IA Scholar Query: Generating Matrix Identities and Proof Complexity Lower Bounds.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSat, 31 Dec 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Adversary Models for Mobile Device Authentication
https://scholar.archive.org/work/6vp5xrtuhbg3za2r243pcry4s4
Mobile device authentication has been a highly active research topic for over 10 years, with a vast range of methods proposed and analyzed. In related areas, such as secure channel protocols, remote authentication, or desktop user authentication, strong, systematic, and increasingly formal threat models have been established and are used to qualitatively compare different methods. However, the analysis of mobile device authentication is often based on weak adversary models, suggesting overly optimistic results on their respective security. In this article, we introduce a new classification of adversaries to better analyze and compare mobile device authentication methods. We apply this classification to a systematic literature survey. The survey shows that security is still an afterthought and that most proposed protocols lack a comprehensive security analysis. The proposed classification of adversaries provides a strong and practical adversary model that offers a comparable and transparent classification of security properties in mobile device authentication.René Mayrhofer, Stephan Siggwork_6vp5xrtuhbg3za2r243pcry4s4Sat, 31 Dec 2022 00:00:00 GMTApproximating Decision Diagrams for Quantum Circuit Simulation
https://scholar.archive.org/work/eiqp5phe5femrpc3swzmghoi3q
Quantum computers promise to solve important problems faster than conventional computers ever could. Underneath is a fundamentally different computational primitive that introduces new challenges for the development of software tools that aid designers of corresponding quantum algorithms. The different computational primitives render classical simulation of quantum circuits particularly challenging. While the logic simulation of conventional circuits is comparatively simple with linear complexity with respect to the number of gates, quantum circuit simulation has to deal with the exponential memory requirements to represent quantum states on non-quantum hardware with respect to the number of qubits. Decision Diagrams (DDs) address this challenge through exploitation of redundancies in matrices and vectors to provide significantly more compact representations in many cases. Moreover, the probabilistic nature of quantum computations enables another angle to tackle the complexity: Quantum algorithms are resistant to some degree against small inaccuracies in the quantum state as these only lead to small changes in the outcome probabilities. We propose to exploit this resistance against (small) errors to gain even more compact decision diagrams. In this work, we investigate the potential of approximation in quantum circuit simulation in detail. To this end, we first present four dedicated schemes that exploit the error resistance and efficiently approximate quantum states represented by decision diagrams. Subsequently, we propose two simulation strategies that utilize those approximations schemes in order to improve the efficiency of DD-based quantum circuit simulation, while, at the same time, allowing the user to control the resulting degradation in accuracy. We empirically show that the proposed approximation schemes reduce the size of decision diagrams substantially and also analytically prove the effect of multiple approximations on the attained accuracy. Eventually, this enables speed-ups of the resulting approximate quantum circuit simulation of up to several orders of magnitudes—again, while controlling the fidelity of the result.Stefan Hillmich, Alwin Zulehner, Richard Kueng, Igor L. Markov, Robert Willework_eiqp5phe5femrpc3swzmghoi3qSat, 31 Dec 2022 00:00:00 GMTAsymptotically equivalent prediction in multivariate geostatistics
https://scholar.archive.org/work/rkcfyrcxjzgqdlqv3muo5m4x2q
Cokriging is the common method of spatial interpolation (best linear unbiased prediction) in multivariate geostatistics. While best linear prediction has been well understood in univariate spatial statistics, the literature for the multivariate case has been elusive so far. The new challenges provided by modern spatial datasets, being typically multivariate, call for a deeper study of cokriging. In particular, we deal with the problem of misspecified cokriging prediction within the framework of fixed domain asymptotics. Specifically, we provide conditions for equivalence of measures associated with multivariate Gaussian random fields, with index set in a compact set of a d-dimensional Euclidean space. Such conditions have been elusive for over about 50 years of spatial statistics. We then focus on the multivariate Matérn and Generalized Wendland classes of matrix valued covariance functions, that have been very popular for having parameters that are crucial to spatial interpolation, and that control the mean square differentiability of the associated Gaussian process. We provide sufficient conditions, for equivalence of Gaussian measures, relying on the covariance parameters of these two classes. This enables to identify the parameters that are crucial to asymptotically equivalent interpolation in multivariate geostatistics. Our findings are then illustrated through simulation studies.François Bachoc, Emilio Porcu, Moreno Bevilacqua, Reinhard Furrer, Tarik Faouziwork_rkcfyrcxjzgqdlqv3muo5m4x2qTue, 01 Nov 2022 00:00:00 GMTDescriptive Combinatorics and Distributed Algorithms
https://scholar.archive.org/work/pjgjlnfrkzd5vmd7p7c5yr66pe
In this article we shall explore a fascinating area called descriptive combinatorics and its recently discovered connections to distributed algorithms-a fundamental part of computer science that is becoming increasingly important in the modern era of decentralized computation. The interdisciplinary nature of these connections means that there is very little common background shared by the researchers who are interested in them. With this in mind, this article was written under the assumption that the reader would have close to no background in either descriptive set theory or computer science. The reader will judge to what degree this endeavor was successful. The article comprises two parts. In the first part we give a brief introduction to some of the central notions and problems of descriptive combinatorics. The second part is devoted to a survey of some of the results concerning theAnton Bernshteynwork_pjgjlnfrkzd5vmd7p7c5yr66peSat, 01 Oct 2022 00:00:00 GMTConstitutive modelling of fibre networks with stretch distributions. Part I: Theory and illustration
https://scholar.archive.org/work/2qz4vqtsafarbnhyfrh64rrkpu
Ben R. Britt, Alexander Edmund Ehretwork_2qz4vqtsafarbnhyfrh64rrkpuSat, 01 Oct 2022 00:00:00 GMTIsadore M. Singer (1924–2021) In Memoriam Part 1: Scientific Works
https://scholar.archive.org/work/aejx3oq2lvch5gdpwoqpzzlbqe
Robert Bryant, Jean-Michel Bismut, Jeff Cheeger, Phillip Griffiths, Simon Donaldson, Nigel Hitchin, H Blaine Lawson, Michail Gromov, Adam Marcus, Daniel Spielman, Nikhil Srivastava, Edward Wittenwork_aejx3oq2lvch5gdpwoqpzzlbqeSat, 01 Oct 2022 00:00:00 GMTTinyKG: Memory-Efficient Training Framework for Knowledge Graph Neural Recommender Systems
https://scholar.archive.org/work/4hmccbfivnaw5analvlri6c66y
There has been an explosion of interest in designing various Knowledge Graph Neural Networks (KGNNs), which achieve state-ofthe-art performance and provide great explainability for recommendation. The promising performance is mainly resulting from their capability of capturing high-order proximity messages over the knowledge graphs. However, training KGNNs at scale is challenging due to the high memory usage. In the forward pass, the automatic differentiation engines (e.g., TensorFlow/PyTorch) generally need to cache all intermediate activation maps in order to compute gradients in the backward pass, which leads to a large GPU memory footprint. Existing work solves this problem by utilizing multi-GPU distributed frameworks. Nonetheless, this poses a practical challenge when seeking to deploy KGNNs in memoryconstrained environments, especially for industry-scale graphs. Here we present TinyKG, a memory-efficient GPU-based training framework for KGNNs for the tasks of recommendation. Specifically, TinyKG uses exact activations in the forward pass while storing a quantized version of activations in the GPU buffers. During the backward pass, these low-precision activations are dequantized back to full-precision tensors, in order to compute gradients. To reduce the quantization errors, TinyKG applies a simple yet effective quantization algorithm to compress the activations, which ensures unbiasedness with low variance. As such, the training memory footprint of KGNNs is largely reduced with negligible accuracy loss. To evaluate the performance of our TinyKG, we conduct comprehensive experiments on real-world datasets. We found that our TinyKG with INT2 quantization aggressively reduces the memoryHuiyuan Chen, Xiaoting Li, Kaixiong Zhou, Xia Hu, Chin-Chia Michael Yeh, Yan Zheng, Hao Yangwork_4hmccbfivnaw5analvlri6c66ySun, 18 Sep 2022 00:00:00 GMTNon-stationary Bandits and Meta-Learning with a Small Set of Optimal Arms
https://scholar.archive.org/work/jwulq5sndzau7g2524ejcy2nie
We study a sequential decision problem where the learner faces a sequence of K-armed stochastic bandit tasks. An adversary may design the tasks, but the adversary is constrained to choose the optimal arm of each task in a smaller (but unknown) subset of M arms. The task boundaries might be known (the bandit meta-learning setting), or unknown (the non-stationary bandit setting). We design an algorithm based on a reduction to bandit submodular maximization and show that, in the regime of large number of tasks and small number of optimal arms, its regret in both settings is smaller than the simple baseline of Õ(√(KNT)) that can be obtained by using standard algorithms designed for non-stationary bandit problems. For the bandit meta-learning problem with fixed task length τ, we show that the regret of the algorithm is bounded as Õ(NM√(M τ)+N^2/3Mτ). Under additional assumptions on the identifiability of the optimal arms in each task, we show a bandit meta-learning algorithm with an improved Õ(N√(M τ)+N^1/2√(M K τ)) regret.MohammadJavad Azizi, Thang Duong, Yasin Abbasi-Yadkori, András György, Claire Vernade, Mohammad Ghavamzadehwork_jwulq5sndzau7g2524ejcy2nieFri, 16 Sep 2022 00:00:00 GMTRoot of unity quantum cluster algebras and Cayley-Hamilton algebras
https://scholar.archive.org/work/mntuqgaarnhwbolvzlfzwfrr2u
We prove that large classes of algebras in the framework of root of unity quantum cluster algebras have the structures of maximal orders in central simple algebras and Cayley-Hamilton algebras in the sense of Procesi. We show that every root of unity upper quantum cluster algebra is a maximal order and obtain an explicit formula for its reduced trace. Under mild assumptions, inside each such algebra we construct a canonical central subalgebra isomorphic to the underlying upper cluster algebra, such that the pair is a Cayley-Hamilton algebra; its fully Azumaya locus is shown to contain a copy of the underlying cluster 𝒜-variety. Both results are proved in the wider generality of intersections of mixed quantum tori over subcollections of seeds. Furthermore, we prove that all monomial subalgebras of root of unity quantum tori are Cayley-Hamilton algebras and classify those ones that are maximal orders. Arbitrary intersections of those over subsets of seeds are also proved to be Cayley-Hamilton algebras. Previous approaches to constructing maximal orders relied on filtration and homological methods. We use new methods based on cluster algebras.Shengnan Huang, Thang T. Q. Lê, Milen Yakimovwork_mntuqgaarnhwbolvzlfzwfrr2uFri, 16 Sep 2022 00:00:00 GMTLow-rank matrix estimation in multi-response regression with measurement errors: Statistical and computational guarantees
https://scholar.archive.org/work/7vxp62u32fhovfbkh54o4ee5de
In this paper, we investigate the matrix estimation problem in the multi-response regression model with measurement errors. A nonconvex error-corrected estimator based on a combination of the amended loss function and the nuclear norm regularizer is proposed to estimate the matrix parameter. Then under the (near) low-rank assumption, we analyse statistical and computational theoretical properties of global solutions of the nonconvex regularized estimator from a general point of view. In the statistical aspect, we establish the nonasymptotic recovery bound for any global solution of the nonconvex estimator, under restricted strong convexity on the loss function. In the computational aspect, we solve the nonconvex optimization problem via the proximal gradient method. The algorithm is proved to converge to a near-global solution and achieve a linear convergence rate. In addition, we also verify sufficient conditions for the general results to be held, in order to obtain probabilistic consequences for specific types of measurement errors, including the additive noise and missing data. Finally, theoretical consequences are demonstrated by several numerical experiments on corrupted errors-in-variables multi-response regression models. Simulation results reveal excellent consistency with our theory under high-dimensional scaling.Xin Li, Dongya Wuwork_7vxp62u32fhovfbkh54o4ee5deFri, 16 Sep 2022 00:00:00 GMTThe ICON-A model for direct QBO simulations on GPUs (version icon-cscs:baf28a514)
https://scholar.archive.org/work/smkqemns4jh35bzblqqem2b5hu
Abstract. Classical numerical models for the global atmosphere, as used for numerical weather forecasting or climate research, have been developed for conventional central processing unit (CPU) architectures. This hinders the employment of such models on current top-performing supercomputers, which achieve their computing power with hybrid architectures, mostly using graphics processing units (GPUs). Thus also scientific applications of such models are restricted to the lesser computer power of CPUs. Here we present the development of a GPU-enabled version of the ICON atmosphere model (ICON-A), motivated by a research project on the quasi-biennial oscillation (QBO), a global-scale wind oscillation in the equatorial stratosphere that depends on a broad spectrum of atmospheric waves, which originates from tropical deep convection. Resolving the relevant scales, from a few kilometers to the size of the globe, is a formidable computational problem, which can only be realized now on top-performing supercomputers. This motivated porting ICON-A, in the specific configuration needed for the research project, in a first step to the GPU architecture of the Piz Daint computer at the Swiss National Supercomputing Centre and in a second step to the JUWELS Booster computer at the Forschungszentrum Jülich. On Piz Daint, the ported code achieves a single-node GPU vs. CPU speedup factor of 6.4 and allows for global experiments at a horizontal resolution of 5 km on 1024 computing nodes with 1 GPU per node with a turnover of 48 simulated days per day. On JUWELS Booster, the more modern hardware in combination with an upgraded code base allows for simulations at the same resolution on 128 computing nodes with 4 GPUs per node and a turnover of 133 simulated days per day. Additionally, the code still remains functional on CPUs, as is demonstrated by additional experiments on the Levante compute system at the German Climate Computing Center. While the application shows good weak scaling over the tested 16-fold increase in grid size and node count, making also higher resolved global simulations possible, the strong scaling on GPUs is relatively poor, which limits the options to increase turnover with more nodes. Initial experiments demonstrate that the ICON-A model can simulate downward-propagating QBO jets, which are driven by wave–mean flow interaction.Marco A. Giorgetta, William Sawyer, Xavier Lapillonne, Panagiotis Adamidis, Dmitry Alexeev, Valentin Clément, Remo Dietlicher, Jan Frederik Engels, Monika Esch, Henning Franke, Claudia Frauen, Walter M. Hannah, Benjamin R. Hillman, Luis Kornblueh, Philippe Marti, Matthew R. Norman, Robert Pincus, Sebastian Rast, Daniel Reinert, Reiner Schnur, Uwe Schulzweida, Bjorn Stevenswork_smkqemns4jh35bzblqqem2b5huFri, 16 Sep 2022 00:00:00 GMTBreaking the Sample Size Barrier in Model-Based Reinforcement Learning with a Generative Model
https://scholar.archive.org/work/p5jmuugmhzhqzjecxueuaruptu
This paper is concerned with the sample efficiency of reinforcement learning, assuming access to a generative model (or simulator). We first consider γ-discounted infinite-horizon Markov decision processes (MDPs) with state space 𝒮 and action space 𝒜. Despite a number of prior works tackling this problem, a complete picture of the trade-offs between sample complexity and statistical accuracy is yet to be determined. In particular, all prior results suffer from a severe sample size barrier, in the sense that their claimed statistical guarantees hold only when the sample size exceeds at least |𝒮||𝒜|/(1-γ)^2. The current paper overcomes this barrier by certifying the minimax optimality of two algorithms – a perturbed model-based algorithm and a conservative model-based algorithm – as soon as the sample size exceeds the order of |𝒮||𝒜|/1-γ (modulo some log factor). Moving beyond infinite-horizon MDPs, we further study time-inhomogeneous finite-horizon MDPs, and prove that a plain model-based planning algorithm suffices to achieve minimax-optimal sample complexity given any target accuracy level. To the best of our knowledge, this work delivers the first minimax-optimal guarantees that accommodate the entire range of sample sizes (beyond which finding a meaningful policy is information theoretically infeasible).Gen Li, Yuting Wei, Yuejie Chi, Yuxin Chenwork_p5jmuugmhzhqzjecxueuaruptuFri, 16 Sep 2022 00:00:00 GMTThe hp-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect
https://scholar.archive.org/work/vqslv4wnbvfahcmobuzwc6bape
We consider approximation of the variable-coefficient Helmholtz equation in the exterior of a Dirichlet obstacle using perfectly-matched-layer (PML) truncation; it is well known that this approximation is exponentially accurate in the PML width and the scaling angle, and the approximation was recently proved to be exponentially accurate in the wavenumber k in [Galkowski, Lafontaine, Spence, 2021]. We show that the hp-FEM applied to this problem does not suffer from the pollution effect, in that there exist C_1,C_2>0 such that if hk/p≤ C_1 and p ≥ C_2 log k then the Galerkin solutions are quasioptimal (with constant independent of k), under the following two conditions (i) the solution operator of the original Helmholtz problem is polynomially bounded in k (which occurs for "most" k by [Lafontaine, Spence, Wunsch, 2021]), and (ii) either there is no obstacle and the coefficients are smooth or the obstacle is analytic and the coefficients are analytic in a neighbourhood of the obstacle and smooth elsewhere. This hp-FEM result is obtained via a decomposition of the PML solution into "high-" and "low-frequency" components, analogous to the decomposition for the original Helmholtz solution recently proved in [Galkowski, Lafontaine, Spence, Wunsch, 2022]. The decomposition is obtained using tools from semiclassical analysis (i.e., the PDE techniques specifically designed for studying Helmholtz problems with large k).Jeffrey Galkowski, David Lafontaine, Euan A. Spence, Jared Wunschwork_vqslv4wnbvfahcmobuzwc6bapeFri, 16 Sep 2022 00:00:00 GMTJoint Optimization for RIS-Assisted Wireless Communications: From Physical and Electromagnetic Perspectives
https://scholar.archive.org/work/cpqd7hsvrrfadix44qblcxv4ge
Reconfigurable intelligent surfaces (RISs) are envisioned to be a disruptive wireless communication technique that is capable of reconfiguring the wireless propagation environment. In this paper, we study a free-space RIS-assisted multiple-input single-output (MISO) communication system in far-field operation. To maximize the received power from the physical and electromagnetic nature point of view, a comprehensive optimization, including beamforming of the transmitter, phase shifts of the RIS, orientation and position of the RIS is formulated and addressed. After exploiting the property of line-of-sight (LoS) links, we derive closed-form solutions of beamforming and phase shifts. For the non-trivial RIS position optimization problem in arbitrary three-dimensional space, a dimensional-reducing theory is proved. The simulation results show that the proposed closed-form beamforming and phase shifts approach the upper bound of the received power. The robustness of our proposed solutions in terms of the perturbation is also verified. Moreover, the RIS significantly enhances the performance of the mmWave/THz communication system.Xin Cheng, Yan Lin, Weiping Shi, Jiayu Li, Cunhua Pan, Feng Shu, Yongpeng Wu, Jiangzhou Wangwork_cpqd7hsvrrfadix44qblcxv4geFri, 16 Sep 2022 00:00:00 GMTMitigating the Effects of Non-Identifiability on Inference for Bayesian Neural Networks with Latent Variables
https://scholar.archive.org/work/t43ckepn5bfenhi6wopmuujxxm
Bayesian Neural Networks with Latent Variables (BNN+LVs) capture predictive uncertainty by explicitly modeling model uncertainty (via priors on network weights) and environmental stochasticity (via a latent input noise variable). In this work, we first show that BNN+LV suffers from a serious form of non-identifiability: explanatory power can be transferred between the model parameters and latent variables while fitting the data equally well. We demonstrate that as a result, in the limit of infinite data, the posterior mode over the network weights and latent variables is asymptotically biased away from the ground-truth. Due to this asymptotic bias, traditional inference methods may in practice yield parameters that generalize poorly and misestimate uncertainty. Next, we develop a novel inference procedure that explicitly mitigates the effects of likelihood non-identifiability during training and yields high-quality predictions as well as uncertainty estimates. We demonstrate that our inference method improves upon benchmark methods across a range of synthetic and real data-sets.Yaniv Yacoby, Weiwei Pan, Finale Doshi-Velezwork_t43ckepn5bfenhi6wopmuujxxmFri, 16 Sep 2022 00:00:00 GMTThe Global Well-Posedness of the Relativistic Boltzmann Equation with Diffuse Reflection Boundary Condition in Bounded Domains
https://scholar.archive.org/work/3oog3yrwqraani5drklaepon5u
The relativistic Boltzmann equation in bounded domains has been widely used in physics and engineering, for example, Tokamak devices in fusion reactors.In spite of its importance, there has, to the best of our knowledge, been no mathematical theory on the global existence of solutions to the relativistic Boltzmann equation in bounded domains. In the present paper, assuming that the motion of single-species relativistic particles in a bounded domain is governed by the relativistic Boltzmann equation with diffuse reflection boundary conditions of non-isothermal wall temperature of small variations around a positive constant, and regarding the speed of light 𝔠 as a large parameter, we first construct a unique non-negative stationary solution F_*, and further establish the dynamical stability of such stationary solution with exponential time decay rate. We point out that the L^∞-bound of perturbations for both steady and non-steady solutions are independent of the speed of light 𝔠, and such uniform in 𝔠 estimates will be useful in the study of Newtonian limit in the future.Yong Wang, Changguo Xiaowork_3oog3yrwqraani5drklaepon5uFri, 16 Sep 2022 00:00:00 GMTEfficient extraction of resonant states in systems with defects
https://scholar.archive.org/work/ebhfmz4rhvhonkynv2khmcn56m
We introduce a new numerical method to compute resonances induced by localized defects in crystals. This method solves an integral equation in the defect region to compute analytic continuations of resolvents. Such an approach enables one to express the resonance in terms of a "resonance source", a function that is strictly localized within the defect region. The kernel of the integral equation, to be applied on such a source term, is the Green function of the perfect crystal, which we show can be computed efficiently by a complex deformation of the Brillouin zone, named Brillouin Complex Deformation (BCD), thereby extending to reciprocal space the concept of complex coordinate transformations.Ivan Duchemin, Luigi Genovese, Eloïse Letournel, Antoine Levitt, Simon Rugetwork_ebhfmz4rhvhonkynv2khmcn56mFri, 16 Sep 2022 00:00:00 GMTMaximum Likelihood Training of Implicit Nonlinear Diffusion Models
https://scholar.archive.org/work/k64upvy2wzgqblskd3cidurryi
Whereas diverse variations of diffusion models exist, expanding the linear diffusion into a nonlinear diffusion process is investigated only by a few works. The nonlinearity effect has been hardly understood, but intuitively, there would be more promising diffusion patterns to optimally train the generative distribution towards the data distribution. This paper introduces such a data-adaptive and nonlinear diffusion process for score-based diffusion models. The proposed Implicit Nonlinear Diffusion Model (INDM) learns the nonlinear diffusion process by combining a normalizing flow and a diffusion process. Specifically, INDM implicitly constructs a nonlinear diffusion on the data space by leveraging a linear diffusion on the latent space through a flow network. This flow network is the key to forming a nonlinear diffusion as the nonlinearity fully depends on the flow network. This flexible nonlinearity is what improves the learning curve of INDM to nearly Maximum Likelihood Estimation (MLE) training, against the non-MLE training of DDPM++, which turns out to be a special case of INDM with the identity flow. Also, training the nonlinear diffusion yields the sampling robustness by the discretization step sizes. In experiments, INDM achieves the state-of-the-art FID on CelebA.Dongjun Kim, Byeonghu Na, Se Jung Kwon, Dongsoo Lee, Wanmo Kang, Il-Chul Moonwork_k64upvy2wzgqblskd3cidurryiFri, 16 Sep 2022 00:00:00 GMTA Spectral Method for Joint Community Detection and Orthogonal Group Synchronization
https://scholar.archive.org/work/24kn4kf3l5djpmotk4nxpmch6q
Community detection and orthogonal group synchronization are both fundamental problems with a variety of important applications in science and engineering. In this work, we consider the joint problem of community detection and orthogonal group synchronization which aims to recover the communities and perform synchronization simultaneously. To this end, we propose a simple algorithm that consists of a spectral decomposition step followed by a blockwise column pivoted QR factorization (CPQR). The proposed algorithm is efficient and scales linearly with the number of edges in the graph. We also leverage the recently developed 'leave-one-out' technique to establish a near-optimal guarantee for exact recovery of the cluster memberships and stable recovery of the orthogonal transforms. Numerical experiments demonstrate the efficiency and efficacy of our algorithm and confirm our theoretical characterization of it.Yifeng Fan, Yuehaw Khoo, Zhizhen Zhaowork_24kn4kf3l5djpmotk4nxpmch6qFri, 16 Sep 2022 00:00:00 GMTUnderstanding the Relative Strength of QBF CDCL Solvers and QBF Resolution
https://scholar.archive.org/work/kte23ktd65brhpv6attwvzqfvm
QBF solvers implementing the QCDCL paradigm are powerful algorithms that successfully tackle many computationally complex applications. However, our theoretical understanding of the strength and limitations of these QCDCL solvers is very limited. In this paper we suggest to formally model QCDCL solvers as proof systems. We define different policies that can be used for decision heuristics and unit propagation and give rise to a number of sound and complete QBF proof systems (and hence new QCDCL algorithms). With respect to the standard policies used in practical QCDCL solving, we show that the corresponding QCDCL proof system is incomparable (via exponential separations) to Q-resolution, the classical QBF resolution system used in the literature. This is in stark contrast to the propositional setting where CDCL and resolution are known to be p-equivalent. This raises the question what formulas are hard for standard QCDCL, since Q-resolution lower bounds do not necessarily apply to QCDCL as we show here. In answer to this question we prove several lower bounds for QCDCL, including exponential lower bounds for a large class of random QBFs. We also introduce a strengthening of the decision heuristic used in classical QCDCL, which does not necessarily decide variables in order of the prefix, but still allows to learn asserting clauses. We show that with this decision policy, QCDCL can be exponentially faster on some formulas. We further exhibit a QCDCL proof system that is p-equivalent to Q-resolution. In comparison to classical QCDCL, this new QCDCL version adapts both decision and unit propagation policies.Olaf Beyersdorff, Benjamin Böhmwork_kte23ktd65brhpv6attwvzqfvmFri, 16 Sep 2022 00:00:00 GMT