1 research outputs found
On countings and enumerations of block-parallel automata networks
When we focus on finite dynamical systems from both the
computability/complexity and the modelling standpoints, automata networks seem
to be a particularly appropriate mathematical model on which theory shall be
developed. In this paper, automata networks are finite collections of entities
(the automata), each automaton having its own set of possible states, which
interact with each other over discrete time, interactions being defined as
local functions allowing the automata to change their state according to the
states of their neighbourhoods. The studies on this model of computation have
underlined the very importance of the way (i.e. the schedule) according to
which the automata update their states, namely the update modes which can be
deterministic, periodic, fair, or not. Indeed, a given network may admit
numerous underlying dynamics, these latter depending highly on the update modes
under which we let the former evolve. In this paper, we pay attention to a new
kind of deterministic, periodic and fair update mode family introduced recently
in a modelling framework, called the block-parallel update modes by duality
with the well-known and studied block-sequential update modes. More precisely,
in the general context of automata networks, this work aims at presenting what
distinguish block-parallel update modes from block-sequential ones, and at
counting and enumerating them: in absolute terms, by keeping only
representatives leading to distinct dynamics, and by keeping only
representatives giving rise to distinct isomorphic limit dynamics. Put
together, this paper constitutes a first theoretical analysis of these update
modes and their impact on automata networks dynamics