29,968 research outputs found
Are Short Proofs Narrow? QBF Resolution is not so Simple
The ground-breaking paper “Short Proofs Are Narrow -- Resolution Made Simple” by Ben-Sasson and Wigderson (J. ACM 2001) introduces what is today arguably the main technique to obtain resolution lower bounds: to show a lower bound for the width of proofs. Another important measure for resolution is space, and in their fundamental work, Atserias and Dalmau (J. Comput. Syst. Sci. 2008) show that lower bounds for space again can be obtained via lower bounds for width. In this article, we assess whether similar techniques are effective for resolution calculi for quantified Boolean formulas (QBFs). There are a number of different QBF resolution calculi like Q-resolution (the classical extension of propositional resolution to QBF) and the more recent calculi ∀Exp+Res and IR-calc. For these systems, a mixed picture emerges. Our main results show that the relations both between size and width and between space and width drastically fail in Q-resolution, even in its weaker tree-like version. On the other hand, we obtain positive results for the expansion-based resolution systems ∀Exp+Res and IR-calc, however, only in the weak tree-like models. Technically, our negative results rely on showing width lower bounds together with simultaneous upper bounds for size and space. For our positive results, we exhibit space and width-preserving simulations between QBF resolution calculi
Are Short Proofs Narrow? QBF Resolution is not Simple.
The groundbreaking paper ‘Short proofs are narrow – resolution made simple’ by Ben-Sasson and Wigderson (J. ACM 2001) introduces what is today arguably the main technique to obtain resolution lower bounds: to show a lower bound for the width of proofs. Another important measure for resolution is space, and in their fundamental work, Atserias and Dalmau (J. Comput. Syst. Sci. 2008) show that space lower bounds again can be obtained via width lower bounds. Here we assess whether similar techniques are effective for resolution calculi for quantified Boolean formulas (QBF). A mixed picture emerges. Our main results show that both the relations between size and width as well as between space and width drastically fail in Q-resolution, even in its weaker tree-like version. On the other hand, we obtain positive results for the expansion-based resolution systems Exp+Res and IR-calc, however only in the weak tree-like models. Technically, our negative results rely on showing width lower bounds together with simultaneous upper bounds for size and space. For our positive results we exhibit space and width preserving simulations between QBF resolution calculi
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 and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. The lower bound does not extend all the way to Lasserre, however, since we show that there the formulas we study have proofs of constant rank and size polynomial in both n and w.Peer ReviewedPostprint (author's final draft
Optimal binning of X-ray spectra and response matrix design
A theoretical framework is developed to estimate the optimal binning of X-ray
spectra. We derived expressions for the optimal bin size for model spectra as
well as for observed data using different levels of sophistication. It is shown
that by taking into account both the number of photons in a given spectral
model bin and their average energy over the bin size, the number of model
energy bins and the size of the response matrix can be reduced by a factor of
. The response matrix should then contain the response at the bin
centre as well as its derivative with respect to the incoming photon energy. We
provide practical guidelines for how to construct optimal energy grids as well
as how to structure the response matrix. A few examples are presented to
illustrate the present methods.Comment: 16 pages, 7 figures, accepted for publication in Astronomy and
Astrophysic
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
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
Spectrophotometric calibration of low-resolution spectra
Low-resolution spectroscopy is a frequently used technique. Aperture prism
spectroscopy in particular is an important tool for large-scale survey
observations. The ongoing ESA space mission Gaia is the currently most relevant
example. In this work we analyse the fundamental limitations of the calibration
of low-resolution spectrophotometric observations and introduce a calibration
method that avoids simplifying assumptions on the smearing effects of the line
spread functions. To this aim, we developed a functional analytic mathematical
formulation of the problem of spectrophotometric calibration. In this
formulation, the calibration process can be described as a linear mapping
between two suitably constructed Hilbert spaces, independently of the
resolution of the spectrophotometric instrument. The presented calibration
method can provide a formally unusual but precise calibration of low-resolution
spectrophotometry with non-negligible widths of line spread functions. We used
the Gaia spectrophotometric instruments to demonstrate that the calibration
method of this work can potentially provide a significantly better calibration
than methods neglecting the smearing effects of the line spread functions.Comment: Final versio
An edge-on translucent dust disk around the nearest AGB star L2 Puppis - VLT/NACO spectro-imaging from 1.04 to 4.05 microns and VLTI interferometry
As the nearest known AGB star (d=64pc) and one of the brightest (mK-2), L2
Pup is a particularly interesting benchmark object to monitor the final stages
of stellar evolution. We report new lucky imaging observations of this star
with the VLT/NACO adaptive optics system in twelve narrow band filters covering
the 1.0-4.0 microns wavelength range. These diffraction limited images reveal
an extended circumstellar dust lane in front of the star, that exhibits a high
opacity in the J band and becomes translucent in the H and K bands. In the L
band, extended thermal emission from the dust is detected. We reproduce these
observations using Monte-Carlo radiative transfer modeling of a dust disk with
the RADMC-3D code. We also present new interferometric observations with the
VLTI/VINCI and MIDI instruments. We measure in the K band an upper limit to the
limb-darkened angular diameter of theta_LD = 17.9 +/- 1.6 mas, converting to a
maximum linear radius of R = 123 +/- 14 Rsun. Considering the geometry of the
extended K band emission in the NACO images, this upper limit is probably close
to the actual angular diameter of the star. The position of L2 Pup in the
Herzsprung-Russell diagram indicates that this star has a mass around 2 Msun
and is probably experiencing an early stage of the asymptotic giant branch. We
do not detect any stellar companion of L2 Pup in our adaptive optics and
interferometric observations, and we attribute its apparent astrometric wobble
in the Hipparcos data to variable lighting effects on its circumstellar
material. We however do not exclude the presence of a binary companion, as the
large loop structure extending to more than 10 AU to the North-East of the disk
in our L band images may be the result of interaction between the stellar wind
of L2 Pup and a hidden secondary object. The geometric configuration that we
propose, with a large dust disk seen almost edge-on, appears particularly
favorable to test and develop our understanding of the formation of bipolar
nebulae.Comment: 16 pages, 15 figure
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