reversibility of a symmetric linear cellular automata
TRANSCRIPT
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International Journal of Modern Physics CVol. 20, No. 7 (2009) 1081–1086c© World Scientific Publishing Company
REVERSIBILITY OF A SYMMETRIC LINEAR
CELLULAR AUTOMATA
A. MARTIN DEL REY
Department of Applied MathematicsE. P. S. de Avila, Universidad de SalamancaC/ Hornos Caleros 50, 05003-Avila, Spain
G. RODRIGUEZ SANCHEZ
Department of Applied MathematicsE. P. S. de Zamora, Universidad de Salamanca
Avda. Cardenal Cisneros 34, 49022-Zamora, [email protected]
Received 13 February 2009Accepted 30 March 2009
The characterization of the size of the cellular space of a particular type of reversiblesymmetric linear cellular automata is introduced in this paper. Specifically, it is shownthat those symmetric linear cellular with 2k + 1 cells, and whose transition matrix is ak-diagonal square band matrix with nonzero entries equal to 1 are reversible. Further-more, in this case the inverse cellular automata are explicitly computed. Moreover, thereversibility condition is also studied for a general number of cells.
Keywords: Cellular automata; symmetric neighborhoods; reversibility; matrix theory.
PACS Nos.: 05.65.+b, 89.75.-k.
1. Introduction and Mathematical Preliminaries
Linear cellular automata are a special type of finite state machines formed by n
memory units called cells, which are arranged uniformly in a one-dimensional space.
Each cell is endowed with a state from the finite state set F2 = {0, 1}, that changes
at every discrete step of time according to a linear transition function. The state of
the ith cell at time t is denoted by sti ∈ F2, being the vector Ct = (st
1, . . . , stn) ∈ F
n2
the configuration of the LCA at time t, where C0 is the initial configuration.
The neighborhood of a cell is the set of all cells whose states at time t determines
the state of the main cell at time t + 1 by means of the local transition function.
A neighborhood is defined by a finite subset of indices V = {α1, . . . , αm} ⊂ Z,
such that the neighborhood of the ith cell is formed by the cells at positions i +
α1, . . . , i + αm. Symmetric linear cellular automata (SLCA) of order k are linear
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1082 A. M. del Rey & G. R. Sanchez
cellular automata for which symmetric neighborhood of radius k is considered: it is
defined by V = {−k, . . . , 0, . . . , k}. It is denoted by An,k.
An,k evolves deterministically in discrete timesteps changing the states of all
cells according to a local linear transition function f : F2k+12 → F2. The updated
state of the ith cell depends on the 2k + 1 variables of the linear function, which
are the previous states of the cells constituting its neighborhood, that is,
st+1i = f(st
i−k, . . . , sti, . . . , s
ti+k)
= α1sti−k + · · · + αk+1s
ti + · · · + α2k+1s
ti+k (mod 2) ,
where αi ∈ F2 for 1 ≤ i ≤ 2k + 1.
This paper deals with a particular type of symmetric linear cellular automata:
those which are defined by α1 = · · · = α2k+1 = 1. For the sake of notation we will
also denote this special type by An,k.
As the cellular space is finite, some type of boundary conditions must be consid-
ered to assure the well-defined dynamics of An,k. Usually null boundary conditions
are used: sti = 0 for i /∈ {1, 2, . . . , n}.
An,k has a very important property: Its evolution can be interpreted by means
of matrix algebraic tools1: the next configuration of the LCA can be generated
by multiplying the present configuration by a fixed band matrix under modulo-2
addition, that is:
(Ct+1)T = Mn,k · (Ct)T (mod 2) ,
where Mn,k is the characteristic matrix and (Ct)T is the transpose of Ct. Note that
Mn,k is a k-diagonal square band matrix of order n, whose nonzero coefficients are
all equal to 1, that is:
Mn,k =
1(k+1)· · · 1 0
(n−k−1)· · · 0
.... . .
. . .. . .
...
1. . .
. . . 0
0. . .
. . . 1
.... . .
. . ....
0(n−k−1)
· · · 0 1(k+1)· · · 1
.
A cellular automaton is said to be reversible if the evolution backwards is possi-
ble, that is, the cellular automaton is deterministic in both directions of time.3,4 In
the particular case of An,k, the reversibility property means that the characteristic
matrix of An,k will be non-singular. Note that in this case, the inverse evolution is
given by the following equation:
(Ct)T = M−1n,k · (Ct+1)T (mod 2) .
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Reversibility of a Symmetric Linear Cellular Automata 1083
The reversibility property is very important due to its several applications to
coding theory, image processing, etc.5 As a consequence it is an important challenge
to determine in an explicit way when a certain cellular automaton is reversible.
This problem has been successfully tackled2 for k = 1. In this paper a generalized
criterion for any k is presented and the inverse cellular automaton is explicitly given
when n = 2k + 1.
2. The Reversibility Theorem
Let Mn,k be the characteristic matrix of the kth symmetric linear cellular automa-
ton An,k.
Lemma 1. If n = 2k + 1, then |M2k+1,k| = 1 (mod 2).
Proof. Let
C =
P Q R
QT 1 S
RT St P
be a square matrix of order 2k + 1 such that: P is the null square matrix of order
k, Q is the kth order column matrix
Q =
1
0
...
0
,
S is the kth order row matrix S = (0 · · · 0 1), and R is the kth order square
matrix:
R =
1 0 · · · · · · 0
1. . .
. . ....
0. . .
. . .. . .
...
.... . .
. . .. . . 0
0 · · · 0 1 1
.
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1084 A. M. del Rey & G. R. Sanchez
As
M2k+1,k =
1(k+1)· · · 1 0
(k)· · · 0
.... . .
. . .. . .
...
1. . .
. . . 0
0. . .
. . . 1
.... . .
. . .. . .
...
0(k)· · · 0 1
(k+1)· · · 1
,
a simple calculus shows that M2k+1,k · C = C · M2k+1,k = Id (mod 2), thus C =
M−1(2k+1),k. As a consequence M(2k+1),k is non-singular and |M(2k+1),k| 6= 0; then
|M(2k+1),k| = 1 as the arithmetic is performed modulo 2.
As a consequence, the following result holds:
Proposition 1. The cellular automaton A2k+1,k is reversible and its inverse linear
cellular automaton is given by the following transition matrix :
M−12k+1,k =
0(k)· · · 0 1 1 0
(k−1)· · · 0
.... . .
. . .. . .
. . .. . .
...
0 0. . .
. . .. . . 0
1. . . 1
. . .. . . 1
1. . .
. . . 0. . . 1
0. . .
. . .. . .
. . . 0
.... . .
. . .. . .
. . .. . .
...
0(k−1)· · · 0 1 1 0
(k)· · · 0
.
Lemma 2. The determinant of Mn,k satisfies the following recurrence relation:
|Mn,k| = |Mn−(2k+1),k| (mod 2) .
Proof. The matrix Mn,k can be written as follows:
Mn,k =
(
M2k+1,k A
At Mn−(2k+1),k
)
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Reversibility of a Symmetric Linear Cellular Automata 1085
where
A =
0 · · · · · · · · · · · · 0
......
0...
1. . .
...
.... . .
. . ....
1(k)· · · 1 0
(n−2k−1−k)· · · 0
is a matrix of order (2k + 1) × (n − 2k − 1). Then:
|Mn,k| = |M2k+1,k| · |Mn−(2k+1),k − At · M−12k+1,k · A| . (1)
Taking into account Lemma 1, |M2k+1,k| = 1 (mod 2), and (1) yields:
|Mn,k| = |Mn−(2k+1),k − At · M−12k+1,k · A| .
Now, a simple computation shows that At · M−12k+1,k · A = 0 (mod 2), and conse-
quently:
|Mn,k| = |Mn−(2k+1),k − At · M−12k+1,k · A| = |Mn−(2k+1),k| (mod 2) .
Proposition 2. The determinant of Mn,k satisfies:
|Mn,k| (mod 2) =
{
1 , if n = (2k + 1)m or n = (2k + 1)m + 1, with m ∈ Z+
0 , otherwise .
Proof. Set n = (2k + 1)m + p where m ∈ N, and 0 ≤ p ≤ 2k. Then, taking into
account Lemma 2, it is:
|Mn,k| = |M(2k+1)m+p,k| = |M(2k+1)(m−1)+p,k|
= |M(2k+1)(m−2)+p,k| = · · · = |M(2k+1)+p,k| .
A simple computation shows that:
|M(2k+1)+p,k| (mod 2) =
{
1 , if p = 0, 1
0 , if 2 ≤ p ≤ 2k
thus finishing.
As a consequence and taking into account the last two results, the following
theorem is stated:
Theorem 1. The symmetric linear cellular automaton An,k is reversible if and
only if n ≡ 0, 1 (mod(2k + 1)).
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1086 A. M. del Rey & G. R. Sanchez
Proof. As is well-known, An,k is reversible iff its characteristic matrix is non-
singular. As its characteristic matrix is Mn,k, following the Proposition 2, this
occurs when n = (2k + 1)m or (2k + 1)m + 1 with m ∈ Z+, that is, when n ≡ 0, 1
(mod(2k + 1)).
Acknowledgments
This work has been supported by Ministerio de Ciencia e Innovacion (Spain) under
Grant MTM2008-02773.
References
1. P. P. Chaudhuri, D. R. Chowdhury, S. Nandi and S. Chattopadhyay, Additive Cellu-
lar Automata: Theory and Applications, Vol. 1 (IEEE Computer Society Press, LosAlamitos, CA, 1997).
2. A. M. del Rey and G. R. Sanchez, Int. J. Mod. Phys. C 17, 975 (2006).3. T. Toffoli and N. Margolus, Physica D 45, 229 (1990).4. D. Richardson, J. Comput. Syst. Sci. 6, 373 (1972).5. S. Wolfram, A New Kind of Science (Wolfram Media Inc., 2002).
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