4 research outputs found
On simulation in automata networks
An automata network is a finite graph where each node holds a state from some
finite alphabet and is equipped with an update function that changes its state
according to the configuration of neighboring states. More concisely, it is
given by a finite map . In this paper we study how some
(sets of) automata networks can be simulated by some other (set of) automata
networks with prescribed update mode or interaction graph. Our contributions
are the following. For non-Boolean alphabets and for any network size, there
are intrinsically non-sequential transformations (i.e. that can not be obtained
as composition of sequential updates of some network). Moreover there is no
universal automaton network that can produce all non-bijective functions via
compositions of asynchronous updates. On the other hand, we show that there are
universal automata networks for sequential updates if one is allowed to use a
larger alphabet and then use either projection onto or restriction to the
original alphabet. We also characterize the set of functions that are generated
by non-bijective sequential updates. Following Tchuente, we characterize the
interaction graphs whose semigroup of transformations is the full semigroup
of transformations on , and we show that they are the same if we force
either sequential updates only, or all asynchronous updates