6,315 research outputs found
Formal Languages in Dynamical Systems
We treat here the interrelation between formal languages and those dynamical
systems that can be described by cellular automata (CA). There is a well-known
injective map which identifies any CA-invariant subshift with a central formal
language. However, in the special case of a symbolic dynamics, i.e. where the
CA is just the shift map, one gets a stronger result: the identification map
can be extended to a functor between the categories of symbolic dynamics and
formal languages. This functor additionally maps topological conjugacies
between subshifts to empty-string-limited generalized sequential machines
between languages. If the periodic points form a dense set, a case which arises
in a commonly used notion of chaotic dynamics, then an even more natural map to
assign a formal language to a subshift is offered. This map extends to a
functor, too. The Chomsky hierarchy measuring the complexity of formal
languages can be transferred via either of these functors from formal languages
to symbolic dynamics and proves to be a conjugacy invariant there. In this way
it acquires a dynamical meaning. After reviewing some results of the complexity
of CA-invariant subshifts, special attention is given to a new kind of
invariant subshift: the trapped set, which originates from the theory of
chaotic scattering and for which one can study complexity transitions.Comment: 23 pages, LaTe
Descriptional complexity of cellular automata and decidability questions
We study the descriptional complexity of cellular automata (CA), a parallel model of computation. We show that between one of the simplest cellular models, the realtime-OCA. and "classical" models like deterministic finite automata (DFA) or pushdown automata (PDA), there will be savings concerning the size of description not bounded by any recursive function, a so-called nonrecursive trade-off. Furthermore, nonrecursive trade-offs are shown between some restricted classes of cellular automata. The set of valid computations of a Turing machine can be recognized by a realtime-OCA. This implies that many decidability questions are not even semi decidable for cellular automata. There is no pumping lemma and no minimization algorithm for cellular automata
Bounded Languages Meet Cellular Automata with Sparse Communication
Cellular automata are one-dimensional arrays of interconnected interacting
finite automata. We investigate one of the weakest classes, the real-time
one-way cellular automata, and impose an additional restriction on their
inter-cell communication by bounding the number of allowed uses of the links
between cells. Moreover, we consider the devices as acceptors for bounded
languages in order to explore the borderline at which non-trivial decidability
problems of cellular automata classes become decidable. It is shown that even
devices with drastically reduced communication, that is, each two neighboring
cells may communicate only constantly often, accept bounded languages that are
not semilinear. If the number of communications is at least logarithmic in the
length of the input, several problems are undecidable. The same result is
obtained for classes where the total number of communications during a
computation is linearly bounded
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
Upper Bound on the Products of Particle Interactions in Cellular Automata
Particle-like objects are observed to propagate and interact in many
spatially extended dynamical systems. For one of the simplest classes of such
systems, one-dimensional cellular automata, we establish a rigorous upper bound
on the number of distinct products that these interactions can generate. The
upper bound is controlled by the structural complexity of the interacting
particles---a quantity which is defined here and which measures the amount of
spatio-temporal information that a particle stores. Along the way we establish
a number of properties of domains and particles that follow from the
computational mechanics analysis of cellular automata; thereby elucidating why
that approach is of general utility. The upper bound is tested against several
relatively complex domain-particle cellular automata and found to be tight.Comment: 17 pages, 12 figures, 3 tables,
http://www.santafe.edu/projects/CompMech/papers/ub.html V2: References and
accompanying text modified, to comply with legal demands arising from
on-going intellectual property litigation among third parties. V3: Accepted
for publication in Physica D. References added and other small changes made
per referee suggestion
Complete Symmetry in D2L Systems and Cellular Automata
We introduce completely symmetric D2L systems and cellular automata by means of an additional restriction on the corresponding symmetric devices. Then we show that completely symmetric D2L systems and cellular automata are still able to simulate Turing machine computations. As corollaries we obtain new characterizations of the recursively enumerable languages and of some space-bounded complexity classes
On the descriptional complexity of iterative arrays
The descriptional complexity of iterative arrays (lAs) is studied. Iterative arrays are a parallel computational model with a sequential processing of the input. It is shown that lAs when compared to deterministic finite automata or pushdown automata may provide savings in size which are not bounded by any recursive function, so-called non-recursive trade-offs. Additional non-recursive trade-offs are proven to exist between lAs working in linear time and lAs working in real time. Furthermore, the descriptional complexity of lAs is compared with cellular automata (CAs) and non-recursive trade-offs are proven between two restricted classes. Finally, it is shown that many decidability questions for lAs are undecidable and not semidecidable
Transductions Computed by One-Dimensional Cellular Automata
Cellular automata are investigated towards their ability to compute
transductions, that is, to transform inputs into outputs. The families of
transductions computed are classified with regard to the time allowed to
process the input and to compute the output. Since there is a particular
interest in fast transductions, we mainly focus on the time complexities real
time and linear time. We first investigate the computational capabilities of
cellular automaton transducers by comparing them to iterative array
transducers, that is, we compare parallel input/output mode to sequential
input/output mode of massively parallel machines. By direct simulations, it
turns out that the parallel mode is not weaker than the sequential one.
Moreover, with regard to certain time complexities cellular automaton
transducers are even more powerful than iterative arrays. In the second part of
the paper, the model in question is compared with the sequential devices
single-valued finite state transducers and deterministic pushdown transducers.
It turns out that both models can be simulated by cellular automaton
transducers faster than by iterative array transducers.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
The Computational Complexity of Symbolic Dynamics at the Onset of Chaos
In a variety of studies of dynamical systems, the edge of order and chaos has
been singled out as a region of complexity. It was suggested by Wolfram, on the
basis of qualitative behaviour of cellular automata, that the computational
basis for modelling this region is the Universal Turing Machine. In this paper,
following a suggestion of Crutchfield, we try to show that the Turing machine
model may often be too powerful as a computational model to describe the
boundary of order and chaos. In particular we study the region of the first
accumulation of period doubling in unimodal and bimodal maps of the interval,
from the point of view of language theory. We show that in relation to the
``extended'' Chomsky hierarchy, the relevant computational model in the
unimodal case is the nested stack automaton or the related indexed languages,
while the bimodal case is modeled by the linear bounded automaton or the
related context-sensitive languages.Comment: 1 reference corrected, 1 reference added, minor changes in body of
manuscrip
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