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Surjective linear cellular automata over Z m
Informqtion l$z4zzing Information
Processing
Letters 66 ( 1998) 10 I-
IO4
Surjective linear cellular automata over Z, Tadakazu Sato Department
of Information
and
Computer Science,
Toyo Univemity
2100, Kujirai, Kawagoe, Saifama, Japan
Received 17 September 1997: revised 8 February Communicated by T. Asano
Keywords: Linear cellular automaton;
1998
Local map; Parallel map: Theory of computation
1. Introduction Injectivity or surjectivity of parallel maps of cellular automata has been discussed in many papers [l-7]. In our previous paper, we gave an algorithm for deciding injective local maps or surjective local maps of linear cellular automata over finite commutative rings. In this paper, we consider d-dimensional linear cellular automata over Z m and obtain the following main results. Let m = $, py be the factorization in prime numbers. Then any surjective parallel map induced by local map over Z, is a constant-to-one map, however, in one-dimensional cell space only, its constant is finite and takes the value of the form nr=, (P~)~J where nj is a non-negative integer.
and let N = {VI...., u, ). N represents the neighbor frame at each cell. A configuration over Z, is a map of Zd into Z, and the set of all configurations over Z, is denoted by C(Z,). For a linear local map f with a neighbour frame N, we define a map fm :C(Z,) -+ C(Z,) as follows: for u, w E C(Z,),
fm(l.4.) = W*
W(T)
=
CajU(r
+
Vj)
for all r E Zd wherer+ul,..., r + v,, are called the neighbourhoods of the cell r E Zd. fm is called a linear parallel map induced by f. For s E iZd, we define a shift map gS : C(Z,) + C(Z,) as follows: for II, w E C(Z,), a”(u) = w * W(T) = u(r + S)
for all r E Zd.
2. Preliminary
Now let’s give some definitions with respect to local maps which are needed in the following.
A linear cellular automaton over Z, is given by a quardruplet (Zd, Z,, f,N) where Z is the set of all integers, Zd is the set of all d-tuples of integers called a d-dimensional cell space. For a positive integer m, Z, is the set of integers modulo m and denotes a state set of each cell. f is a linear map of Zk into Z, called a linear local map over Z, with scope II, that is, f(xl,...,x,)=Cajxj. NisafinitesubsetofZd
Definition 1. (1) f is injective on Z, if foeis injective on C(Z,). (2) f is surjective on Z, if fm is surjective on C(&n). (3) fm has finite order on C(Z,) if some power of fm equals an identity map on C(Z,). (4) foeis k-to-one map on C(Z,) if 1f;'(u)1 = k for all II E C(iZ,).
OO20-0190/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved. PII SOO20-0190(98)00035-O
7: Sara /Infomrion
‘02
Processing Letters 66 (1998) 101-104
For a local map f = zajxj with a neighbour frame N 5 Zd, let F(X, Y) denote the polynomial representation of foe. (See [3].) That is
fee= z#jU’j
e F(X, Y) =
CajX*,
Y’I,
where the summation is taken over all vectors Uj = (xi, yj) in N. The importance of the polynomial representation is that if F(X, Y) and G(X, Y) are the polynomial representations of foe and g,, respectively, then that of the composition of f, and g, equals the product of F(X, Y) and G(X, Y). For example, if foe has finite order on C(Z,), then F(X, Y)” = 1 for some positive integer n. By 0, we denote a quiescent configuration, that is, 8(r) = 0 for all points r E Zd. Let ker f, denote the subset of C(Z,) defined by f-’co Co>.
Proposition 2. The following statements are equivalent. (1) foe is surjective on C(Z,). (2) Foranyx E C(Z,), IfG’(x)l=
Lemma 4. Let f be a local map over Z,. Then we have foreachn>O.
(kerf&(=Ikerf,(”
Proof. The proof is given by induction on n. It is valid fern= 1. Assume that it is also valid for any integer n such that 1 6 n 6 s. We show that ( ker fiso+’( = Ikerf,lS+‘.Letkerf&={x’,...,x,].Then ker fz’
= f&“‘“+‘)(o) =f~‘~x’,...,x~l =
0
fG’(Xj).
U j=’
Sincef&‘(xi)nf;‘(xj)=0fori#j,wehave (kerfz’)
= c
If&‘(xi)l
j=l
=t(kerf,l
Ikerfool.
=)kerf&I.Ikerf,l
Proof. Since foe is linear, ker foe contains a quiescent
= ) ker foe IS. ) ker foe )
configuration 0. Then we have 1ker foe I # 0. So, it suffices to show that (1) implies (2). For any x E C(Z,), let y be an arbitrary point in C(Z,) such that y E fd;;’ (x). Then we show that
= ) ker fools+‘.
f;‘(X)=y+kerf,={y+u[uEkerf,}. Since fdy + u> = foe(y) + f&> = foe(y) = x for any u E ker foe. This shows that y + u E f;’ (x). Conversely, for any z E fd;i’(x), we have fW(z> = foe(y). Then foo(z - y) = 0. We have z -y E kerf,. 0 So, z E y + ker foe. This completes the proof. In the following, we shall show that in one-dimensional cell spaces, we have 1ker fool < 00, but in two or higher-dimensional cell spaces, we have 1ker f, I = cm. First, we begin with the one-dimensional case.
Lemma 3 (Theorem 6.7 in [2]). Let f = xajxj be a local map with neighbourframe N and let F(X) be the polynomial representation. If the coeficients with the largest power of X in F(X) and the smallest such power are invertible elements of Z,, then we have 1ker fool = m” where n is the dzxerence between the largest power of X and the smallest such power:
This completes the proof.
(by assumption)
R
Lemma 5 [7]. Let p be a prime and r be a positive integer: Then we have [A(X, Y) + p’B(X)lP’-’
= A(X, Y)P’-’ (mod p’),
where A(X, Y) and B(X, Y) are arbitrary polynomials over Z. In the following, we use the notation IIF(X)II insteadof I(ker foolI.
IIfooII or
Lemma 6. Let fj be a surjective local map over Z’d, and let Fj (X) be the polynomial representation of fj with neighbour frame Nj. Let nj be the difSerence between the largest power of X and the smallest such power of Fj (X). Then we have
II(fj>cc
II =
(P:i)ni.
Proof. Consider Gj(X)
+ Hi(X)
the decomposition of Fj(X) into such that all coefficients of Gj(X)
7: Sat0 /Information
are invertible
elements of Z;j
Processing
and all ones of Hj(X)
are nilpotent elements of I$, . Since is sujective, we have Gj(X) # 0. From Lemma 5, there exists a positive integer t such that fj
Letters 66 (1998) 101-104
Theorem 8. Let m = l$=,
py be the factorization in prime numbers and let f = (fl , . . , fk). Then we have k
llfcoll Fj(X)’
= (Gj(X)
+ Hj(X))’
= Gj(X)‘.
(fromLemma4) F,(M)
IlGj(X)‘II
= ]]Gj (X) ]I’ (from Lemma 4). This shows that ((Fj(X)II = IlGj(X)II. hand, since all coefficients of Gj(X) of Z;, , it follows
from
On the other are invertible Lemma
mod pj.
Proof. Let Fj(X)
3 that
]]Gj(X)]] = (pl’)“J where nj is the difference between the largest power of X and the smallest such power of G j (X). This completes the proof. q Let k 172=
(Py’?
where nj denotes the difference between the largest power of X and the smallest such power of
]]Fj(X)]]’ = ]]Fj(X)‘](
elements
= n j=l
So, we have
=
103
P>’ I-l j=l
be the factorization in prime numbers. Then we have Z, 2: Z’d, x . . . x L?L$. This induces naturally the decomposition of f into (fl, . . , fk) and C(Z,) 2 C(Z’,‘,) x ... x C(Z’ak). See [l].
Lemma 7. Let m = nr=, py be the factorization prime numbers. Then we have
in
= Gj(X) + Hi(X) in the proof of Lemma 6 above. Then nj is equal to the difference between the largest power of X and the smallest such power of Gj(X). Therefore it follows directly from Lemmas 6 and 7 that the theorem holds. q
Corollary 9. Let m = $,
py be the factorization in prime numbers. Let f = (fl , . . . , ,fk). Then the following statements are all equivalent. (1) f is injective on Z,. (2) For each j (1 < j 6 k), fj is injective on Z’,‘, . (3) For each j (1 < j < k), F,(X) (mod pi) is a monomial and does not vanish.
Remark 10. Above corollary holds for two or higherdimensional
cell spaces.
Example 11. Let foe = 1 + 100 (mod 12), where 0 denotes a left shift map on C(Z,) in the onedimensional cell space. Then we have fm = (( fl)m, (f&J
where (fr )m = 1 + 20 (mod 4) and (f& = 1 + o (mod 3). Hence, we have Il(fl)mll = 1 and I](fz)co )I = 3. From Theorem 8, we have IIfoe /I = 3.
k
j=l
We next consider
Proof. It suffices to consider the case for k = 2. Let f = (fl,
ker(.f&
f2). Let ker(ft)oo = {xt,...,x,] = (YI , . . . , yr). Clearly, we have
s
I
kerfoo = U
IJ
i=l
and let
((xiyyj)].
Lemma 12. Let m = nr=, py be the factorization
j=l
Then IIfm II = Il(fl loo II . II (fzh proof. 0
two or higher-dimensional cell we denote the cardinality of the spaces. BY IIfco II,,, kernel of the parallel map induced by the local map f (mod p) which is the one over Z, with coefficients obtained from ,f by mod p.
II. This completes
the
The following theorem shows that I],foo I] takes the value of the form nr=, (py )“i.
in prime numbers. Let f be a surjective local map on Z,, then we have Ilfoollp, 6 lIfmll, for each j (1 < j 6 k).
Proof. Let fj = f (mod pj). For x E C(Z,), let x (mod pj) denote a configuration over C(Z,,) whose
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Processing Letters 66 (1998) 101-104
value at a point r E Z2 is obtained from x(r) by mod pj . Then it is easy to see that if x E ker foe, then n (mod pj) E ker(fj)oo. In this case, the different configurations in ker foe may fall in the same one in ker(fj)co by mod pi. This proves the Lemma 12. q Theorem 13. In the two or more higher-dimensional cell spaces, let foe be surjective on C(Z,). Then the following statements are equivalent. (1) foe is injective on C(Z,). (2)
IIfca II< CQ
Proof. It suffices to show that (2) implies (1). We consider the case for two-dimensional cell spaces. The proofs for higher-dimensional cell spaces are similar. Suppose that foe is not injective on C(Z,). Let F(X, Y) denote the polynomial representation of foe. First, consider the case when F(X, Y) can be written in the form CUjX j. Regarding it as the polynomial representation in one-dimensional cell spaces, we have 1 < I]CajXj ]I < 00. However, regarding it as the representation in two-dimensional cell spaces, I]F(X, Y) II must have the cardinality of continuum. Hence, 11 foe 11= co. Next, consider the case for which F(X, Y) can be written in the form C ajj Xi Yj. From Lemma 6, it suffices to show that ]]foolIPi = 00. Since f (mod pj) is surjective on ZPj , all coefficients of f
are invertible elements of ZPj. This shows that it is permutive at each variables [2]. So, it is easy to find a configuration u such that ~(0, 0) = 1, U(S,0) = 0 for s # 0 and foe(u) = 0. Clearly, u is not periodic in x-direction, that is, o.$ # u for all t # 0, where a, denotes a shift map such that a,u(i, j) = u(i + 1, j). Then we have f,(a:u) = aifoo(u) = ajo = 0 for all t E Z. This shows that ( kerf,( = 00. The proof is completed. q
References [l] H. Aso, H. Honda, Dynamical characteristics of linear cellular automata, J. Comput. System Sci. 30 (3) (1985) 291-317. [2] GA. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory 3 (1969) 320-375. [3] M. Ito, N. Osato, M. Nasu, Linear cellular automata over Z,, J. Comput. System Sci. 27 (2) (1983) 125-140. [4] A. Maruoka, M. Kimura, Condition for injectivity of global maps for tessellation automata, Inform. Control 32 (1976) 158162. [5] T. Sato, N. Honda, Certain relations between properties of maps of tessellation automata, J. Comput. System Sci. 15 (2) (1977) 121-145. [6] T. Sato, Decidability for some problems of linear cellular automata over finite commutative rings, Inform. Process. Lett. 46 (1993) 151-155. [7] T. Sato, Group structured linear cellular automata over Z,, J. Comput. System Sci. 49 (1) (1994) 18-23.
Processing
Letters 66 ( 1998) 10 I-
IO4
Surjective linear cellular automata over Z, Tadakazu Sato Department
of Information
and
Computer Science,
Toyo Univemity
2100, Kujirai, Kawagoe, Saifama, Japan
Received 17 September 1997: revised 8 February Communicated by T. Asano
Keywords: Linear cellular automaton;
1998
Local map; Parallel map: Theory of computation
1. Introduction Injectivity or surjectivity of parallel maps of cellular automata has been discussed in many papers [l-7]. In our previous paper, we gave an algorithm for deciding injective local maps or surjective local maps of linear cellular automata over finite commutative rings. In this paper, we consider d-dimensional linear cellular automata over Z m and obtain the following main results. Let m = $, py be the factorization in prime numbers. Then any surjective parallel map induced by local map over Z, is a constant-to-one map, however, in one-dimensional cell space only, its constant is finite and takes the value of the form nr=, (P~)~J where nj is a non-negative integer.
and let N = {VI...., u, ). N represents the neighbor frame at each cell. A configuration over Z, is a map of Zd into Z, and the set of all configurations over Z, is denoted by C(Z,). For a linear local map f with a neighbour frame N, we define a map fm :C(Z,) -+ C(Z,) as follows: for u, w E C(Z,),
fm(l.4.) = W*
W(T)
=
CajU(r
+
Vj)
for all r E Zd wherer+ul,..., r + v,, are called the neighbourhoods of the cell r E Zd. fm is called a linear parallel map induced by f. For s E iZd, we define a shift map gS : C(Z,) + C(Z,) as follows: for II, w E C(Z,), a”(u) = w * W(T) = u(r + S)
for all r E Zd.
2. Preliminary
Now let’s give some definitions with respect to local maps which are needed in the following.
A linear cellular automaton over Z, is given by a quardruplet (Zd, Z,, f,N) where Z is the set of all integers, Zd is the set of all d-tuples of integers called a d-dimensional cell space. For a positive integer m, Z, is the set of integers modulo m and denotes a state set of each cell. f is a linear map of Zk into Z, called a linear local map over Z, with scope II, that is, f(xl,...,x,)=Cajxj. NisafinitesubsetofZd
Definition 1. (1) f is injective on Z, if foeis injective on C(Z,). (2) f is surjective on Z, if fm is surjective on C(&n). (3) fm has finite order on C(Z,) if some power of fm equals an identity map on C(Z,). (4) foeis k-to-one map on C(Z,) if 1f;'(u)1 = k for all II E C(iZ,).
OO20-0190/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved. PII SOO20-0190(98)00035-O
7: Sara /Infomrion
‘02
Processing Letters 66 (1998) 101-104
For a local map f = zajxj with a neighbour frame N 5 Zd, let F(X, Y) denote the polynomial representation of foe. (See [3].) That is
fee= z#jU’j
e F(X, Y) =
CajX*,
Y’I,
where the summation is taken over all vectors Uj = (xi, yj) in N. The importance of the polynomial representation is that if F(X, Y) and G(X, Y) are the polynomial representations of foe and g,, respectively, then that of the composition of f, and g, equals the product of F(X, Y) and G(X, Y). For example, if foe has finite order on C(Z,), then F(X, Y)” = 1 for some positive integer n. By 0, we denote a quiescent configuration, that is, 8(r) = 0 for all points r E Zd. Let ker f, denote the subset of C(Z,) defined by f-’co Co>.
Proposition 2. The following statements are equivalent. (1) foe is surjective on C(Z,). (2) Foranyx E C(Z,), IfG’(x)l=
Lemma 4. Let f be a local map over Z,. Then we have foreachn>O.
(kerf&(=Ikerf,(”
Proof. The proof is given by induction on n. It is valid fern= 1. Assume that it is also valid for any integer n such that 1 6 n 6 s. We show that ( ker fiso+’( = Ikerf,lS+‘.Letkerf&={x’,...,x,].Then ker fz’
= f&“‘“+‘)(o) =f~‘~x’,...,x~l =
0
fG’(Xj).
U j=’
Sincef&‘(xi)nf;‘(xj)=0fori#j,wehave (kerfz’)
= c
If&‘(xi)l
j=l
=t(kerf,l
Ikerfool.
=)kerf&I.Ikerf,l
Proof. Since foe is linear, ker foe contains a quiescent
= ) ker foe IS. ) ker foe )
configuration 0. Then we have 1ker foe I # 0. So, it suffices to show that (1) implies (2). For any x E C(Z,), let y be an arbitrary point in C(Z,) such that y E fd;;’ (x). Then we show that
= ) ker fools+‘.
f;‘(X)=y+kerf,={y+u[uEkerf,}. Since fdy + u> = foe(y) + f&> = foe(y) = x for any u E ker foe. This shows that y + u E f;’ (x). Conversely, for any z E fd;i’(x), we have fW(z> = foe(y). Then foo(z - y) = 0. We have z -y E kerf,. 0 So, z E y + ker foe. This completes the proof. In the following, we shall show that in one-dimensional cell spaces, we have 1ker fool < 00, but in two or higher-dimensional cell spaces, we have 1ker f, I = cm. First, we begin with the one-dimensional case.
Lemma 3 (Theorem 6.7 in [2]). Let f = xajxj be a local map with neighbourframe N and let F(X) be the polynomial representation. If the coeficients with the largest power of X in F(X) and the smallest such power are invertible elements of Z,, then we have 1ker fool = m” where n is the dzxerence between the largest power of X and the smallest such power:
This completes the proof.
(by assumption)
R
Lemma 5 [7]. Let p be a prime and r be a positive integer: Then we have [A(X, Y) + p’B(X)lP’-’
= A(X, Y)P’-’ (mod p’),
where A(X, Y) and B(X, Y) are arbitrary polynomials over Z. In the following, we use the notation IIF(X)II insteadof I(ker foolI.
IIfooII or
Lemma 6. Let fj be a surjective local map over Z’d, and let Fj (X) be the polynomial representation of fj with neighbour frame Nj. Let nj be the difSerence between the largest power of X and the smallest such power of Fj (X). Then we have
II(fj>cc
II =
(P:i)ni.
Proof. Consider Gj(X)
+ Hi(X)
the decomposition of Fj(X) into such that all coefficients of Gj(X)
7: Sat0 /Information
are invertible
elements of Z;j
Processing
and all ones of Hj(X)
are nilpotent elements of I$, . Since is sujective, we have Gj(X) # 0. From Lemma 5, there exists a positive integer t such that fj
Letters 66 (1998) 101-104
Theorem 8. Let m = l$=,
py be the factorization in prime numbers and let f = (fl , . . , fk). Then we have k
llfcoll Fj(X)’
= (Gj(X)
+ Hj(X))’
= Gj(X)‘.
(fromLemma4) F,(M)
IlGj(X)‘II
= ]]Gj (X) ]I’ (from Lemma 4). This shows that ((Fj(X)II = IlGj(X)II. hand, since all coefficients of Gj(X) of Z;, , it follows
from
On the other are invertible Lemma
mod pj.
Proof. Let Fj(X)
3 that
]]Gj(X)]] = (pl’)“J where nj is the difference between the largest power of X and the smallest such power of G j (X). This completes the proof. q Let k 172=
(Py’?
where nj denotes the difference between the largest power of X and the smallest such power of
]]Fj(X)]]’ = ]]Fj(X)‘](
elements
= n j=l
So, we have
=
103
P>’ I-l j=l
be the factorization in prime numbers. Then we have Z, 2: Z’d, x . . . x L?L$. This induces naturally the decomposition of f into (fl, . . , fk) and C(Z,) 2 C(Z’,‘,) x ... x C(Z’ak). See [l].
Lemma 7. Let m = nr=, py be the factorization prime numbers. Then we have
in
= Gj(X) + Hi(X) in the proof of Lemma 6 above. Then nj is equal to the difference between the largest power of X and the smallest such power of Gj(X). Therefore it follows directly from Lemmas 6 and 7 that the theorem holds. q
Corollary 9. Let m = $,
py be the factorization in prime numbers. Let f = (fl , . . . , ,fk). Then the following statements are all equivalent. (1) f is injective on Z,. (2) For each j (1 < j 6 k), fj is injective on Z’,‘, . (3) For each j (1 < j < k), F,(X) (mod pi) is a monomial and does not vanish.
Remark 10. Above corollary holds for two or higherdimensional
cell spaces.
Example 11. Let foe = 1 + 100 (mod 12), where 0 denotes a left shift map on C(Z,) in the onedimensional cell space. Then we have fm = (( fl)m, (f&J
where (fr )m = 1 + 20 (mod 4) and (f& = 1 + o (mod 3). Hence, we have Il(fl)mll = 1 and I](fz)co )I = 3. From Theorem 8, we have IIfoe /I = 3.
k
j=l
We next consider
Proof. It suffices to consider the case for k = 2. Let f = (fl,
ker(.f&
f2). Let ker(ft)oo = {xt,...,x,] = (YI , . . . , yr). Clearly, we have
s
I
kerfoo = U
IJ
i=l
and let
((xiyyj)].
Lemma 12. Let m = nr=, py be the factorization
j=l
Then IIfm II = Il(fl loo II . II (fzh proof. 0
two or higher-dimensional cell we denote the cardinality of the spaces. BY IIfco II,,, kernel of the parallel map induced by the local map f (mod p) which is the one over Z, with coefficients obtained from ,f by mod p.
II. This completes
the
The following theorem shows that I],foo I] takes the value of the form nr=, (py )“i.
in prime numbers. Let f be a surjective local map on Z,, then we have Ilfoollp, 6 lIfmll, for each j (1 < j 6 k).
Proof. Let fj = f (mod pj). For x E C(Z,), let x (mod pj) denote a configuration over C(Z,,) whose
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104
Processing Letters 66 (1998) 101-104
value at a point r E Z2 is obtained from x(r) by mod pj . Then it is easy to see that if x E ker foe, then n (mod pj) E ker(fj)oo. In this case, the different configurations in ker foe may fall in the same one in ker(fj)co by mod pi. This proves the Lemma 12. q Theorem 13. In the two or more higher-dimensional cell spaces, let foe be surjective on C(Z,). Then the following statements are equivalent. (1) foe is injective on C(Z,). (2)
IIfca II< CQ
Proof. It suffices to show that (2) implies (1). We consider the case for two-dimensional cell spaces. The proofs for higher-dimensional cell spaces are similar. Suppose that foe is not injective on C(Z,). Let F(X, Y) denote the polynomial representation of foe. First, consider the case when F(X, Y) can be written in the form CUjX j. Regarding it as the polynomial representation in one-dimensional cell spaces, we have 1 < I]CajXj ]I < 00. However, regarding it as the representation in two-dimensional cell spaces, I]F(X, Y) II must have the cardinality of continuum. Hence, 11 foe 11= co. Next, consider the case for which F(X, Y) can be written in the form C ajj Xi Yj. From Lemma 6, it suffices to show that ]]foolIPi = 00. Since f (mod pj) is surjective on ZPj , all coefficients of f
are invertible elements of ZPj. This shows that it is permutive at each variables [2]. So, it is easy to find a configuration u such that ~(0, 0) = 1, U(S,0) = 0 for s # 0 and foe(u) = 0. Clearly, u is not periodic in x-direction, that is, o.$ # u for all t # 0, where a, denotes a shift map such that a,u(i, j) = u(i + 1, j). Then we have f,(a:u) = aifoo(u) = ajo = 0 for all t E Z. This shows that ( kerf,( = 00. The proof is completed. q
References [l] H. Aso, H. Honda, Dynamical characteristics of linear cellular automata, J. Comput. System Sci. 30 (3) (1985) 291-317. [2] GA. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory 3 (1969) 320-375. [3] M. Ito, N. Osato, M. Nasu, Linear cellular automata over Z,, J. Comput. System Sci. 27 (2) (1983) 125-140. [4] A. Maruoka, M. Kimura, Condition for injectivity of global maps for tessellation automata, Inform. Control 32 (1976) 158162. [5] T. Sato, N. Honda, Certain relations between properties of maps of tessellation automata, J. Comput. System Sci. 15 (2) (1977) 121-145. [6] T. Sato, Decidability for some problems of linear cellular automata over finite commutative rings, Inform. Process. Lett. 46 (1993) 151-155. [7] T. Sato, Group structured linear cellular automata over Z,, J. Comput. System Sci. 49 (1) (1994) 18-23.