Emergence of Macro Spatial Structures in Dissipative Cellular Automata - Cellular Automata - page 147

As an example, Fig. 1 and Fig. 2 show the steady states reached by a syn-
chronous and an asynchronous CA, starting from the same initial (random)state.
These are characterized by a Moore neighborhood structure (the neighbors of a
cell are the 8 one defining a 3 × 3 square around the cell itself)and the following
transition rule:
f = { a dead cell gets alive iff it has 2 neighbors alive; a living cells lives iff it
has 1 or 2 neighbors alive } .
Under asynchronous regime, CA usually reaches a fixed point that its syn-
chronous counterpart has never been observed to be able to reach in all the
experiments we performed.
2.2 Openness
Most of CA studied so far are closed systems, as they do not take into account
the interaction between the CA and an environment. Instead, the class of CA
that we have studied is, in addition to asynchronous, open : the dynamic behavior
of the CA can be influenced by the external environment.
From an operative point of view, the openness of the CA implies that some
cell can be forced from the external to change its state, independently of the
cell having evaluated its state and independently of the transition function (see
Fig. 3).
From a thermodynamic perspective, one can consider this manifestation of
the external environment in terms of energy flows: forcing a cell to change its
state can be considered as a manifestation of energy flowing into the system and
influencing it [11]. This similarity, together with the fact that the activities of
the cells are intrinsically asynchronous and that the externally forced changes in
the state of cells perturb the CA in an irreversible way, made us call this kind
of CA as Dissipative Cellular Automata (DCA).
From a more formal point of view, a DCA can be defined as follows:
- A =( S, d, V, f ),
- asynchronous time-driven dynamics (with probability λ a ),
- a perturbation action ϕ ( α, D,λ e ).
where A is the quadruple defining the CA, the dynamics is the one already
discussed in Subsect. 2.1, and the perturbation action ϕ is a transition function
which acts concurrently with f and can change the state of any of the CA
cells to a given state α ∈ V depending on some probabilistic distribution D ,
independently of the current state of the cells and of their neighbors. Specifically,
in our experiments α = 1 (i.e., the cell if forced to be “alive”)and D is a
distribution such that each cell has probability λ e to be perturbed.