IA Scholar Query: Approximation for Maximum Surjective Constraint Satisfaction Problems
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgFri, 18 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Robotic manipulators for in situ inspections of jet engines
https://scholar.archive.org/work/akn6hclqbnggrcnwxuhvh2oiby
Jet engines need to be inspected periodically and, in some instances, repaired. Currently, some of these maintenance operations require the engine to be removed from the wing and dismantled, which has a significant associated cost. The capability of performing some of these inspections and repairs while the engine is on-wing could lead to important cost savings. However, existing technology for on-wing operations is limited, and does not suffice to satisfy some of the needs. In this work, the problem of performing on-wing operations such as inspection and repair is analysed, and after an extensive literature review, a novel robotic system for the on-wing insertion and deployment of probes or other tools is proposed. The system consists of a fine-positioner, which is a miniature and dexterous robotic manipulator; a gross-positioner, which is a device to insert the fine-positioner to the engine region of interest; an end-effector, such as a probe; a deployment mechanism, which is a passive device to ensure correct contact between probe and component; and a feedback system that provides information about the robot state for control. The research and development work conducted to address the main challenges to create this robotic system is presented in this thesis. The work is focussed on the fine-positioner, as it is the most relevant and complex part of the system. After a literature review of relevant work, and as part of the exploration of potential robot concepts for the system, the kinematic capabilities of concentric tube robots (CTRs) are first investigated. The complete set of stable trajectories that can be traced in follow-the-leader motion is discovered. A case study involving simulations and an experiment is then presented to showcase and verify the work. The research findings indicate that CTRs are not suitable for the fine-positioner. However, they show that CTRs with non-annular cross section can be used for the gross-positioner. In addition, the new trajectories discovered show promise in minimally i [...]Arnau Garriga Casanovas, Ferdinando Rodriguez Y Baena, Engineering And Physical Sciences Research Council, Rolls-Royce Group Plc.work_akn6hclqbnggrcnwxuhvh2oibyFri, 18 Nov 2022 00:00:00 GMTHierarchies of Minion Tests for PCSPs through Tensors
https://scholar.archive.org/work/7y6by4r2k5d7tnw7pvhjp3dvha
We provide a unified framework to study hierarchies of relaxations for Constraint Satisfaction Problems and their Promise variant. The idea is to split the description of a hierarchy into an algebraic part, depending on a minion capturing the "base level" of the hierarchy, and a geometric part -- which we call tensorisation -- inspired by multilinear algebra. We show that the hierarchies of minion tests obtained in this way are general enough to capture the (combinatorial) bounded width and also the Sherali-Adams LP, Sum-of-Squares SDP, and affine IP hierarchies. We exploit the geometry of the tensor spaces arising from our construction to prove general properties of such hierarchies. We identify certain classes of minions, which we call linear and conic, whose corresponding hierarchies have particularly fine features. Finally, in order to analyse the Sum-of-Squares SDP hierarchy we also characterise the solvability of the standard SDP relaxation through a new minion.Lorenzo Ciardo, Stanislav Živnýwork_7y6by4r2k5d7tnw7pvhjp3dvhaThu, 10 Nov 2022 00:00:00 GMTThe Sherali-Adams Hierarchy for Promise CSPs through Tensors
https://scholar.archive.org/work/x5l5pytpqvafzm4ztjqtqnuf7a
We study the Sherali-Adams linear programming hierarchy in the context of promise constraint satisfaction problems (PCSPs). We characterise when a level of the hierarchy accepts an instance in terms of a homomorphism problem for an appropriate multilinear structure obtained through a tensor power of the constraint language. The geometry of this structure, which consists in a space of tensors satisfying certain symmetries, allows then to establish non-solvability of the approximate graph colouring problem via constantly many levels of Sherali-Adams. Besides this primary application, our tensorisation construction introduces a new tool to the study of hierarchies of algorithmic relaxations for computational problems within (and, possibly, beyond) the context of constraint satisfaction. In particular, we see it as a key step towards the algebraic characterisation of the power of Sherali-Adams for PCSPs.Lorenzo Ciardo, Stanislav Živnýwork_x5l5pytpqvafzm4ztjqtqnuf7aThu, 10 Nov 2022 00:00:00 GMTPTAS for Sparse General-Valued CSPs
https://scholar.archive.org/work/p2stl6vypbdb5ks5jn5jhaqyya
We study polynomial-time approximation schemes (PTASes) for constraint satisfaction problems (CSPs) such as Maximum Independent Set or Minimum Vertex Cover on sparse graph classes. Baker's approach gives a PTAS on planar graphs, excluded-minor classes, and beyond. For Max-CSPs, and even more generally, maximisation finite-valued CSPs (where constraints are arbitrary non-negative functions), Romero, Wrochna, and Živný [SODA'21] showed that the Sherali-Adams LP relaxation gives a simple PTAS for all fractionally-treewidth-fragile classes, which is the most general "sparsity" condition for which a PTAS is known. We extend these results to general-valued CSPs, which include "crisp" (or "strict") constraints that have to be satisfied by every feasible assignment. The only condition on the crisp constraints is that their domain contains an element which is at least as feasible as all the others (but possibly less valuable). For minimisation general-valued CSPs with crisp constraints, we present a PTAS for all Baker graph classes – a definition by Dvořák [SODA'20] which encompasses all classes where Baker's technique is known to work, except possibly for fractionally-treewidth-fragile classes. While this is standard for problems satisfying a certain monotonicity condition on crisp constraints, we show this can be relaxed to diagonalisability – a property of relational structures connected to logics, statistical physics, and random CSPs.Balázs F. Mezei, Marcin Wrochna, Stanislav Živnýwork_p2stl6vypbdb5ks5jn5jhaqyyaThu, 27 Oct 2022 00:00:00 GMTDifferential privacy for metric spaces: information-theoretic models for privacy and utility with new applications to metric domains
https://scholar.archive.org/work/k7ua6tg73zhgfcumtuf5g56s2m
The problem of data privacy – protecting sensitive or personal data from discovery – has been a long-standing research issue. In this regard, differential privacy, introduced in 2006, is considered to be the gold standard. Differential privacy was designed to protect the privacy of individuals in statistical datasets such as census datasets. Its widespread popularity has led to interest in applying differential privacy to new domains for which it was not originally designed, such as text documents. This raises questions regarding the interpretability of differential privacy's guarantees, which are usually expressed in the language of statistical disclosure control. In addition, it escalates the need for answers to core issues currently debated in the differential privacy community: how does the application of differential privacy protect against inference attacks? How can the use of noise-adding mechanisms guarantee the release of useful information? And how can this privacy-utility balance be achieved? The goal of this thesis is to address these foundational questions. Firstly, we approach the problem of interpretability by exploring a generalisation of differential privacy for metric domains known as metric differential privacy or d-privacy. Metric differential privacy abstracts away from the particulars of statistical databases and permits reasoning about privacy on more general domains endowed with a metric. This allows differential privacy's guarantees to be understood in more general terms which can be applied to arbitrary domains of interest, including text documents. Secondly, we propose to study the key questions surrounding privacy and utility in differential privacy using the Quantitative Information Flow (QIF) framework, an information-theoretic framework previously used to analyse threats to secure systems. In this thesis, we repurpose QIF to analyse the privacy and utility guarantees provided by differentially private systems modelled as probabilistic channels. Using information flo [...]Natasha Fernandeswork_k7ua6tg73zhgfcumtuf5g56s2mWed, 19 Oct 2022 00:00:00 GMTEstimation under group actions: recovering orbits from invariants
https://scholar.archive.org/work/fqfsejx2cre33febqbuhxfjd5y
We study a class of orbit recovery problems in which we observe independent copies of an unknown element of ℝ^p, each linearly acted upon by a random element of some group (such as ℤ/p or SO(3)) and then corrupted by additive Gaussian noise. We prove matching upper and lower bounds on the number of samples required to approximately recover the group orbit of this unknown element with high probability. These bounds, based on quantitative techniques in invariant theory, give a precise correspondence between the statistical difficulty of the estimation problem and algebraic properties of the group. Furthermore, we give computer-assisted procedures to certify these properties that are computationally efficient in many cases of interest. The model is motivated by geometric problems in signal processing, computer vision, and structural biology, and applies to the reconstruction problem in cryo-electron microscopy (cryo-EM), a problem of significant practical interest. Our results allow us to verify (for a given problem size) that if cryo-EM images are corrupted by noise with variance σ^2, the number of images required to recover the molecule structure scales as σ^6. We match this bound with a novel (albeit computationally expensive) algorithm for ab initio reconstruction in cryo-EM, based on invariant features of degree at most 3. We further discuss how to recover multiple molecular structures from mixed (or heterogeneous) cryo-EM samples.Afonso S. Bandeira, Ben Blum-Smith, Joe Kileel, Amelia Perry, Jonathan Weed, Alexander S. Weinwork_fqfsejx2cre33febqbuhxfjd5yThu, 13 Oct 2022 00:00:00 GMTNotes on CSPs and Polymorphisms
https://scholar.archive.org/work/kouwgol6o5h55lxjkqyjupnv2i
These are notes from a multi-year learning seminar on the algebraic approach to Constraint Satisfaction Problems (CSPs). The main topics covered are the theory of algebraic structures with few subpowers, the theory of absorbing subalgebras and its applications to studying CSP templates which can be solved by local consistency methods, and the dichotomy theorem for conservative CSP templates. Subsections and appendices cover supplementary material.Zarathustra Bradywork_kouwgol6o5h55lxjkqyjupnv2iThu, 13 Oct 2022 00:00:00 GMTBig Bang Singularity Resolution In Quantum Cosmology
https://scholar.archive.org/work/bapxjdb2bvc5pecyoojnst53cy
We evaluate the physical viability and logical strength of an array of putative criteria for big bang singularity resolution in quantum cosmology. Based on this analysis, we propose a mutually consistent set of constitutive conditions, which we argue should be taken to jointly define 'global' and 'local' big bang singularity resolution in this context. Whilst the present article will focus exclusively on evaluating resolution criteria for big bang singularities in the context of finite dimensional models of quantum cosmology, it is also hoped that the core features of our analysis will be extendable to a more general analysis of criteria for quantum singularity resolution in cosmology and black hole physics.Karim P. Y. Thebaultwork_bapxjdb2bvc5pecyoojnst53cyTue, 13 Sep 2022 00:00:00 GMTIt's Not Fairness, and It's Not Fair: The Failure of Distributional Equality and the Promise of Relational Equality in Complete-Information Hiring Games
https://scholar.archive.org/work/xkd6gmnxrrce3eql5v2unufcrq
Existing efforts to formulate computational definitions of fairness have largely focused on distributional notions of equality, where equality is defined by the resources or decisions given to individuals in the system. Yet existing discrimination and injustice is often the result of unequal social relations, rather than an unequal distribution of resources. Here, we show how optimizing for existing computational and economic definitions of fairness and equality fail to prevent unequal social relations. To do this, we provide an example of a self-confirming equilibrium in a simple hiring market that is relationally unequal but satisfies existing distributional notions of fairness. In doing so, we introduce a notion of blatant relational unfairness for complete-information games, and discuss how this definition helps initiate a new approach to incorporating relational equality into computational systems.Benjamin Fish, Luke Starkwork_xkd6gmnxrrce3eql5v2unufcrqMon, 12 Sep 2022 00:00:00 GMTInapproximability of Counting Hypergraph Colourings
https://scholar.archive.org/work/isjlzjxaznakjgbquvxctzsrom
Recent developments in approximate counting have made startling progress in developing fast algorithmic methods for approximating the number of solutions to constraint satisfaction problems (CSPs) with large arities, using connections to the Lovász Local Lemma. Nevertheless, the boundaries of these methods for CSPs with non-Boolean domain are not well-understood. Our goal in this paper is to fill in this gap and obtain strong inapproximability results by studying the prototypical problem in this class of CSPs, hypergraph colourings. More precisely, we focus on the problem of approximately counting q -colourings on K -uniform hypergraphs with bounded degree Δ . An efficient algorithm exists if \(\Delta \lesssim \frac{q^{K/3-1}}{4^KK^2} \) (Jain, Pham, and Vuong, 2021; He, Sun, and Wu, 2021). Somewhat surprisingly however, a hardness bound is not known even for the easier problem of finding colourings. For the counting problem, the situation is even less clear and there is no evidence of the right constant controlling the growth of the exponent in terms of K . To this end, we first establish that for general q computational hardness for finding a colouring on simple/linear hypergraphs occurs at Δ ≳ Kq K , almost matching the algorithm from the Lovász Local Lemma. Our second and main contribution is to obtain a far more refined bound for the counting problem that goes well beyond the hardness of finding a colouring and which we conjecture is asymptotically tight (up to constant factors). We show in particular that for all even q ≥ 4 it is NP -hard to approximate the number of colourings when Δ ≳ q K /2 . Our approach is based on considering an auxiliary weighted binary CSP model on graphs, which is obtained by "halving" the K -ary hypergraph constraints. This allows us to utilise reduction techniques available for the graph case, which hinge upon understanding the behaviour on random regular bipartite graphs that serve as gadgets in the reduction. The major challenge in our setting is to analyse the induced matrix norm of the interaction matrix of the new CSP which captures the most likely solutions of the system. In contrast to previous analyses in the literature, the auxiliary CSP demonstrates both symmetry and asymmetry, making the analysis of the optimisation problem severely more complicated and demanding the combination of delicate perturbation arguments and careful asymptotic estimates.Andreas Galanis, Heng Guo, Jiaheng Wangwork_isjlzjxaznakjgbquvxctzsromFri, 02 Sep 2022 00:00:00 GMTConjunctive Queries: Unique Characterizations and Exact Learnability
https://scholar.archive.org/work/omorpydbxnco3elpqbqwm5qlje
We answer the question of which conjunctive queries are uniquely characterized by polynomially many positive and negative examples, and how to construct such examples efficiently. As a consequence, we obtain a new efficient exact learning algorithm for a class of conjunctive queries. At the core of our contributions lie two new polynomial-time algorithms for constructing frontiers in the homomorphism lattice of finite structures. We also discuss implications for the unique characterizability and learnability of schema mappings and of description logic concepts.Balder ten Cate, Victor Dalmauwork_omorpydbxnco3elpqbqwm5qljeWed, 31 Aug 2022 00:00:00 GMTLifted edges as connectivity priors for multicut and disjoint paths
https://scholar.archive.org/work/edizj43isvflhhihrsapdwjlhu
This work studies graph decompositions and their representation by 0/1 labeling of edges. We study two problems. The first is multicut (MC) which represents decompositions of undirected graphs (clustering of nodes into connected components). The second is disjoint paths (DP) in directed acyclic graphs where the clusters correspond to nodedisjoint paths. Unlike an alternative representation by node labeling, the number of clusters is not part of the input but is fully determined by the costs of edges. I would like to thank all my co-authors for a pleasant and constructive cooperation. Besides my supervisor Paul Swoboda, I would like to name especially Roberto Henschel, Timo Kaiser, Bjoern Andres, and Jan-Hendrik Lange for their major contribution to the shared publications that are part of this thesis. The publications could not be realized without their part of the work. I would like to thank Bjoern Andres for his supervision and help during the work on our common paper. I would like to mention also Michal Rolinek who helped us with our latest publication. I would like to thank Jiles Vreeken, Marcel Schulz and Markus List who cooperated with me on a research project that is not part of this thesis. I am very grateful to Bernt Schiele, the director of our department, who provided me with good working conditions, fully supported me in combining my working duties with family, and found a solution in the difficult stage of my PhD study by finding a new supervisor. Also, other people at MPI and Saarland University helped me to organize my work and family life and helped me with administrative issues.Andrea Hornakova, Universität Des Saarlandeswork_edizj43isvflhhihrsapdwjlhuMon, 29 Aug 2022 00:00:00 GMTA Proximal Linearization-based Decentralized Method for Nonconvex Problems with Nonlinear Constraints
https://scholar.archive.org/work/6igwqfzuhrfclezho4futqh4eq
Decentralized optimization for non-convex problems are now demanding by many emerging applications (e.g., smart grids, smart building, etc.). Though dramatic progress has been achieved in convex problems, the results for non-convex cases, especially with non-linear constraints, are still largely unexplored. This is mainly due to the challenges imposed by the non-linearity and non-convexity, which makes establishing the convergence conditions bewildered. This paper investigates decentralized optimization for a class of structured non-convex problems characterized by: (i) nonconvex global objective function (possibly nonsmooth) and (ii) coupled nonlinear constraints and local bounded convex constraints w.r.t. the agents. For such problems, a decentralized approach called Proximal Linearizationbased Decentralized Method (PLDM) is proposed. Different from the traditional (augmented) Lagrangian-based methods which usually require the exact (local) optima at each iteration, the proposed method leverages a proximal linearization-based technique to update the decision variables iteratively, which makes it computationally efficient and viable for the non-linear cases. Under some standard conditions, the PLDM global convergence and local convergence rate to the epsilon-critical points are studied based on the Kurdyka-Lojasiewicz property which holds for most analytical functions. Finally, the performance and efficacy of the method are illustrated through a numerical example and an application to multi-zone heating, ventilation and air-conditioning (HVAC) control.Yu Yang, Guoqiang Hu, Costas J. Spanoswork_6igwqfzuhrfclezho4futqh4eqSat, 27 Aug 2022 00:00:00 GMTConjunctive Queries: Unique Characterizations and Exact Learnability
https://scholar.archive.org/work/vhetqhsphferndefb4dc7wer5q
We answer the question which conjunctive queries are uniquely characterized by polynomially many positive and negative examples, and how to construct such examples efficiently. As a consequence, we obtain a new efficient exact learning algorithm for a class of conjunctive queries. At the core of our contributions lie two new polynomial-time algorithms for constructing frontiers in the homomorphism lattice of finite structures. We also discuss implications for the unique characterizability and learnability of schema mappings and of description logic concepts.Balder ten Cate, Victor Dalmauwork_vhetqhsphferndefb4dc7wer5qWed, 24 Aug 2022 00:00:00 GMTQuadratic Integral Penalty Methods for Numerical Trajectory Optimization
https://scholar.archive.org/work/bohzi5azm5cxrbqdlal5h6j3ey
This thesis presents new mathematical algorithms for the numerical solution of a mathematical problem class called dynamic optimization problems. These are mathematical optimization problems, i.e., problems in which numbers are sought that minimize an expression subject to obeying equality and inequality constraints. Dynamic optimization problems are distinct from non-dynamic problems in that the sought numbers may vary over one independent variable. This independent variable can be thought of as, e.g., time. This thesis presents three methods, with emphasis on algorithms, convergence analysis, and computational demonstrations. The first method is a direct transcription method that is based on an integral quadratic penalty term. The purpose of this method is to avoid numerical artifacts such as ringing or erroneous/spurious solutions that may arise in direct collocation methods. The second method is a modified augmented Lagrangian method that leverages ideas from augmented Lagrangian methods for the solution of optimization problems with large quadratic penalty terms, such as they arise from the prior direct transcription method. Lastly, we present a direct transcription method with integral quadratic penalties and integral logarithmic barriers. All methods are motivated with applications and examples, analyzed with complete proofs for their convergence, and practically verified with numerical experiments.Martin Peter Neuenhofenwork_bohzi5azm5cxrbqdlal5h6j3eyFri, 19 Aug 2022 00:00:00 GMTOn the connection of probabilistic model checking, planning, and learning for system verification
https://scholar.archive.org/work/elsht7dndnf7jlue3wo3zte7l4
This thesis presents approaches using techniques from the model checking, planning, and learning community to make systems more reliable and perspicuous. First, two heuristic search and dynamic programming algorithms are adapted to be able to check extremal reachability probabilities, expected accumulated rewards, and their bounded versions, on general Markov decision processes (MDPs). Thereby, the problem space originally solvable by these algorithms is enlarged considerably. Correctness and optimality proofs for the adapted algorithms are given, and in a comprehensive case study on established benchmarks it is shown that the implementation, called Modysh, is competitive with state-of-the-art model checkers and even outperforms them on very large state spaces. Second, Deep Statistical Model Checking (DSMC) is introduced, usable for quality assessment and learning pipeline analysis of systems incorporating trained decision-making agents, like neural networks (NNs). The idea of DSMC is to use statistical model checking to assess NNs resolving nondeterminism in systems modeled as MDPs. The versatility of DSMC is exemplified in a number of case studies on Racetrack, an MDP benchmark designed for this purpose, flexibly modeling the autonomous driving challenge. In a comprehensive scalability study it is demonstrated that DSMC is a lightweight technique tackling the complexity of NN analysis in combination with the state space explosion problem. III Zusammenfassung Diese Arbeit präsentiert Ansätze, die Techniken aus dem Model Checking, Planning und Learning Bereich verwenden, um Systeme verlässlicher und klarer verständlich zu machen. Zuerst werden zwei Algorithmen für heuristische Suche und dynamisches Programmieren angepasst, um Extremwerte für Erreichbarkeitswahrscheinlichkeiten, Erwartungswerte für Kosten und beschränkte Varianten davon, auf generellen Markov Entscheidungsprozessen (MDPs) zu untersuchen. Damit wird der Problemraum, der ursprünglich mit diesen Algorithmen gelöst wurde, deutlich erweitert. Korrektheits-und Optimalitätsbeweise für die angepassten Algorithmen werden gegeben und in einer umfassenden Fallstudie wird gezeigt, dass die Implementierung, namens Modysh, konkurrenzfähig mit den modernsten Model Checkern ist und deren Leistung auf sehr großen Zustandsräumen sogar übertrifft. Als Zweites wird Deep Statistical Model Checking (DSMC) für die Qualitätsbewertung und Lernanalyse von Systemen mit integrierten trainierten Entscheidungsgenten, wie z.B. neuronalen Netzen (NN), eingeführt. Die Idee von DSMC ist es, statistisches Model Checking zur Bewertung von NNs zu nutzen, die Nichtdeterminismus in Systemen, die als MDPs modelliert sind, auflösen. Die Vielseitigkeit des Ansatzes wird in mehreren Fallbeispielen auf Racetrack gezeigt, einer MDP Benchmark, die zu diesem Zweck entwickelt wurde und die Herausforderung des autonomen Fahrens flexibel modelliert. In einer umfassenden Skalierbarkeitsstudie wird demonstriert, dass DSMC eine leichtgewichtige Technik ist, die die Komplexität der NN-Analyse in Kombination mit dem State Space Explosion Problem bewältigt. V Some Personal Words My endeavor of pursuing a PhD at the Dependable Systems and Software chair started already in August 2017 before even signing a contract. After several tutor positions and a lot of fun and work in the mathematics preparatory course team, I already knew Felix, Gereon, and Sebastian who promised me that working here would be a lot of fun. To get to know also the others in the team, Holger offered me the opportunity to take part in the chair's retreat in Tanna, a very small city in Thüringen. Nobody expected that this trip would turn into a literal crash landing for me. But after surviving this first shock, returning home without getting lost at a toilet on a service station, and knowing that Vahid has a big heart for goats, I started officially in November and shared my office with Gilles and Yuliya. Dear Gilles, I have to confess, that at first glance I was misled like the grandma who changed to the other side of the street but very soon we turned into office mates who know the habits of each other very well. We had quite some philosophical and deep discussions about our work and doing a PhD in general. All this helped me so much in overcoming some doubts and I learned a lot from you. Thank you very much! Soon we established Game and Movie Nights in our new leisure room with the very extravagant orange zebra sofa which is exactly 1 m(ichaela) long. At CONFESTA 2018 I got the chance to go with Daniel, Gereon, and Sebastian on the greatest trip ever. We visited Beijing for the conference and the final CAP workshop. We made it without getting hacked, or thrown into prison, but with hurting legs from all the sightseeing, climbing up the wall, and especially from the soccer match Gereon, Rob van Glabbeek, Uwe Nestmann, and me won against Kim Larsen.Michaela Klauck, Universität Des Saarlandeswork_elsht7dndnf7jlue3wo3zte7l4Mon, 18 Jul 2022 00:00:00 GMTFoundations for the Analysis of Surreal-Valued Genetic Functions
https://scholar.archive.org/work/zs2fyujmbbfozkhzlw4p6jlasa
In this thesis we systematize earlier results from the literature of functions on surreal num- bers and consider the generalization of results of Ehrlich and van den Dries regarding models of real-closed fields with exponentiation to the wider class of genetic functions, which includes many examples of interest such as the class of restricted analytic functions, exp, and log, as well as the ω map and other recursively definable functions. We do so by first amending the construction of arbitrary genetic functions found in the literature, so that we may properly compose functions, and so that one can easily recover the definition of exp. We then analyze our newly proposed inductive construction with two natural notions of complexity - that of generation, which tracks the dependence on earlier genetic functions, and that of Veblen rank, which describes the complexity of subtrees closed under a genetic function - in order to characterize the ordinals α such that the surreal numbers below height α will correspond to models satisfying the cofinality conditions and the axioms of real closed fields. After recovering fundamental analytic results for general surreal-valued functions, we further prove that every genetic function has a Veblen rank corresponding to an ordinal, and that our notion of Veblen rank behaves well under addition, multiplication, and composition, and in turn can be extended to arbitrary sets closed under said operations. In particular, the Veblen rank of a genetic function g identifies the largest subclass of epsilon numbers α such that sets of surreal numbers of height below α form a real closed field closed under g. From this, we establish many important functions, such as exp and log will have minimal Veblen rank, while the lambda and kappa maps used to define the Berarducci-Mantova derivative have non-trivial Veblen rank. As a further consequence of our Veblen rank bound, we establish that every entire genetic function is strictly tame in the sense of Fornasiero [4]. Afterwards, with G denoting a [...]Alexander Michael Berenbeimwork_zs2fyujmbbfozkhzlw4p6jlasaThu, 07 Jul 2022 00:00:00 GMTModeling and Control of Morphing Covers for the Adaptive Morphology of Humanoid Robots
https://scholar.archive.org/work/jpfa6lqyrzfr5l5i24fjjbea64
This article takes a step to provide humanoid robots with adaptive morphology abilities. We present a systematic approach for enabling robotic covers to morph their shape, with an overall size fitting the anthropometric dimensions of a humanoid robot. More precisely, we present a cover concept consisting of two main components: a skeleton, which is a repetition of a basic element called node, and a soft membrane, which encloses the cover and deforms with its motion. This article focuses on the cover skeleton and addresses the challenging problems of node design, system modeling, motor positioning, and control design of the morphing system. The cover modeling focuses on kinematics, and a systematic approach for defining the system kinematic constraints is presented. Then, we apply genetic algorithms to find the motor locations so that the morphing cover is fully actuated. Finally, we present control algorithms that allow the cover to morph into a time-varying shape. The entire approach is validated by performing kinematic simulations with four different covers of square dimensions and having 3x3, 4x8, 8x8, and 20x20 nodes, respectively. For each cover, we apply the genetic algorithms to choose the motor locations and perform simulations for tracking a desired shape. The simulation results show that the presented approach ensures the covers to track a desired shape with good tracking performances.Fabio Bergonti, Gabriele Nava, Luca Fiorio, Giuseppe L'Erario, Daniele Pucciwork_jpfa6lqyrzfr5l5i24fjjbea64Sun, 03 Jul 2022 00:00:00 GMTCohomology in Constraint Satisfaction and Structure Isomorphism
https://scholar.archive.org/work/3e7gfqwegnfrfnivwf7r5ei2sy
Constraint satisfaction (CSP) and structure isomorphism (SI) are among the most well-studied computational problems in Computer Science. While neither problem is thought to be in , much work is done on approximations to both problems. Two such historically important approximations are the k-consistency algorithm for CSP and the k-Weisfeiler-Leman algorithm for SI, both of which are based on propagating local partial solutions. The limitations of these algorithms are well-known; k-consistency can solve precisely those CSPs of bounded width and k-Weisfeiler-Leman can only distinguish structures which differ on properties definable in C^k. In this paper, we introduce a novel sheaf-theoretic approach to CSP and SI and their approximations. We show that both problems can be viewed as deciding the existence of global sections of presheaves, ℋ_k(A,B) and ℐ_k(A,B) and that the success of the k-consistency and k-Weisfeiler-Leman algorithms correspond to the existence of certain efficiently computable subpresheaves of these. Furthermore, building on work of Abramsky and others in quantum foundations, we show how to use Čech cohomology in ℋ_k(A,B) and ℐ_k(A,B) to detect obstructions to the existence of the desired global sections and derive new efficient cohomological algorithms extending k-consistency and k-Weisfeiler-Leman. We show that cohomological k-consistency can solve systems of equations over all finite rings and that cohomological Weisfeiler-Leman can distinguish positive and negative instances of the Cai-Fürer-Immerman property over several important classes of structures.Adam Ó Conghailework_3e7gfqwegnfrfnivwf7r5ei2syThu, 30 Jun 2022 00:00:00 GMTPolynomial-Time Approximation of Zero-Free Partition Functions
https://scholar.archive.org/work/doemrovqinalfmht5gxgbhomkq
Zero-free based algorithms are a major technique for deterministic approximate counting. In Barvinok's original framework [Barvinok, 2017], by calculating truncated Taylor expansions, a quasi-polynomial time algorithm was given for estimating zero-free partition functions. Patel and Regts [Patel and Regts, 2017] later gave a refinement of Barvinok's framework, which gave a polynomial-time algorithm for a class of zero-free graph polynomials that can be expressed as counting induced subgraphs in bounded-degree graphs. In this paper, we give a polynomial-time algorithm for estimating classical and quantum partition functions specified by local Hamiltonians with bounded maximum degree, assuming a zero-free property for the temperature. Consequently, when the inverse temperature is close enough to zero by a constant gap, we have a polynomial-time approximation algorithm for all such partition functions. Our result is based on a new abstract framework that extends and generalizes the approach of Patel and Regts.Penghui Yao, Yitong Yin, Xinyuan Zhang, Mikołaj Bojańczyk, Emanuela Merelli, David P. Woodruffwork_doemrovqinalfmht5gxgbhomkqTue, 28 Jun 2022 00:00:00 GMT