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

Towards Verifying Nonlinear Integer Arithmetic

We eliminate a key roadblock to efficient verification of nonlinear integer arithmetic using CDCL SAT solvers, by showing how to construct short resolution proofs for many properties of the most widely used multiplier circuits. Such short proofs were conjectured not to exist. More precisely, we give n^{O(1)} size regular resolution proofs for arbitrary degree 2 identities on array, diagonal, and Booth multipliers and quasipolynomial- n^{O(\log n)} size proofs for these identities on Wallace tree multipliers.Comment: Expanded and simplified with improved result

Hardness measures and resolution lower bounds

Various "hardness" measures have been studied for resolution, providing theoretical insight into the proof complexity of resolution and its fragments, as well as explanations for the hardness of instances in SAT solving. In this report we aim at a unified view of a number of hardness measures, including different measures of width, space and size of resolution proofs. We also extend these measures to all clause-sets (possibly satisfiable).Comment: 43 pages, preliminary version (yet the application part is only sketched, with proofs missing

A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds

We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. '99] also on proof size. [Alekhnovich and Razborov '03] established that if the clause-variable incidence graph of a CNF formula F is a good enough expander, then proving that F is unsatisfiable requires high PC/PCR degree. We further develop the techniques in [AR03] to show that if one can "cluster" clauses and variables in a way that "respects the structure" of the formula in a certain sense, then it is sufficient that the incidence graph of this clustered version is an expander. As a corollary of this, we prove that the functional pigeonhole principle (FPHP) formulas require high PC/PCR degree when restricted to constant-degree expander graphs. This answers an open question in [Razborov '02], and also implies that the standard CNF encoding of the FPHP formulas require exponential proof size in polynomial calculus resolution. Thus, while Onto-FPHP formulas are easy for polynomial calculus, as shown in [Riis '93], both FPHP and Onto-PHP formulas are hard even when restricted to bounded-degree expanders.Comment: Full-length version of paper to appear in Proceedings of the 30th Annual Computational Complexity Conference (CCC '15), June 201

An Improved Separation of Regular Resolution from Pool Resolution and Clause Learning

We prove that the graph tautology principles of Alekhnovich, Johannsen, Pitassi and Urquhart have polynomial size pool resolution refutations that use only input lemmas as learned clauses and without degenerate resolution inferences. We also prove that these graph tautology principles can be refuted by polynomial size DPLL proofs with clause learning, even when restricted to greedy, unit-propagating DPLL search

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

Graph Isomorphism and the Lasserre Hierarchy

In this paper we show lower bounds for a certain large class of algorithms solving the Graph Isomorphism problem, even on expander graph instances. Spielman [25] shows an algorithm for isomorphism of strongly regular expander graphs that runs in time exp(O(n^(1/3)) (this bound was recently improved to expf O(n^(1/5) [5]). It has since been an open question to remove the requirement that the graph be strongly regular. Recent algorithmic results show that for many problems the Lasserre hierarchy works surprisingly well when the underlying graph has expansion properties. Moreover, recent work of Atserias and Maneva [3] shows that k rounds of the Lasserre hierarchy is a generalization of the k-dimensional Weisfeiler-Lehman algorithm for Graph Isomorphism. These two facts combined make the Lasserre hierarchy a good candidate for solving graph isomorphism on expander graphs. Our main result rules out this promising direction by showing that even Omega(n) rounds of the Lasserre semidefinite program hierarchy fail to solve the Graph Isomorphism problem even on expander graphs.Comment: 22 pages, 3 figures, submitted to CC

CO ro-vibrational lines in HD100546: A search for disc asymmetries and the role of fluorescence

We have studied the emission of CO ro-vibrational lines in the disc around the Herbig Be star HD100546 with the final goal of using these lines as a diagnostic to understand inner disc structure in the context of planet formation. High-resolution IR spectra of CO ro-vibrational emission at eight different position angles were taken with CRIRES at the VLT. From these spectra flux tables, CO ro-vibrational line profiles, and population diagrams were produced. We have investigated variations in the line profile shapes and line strengths as a function of slit position angle. We used the thermochemical disc modelling code ProDiMo based on the chemistry, radiation field, and temperature structure of a previously published model for HD100546. Comparing observations and the model, we investigated the possibility of disc asymmetries, the excitation mechanism (UV fluorescence), the geometry, and physical conditions of the inner disc. The observed CO ro-vibrational lines are largely emitted from the inner rim of the outer disc at 10-13 AU. The line shapes are similar for all v levels and line fluxes from all vibrational levels vary only within one order of magnitude. All line profile asymmetries and variations can be explained with a symmetric disc model to which a slit correction and pointing offset is applied. Because the angular size of the CO emitting region (10-13 AU) and the slit width are comparable the line profiles are very sensitive to the placing of the slit. The model reproduces the line shapes and the fluxes of the v=1-0 lines as well as the spatial extent of the CO ro-vibrational emission. It does not reproduce the observed band ratios of 0.5-0.2 with higher vibrational bands. We find that lower gas volume densities at the surface of the inner rim of the outer disc can make the fluorescence pumping more effcient and reproduce the observed band ratios.Comment: 20 pages, 21 figure

Ceres' opposition effect observed by the Dawn framing camera

The surface reflectance of planetary regoliths may increase dramatically towards zero phase angle, a phenomenon known as the opposition effect (OE). Two physical processes that are thought to be the dominant contributors to the brightness surge are shadow hiding (SH) and coherent backscatter (CB). The occurrence of shadow hiding in planetary regoliths is self-evident, but it has proved difficult to unambiguously demonstrate CB from remote sensing observations. One prediction of CB theory is the wavelength dependence of the OE angular width. The Dawn spacecraft observed the OE on the surface of dwarf planet Ceres. We characterize the OE over the resolved surface, including the bright Cerealia Facula, and to find evidence for SH and/or CB. We analyze images of the Dawn framing camera by means of photometric modeling of the phase curve. We find that the OE of most of the investigated surface has very similar characteristics, with an enhancement factor of 1.4 and a FWHM of 3{\deg} (broad OE). A notable exception are the fresh ejecta of the Azacca crater, which display a very narrow brightness enhancement that is restricted to phase angles $< 0.5${\deg} (narrow OE); suggestively, this is in the range in which CB is thought to dominate. We do not find a wavelength dependence for the width of the broad OE, and lack the data to investigate the dependence for the narrow OE. The prediction of a wavelength-dependent CB width is rather ambiguous. The zero-phase observations allow us to determine Ceres' visible geometric albedo as $p_V = 0.094 \pm 0.005$. A comparison with other asteroids suggests that Ceres' broad OE is typical for an asteroid of its spectral type, with characteristics that are primarily linked to surface albedo. Our analysis suggests that CB may occur on the dark surface of Ceres in a highly localized fashion.Comment: Credit: Schr\"oder et al, A&A in press, 2018, reproduced with permission, \copyright ES

A low H I column density filament in NGC 2403 : signature of interaction or accretion

Date of acceptance: 12/07/2014Observed H i accretion around nearby galaxies can only account for a fraction of the gas supply needed to sustain the currently observed star formation rates. It is possible that additional accretion occurs in the form of low column density cold flows, as predicted by numerical simulations of galaxy formation. To constrain the presence and properties of such flows, we present deep H i observations obtained with the NRAO Green Bank Telescope of an area measuring 4° × 4° around NGC 2403. These observations, with a 5σ detection limit of 2.4 × 1018 cm-2 over a 20 km s-1 linewidth, reveal a low column density, extended cloud outside the main H i disk, about 17′ (~ 16 kpc or ~ 2 R25) to the NW of the center of the galaxy. The total H i mass of the cloud is 6.3 × 106 M⊙, or 0.15 percent of the total H i mass of NGC 2403. The cloud is associated with an 8 kpc anomalous-velocity H i filament in the inner disk, that was previously observed in deep VLA observations. We discuss several scenarios for the origin of the cloud, and conclude that it is either accreting from the intergalactic medium, or is the result of a minor interaction with a neigboring dwarf galaxyPeer reviewe