5,010 research outputs found
Computation in Finitary Stochastic and Quantum Processes
We introduce stochastic and quantum finite-state transducers as
computation-theoretic models of classical stochastic and quantum finitary
processes. Formal process languages, representing the distribution over a
process's behaviors, are recognized and generated by suitable specializations.
We characterize and compare deterministic and nondeterministic versions,
summarizing their relative computational power in a hierarchy of finitary
process languages. Quantum finite-state transducers and generators are a first
step toward a computation-theoretic analysis of individual, repeatedly measured
quantum dynamical systems. They are explored via several physical systems,
including an iterated beam splitter, an atom in a magnetic field, and atoms in
an ion trap--a special case of which implements the Deutsch quantum algorithm.
We show that these systems' behaviors, and so their information processing
capacity, depends sensitively on the measurement protocol.Comment: 25 pages, 16 figures, 1 table; http://cse.ucdavis.edu/~cmg; numerous
corrections and update
Local Causal States and Discrete Coherent Structures
Coherent structures form spontaneously in nonlinear spatiotemporal systems
and are found at all spatial scales in natural phenomena from laboratory
hydrodynamic flows and chemical reactions to ocean, atmosphere, and planetary
climate dynamics. Phenomenologically, they appear as key components that
organize the macroscopic behaviors in such systems. Despite a century of
effort, they have eluded rigorous analysis and empirical prediction, with
progress being made only recently. As a step in this, we present a formal
theory of coherent structures in fully-discrete dynamical field theories. It
builds on the notion of structure introduced by computational mechanics,
generalizing it to a local spatiotemporal setting. The analysis' main tool
employs the \localstates, which are used to uncover a system's hidden
spatiotemporal symmetries and which identify coherent structures as
spatially-localized deviations from those symmetries. The approach is
behavior-driven in the sense that it does not rely on directly analyzing
spatiotemporal equations of motion, rather it considers only the spatiotemporal
fields a system generates. As such, it offers an unsupervised approach to
discover and describe coherent structures. We illustrate the approach by
analyzing coherent structures generated by elementary cellular automata,
comparing the results with an earlier, dynamic-invariant-set approach that
decomposes fields into domains, particles, and particle interactions.Comment: 27 pages, 10 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/dcs.ht
Coding-theorem Like Behaviour and Emergence of the Universal Distribution from Resource-bounded Algorithmic Probability
Previously referred to as `miraculous' in the scientific literature because
of its powerful properties and its wide application as optimal solution to the
problem of induction/inference, (approximations to) Algorithmic Probability
(AP) and the associated Universal Distribution are (or should be) of the
greatest importance in science. Here we investigate the emergence, the rates of
emergence and convergence, and the Coding-theorem like behaviour of AP in
Turing-subuniversal models of computation. We investigate empirical
distributions of computing models in the Chomsky hierarchy. We introduce
measures of algorithmic probability and algorithmic complexity based upon
resource-bounded computation, in contrast to previously thoroughly investigated
distributions produced from the output distribution of Turing machines. This
approach allows for numerical approximations to algorithmic
(Kolmogorov-Chaitin) complexity-based estimations at each of the levels of a
computational hierarchy. We demonstrate that all these estimations are
correlated in rank and that they converge both in rank and values as a function
of computational power, despite fundamental differences between computational
models. In the context of natural processes that operate below the Turing
universal level because of finite resources and physical degradation, the
investigation of natural biases stemming from algorithmic rules may shed light
on the distribution of outcomes. We show that up to 60\% of the
simplicity/complexity bias in distributions produced even by the weakest of the
computational models can be accounted for by Algorithmic Probability in its
approximation to the Universal Distribution.Comment: 27 pages main text, 39 pages including supplement. Online complexity
calculator: http://complexitycalculator.com
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
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
Graph Kernels
We present a unified framework to study graph kernels, special cases of which include the random
walk (Gärtner et al., 2003; Borgwardt et al., 2005) and marginalized (Kashima et al., 2003, 2004;
Mahé et al., 2004) graph kernels. Through reduction to a Sylvester equation we improve the time
complexity of kernel computation between unlabeled graphs with n vertices from O(n^6) to O(n^3).
We find a spectral decomposition approach even more efficient when computing entire kernel matrices.
For labeled graphs we develop conjugate gradient and fixed-point methods that take O(dn^3)
time per iteration, where d is the size of the label set. By extending the necessary linear algebra to
Reproducing Kernel Hilbert Spaces (RKHS) we obtain the same result for d-dimensional edge kernels,
and O(n^4) in the infinite-dimensional case; on sparse graphs these algorithms only take O(n^2)
time per iteration in all cases. Experiments on graphs from bioinformatics and other application
domains show that these techniques can speed up computation of the kernel by an order of magnitude
or more. We also show that certain rational kernels (Cortes et al., 2002, 2003, 2004) when
specialized to graphs reduce to our random walk graph kernel. Finally, we relate our framework to
R-convolution kernels (Haussler, 1999) and provide a kernel that is close to the optimal assignment
kernel of Fröhlich et al. (2006) yet provably positive semi-definite
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