PHYSICAL REVIEW E 107, 024211 (2023)
Seven-state rotation-symmetric number-conserving cellular automaton that is not isomorphic to
any septenary one
Barbara Wolnik ,
1, 2
Anna Nenca ,
3, *
Adam Dzedzej,
1
and Bernard De Baets
2
1
Institute of Mathematics, Faculty of Mathematics, Physics and Informatics, University of Gda´ nsk, 80-308 Gda ´ nsk, Poland
2
KERMIT, Department of Data Analysis and Mathematical Modelling, Faculty of Bioscience Engineering, Ghent University,
Coupure links 653, B-9000 Gent, Belgium
3
Institute of Informatics, Faculty of Mathematics, Physics and Informatics, University of Gda´ nsk, 80-308 Gda ´ nsk, Poland
(Received 16 October 2022; accepted 1 February 2023; published 21 February 2023)
We consider two-dimensional cellular automata with the von Neumann neighborhood that satisfy two proper-
ties of interest from a modeling viewpoint: rotation symmetry (i.e., the local rule is invariant under rotation of
the neighborhood by 90
◦
) and number conservation (i.e., the sum of all the cell states is conserved upon every
update). It is known that if the number of states k is smaller than or equal to six, then each rotation-symmetric
number-conserving cellular automaton is isomorphic to some k-ary one, i.e., one with state set {0, 1,..., k − 1}.
In this paper, we exhibit an example of a seven-state rotation-symmetric number-conserving cellular automaton
that is not isomorphic to any septenary one. This example strongly supports our plea that research into multistate
cellular automata should not only focus on those that have {0, 1,..., k − 1} as a state set.
DOI: 10.1103/PhysRevE.107.024211
I. INTRODUCTION
Cellular automata (CAs) are discrete dynamical systems
that evolve according to a simple local rule, but can never-
theless produce very complex dynamics (see [1] or [2], for
instance). For this reason, such systems for several decades
are willingly used to build models of various phenomena, in
particular physical ones (see, for example, [3] or [4]).
The definition of a CA requires the choice of a cellular
space C , a neighborhood N of every cell, a state set Q, and
a local rule f . Although the specification of the quadruple
(C , N , Q, f ) has a great influence on the resulting model,
usually little attention is paid to what elements Q contains, but
only to how many there are. Therefore, theoretical research—
if it does not consider some general hypothetical state set
Q—mostly concerns the so-called k -ary CAs (binary, ternary,
quaternary, quinary, and so on), for which the state set equals
{0, 1,..., k − 1} for some integer k 2 (see, for instance,
[5–8]).
The above approach seems justified for the following
reason. Suppose that C and N are fixed and that two finite
equinumerous sets Q and
Q are given. Then for each CA
A = (C , N , Q, f ), using any bijection φ : Q →
Q, one can
define an isomorphic CA
A = (C , N ,
Q,
f ), which means
that when identifying the states q and φ (q ) (for any q ∈ Q),
the CAs A and
A are indistinguishable (for the strict meaning,
see Definition 4). For example, in the one-dimensional
case, where the dynamics of a given cellular automaton
is usually represented by a space-time diagram, the fact
that two CAs A = (C , N , Q, f ) and
A = (C , N ,
Q,
f ) are
isomorphic simply means that if we choose some collection
*
Corresponding author: anna.nenca@ug.edu.pl
of colors to represent the states of
A and then for every
state q ∈ Q we use the same color as for the state φ (q ),
then the corresponding space-time diagrams for A
and
A will be exactly the same. Perhaps the most common
situation is when considering binary CAs and the set of states
can be named {0, 1} or {white, black } or {dead , alive} or
{healthy, sick } or {empty, occupied }, to name but a few.
However, in many situations, only CAs having some addi-
tional property P are considered (for instance, resulting from
the properties of the model being built). Since isomorphy
does not need to preserve P, for two equinumerous sets Q
and
Q, the family of all Q-state CAs satisfying P may differ
significantly from the family of all
Q-state CAs satisfying
P. For example, when modeling various kinds of physical
phenomena, very often some additional condition is imposed
on the local rule f related to the symmetries of the neigh-
borhood N (for instance, rotation invariance) or related to
some conservation laws (of mass, energy, and so on). While in
the former case we still do not have to worry about the choice
of Q, in the latter case, it is known that selecting Q may be
important.
We are encountering such a situation in the case of
two-dimensional rotation-symmetric number-conserving CAs
(RSNCCAs) with the von Neumann neighborhood (see Sec. II
for definitions). Tanimoto et al. [9] proved that such CAs
with at most four states are all trivial, regardless of Q, i.e.,
if just |Q| 4, then the identity is the only RSNCCA with
such a state set. Later, Imai et al. [10] showed that the case
of five states is a bit more complicated: if the elements of a
given five-element set Q form an arithmetic progression, then
there are exactly four RSNCCAs with this state set Q; if the
elements of Q do not form an arithmetic progression, then
there is only one RSNCCA with this state set (the identity).
Thus the choice of Q has a big influence on the number of such
2470-0045/2023/107(2)/024211(4) 024211-1 ©2023 American Physical Society