24,461 research outputs found
Computing by nowhere increasing complexity
A cellular automaton is presented whose governing rule is that the Kolmogorov
complexity of a cell's neighborhood may not increase when the cell's present
value is substituted for its future value. Using an approximation of this
two-dimensional Kolmogorov complexity the underlying automaton is shown to be
capable of simulating logic circuits. It is also shown to capture trianry logic
described by a quandle, a non-associative algebraic structure. A similar
automaton whose rule permits at times the increase of a cell's neighborhood
complexity is shown to produce animated entities which can be used as
information carriers akin to gliders in Conway's game of life
Deciding first-order properties of nowhere dense graphs
Nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez,
form a large variety of classes of "sparse graphs" including the class of
planar graphs, actually all classes with excluded minors, and also bounded
degree graphs and graph classes of bounded expansion.
We show that deciding properties of graphs definable in first-order logic is
fixed-parameter tractable on nowhere dense graph classes. At least for graph
classes closed under taking subgraphs, this result is optimal: it was known
before that for all classes C of graphs closed under taking subgraphs, if
deciding first-order properties of graphs in C is fixed-parameter tractable,
then C must be nowhere dense (under a reasonable complexity theoretic
assumption).
As a by-product, we give an algorithmic construction of sparse neighbourhood
covers for nowhere dense graphs. This extends and improves previous
constructions of neighbourhood covers for graph classes with excluded minors.
At the same time, our construction is considerably simpler than those. Our
proofs are based on a new game-theoretic characterisation of nowhere dense
graphs that allows for a recursive version of locality-based algorithms on
these classes. On the logical side, we prove a "rank-preserving" version of
Gaifman's locality theorem.Comment: 30 page
Nomographic Functions: Efficient Computation in Clustered Gaussian Sensor Networks
In this paper, a clustered wireless sensor network is considered that is
modeled as a set of coupled Gaussian multiple-access channels. The objective of
the network is not to reconstruct individual sensor readings at designated
fusion centers but rather to reliably compute some functions thereof. Our
particular attention is on real-valued functions that can be represented as a
post-processed sum of pre-processed sensor readings. Such functions are called
nomographic functions and their special structure permits the utilization of
the interference property of the Gaussian multiple-access channel to reliably
compute many linear and nonlinear functions at significantly higher rates than
those achievable with standard schemes that combat interference. Motivated by
this observation, a computation scheme is proposed that combines a suitable
data pre- and post-processing strategy with a nested lattice code designed to
protect the sum of pre-processed sensor readings against the channel noise.
After analyzing its computation rate performance, it is shown that at the cost
of a reduced rate, the scheme can be extended to compute every continuous
function of the sensor readings in a finite succession of steps, where in each
step a different nomographic function is computed. This demonstrates the
fundamental role of nomographic representations.Comment: to appear in IEEE Transactions on Wireless Communication
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