Anytime Algorithms for ROBDD Symmetry Detection and Approximation

Reduced Ordered Binary Decision Diagrams (ROBDDs) provide a dense and memory efficient representation of Boolean functions. When ROBDDs are applied in logic synthesis, the problem arises of detecting both classical and generalised symmetries. State-of-the-art in symmetry detection is represented by Mishchenko's algorithm. Mishchenko showed how to detect symmetries in ROBDDs without the need for checking equivalence of all co-factor pairs. This work resulted in a practical algorithm for detecting all classical symmetries in an ROBDD in O(|G|3) set operations where |G| is the number of nodes in the ROBDD. Mishchenko and his colleagues subsequently extended the algorithm to find generalised symmetries. The extended algorithm retains the same asymptotic complexity for each type of generalised symmetry. Both the classical and generalised symmetry detection algorithms are monolithic in the sense that they only return a meaningful answer when they are left to run to completion. In this thesis we present efficient anytime algorithms for detecting both classical and generalised symmetries, that output pairs of symmetric variables until a prescribed time bound is exceeded. These anytime algorithms are complete in that given sufficient time they are guaranteed to find all symmetric pairs. Theoretically these algorithms reside in O(n3+n|G|+|G|3) and O(n3+n2|G|+|G|3) respectively, where n is the number of variables, so that in practice the advantage of anytime generality is not gained at the expense of efficiency. In fact, the anytime approach requires only very modest data structure support and offers unique opportunities for optimisation so the resulting algorithms are very efficient. The thesis continues by considering another class of anytime algorithms for ROBDDs that is motivated by the dearth of work on approximating ROBDDs. The need for approximation arises because many ROBDD operations result in an ROBDD whose size is quadratic in the size of the inputs. Furthermore, if ROBDDs are used in abstract interpretation, the running time of the analysis is related not only to the complexity of the individual ROBDD operations but also the number of operations applied. The number of operations is, in turn, constrained by the number of times a Boolean function can be weakened before stability is achieved. This thesis proposes a widening that can be used to both constrain the size of an ROBDD and also ensure that the number of times that it is weakened is bounded by some given constant. The widening can be used to either systematically approximate an ROBDD from above (i.e. derive a weaker function) or below (i.e. infer a stronger function). The thesis also considers how randomised techniques may be deployed to improve the speed of computing an approximation by avoiding potentially expensive ROBDD manipulation

Anytime algorithms for ROBDD symmetry detection and approximation

Reduced Ordered Binary Decision Diagrams (ROBDDs) provide a dense and memory efficient representation of Boolean functions. When ROBDDs are applied in logic synthesis, the problem arises of detecting both classical and generalised symmetries. State-of-the-art in symmetry detection is represented by Mishchenko's algorithm. Mishchenko showed how to detect symmetries in ROBDDs without the need for checking equivalence of all co-factor pairs. This work resulted in a practical algorithm for detecting all classical symmetries in an ROBDD in O(|G|³) set operations where |G| is the number of nodes in the ROBDD. Mishchenko and his colleagues subsequently extended the algorithm to find generalised symmetries. The extended algorithm retains the same asymptotic complexity for each type of generalised symmetry. Both the classical and generalised symmetry detection algorithms are monolithic in the sense that they only return a meaningful answer when they are left to run to completion. In this thesis we present efficient anytime algorithms for detecting both classical and generalised symmetries, that output pairs of symmetric variables until a prescribed time bound is exceeded. These anytime algorithms are complete in that given sufficient time they are guaranteed to find all symmetric pairs. Theoretically these algorithms reside in O(n³+n|G|+|G|³) and O(n³+n²|G|+|G|³) respectively, where n is the number of variables, so that in practice the advantage of anytime generality is not gained at the expense of efficiency. In fact, the anytime approach requires only very modest data structure support and offers unique opportunities for optimisation so the resulting algorithms are very efficient. The thesis continues by considering another class of anytime algorithms for ROBDDs that is motivated by the dearth of work on approximating ROBDDs. The need for approximation arises because many ROBDD operations result in an ROBDD whose size is quadratic in the size of the inputs. Furthermore, if ROBDDs are used in abstract interpretation, the running time of the analysis is related not only to the complexity of the individual ROBDD operations but also the number of operations applied. The number of operations is, in turn, constrained by the number of times a Boolean function can be weakened before stability is achieved. This thesis proposes a widening that can be used to both constrain the size of an ROBDD and also ensure that the number of times that it is weakened is bounded by some given constant. The widening can be used to either systematically approximate an ROBDD from above (i.e. derive a weaker function) or below (i.e. infer a stronger function). The thesis also considers how randomised techniques may be deployed to improve the speed of computing an approximation by avoiding potentially expensive ROBDD manipulation

Universal Smart Grid Agent for Distributed Power Generation Management

"Somewhere, there is always wind blowing or the sun shining." This maxim could lead the global shift from fossil to renewable energy sources, suggesting that there is enough energy available to be turned into electricity. But the already impressive numbers that are available today, along with the European Union's 20-20-20 goal – to power 20% of the EU energy consumption from renewables until 2020 –, might mislead us over the problem that the go-to renewables readily available rely on a primary energy source mankind cannot control: the weather. At the same time, the notion of the smart grid introduces a vast array of new data coming from sensors in the power grid, at wind farms, power plants, transformers, and consumers. The new wealth of information might seem overwhelming, but can help to manage the different actors in the power grid. This book proposes to view the problem of power generation and distribution in the face of increased volatility as a problem of information distribution and processing. It enhances the power grid by turning its nodes into agents that forecast their local power balance from historical data, using artificial neural networks and the multi-part evolutionary training algorithm described in this book. They pro-actively communicate power demand and supply, adhering to a set of behavioral rules this book defines, and finally solve the 0-1 knapsack problem of choosing offers in such a way that not only solves the disequilibrium, but also minimizes line loss, by elegant modeling in the Boolean domain. The book shows that the Divide-et-Impera approach of a distributed grid control can lead to an efficient, reliable integration of volatile renewable energy sources into the power grid

Anytime algorithms for ROBDD symmetry detection and approximation

Reduced Ordered Binary Decision Diagrams (ROBDDs) provide a dense and memory efficient representation of Boolean functions. When ROBDDs are applied in logic synthesis, the problem arises of detecting both classical and generalised symmetries. State-of-the-art in symmetry detection is represented by Mishchenko's algorithm. Mishchenko showed how to detect symmetries in ROBDDs without the need for checking equivalence of all co-factor pairs. This work resulted in a practical algorithm for detecting all classical symmetries in an ROBDD in O(|G|³) set operations where |G| is the number of nodes in the ROBDD. Mishchenko and his colleagues subsequently extended the algorithm to find generalised symmetries. The extended algorithm retains the same asymptotic complexity for each type of generalised symmetry. Both the classical and generalised symmetry detection algorithms are monolithic in the sense that they only return a meaningful answer when they are left to run to completion. In this thesis we present efficient anytime algorithms for detecting both classical and generalised symmetries, that output pairs of symmetric variables until a prescribed time bound is exceeded. These anytime algorithms are complete in that given sufficient time they are guaranteed to find all symmetric pairs. Theoretically these algorithms reside in O(n³+n|G|+|G|³) and O(n³+n²|G|+|G|³) respectively, where n is the number of variables, so that in practice the advantage of anytime generality is not gained at the expense of efficiency. In fact, the anytime approach requires only very modest data structure support and offers unique opportunities for optimisation so the resulting algorithms are very efficient. The thesis continues by considering another class of anytime algorithms for ROBDDs that is motivated by the dearth of work on approximating ROBDDs. The need for approximation arises because many ROBDD operations result in an ROBDD whose size is quadratic in the size of the inputs. Furthermore, if ROBDDs are used in abstract interpretation, the running time of the analysis is related not only to the complexity of the individual ROBDD operations but also the number of operations applied. The number of operations is, in turn, constrained by the number of times a Boolean function can be weakened before stability is achieved. This thesis proposes a widening that can be used to both constrain the size of an ROBDD and also ensure that the number of times that it is weakened is bounded by some given constant. The widening can be used to either systematically approximate an ROBDD from above (i.e. derive a weaker function) or below (i.e. infer a stronger function). The thesis also considers how randomised techniques may be deployed to improve the speed of computing an approximation by avoiding potentially expensive ROBDD manipulation.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

VLSI Design

This book provides some recent advances in design nanometer VLSI chips. The selected topics try to present some open problems and challenges with important topics ranging from design tools, new post-silicon devices, GPU-based parallel computing, emerging 3D integration, and antenna design. The book consists of two parts, with chapters such as: VLSI design for multi-sensor smart systems on a chip, Three-dimensional integrated circuits design for thousand-core processors, Parallel symbolic analysis of large analog circuits on GPU platforms, Algorithms for CAD tools VLSI design, A multilevel memetic algorithm for large SAT-encoded problems, etc

Preimages for SHA-1

This research explores the problem of finding a preimage — an input that, when passed through a particular function, will result in a pre-specified output — for the compression function of the SHA-1 cryptographic hash. This problem is much more difficult than the problem of finding a collision for a hash function, and preimage attacks for very few popular hash functions are known. The research begins by introducing the field and giving an overview of the existing work in the area. A thorough analysis of the compression function is made, resulting in alternative formulations for both parts of the function, and both statistical and theoretical tools to determine the difficulty of the SHA-1 preimage problem. Different representations (And- Inverter Graph, Binary Decision Diagram, Conjunctive Normal Form, Constraint Satisfaction form, and Disjunctive Normal Form) and associated tools to manipulate and/or analyse these representations are then applied and explored, and results are collected and interpreted. In conclusion, the SHA-1 preimage problem remains unsolved and insoluble for the foreseeable future. The primary issue is one of efficient representation; despite a promising theoretical difficulty, both the diffusion characteristics and the depth of the tree stand in the way of efficient search. Despite this, the research served to confirm and quantify the difficulty of the problem both theoretically, using Schaefer's Theorem, and practically, in the context of different representations

Symmetric and efficient synthesis

Since the formulation of the synthesis problem for reactive systems by Church in the 60s, research on synthesis has lead to both theoretical insights and practical approaches for automatically constructing systems from their specifications. While the first solution of the problem was given by Büchi as early as 1969, only very recently, focus has shifted towards identifying ways to exploit the structure in reactive system specifications in order to lift the scalability of synthesis to industrial-sized designs. The recent progress in synthesis not only lead to a renewed interest in the subject, but also shed light onto the downsides of current synthesis approaches. In the original formulation of the problem, the structure of the produced solutions was not a concern. Experiments with current synthesis approaches has however shown that the computed implementations are usually very hard to understand and have little of the structure that manually constructed implementations have. Furthermore, the scalability of current synthesis approaches is still deemed to be insufficient for many industrial application scenarios, which prevents the introduction of reactive synthesis technology into industrial design flows. In this thesis, we tackle both of these problems for reactive synthesis. To counter the insufficient structure in the solutions, we analyse the problem of symmetric synthesis. In this alternative synthesis problem, the aim is to compute a solution that consists of multiple copies of the same process such that the overall system satisfies the specification. Such systems have no centralised control units, and are considered to be more robust and easier to maintain. We characterise undecidable and decidable cases of the problem, and provide a synthesis algorithm for rotation-symmetric architectures, which capture many cases of practical relevance. To improve the scalability in synthesis, we start with a simple but scalable approach to reactive synthesis that has shown its principal applicability in the field, and extend its main idea both in terms of scope and usability. We enhance its expressivity in a way that allows to synthesise robust systems, and remove its limitation to specifications of a very special form. Both improvements yield theoretical insights into the synthesis problem: we characterise which specification classes can be supported in synthesis approaches that use parity games with a fixed number of colours as the underlying computation model, and examine the properties of universal very-weak automata, on which we base a synthesis workflow that combines ease of specification with a low complexity of the underlying game solving step. As a side-result, we also obtain the first procedure to translate a formula in linear-time temporal logic (LTL) to a computation tree logic (CTL) formula with only universal path quantifiers, whenever possible. The new results on symmetric and efficient reactive synthesis are complemented by an easily accessible introductory chapter to the field of reactive synthesis that can also be read in isolation.paddle apparatus with membrane holder were identified.Trotz der Vorzüge der Synthese reaktiver Systeme gegenüber der manuellen Konstruktion solcher Systeme ist Synthese noch nicht als Teil industrieller Vorgehensmodelle etabliert. Als Hauptgrund für diese Diskrepanz gilt allgemein, dass sowohl die Qualität der synthetisierten Systeme bei Anwendung bisheriger Methoden unzureichend ist, als auch die Skalierbarkeit aktueller Syntheseverfahren der Verbesserung bedarf. Diese Dissertation behandelt beide diese Probleme der Synthese reaktiver Systeme auf breiter Front. Zur Verbesserung der Qualität synthetisierter Systeme wird die Synthese von strukturierten Systemen betrachtet. Experimente mit aktuellen Syntheseverfahren haben gezeigt, dass die erzeugten Implementierungen oft schwer zu verstehen sind und anders als handgeschriebene Implementierungen kaum Struktur haben. Abhilfe verschafft die Beschränkung auf die Erzeugung symmetrischer Systeme, die aus mehreren Kopien des selben Prozesses bestehen, so dass das Gesamtsystem die Spezifikation erfüllt. Solche Systeme haben keine zentrale Koordinationskomponente und werden allgemein als robuster und einfacher zu warten eingestuft. In dieser Dissertation werden entscheidbare und unentscheidbare Fälle des symmetrischen Syntheseproblems identifiziert und ein Synthesealgorithmus für rotationssymmetrische Systeme beschrieben. Diese Systemklasse deckt viele praktisch relevante Architekturen ab. Um das Problem der mangelnden Skalierbarkeit anzugehen, wird die Hauptidee des Generalised Reactivity(1) Syntheseansatzes, welcher seine praktische Anwendbarkeit bereits unter Beweis gestellt hat, aufgegriffen und sowohl bezüglich der Expressivität als auch der Benutzbarkeit vervollständigt. Die Erweiterung der Expressivität ermöglicht es, den resultierenden Ansatz für die Synthese robuster Systeme zu nutzen, während die Benutzbarkeit für industrielle Anwendungen durch die Aufhebung der Beschränkung, dass die Spezifikation eine sehr spezielle Form haben muss, erreicht wird. Beide Erweiterungen geben Einsicht in die Theorie der Synthese: Zum einen wird die Klasse der Spezifikationen, die in Syntheseansätzen verwendet werden können, die auf dem Lösen von Paritätsspielen mit einer vordefinierten Anzahl von Farben basieren, charakterisiert. Zum anderen wird Einsicht in die Eigenschaften universeller sehr schwacher Automaten gegeben. Ein Nebenprodukt der neuen Syntheseverfahren ist die erste Prozedur, um einen Ausdruck in linear-time temporal logic (LTL) in computation tree logic mit universellen Pfadquantoren (ACTL) zu übersetzen, wann immer dies möglich ist. Die Resultate zur symmetrischen und effizienten reaktiven Synthese werden von einer didaktisch aufbereiteten Einführung in das Gebiet der reaktiven Synthese begleitet, welche auch unabhängig von den übrigen Teilen der Dissertation gelesen werden kann

Polynomial systems : graphical structure, geometry, and applications

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2018.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages 199-208).Solving systems of polynomial equations is a foundational problem in computational mathematics, that has several applications in the sciences and engineering. A closely related problem, also prevalent in applications, is that of optimizing polynomial functions subject to polynomial constraints. In this thesis we propose novel methods for both of these tasks. By taking advantage of the graphical and geometrical structure of the problem, our methods can achieve higher efficiency, and we can also prove better guarantees. Various problems in areas such as robotics, power systems, computer vision, cryptography, and chemical reaction networks, can be modeled by systems of polynomial equations, and in many cases the resulting systems have a simple sparsity structure. In the first part of this thesis we represent this sparsity structure with a graph, and study the algorithmic and complexity consequences of this graphical abstraction. Our main contribution is the introduction of a novel data structure, chordal networks, that always preserves the underlying graphical structure of the system. Remarkably, many interesting families of polynomial systems admit compact chordal network representations (of size linear in the number of variables), even though the number of components is exponentially large. Our methods outperform existing techniques by orders of magnitude in applications from algebraic statistics and vector addition systems. We then turn our attention to the study of graphical structure in the computation of matrix permanents, a classical problem from computer science. We provide a novel algorithm that requires Õ(n 2[superscript w]) arithmetic operations, where [superscript w] is the treewidth of its bipartite adjacency graph. We also investigate the complexity of some related problems, including mixed discriminants, hyperdeterminants, and mixed volumes. Although seemingly unrelated to polynomial systems, our results have natural implications on the complexity of solving sparse systems. The second part of this thesis focuses on the problem of minimizing a polynomial function subject to polynomial equality constraints. This problem captures many important applications, including Max-Cut, tensor low rank approximation, the triangulation problem, and rotation synchronization. Although these problems are nonconvex, tractable semidefinite programming (SDP) relaxations have been proposed. We introduce a methodology to derive more efficient (smaller) relaxations, by leveraging the geometrical structure of the underlying variety. The main idea behind our method is to describe the variety with a generic set of samples, instead of relying on an algebraic description. Our methods are particularly appealing for varieties that are easy to sample from, such as SO(n), Grassmannians, or rank k tensors. For arbitrary varieties we can take advantage of the tools from numerical algebraic geometry. Optimization problems from applications usually involve parameters (e.g., the data), and there is often a natural value of the parameters for which SDP relaxations solve the (polynomial) problem exactly. The final contribution of this thesis is to establish sufficient conditions (and quantitative bounds) under which SDP relaxations will continue to be exact as the parameter moves in a neighborhood of the original one. Our results can be used to show that several statistical estimation problems are solved exactly by SDP relaxations in the low noise regime. In particular, we prove this for the triangulation problem, rotation synchronization, rank one tensor approximation, and weighted orthogonal Procrustes.by Diego Cifuentes.Ph. D