17,743 research outputs found
Width and size of regular resolution proofs
This paper discusses the topic of the minimum width of a regular resolution
refutation of a set of clauses. The main result shows that there are examples
having small regular resolution refutations, for which any regular refutation
must contain a large clause. This forms a contrast with corresponding results
for general resolution refutations.Comment: The article was reformatted using the style file for Logical Methods
in Computer Scienc
From Small Space to Small Width in Resolution
In 2003, Atserias and Dalmau resolved a major open question about the
resolution proof system by establishing that the space complexity of CNF
formulas is always an upper bound on the width needed to refute them. Their
proof is beautiful but somewhat mysterious in that it relies heavily on tools
from finite model theory. We give an alternative, completely elementary proof
that works by simple syntactic manipulations of resolution refutations. As a
by-product, we develop a "black-box" technique for proving space lower bounds
via a "static" complexity measure that works against any resolution
refutation---previous techniques have been inherently adaptive. We conclude by
showing that the related question for polynomial calculus (i.e., whether space
is an upper bound on degree) seems unlikely to be resolvable by similar
methods
What grid cells convey about rat location
We characterize the relationship between the simultaneously recorded quantities of rodent grid cell firing and the position of the rat. The formalization reveals various properties of grid cell activity when considered as a neural code for representing and updating estimates of the rat's location. We show that, although the spatially periodic response of grid cells appears wasteful, the code is fully combinatorial in capacity. The resulting range for unambiguous position representation is vastly greater than the ≈1–10 m periods of individual lattices, allowing for unique high-resolution position specification over the behavioral foraging ranges of rats, with excess capacity that could be used for error correction. Next, we show that the merits of the grid cell code for position representation extend well beyond capacity and include arithmetic properties that facilitate position updating. We conclude by considering the numerous implications, for downstream readouts and experimental tests, of the properties of the grid cell code
Narrow Proofs May Be Maximally Long
We prove that there are 3-CNF formulas over n variables that can be refuted
in resolution in width w but require resolution proofs of size n^Omega(w). This
shows that the simple counting argument that any formula refutable in width w
must have a proof in size n^O(w) is essentially tight. Moreover, our lower
bound generalizes to polynomial calculus resolution (PCR) and Sherali-Adams,
implying that the corresponding size upper bounds in terms of degree and rank
are tight as well. Our results do not extend all the way to Lasserre, however,
where the formulas we study have proofs of constant rank and size polynomial in
both n and w
Sub-wavelength focusing meta-lens
We show that planar a plasmonic metamaterial with spatially variable
meta-atom parameters can focus transmitted light into sub-wavelength hot-spots
located beyond the near-field of the metamaterial. By nano-structuring a gold
film we created an array of meta-lenses generating foci of 160 nm
(0.2{\lambda}) in diameter when illuminated by a wavelength of 800 nm. We
attribute the occurrence of sub-wavelength hotspots beyond the near field to
the phenomenon of superoscillation
- …