1,419 research outputs found
Phase Space Invertible Asynchronous Cellular Automata
While for synchronous deterministic cellular automata there is an accepted
definition of reversibility, the situation is less clear for asynchronous
cellular automata. We first discuss a few possibilities and then investigate
what we call phase space invertible asynchronous cellular automata in more
detail. We will show that for each Turing machine there is such a cellular
automaton simulating it, and that it is decidable whether an asynchronous
cellular automaton has this property or not, even in higher dimensions.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
A guided tour of asynchronous cellular automata
Research on asynchronous cellular automata has received a great amount of
attention these last years and has turned to a thriving field. We survey the
recent research that has been carried out on this topic and present a wide
state of the art where computing and modelling issues are both represented.Comment: To appear in the Journal of Cellular Automat
On the decomposition of stochastic cellular automata
In this paper we present two interesting properties of stochastic cellular
automata that can be helpful in analyzing the dynamical behavior of such
automata. The first property allows for calculating cell-wise probability
distributions over the state set of a stochastic cellular automaton, i.e.
images that show the average state of each cell during the evolution of the
stochastic cellular automaton. The second property shows that stochastic
cellular automata are equivalent to so-called stochastic mixtures of
deterministic cellular automata. Based on this property, any stochastic
cellular automaton can be decomposed into a set of deterministic cellular
automata, each of which contributes to the behavior of the stochastic cellular
automaton.Comment: Submitted to Journal of Computation Science, Special Issue on
Cellular Automata Application
Adaptive Complex Contagions and Threshold Dynamical Systems
A broad range of nonlinear processes over networks are governed by threshold
dynamics. So far, existing mathematical theory characterizing the behavior of
such systems has largely been concerned with the case where the thresholds are
static. In this paper we extend current theory of finite dynamical systems to
cover dynamic thresholds. Three classes of parallel and sequential dynamic
threshold systems are introduced and analyzed. Our main result, which is a
complete characterization of their attractor structures, show that sequential
systems may only have fixed points as limit sets whereas parallel systems may
only have period orbits of size at most two as limit sets. The attractor states
are characterized for general graphs and enumerated in the special case of
paths and cycle graphs; a computational algorithm is outlined for determining
the number of fixed points over a tree. We expect our results to be relevant
for modeling a broad class of biological, behavioral and socio-technical
systems where adaptive behavior is central.Comment: Submitted for publicatio
Combining Boolean Networks and Ordinary Differential Equations for Analysis and Comparison of Gene Regulatory Networks
This thesis is concerned with different groups of qualitative models of gene regulatory networks. Four types of models will be considered: interaction graphs, Boolean networks, models based on differential equations and discrete abstractions of differential equations. We will investigate the relations between these modeling frameworks and how they can be used in the analysis of individual models. The focus lies on the mathematical analysis of these models. This thesis makes several contributions in relating these different modeling frameworks. The first approach concerns individual Boolean models and parametrized families of ordinary differential equations (ODEs). To construct ODE models systematically from Boolean models several automatic conversion algorithms have been proposed. In Chapter 2 several such closely related algorithms will be considered. It will be proven that certain invariant sets are preserved during the conversion from a Boolean network to a model based on ODEs. In the second approach the idea of abstracting the dynamics of individual models to relate structure and dynamics will be introduced. This approach will be applied to Boolean models and models based on differential equations. This allows to compare groups of models in these modeling frameworks which have the same structure. We demonstrate that this constitutes an approach to link the interaction graph to the dynamics of certain sets of Boolean networks and models based on differential equations. The abstracted dynamics – or more precisely the restrictions on the abstracted behavior – of such sets of Boolean networks or models based on differential equations will be represented as Boolean state transitions graphs themselves. We will show that these state transition graphs can be considered as asynchronous Boolean networks. Despite the rather theoretical question this thesis tries to answer there are many potential applications of the results. The results in Chapter 2 can be applied to network reduction of ODE models based on Hill kinetics. The results of the second approach in Chapter 4 can be applied to network inference and analysis of Boolean model sets. Furthermore, in the last chapter of this thesis several ideas for applications with respect to experiment design will be considered. This leads to the question how different asynchronous Boolean networks or different behaviours of a single asynchronous Boolean network can be distinguishedDiese Arbeit beschäftigt sich mit unterschiedlichen Typen von qualitativen Modellen genregulatorischer Netzwerke. Vier Typen von Modellen werden betrachtet: Interaktionsgraphen, Boolesche Netzwerke, Modelle, die auf Differentialgleichungen basieren und diskrete Abstraktionen von Differentialgleichungen. Wir werden mehr über die Beziehungen zwischen diesen Modellgruppen lernen und wie diese Beziehungen genutzt werden können, um einzelne Modelle zu analysieren. Der Schwerpunkt liegt hierbei auf der mathematischen Analyse dieser Modellgruppen. In dieser Hinsicht leistet diese Arbeit mehrere Beiträge. Zunächst betrachten wir Boolesche Netzwerke und parametrisierte Familien von gewöhnlichen Differentialgleichungen (ODEs). Um solche ODE-Modelle systematisch aus Booleschen Modellen abzuleiten, wurden in der Vergangenheit verschiedene automatische Konvertierungsalgorithmen vorgeschlagen. In Kapitel 2 werden einige dieser Algorithmen näher untersucht. Wir werden beweisen, dass bestimmte invariante Mengen bei der Konvertierung eines Booleschen Modells in ein ODE-Modell erhalten bleiben. Der zweite Ansatz, der in dieser Arbeit verfolgt wird, beschäftigt sich mit diskreten Abstraktionen der Dynamik von Modellen. Mit Hilfe dieser Abstraktionen ist es möglich, die Struktur – den Interaktionsgraphen – und die Dynamik der zugehörigen Modelle in Bezug zu setzen. Diese Methode wird sowohl auf Boolesche Modelle als auch auf ODE-Modelle angewandt. Gleichzeitig erlaubt dieser Ansatz Mengen von Modellen in unterschiedlichen Modellgruppen zu vergleichen, die dieselbe Struktur haben. Die abstrahierten Dynamiken (genauer die Einschränkungen der abstrahierten Dynamiken) der Booleschen Modellmengen oder ODE-Modellmengen können als Boolesche Zustandsübergangsgraphen repräsentiert werden. Wir werden zeigen, dass diese Zustandsübergangsgraphen wiederum selber als (asynchrone) Boolesche Netzwerke aufgefasst werden können. Trotz der theoretischen Ausgangsfrage werden in dieser Arbeit zahlreiche Anwendungen aufgezeigt. Die Ergebnisse aus Kapitel 2 können zur Modellreduktion benutzt werden, indem die Dynamik der ODE-Modelle auf den zu den Booleschen Netzwerken gehörigen “trap spaces” betrachtet wird. Die Resultate aus Kapitel 4 können zur Netzwerkinferenz oder zur Analyse von Modellmengen genutzt werden. Weiterhin werden im letzten Kapitel dieser Arbeit einige Anwendungsideen im Bezug auf Experimentdesign eingeführt. Dies führt zu der Fragestellung, wie verschiedene asynchrone Boolesche Netzwerke oder unterschiedliche Dynamiken, die mit einem einzelnen Modell vereinbar sind, unterschieden werden können
Asynchronous Networks and Event Driven Dynamics
Real-world networks in technology, engineering and biology often exhibit
dynamics that cannot be adequately reproduced using network models given by
smooth dynamical systems and a fixed network topology. Asynchronous networks
give a theoretical and conceptual framework for the study of network dynamics
where nodes can evolve independently of one another, be constrained, stop, and
later restart, and where the interaction between different components of the
network may depend on time, state, and stochastic effects. This framework is
sufficiently general to encompass a wide range of applications ranging from
engineering to neuroscience. Typically, dynamics is piecewise smooth and there
are relationships with Filippov systems. In the first part of the paper, we
give examples of asynchronous networks, and describe the basic formalism and
structure. In the second part, we make the notion of a functional asynchronous
network rigorous, discuss the phenomenon of dynamical locks, and present a
foundational result on the spatiotemporal factorization of the dynamics for a
large class of functional asynchronous networks
Causal graph dynamics
We extend the theory of Cellular Automata to arbitrary, time-varying graphs.
In other words we formalize, and prove theorems about, the intuitive idea of a
labelled graph which evolves in time - but under the natural constraint that
information can only ever be transmitted at a bounded speed, with respect to
the distance given by the graph. The notion of translation-invariance is also
generalized. The definition we provide for these "causal graph dynamics" is
simple and axiomatic. The theorems we provide also show that it is robust. For
instance, causal graph dynamics are stable under composition and under
restriction to radius one. In the finite case some fundamental facts of
Cellular Automata theory carry through: causal graph dynamics admit a
characterization as continuous functions, and they are stable under inversion.
The provided examples suggest a wide range of applications of this mathematical
object, from complex systems science to theoretical physics. KEYWORDS:
Dynamical networks, Boolean networks, Generative networks automata, Cayley
cellular automata, Graph Automata, Graph rewriting automata, Parallel graph
transformations, Amalgamated graph transformations, Time-varying graphs, Regge
calculus, Local, No-signalling.Comment: 25 pages, 9 figures, LaTeX, v2: Minor presentation improvements, v3:
Typos corrected, figure adde
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