What\u27s So Special About Kruskal\u27s Theorem and the Ordinal \u3cem\u3eT\u3c/em\u3e\u3csub\u3eo\u3c/sub\u3e? A Survey of Some Results in Proof Theory

This paper consists primarily of a survey of results of Harvey Friedman about some proof theoretic aspects of various forms of Krusal\u27s tree theorem, and in particular the connection with the ordinal Ƭo. We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Veblen Hierarchies, some subsystems of second-order logic, slow-growing and fast-growing hierarchies including Girard\u27s result, and Goodstein sequences. The central theme of this paper is a powerful theorem due to Kruskal, the tree theorem , as well as a finite miniaturization of Kruskal\u27s theorem due to Harvey Friedman. These versions of Kruskal\u27s theorem are remarkable from a proof-theoretic point of view because they are not provable in relatively strong logical systems. They are examples of so-called natural independence phenomena , which are considered by more logicians as more natural than the mathematical incompleteness results first discovered by Gödel. Kruskal\u27s tree theorem also plays a fundamental role in computer science, because it is one of the main tools for showing that certain orderings on trees are well founded. These orderings play a crucial role in proving the termination of systems of rewrite rules and the correctness of Knuth-Bandix completion procedures. There is also a close connection between a certain infinite countable ordinal called Ƭoand Kruskal\u27s theorem. Previous definitions of the function involved in this connection are known to be incorrect, in that, the function is not monotonic. We offer a repaired definition of this function, and explore briefly the consequences of its existence

Optimal Routing of Large Scale Marine Seismic Mapping Operations

Industrial Engineering and Managemen

Weighted graph based ordering techniques for preconditioned conjugate gradient methods

We describe the basis of a matrix ordering heuristic for improving the incomplete factorization used in preconditioned conjugate gradient techniques applied to anisotropic PDE's. Several new matrix ordering techniques, derived from well-known algorithms in combinatorial graph theory, which attempt to implement this heuristic, are described. These ordering techniques are tested against a number of matrices arising from linear anisotropic PDE's, and compared with other matrix ordering techniques. A variation of RCM is shown to generally improve the quality of incomplete factorization preconditioners

Learning image segmentation and hierarchies by learning ultrametric distances

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Brain and Cognitive Sciences, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 100-105).In this thesis I present new contributions to the fields of neuroscience and computer science. The neuroscientific contribution is a new technique for automatically reconstructing complete neural networks from densely stained 3d electron micrographs of brain tissue. The computer science contribution is a new machine learning method for image segmentation and the development of a new theory for supervised hierarchy learning based on ultrametric distance functions. It is well-known that the connectivity of neural networks in the brain can have a dramatic influence on their computational function . However, our understanding of the complete connectivity of neural circuits has been quite impoverished due to our inability to image all the connections between all the neurons in biological network. Connectomics is an emerging field in neuroscience that aims to revolutionize our understanding of the function of neural circuits by imaging and reconstructing entire neural circuits. In this thesis, I present an automated method for reconstructing neural circuitry from 3d electron micrographs of brain tissue. The cortical column, a basic unit of cortical microcircuitry, will produce a single 3d electron micrograph measuring many 100s terabytes once imaged and contain neurites from well over 100,000 different neurons. It is estimated that tracing the neurites in such a volume by hand would take several thousand human years. Automated circuit tracing methods are thus crucial to the success of connectomics. In computer vision, the circuit reconstruction problem of tracing neurites is known as image segmentation. Segmentation is a grouping problem where image pixels belonging to the same neurite are clustered together. While many algorithms for image segmentation exist, few have parameters that can be optimized using groundtruth data to extract maximum performance on a specialized dataset. In this thesis, I present the first machine learning method to directly minimize an image segmentation error. It is based the theory of ultrametric distances and hierarchical clustering. Image segmentation is posed as the problem of learning and classifying ultrametric distances between image pixels. Ultrametric distances on point set have the special property that(cont.) they correspond exactly to hierarchical clustering of the set. This special property implies hierarchical clustering can be learned by directly learning ultrametric distances. In this thesis, I develop convolutional networks as a machine learning architecture for image processing. I use this powerful pattern recognition architecture with many tens of thousands of free parameters for predicting affinity graphs and detecting object boundaries in images. When trained using ultrametric learning, the convolutional network based algorithm yields an extremely efficient linear-time segmentation algorithm. In this thesis, I develop methods for assessing the quality of image segmentations produced by manual human efforts or by automated computer algorithms. These methods are crucial for comparing the performance of different segmentation methods and is used through out the thesis to demonstrate the quality of the reconstructions generated by the methods in this thesis.by Srinivas C. Turaga.Ph.D

Applicazioni di analisi ordinale: il teorema di Kruskal.

Si fornisce una caratterizzazione del teorema di Kruskal con la tecnica dell'analisi ordinale, uno dei principali strumenti della teoria della dimostrazione

Gentzenから始まる証明論の50年 : 順序数解析を中心として (証明と計算の理論と応用)

おおよそ1930-80年における証明論の主な結果・アイデアを，順序数解析(ordinal analysis)を中心として述べていく．但しこの期間の問題に関わる限り，90年以降の結果も一部盛り込む．尚，記述や記法は後に整理されたかたちで述べるので原論文のままというわけではない．したがって証明論の通史や学史のようなものをこの原稿に期待しないで頂きたい．ここでは紙幅の制限により証明の詳細は省いてある．sequent calculi（とε-calucliも少々）については[A2020a]をご参照願いたい