25,424 research outputs found
Markovian Testing Equivalence and Exponentially Timed Internal Actions
In the theory of testing for Markovian processes developed so far,
exponentially timed internal actions are not admitted within processes. When
present, these actions cannot be abstracted away, because their execution takes
a nonzero amount of time and hence can be observed. On the other hand, they
must be carefully taken into account, in order not to equate processes that are
distinguishable from a timing viewpoint. In this paper, we recast the
definition of Markovian testing equivalence in the framework of a Markovian
process calculus including exponentially timed internal actions. Then, we show
that the resulting behavioral equivalence is a congruence, has a sound and
complete axiomatization, has a modal logic characterization, and can be decided
in polynomial time
On the Expressiveness of Markovian Process Calculi with Durational and Durationless Actions
Several Markovian process calculi have been proposed in the literature, which
differ from each other for various aspects. With regard to the action
representation, we distinguish between integrated-time Markovian process
calculi, in which every action has an exponentially distributed duration
associated with it, and orthogonal-time Markovian process calculi, in which
action execution is separated from time passing. Similar to deterministically
timed process calculi, we show that these two options are not irreconcilable by
exhibiting three mappings from an integrated-time Markovian process calculus to
an orthogonal-time Markovian process calculus that preserve the behavioral
equivalence of process terms under different interpretations of action
execution: eagerness, laziness, and maximal progress. The mappings are limited
to classes of process terms of the integrated-time Markovian process calculus
with restrictions on parallel composition and do not involve the full
capability of the orthogonal-time Markovian process calculus of expressing
nondeterministic choices, thus elucidating the only two important differences
between the two calculi: their synchronization disciplines and their ways of
solving choices
The Computational Structure of Spike Trains
Neurons perform computations, and convey the results of those computations
through the statistical structure of their output spike trains. Here we present
a practical method, grounded in the information-theoretic analysis of
prediction, for inferring a minimal representation of that structure and for
characterizing its complexity. Starting from spike trains, our approach finds
their causal state models (CSMs), the minimal hidden Markov models or
stochastic automata capable of generating statistically identical time series.
We then use these CSMs to objectively quantify both the generalizable structure
and the idiosyncratic randomness of the spike train. Specifically, we show that
the expected algorithmic information content (the information needed to
describe the spike train exactly) can be split into three parts describing (1)
the time-invariant structure (complexity) of the minimal spike-generating
process, which describes the spike train statistically; (2) the randomness
(internal entropy rate) of the minimal spike-generating process; and (3) a
residual pure noise term not described by the minimal spike-generating process.
We use CSMs to approximate each of these quantities. The CSMs are inferred
nonparametrically from the data, making only mild regularity assumptions, via
the causal state splitting reconstruction algorithm. The methods presented here
complement more traditional spike train analyses by describing not only spiking
probability and spike train entropy, but also the complexity of a spike train's
structure. We demonstrate our approach using both simulated spike trains and
experimental data recorded in rat barrel cortex during vibrissa stimulation.Comment: Somewhat different format from journal version but same conten
The Origins of Computational Mechanics: A Brief Intellectual History and Several Clarifications
The principle goal of computational mechanics is to define pattern and
structure so that the organization of complex systems can be detected and
quantified. Computational mechanics developed from efforts in the 1970s and
early 1980s to identify strange attractors as the mechanism driving weak fluid
turbulence via the method of reconstructing attractor geometry from measurement
time series and in the mid-1980s to estimate equations of motion directly from
complex time series. In providing a mathematical and operational definition of
structure it addressed weaknesses of these early approaches to discovering
patterns in natural systems.
Since then, computational mechanics has led to a range of results from
theoretical physics and nonlinear mathematics to diverse applications---from
closed-form analysis of Markov and non-Markov stochastic processes that are
ergodic or nonergodic and their measures of information and intrinsic
computation to complex materials and deterministic chaos and intelligence in
Maxwellian demons to quantum compression of classical processes and the
evolution of computation and language.
This brief review clarifies several misunderstandings and addresses concerns
recently raised regarding early works in the field (1980s). We show that
misguided evaluations of the contributions of computational mechanics are
groundless and stem from a lack of familiarity with its basic goals and from a
failure to consider its historical context. For all practical purposes, its
modern methods and results largely supersede the early works. This not only
renders recent criticism moot and shows the solid ground on which computational
mechanics stands but, most importantly, shows the significant progress achieved
over three decades and points to the many intriguing and outstanding challenges
in understanding the computational nature of complex dynamic systems.Comment: 11 pages, 123 citations;
http://csc.ucdavis.edu/~cmg/compmech/pubs/cmr.ht
A Potential Foundation for Emergent Space-Time
We present a novel derivation of both the Minkowski metric and Lorentz
transformations from the consistent quantification of a causally ordered set of
events with respect to an embedded observer. Unlike past derivations, which
have relied on assumptions such as the existence of a 4-dimensional manifold,
symmetries of space-time, or the constant speed of light, we demonstrate that
these now familiar mathematics can be derived as the unique means to
consistently quantify a network of events. This suggests that space-time need
not be physical, but instead the mathematics of space and time emerges as the
unique way in which an observer can consistently quantify events and their
relationships to one another. The result is a potential foundation for emergent
space-time.Comment: The paper was originally titled "The Physics of Events: A Potential
Foundation for Emergent Space-Time". We changed the title (and abstract) to
be more direct when the paper was accepted for publication at the Journal of
Mathematical Physics. 24 pages, 15 figure
Towards Quantifying Complexity with Quantum Mechanics
While we have intuitive notions of structure and complexity, the
formalization of this intuition is non-trivial. The statistical complexity is a
popular candidate. It is based on the idea that the complexity of a process can
be quantified by the complexity of its simplest mathematical model - the model
that requires the least past information for optimal future prediction. Here we
review how such models, known as -machines can be further simplified
through quantum logic, and explore the resulting consequences for understanding
complexity. In particular, we propose a new measure of complexity based on
quantum -machines. We apply this to a simple system undergoing
constant thermalization. The resulting quantum measure of complexity aligns
more closely with our intuition of how complexity should behave.Comment: 10 pages, 6 figure, Published in the Focus Point on Quantum
information and complexity edition of EPJ Plu
Complexity Through Nonextensivity
The problem of defining and studying complexity of a time series has
interested people for years. In the context of dynamical systems, Grassberger
has suggested that a slow approach of the entropy to its extensive asymptotic
limit is a sign of complexity. We investigate this idea further by information
theoretic and statistical mechanics techniques and show that these arguments
can be made precise, and that they generalize many previous approaches to
complexity, in particular unifying ideas from the physics literature with ideas
from learning and coding theory; there are even connections of this statistical
approach to algorithmic or Kolmogorov complexity. Moreover, a set of simple
axioms similar to those used by Shannon in his development of information
theory allows us to prove that the divergent part of the subextensive component
of the entropy is a unique complexity measure. We classify time series by their
complexities and demonstrate that beyond the `logarithmic' complexity classes
widely anticipated in the literature there are qualitatively more complex,
`power--law' classes which deserve more attention.Comment: summarizes and extends physics/000707
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