5,181 research outputs found
Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems
We present dichotomy theorems regarding the computational complexity of
counting fixed points in boolean (discrete) dynamical systems, i.e., finite
discrete dynamical systems over the domain {0,1}. For a class F of boolean
functions and a class G of graphs, an (F,G)-system is a boolean dynamical
system with local transitions functions lying in F and graphs in G. We show
that, if local transition functions are given by lookup tables, then the
following complexity classification holds: Let F be a class of boolean
functions closed under superposition and let G be a graph class closed under
taking minors. If F contains all min-functions, all max-functions, or all
self-dual and monotone functions, and G contains all planar graphs, then it is
#P-complete to compute the number of fixed points in an (F,G)-system; otherwise
it is computable in polynomial time. We also prove a dichotomy theorem for the
case that local transition functions are given by formulas (over logical
bases). This theorem has a significantly more complicated structure than the
theorem for lookup tables. A corresponding theorem for boolean circuits
coincides with the theorem for formulas.Comment: 16 pages, extended abstract presented at 10th Italian Conference on
Theoretical Computer Science (ICTCS'2007
Reducing the number of time delays in coupled dynamical systems
When several dynamical systems interact, the transmission of the information
between them necessarily implies a time delay. When the time delay is not
negligible, the study of the dynamics of these interactions deserve a special
treatment. We will show here that under certain assumptions, it is possible to
set to zero a significant amount of time-delayed connections without altering
the global dynamics. We will focus on graphs of interactions with identical
time delays and bidirectional connections. With these premises, it is possible
to find a configuration where a number of time delays have been removed
with , where is the number of dynamical
systems on a connected graph
Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions
We attach to each finite bipartite separated graph (E,C) a partial dynamical
system (\Omega(E,C), F, \theta), where \Omega(E,C) is a zero-dimensional
metrizable compact space, F is a finitely generated free group, and {\theta} is
a continuous partial action of F on \Omega(E,C). The full crossed product
C*-algebra O(E,C) = C(\Omega(E,C)) \rtimes_{\theta} F is shown to be a
canonical quotient of the graph C*-algebra C^*(E,C) of the separated graph
(E,C). Similarly, we prove that, for any *-field K, the algebraic crossed
product L^{ab}_K(E,C) = C_K(\Omega(E,C)) \rtimes_\theta^{alg} F is a canonical
quotient of the Leavitt path algebra L_K(E,C) of (E,C). The monoid
V(L^{ab}_K(E,C)) of isomorphism classes of finitely generated projective
modules over L^{ab}_K(E,C) is explicitly computed in terms of monoids
associated to a canonical sequence of separated graphs. Using this, we are able
to construct an action of a finitely generated free group F on a
zero-dimensional metrizable compact space Z such that the type semigroup S(Z,
F, K) is not almost unperforated, where K denotes the algebra of clopen subsets
of Z. Finally we obtain a characterization of the separated graphs (E,C) such
that the canonical partial action of F on \Omega(E,C) is topologically free.Comment: Final version to appear in Advances in Mathematic
Convex subshifts, separated Bratteli diagrams, and ideal structure of tame separated graph algebras
We introduce a new class of partial actions of free groups on totally
disconnected compact Hausdorff spaces, which we call convex subshifts. These
serve as an abstract framework for the partial actions associated with finite
separated graphs in much the same way as classical subshifts generalize the
edge shift of a finite graph. We define the notion of a finite type convex
subshift and show that any such subshift is Kakutani equivalent to the partial
action associated with a finite bipartite separated graph. We then study the
ideal structure of both the full and the reduced tame graph C*-algebras,
and , of a separated graph , and
of the abelianized Leavitt path algebra as well. These
algebras are the (reduced) crossed products with respect to the above-mentioned
partial actions, and we prove that there is a lattice isomorphism between the
lattice of induced ideals and the lattice of hereditary -saturated
subsets of a certain infinite separated graph built
from , called the separated Bratteli diagram of . We finally use
these tools to study simplicity and primeness of the tame separated graph
algebras.Comment: 60 page
Cycle Equivalence of Graph Dynamical Systems
Graph dynamical systems (GDSs) can be used to describe a wide range of
distributed, nonlinear phenomena. In this paper we characterize cycle
equivalence of a class of finite GDSs called sequential dynamical systems SDSs.
In general, two finite GDSs are cycle equivalent if their periodic orbits are
isomorphic as directed graphs. Sequential dynamical systems may be thought of
as generalized cellular automata, and use an update order to construct the
dynamical system map.
The main result of this paper is a characterization of cycle equivalence in
terms of shifts and reflections of the SDS update order. We construct two
graphs C(Y) and D(Y) whose components describe update orders that give rise to
cycle equivalent SDSs. The number of components in C(Y) and D(Y) is an upper
bound for the number of cycle equivalence classes one can obtain, and we
enumerate these quantities through a recursion relation for several graph
classes. The components of these graphs encode dynamical neutrality, the
component sizes represent periodic orbit structural stability, and the number
of components can be viewed as a system complexity measure
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