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 $10-100$. 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