 4 hold.C 1 : ∀υ ∈ Left(S n m k j as associated matrix.Since F is positively expansive if and only if every LCA F j is too, by Lemma 2 and Theorem 1, it follows that F is positively expansive if and only if every matrix (A mod p j ) ∈ L n×n It is clear that the matrix A is expansive if and only if its transpose A T is expansive if and only if χ A n having A as associated matrix.The LCA F is positively expansive if and only if A is expansive if and only if A T is expansive if and only if α 0 + . . .+ α n−1 t n−1 + t n is expansive.Proof: 11: Let C ∈ L n×n p be any matrix in rational canonical form, where p and n are any two naturals such that p is prime and n > 1, and let G be the LCA over (Z/pZ) n having C as associated matrix.The LCA G is positively expansive if and only if C is expansive.Proof:Let C π 1 ∈ L n 1 ×n 1 C π s ∈ L n s ×n s p with n 1 + . ..+ n s = n be the diagonal blocks inside C, where each π i is the monic polynomial defining C π i , i.e., C π i is the companion matrix of π i .Since χ C (t) = s i=1 π i (t) and G is just the product G 1 × . ..×G s , where each G i is the LCA over (Z/pZ) n i having C π i ∈ L n i×n i p as associated matrix, it follows that G is positively expansive if and only if every G i is positively expansive if and only if, by Lemma 9 and 10, every C π i is expansive, i.e., by Definition 2, if and only if every π i is expansive, i.e., by Lemma 1, if and only if χ C is expansive, i.e., if and only if C is expansive.□ We are now able to prove Theorem 1. Proof of Theorem 1: Let A ∈ L n×n p be the matrix associated with F. By Lemma 6, there exist two matrices Q, C ∈ L n×n p such that det(Q) ̸ = 0, C is in rational canonical form, QA = CQ, and χ A = χ C .Let G be the LCA over (Z/pZ) n having C as associated matrix.By Lemma 7, F is positively expansive if and only if G is positively expansive, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.i.e., by Lemma 11, if and only if C is expansive, i.e., since χ A = χ C , if and only if A is expansive.□ p , . ..,V.CONCLUSION