581 research outputs found
The Lattice of integer partitions and its infinite extension
In this paper, we use a simple discrete dynamical system to study the
integers partitions and their lattice. The set of the reachable configurations
equiped with the order induced by the transitions of the system is exactly the
lattice of integer partitions equiped with the dominance ordering. We first
explain how this lattice can be constructed, by showing its strong
self-similarity property. Then, we define a natural extension of the system to
infinity. Using a self-similar tree, we obtain an efficient coding of the
obtained lattice. This approach gives an interesting recursive formula for the
number of partitions of an integer, where no closed formula have ever been
found. It also gives informations on special sets of partitions, such as length
bounded partitions.Comment: To appear in LNCS special issue, proceedings of ORDAL'99. See
http://www.liafa.jussieu.fr/~latap
The Lattice structure of Chip Firing Games and Related Models
In this paper, we study a famous discrete dynamical system, the Chip Firing
Game, used as a model in physics, economics and computer science. We use order
theory and show that the set of reachable states (i.e. the configuration space)
of such a system started in any configuration is a lattice, which implies
strong structural properties. The lattice structure of the configuration space
of a dynamical system is of great interest since it implies convergence (and
more) if the configuration space is finite. If it is infinite, this property
implies another kind of convergence: all the configurations reachable from two
given configurations are reachable from their infimum. In other words, there is
a unique first configuration which is reachable from two given configurations.
Moreover, the Chip Firing Game is a very general model, and we show how known
models can be encoded as Chip Firing Games, and how some results about them can
be deduced from this paper. Finally, we define a new model, which is a
generalization of the Chip Firing Game, and about which many interesting
questions arise.Comment: See http://www.liafa.jussieu.fr/~latap
On Conservative and Monotone One-dimensional Cellular Automata and Their Particle Representation
Number-conserving (or {\em conservative}) cellular automata have been used in
several contexts, in particular traffic models, where it is natural to think
about them as systems of interacting particles. In this article we consider
several issues concerning one-dimensional cellular automata which are
conservative, monotone (specially ``non-increasing''), or that allow a weaker
kind of conservative dynamics. We introduce a formalism of ``particle
automata'', and discuss several properties that they may exhibit, some of
which, like anticipation and momentum preservation, happen to be intrinsic to
the conservative CA they represent. For monotone CA we give a characterization,
and then show that they too are equivalent to the corresponding class of
particle automata. Finally, we show how to determine, for a given CA and a
given integer , whether its states admit a -neighborhood-dependent
relabelling whose sum is conserved by the CA iteration; this can be used to
uncover conservative principles and particle-like behavior underlying the
dynamics of some CA. Complements at {\tt http://www.dim.uchile.cl/\verb'
'anmoreir/ncca}Comment: 38 pages, 2 figures. To appear in Theo. Comp. Sc. Several changes
throughout the text; major change in section 4.
Complexity of Langton's Ant
The virtual ant introduced by C. Langton has an interesting behavior, which
has been studied in several contexts. Here we give a construction to calculate
any boolean circuit with the trajectory of a single ant. This proves the
P-hardness of the system and implies, through the simulation of one dimensional
cellular automata and Turing machines, the universality of the ant and the
undecidability of some problems associated to it.Comment: 8 pages, 9 figures. Complements at
http://www.dim.uchile.cl/~agajardo/langto
On the effects of firing memory in the dynamics of conjunctive networks
Boolean networks are one of the most studied discrete models in the context
of the study of gene expression. In order to define the dynamics associated to
a Boolean network, there are several \emph{update schemes} that range from
parallel or \emph{synchronous} to \emph{asynchronous.} However, studying each
possible dynamics defined by different update schemes might not be efficient.
In this context, considering some type of temporal delay in the dynamics of
Boolean networks emerges as an alternative approach. In this paper, we focus in
studying the effect of a particular type of delay called \emph{firing memory}
in the dynamics of Boolean networks. Particularly, we focus in symmetric
(non-directed) conjunctive networks and we show that there exist examples that
exhibit attractors of non-polynomial period. In addition, we study the
prediction problem consisting in determinate if some vertex will eventually
change its state, given an initial condition. We prove that this problem is
{\bf PSPACE}-complete
Communication Complexity and Intrinsic Universality in Cellular Automata
The notions of universality and completeness are central in the theories of
computation and computational complexity. However, proving lower bounds and
necessary conditions remains hard in most of the cases. In this article, we
introduce necessary conditions for a cellular automaton to be "universal",
according to a precise notion of simulation, related both to the dynamics of
cellular automata and to their computational power. This notion of simulation
relies on simple operations of space-time rescaling and it is intrinsic to the
model of cellular automata. Intrinsinc universality, the derived notion, is
stronger than Turing universality, but more uniform, and easier to define and
study. Our approach builds upon the notion of communication complexity, which
was primarily designed to study parallel programs, and thus is, as we show in
this article, particulary well suited to the study of cellular automata: it
allowed to show, by studying natural problems on the dynamics of cellular
automata, that several classes of cellular automata, as well as many natural
(elementary) examples, could not be intrinsically universal
Block-sequential update schedules and Boolean automata circuits
International audienceOur work is set in the framework of complex dynamical systems and, more precisely, that of Boolean automata networks modeling regulation networks. We study how the choice of an update schedule impacts on the dynamics of such a network. To do this, we explain how studying the dynamics of any network updated with an arbitrary block-sequential update schedule can be reduced to the study of the dynamics of a different network updated in parallel. We give special attention to networks whose underlying structure is a circuit, that is, Boolean automata circuits. These particular and simple networks are known to serve as the "engines'' of the dynamics of arbitrary regulation networks containing them as sub-networks in that they are responsible for their variety of dynamical behaviours. We give both the number of attractors of period , and the total number of attractors in the dynamics of Boolean automata circuits updated with any block-sequential update schedule. We also detail the variety of dynamical behaviours that such networks may exhibit according to the update schedule
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