Cubical rectangles and rectangular lattices

Cubical rectangles are being defined and explored here over the $n-$dimensional geometric cube $Q_n.$ They form a new class of geometric objects that includes all the edges and all the squares of the $n-$cube. We enumerate and characterize them here in order to construct new posets, transforming into special lattices that will be called rectangular lattices. We show that rectangular lattices are closely related to the class of cubical lattices, that is, the face lattice of the $n-$cube

Combinatorics

This is the report on the Oberwolfach workshop on Combinatorics, held 1–7 January 2006. Combinatorics is a branch of mathematics studying families of mainly, but not exclusively, finite or countable structures – discrete objects. The discrete objects considered in the workshop were graphs, set systems, discrete geometries, and matrices. The programme consisted of 15 invited lectures, 18 contributed talks, and a problem session focusing on recent developments in graph theory, coding theory, discrete geometry, extremal combinatorics, Ramsey theory, theoretical computer science, and probabilistic combinatorics

Energy-Efficient Digital Circuit Design using Threshold Logic Gates

abstract: Improving energy efficiency has always been the prime objective of the custom and automated digital circuit design techniques. As a result, a multitude of methods to reduce power without sacrificing performance have been proposed. However, as the field of design automation has matured over the last few decades, there have been no new automated design techniques, that can provide considerable improvements in circuit power, leakage and area. Although emerging nano-devices are expected to replace the existing MOSFET devices, they are far from being as mature as semiconductor devices and their full potential and promises are many years away from being practical. The research described in this dissertation consists of four main parts. First is a new circuit architecture of a differential threshold logic flipflop called PNAND. The PNAND gate is an edge-triggered multi-input sequential cell whose next state function is a threshold function of its inputs. Second a new approach, called hybridization, that replaces flipflops and parts of their logic cones with PNAND cells is described. The resulting \hybrid circuit, which consists of conventional logic cells and PNANDs, is shown to have significantly less power consumption, smaller area, less standby power and less power variation. Third, a new architecture of a field programmable array, called field programmable threshold logic array (FPTLA), in which the standard lookup table (LUT) is replaced by a PNAND is described. The FPTLA is shown to have as much as 50% lower energy-delay product compared to conventional FPGA using well known FPGA modeling tool called VPR. Fourth, a novel clock skewing technique that makes use of the completion detection feature of the differential mode flipflops is described. This clock skewing method improves the area and power of the ASIC circuits by increasing slack on timing paths. An additional advantage of this method is the elimination of hold time violation on given short paths. Several circuit design methodologies such as retiming and asynchronous circuit design can use the proposed threshold logic gate effectively. Therefore, the use of threshold logic flipflops in conventional design methodologies opens new avenues of research towards more energy-efficient circuits.Dissertation/ThesisDoctoral Dissertation Computer Science 201

A Homological Theory of Functions: Nonuniform Boolean Complexity Separation and VC Dimension Bound Via Algebraic Topology, and a Homological Farkas Lemma

In computational complexity, a complexity class is given by a set of problems or functions, and a basic challenge is to show separations of complexity classes A != B especially when A is known to be a subset of B. In this paper we introduce a homological theory of functions that can be used to establish complexity separations, while also providing other interesting consequences. We propose to associate a topological space S_A to each class of functions A, such that, to separate complexity classes A from a superclass B\u27, it suffices to observe a change in "the number of holes", i.e. homology, in S_A as a subclass B of B\u27 is added to A. In other words, if the homologies of S_A and S_{A union B} are different, then A != B\u27. We develop the underlying theory of functions based on homological commutative algebra and Stanley-Reisner theory, and prove a "maximal principle" for polynomial threshold functions that is used to recover Aspnes, Beigel, Furst, and Rudich\u27s characterization of the polynomial threshold degree of symmetric functions. A surprising coincidence is demonstrated, where, roughly speaking, the maximal dimension of "holes" in S_A upper bounds the VC dimension of A, with equality for common computational cases such as the class of polynomial threshold functions or the class of linear functionals over the finite field of 2 elements, or common algebraic cases such as when the Stanley-Reisner ring of S_A is Cohen-Macaulay. As another interesting application of our theory, we prove a result that a priori has nothing to do with complexity separation: it characterizes when a vector subspace intersects the positive cone, in terms of homological conditions. By analogy to Farkas\u27 result doing the same with linear conditions, we call our theorem the Homological Farkas Lemma

Linear separability of the vertices of an n-dimensional hypercube.

No abstract available.The original print copy of this thesis may be available here: http://wizard.unbc.ca/record=b131703

Decision procedures for linear arithmetic

In this thesis, we present new decision procedures for linear arithmetic in the context of SMT solvers and theorem provers: 1) CutSat++, a calculus for linear integer arithmetic that combines techniques from SAT solving and quantifier elimination in order to be sound, terminating, and complete. 2) The largest cube test and the unit cube test, two sound (although incomplete) tests that find integer and mixed solutions in polynomial time. The tests are especially efficient on absolutely unbounded constraint systems, which are difficult to handle for many other decision procedures. 3) Techniques for the investigation of equalities implied by a constraint system. Moreover, we present several applications for these techniques. 4) The Double-Bounded reduction and the Mixed-Echelon-Hermite transformation, two transformations that reduce any constraint system in polynomial time to an equisatisfiable constraint system that is bounded. The transformations are beneficial because they turn branch-and-bound into a complete and efficient decision procedure for unbounded constraint systems. We have implemented the above decision procedures (except for Cut- Sat++) as part of our linear arithmetic theory solver SPASS-IQ and as part of our CDCL(LA) solver SPASS-SATT. We also present various benchmark evaluations that confirm the practical efficiency of our new decision procedures.In dieser Arbeit präsentieren wir neue Entscheidungsprozeduren für lineare Arithmetik im Kontext von SMT-Solvern und Theorembeweisern: 1) CutSat++, ein korrekter und vollständiger Kalkül für ganzzahlige lineare Arithmetik, der Techniken zur Entscheidung von Aussagenlogik mit Techniken aus der Quantorenelimination vereint. 2) Der Größte-Würfeltest und der Einheitswürfeltest, zwei korrekte (wenn auch unvollständige) Tests, die in polynomieller Zeit (gemischt-)ganzzahlige Lösungen finden. Die Tests sind besonders effizient auf vollständig unbegrenzten Systemen, welche für viele andere Entscheidungsprozeduren schwer sind. 3) Techniken zur Ermittlung von Gleichungen, die von einem linearen Ungleichungssystem impliziert werden. Des Weiteren präsentieren wir mehrere Anwendungsmöglichkeiten für diese Techniken. 4) Die Beidseitig-Begrenzte-Reduktion und die Gemischte-Echelon-Hermitesche- Transformation, die ein Ungleichungssystem in polynomieller Zeit auf ein erfüllbarkeitsäquivalentes System reduzieren, das begrenzt ist. Vereint verwandeln die Transformationen Branch-and-Bound in eine vollständige und effiziente Entscheidungsprozedur für unbeschränkte Ungleichungssysteme. Wir haben diese Techniken (ausgenommen CutSat++) in SPASS-IQ (unserem theory solver für lineare Arithmetik) und in SPASS-SATT (unserem CDCL(LA) solver) implementiert. Basierend darauf präsentieren wir Benchmark-Evaluationen, die die Effizienz unserer Entscheidungsprozeduren bestätigen

Fuzzy Classifiers and their Relation to Cluster Analysis and Neural Network

In this work, we examine three softcomputing methodologies, i.e. rule based fuzzy classification systems, fuzzy clustering and neural networks. Rulebased fuzzy systems can be more easily interpreted than neural networks, while neural networks can learn from data although being a black box. Fuzzy clustering methods search for similarities to combine the data into clusters. We combine them to use the advantages of each system for classification problems. This work shows that the fuzzy max-min classifier decides locally on the basis of two attributes, i.e. only a separation between classes that is parallel to n-2 coordinates can be represented. Therefore we consider systems using the Lukasiewicz-t-norm, as they can solve arbitrary piecewise linear problems. We geometrically characterize and visualize the Lukasiewicz-classifier. We can visualize results from fuzzy clustering analysis by placing a separating hyperplane between two prototypes. Using these hyperplanes we construct a fuzzy classification system that exactly reproduces the assignment to the clusters. The common method of projection used to derive fuzzy rules form fuzzy clusters often looses information. The rules derived by our method give exactly the same classification as the clusters. We also construct a multilayer perceptron (MLP) with two hidden layers from the clusters. Information derived from fuzzy clusters or from a rulebased fuzzy classification system, that is representing e.g. expert knowledge, can be used for initialising an MLP, that can be trained afterwards. Our methodology can also be used for problems with continuous output. To use MLPs for prediction of delays of arrivals at airports, we cluster weather data, construct an MLP from the clusters and further train it.In dieser Arbeit werden drei Softcomputing-Modelle, regelbasierte Fuzzy Systeme, Neuronalen Netz und Fuzzy Clustering Methoden, miteinander verknüpft, um die jeweiligen Vorteile für Klassifikationsprobleme zu kombinieren. Regelbasierte Fuzzy Systeme sind dabei leichter interpretierbar als Neuronale Netze. Diese wiederum lernen gut aus Daten, verhalten sich aber als Blackbox. Fuzzy Clustering Methoden stellen Ähnlichkeitsstrukturen fest, nach denen die Daten in Cluster unterteilt werden. Es wird gezeigt, dass ein Fuzzy Max-Min-Klassifikator lokal immer auf der Basis von zwei Attributen entscheidet, d.h. dass nur Klassentrennungen, die lokal parallel zu n-2 Koordinaten verlaufen, abgebildet werden können. Hier werden daher Systeme mit Lukasiewicz-t-Norm betrachtet, die beliebige stückweise linear separable Probleme lösen können. Der Lukasiewicz-Klassifikator wird geometrisch charakterisiert und visualisiert. Ergebnisse der Fuzzy Clusteranalyse lassen sich visualisieren, indem zwischen je zwei Prototypen eine Hyperebene zur Trennung eingezogen wird. Mit deren Hilfe lässt sich ein Fuzzy Klassifikator bauen, der die Zuordnung zu den Clustern genau wiedergibt. Das bisher übliche Projektionsverfahren, das aus einem Fuzzy Clustering Ergebnis Fuzzy Regeln bildet, verliert Informationen, während die Regeln nach dem hier entwickelten Verfahren genau die gleiche Klassifizierung wie die Cluster wiedergeben. Aus den Clustern wird ebenfalls ein Multitlayer Perceptron (MLP) mit zwei inneren Schichten konstruiert. Information, die aus einem Fuzzy Clustering Ergebnis oder einem regelbasierten Fuzzy System gezogen wird und die z.B. Expertenwissen repräsentiert, kann zum Initialisieren eines MLPs benutzt werden, das anschließend weiter lernen kann. Die Methodik lässt sich ebenso für kontinuierliche Ausgaben benutzen. Um MLPs zur Vorhersage von Verspätungen beim Anflug auf Flughäfen zu nutzen, wurden Wetterdaten geclustert, daraus ein MLP konstruiert und dieses untersucht