1,125 research outputs found
Complex Networks from Simple Rewrite Systems
Complex networks are all around us, and they can be generated by simple
mechanisms. Understanding what kinds of networks can be produced by following
simple rules is therefore of great importance. We investigate this issue by
studying the dynamics of extremely simple systems where are `writer' moves
around a network, and modifies it in a way that depends upon the writer's
surroundings. Each vertex in the network has three edges incident upon it,
which are colored red, blue and green. This edge coloring is done to provide a
way for the writer to orient its movement. We explore the dynamics of a space
of 3888 of these `colored trinet automata' systems. We find a large variety of
behaviour, ranging from the very simple to the very complex. We also discover
simple rules that generate forms which are remarkably similar to a wide range
of natural objects. We study our systems using simulations (with appropriate
visualization techniques) and analyze selected rules mathematically. We arrive
at an empirical classification scheme which reveals a lot about the kinds of
dynamics and networks that can be generated by these systems
Spatial updating, spatial transients, and regularities of a complex automaton with nonperiodic architecture
We study the dynamics of patterns exhibited by rule 52, a totalistic cellular automaton displaying intricate behaviors and wide regions of active/inactive synchronization patches. Systematic computer simulations involving 230 initial configurations reveal that all complexity in this automaton originates from random juxtaposition of a very small number of interfaces delimiting active/inactive patches. Such interfaces are studied with a sidewise spatial updating algorithm. This novel tool allows us to prove that the interfaces found empirically are the only interfaces possible for these periods, independently of the size of the automata. The spatial updating algorithm provides an alternative way to determine the dynamics of automata of arbitrary size, a way of taking into account the complexity of the connections in the lattice
Multi-Head Finite Automata: Characterizations, Concepts and Open Problems
Multi-head finite automata were introduced in (Rabin, 1964) and (Rosenberg,
1966). Since that time, a vast literature on computational and descriptional
complexity issues on multi-head finite automata documenting the importance of
these devices has been developed. Although multi-head finite automata are a
simple concept, their computational behavior can be already very complex and
leads to undecidable or even non-semi-decidable problems on these devices such
as, for example, emptiness, finiteness, universality, equivalence, etc. These
strong negative results trigger the study of subclasses and alternative
characterizations of multi-head finite automata for a better understanding of
the nature of non-recursive trade-offs and, thus, the borderline between
decidable and undecidable problems. In the present paper, we tour a fragment of
this literature
Forward Analysis and Model Checking for Trace Bounded WSTS
We investigate a subclass of well-structured transition systems (WSTS), the
bounded---in the sense of Ginsburg and Spanier (Trans. AMS 1964)---complete
deterministic ones, which we claim provide an adequate basis for the study of
forward analyses as developed by Finkel and Goubault-Larrecq (Logic. Meth.
Comput. Sci. 2012). Indeed, we prove that, unlike other conditions considered
previously for the termination of forward analysis, boundedness is decidable.
Boundedness turns out to be a valuable restriction for WSTS verification, as we
show that it further allows to decide all -regular properties on the
set of infinite traces of the system
Localized defects in a cellular automaton model for traffic flow with phase separation
We study the impact of a localized defect in a cellular automaton model for
traffic flow which exhibits metastable states and phase separation. The defect
is implemented by locally limiting the maximal possible flow through an
increase of the deceleration probability. Depending on the magnitude of the
defect three phases can be identified in the system. One of these phases shows
the characteristics of stop-and-go traffic which can not be found in the model
without lattice defect. Thus our results provide evidence that even in a model
with strong phase separation stop-and-go traffic can occur if local defects
exist. From a physical point of view the model describes the competition
between two mechanisms of phase separation.Comment: 14 pages, 7 figure
Complexity of Two-Dimensional Patterns
In dynamical systems such as cellular automata and iterated maps, it is often
useful to look at a language or set of symbol sequences produced by the system.
There are well-established classification schemes, such as the Chomsky
hierarchy, with which we can measure the complexity of these sets of sequences,
and thus the complexity of the systems which produce them.
In this paper, we look at the first few levels of a hierarchy of complexity
for two-or-more-dimensional patterns. We show that several definitions of
``regular language'' or ``local rule'' that are equivalent in d=1 lead to
distinct classes in d >= 2. We explore the closure properties and computational
complexity of these classes, including undecidability and L-, NL- and
NP-completeness results.
We apply these classes to cellular automata, in particular to their sets of
fixed and periodic points, finite-time images, and limit sets. We show that it
is undecidable whether a CA in d >= 2 has a periodic point of a given period,
and that certain ``local lattice languages'' are not finite-time images or
limit sets of any CA. We also show that the entropy of a d-dimensional CA's
finite-time image cannot decrease faster than t^{-d} unless it maps every
initial condition to a single homogeneous state.Comment: To appear in J. Stat. Phy
Asynchronous Games over Tree Architectures
We consider the task of controlling in a distributed way a Zielonka
asynchronous automaton. Every process of a controller has access to its causal
past to determine the next set of actions it proposes to play. An action can be
played only if every process controlling this action proposes to play it. We
consider reachability objectives: every process should reach its set of final
states. We show that this control problem is decidable for tree architectures,
where every process can communicate with its parent, its children, and with the
environment. The complexity of our algorithm is l-fold exponential with l being
the height of the tree representing the architecture. We show that this is
unavoidable by showing that even for three processes the problem is
EXPTIME-complete, and that it is non-elementary in general
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