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THE DOTDEPTH AND THE POLYNOMIAL HIERARCHIES CORRESPOND ON THE DELTA LEVELS
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
"... It is wellknown that the Σk and Πklevels of the dotdepth hierarchy and the polynomial hierarchy correspond via leaf languages. We extend this correspondence to the ∆klevels of these hierarchies: LeafP (∆L k) = ∆p k. The same methods are used to give evidence against an earlier conjecture of S ..."
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Cited by 5 (1 self)
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It is wellknown that the Σk and Πklevels of the dotdepth hierarchy and the polynomial hierarchy correspond via leaf languages. We extend this correspondence to the ∆klevels of these hierarchies: LeafP (∆L k) = ∆p k. The same methods are used to give evidence against an earlier conjecture
Boolean Hierarchies inside DotDepth One
, 1999
"... Let B 1/2 denote the class of languages having dotdepth 1=2, i.e., the class of languages that can we written as finite unions of languages u 0 A + u 1 A + \Delta \Delta \Delta un\Gamma1 A + un , where u i 2 A and n 0. A language has dotdepth one if and only if it is in the Boolean clo ..."
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Cited by 3 (1 self)
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closure of B 1/ . We examine the structure of the class of dotdepth one languages with respect to Boolean operations and identify an infinite family of Boolean hierarchies inside this class. In particular, we show that 1. the union of these hierarchies amounts to the Boolean hierarchy over B 1/2 , 2
The Boolean Hierarchy over DotDepth 1/2
, 1999
"... For some fixed alphabet A with jAj 2, a language L A + is in the class B 1=2 of the dotdepth hierachy if and only if it can be written as a finite union of languages u 0 A + u 1 A + u 2 A + un , where u i 2 A and n 0. Using an automatatheoretic approach, we show a membership cri ..."
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For some fixed alphabet A with jAj 2, a language L A + is in the class B 1=2 of the dotdepth hierachy if and only if it can be written as a finite union of languages u 0 A + u 1 A + u 2 A + un , where u i 2 A and n 0. Using an automatatheoretic approach, we show a membership
The Boolean Structure of DotDepth One
 Journal of Automata, Languages and Combinatorics
, 2000
"... By definition, the class B1 of dotdepth one languages is the Boolean closure of the class B 1=2 of languages that can be written as finite unions of u0A + u1 A + un , where u i 2 A . So dotdepth one languages can be described by Boolean combinations of patterns (u0 , u1 , ..., un ) in words which ..."
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Cited by 2 (2 self)
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By definition, the class B1 of dotdepth one languages is the Boolean closure of the class B 1=2 of languages that can be written as finite unions of u0A + u1 A + un , where u i 2 A . So dotdepth one languages can be described by Boolean combinations of patterns (u0 , u1 , ..., un ) in words which
A New Kind of Science
, 2002
"... “Somebody says, ‘You know, you people always say that space is continuous. How do you know when you get to a small enough dimension that there really are enough points in between, that it isn’t just a lot of dots separated by little distances? ’ Or they say, ‘You know those quantum mechanical amplit ..."
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Cited by 850 (0 self)
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“Somebody says, ‘You know, you people always say that space is continuous. How do you know when you get to a small enough dimension that there really are enough points in between, that it isn’t just a lot of dots separated by little distances? ’ Or they say, ‘You know those quantum mechanical
Support Vector Machine Classification and Validation of Cancer Tissue Samples Using Microarray Expression Data
, 2000
"... Motivation: DNA microarray experiments generating thousands of gene expression measurements, are being used to gather information from tissue and cell samples regarding gene expression differences that will be useful in diagnosing disease. We have developed a new method to analyse this kind of data ..."
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Cited by 566 (1 self)
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, and other normal tissues. The dataset consists of expression experiment results for 97 802 cDNAs for each tissue. As a result of computational analysis, a tissue sample is discovered and confirmed to be wrongly labeled. Upon correction of this mistake and the removal of an outlier, perfect classification
Planning Algorithms
, 2004
"... This book presents a unified treatment of many different kinds of planning algorithms. The subject lies at the crossroads between robotics, control theory, artificial intelligence, algorithms, and computer graphics. The particular subjects covered include motion planning, discrete planning, planning ..."
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Cited by 1108 (51 self)
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This book presents a unified treatment of many different kinds of planning algorithms. The subject lies at the crossroads between robotics, control theory, artificial intelligence, algorithms, and computer graphics. The particular subjects covered include motion planning, discrete planning
Graphical models, exponential families, and variational inference
, 2008
"... The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fiel ..."
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Cited by 800 (26 self)
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of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing
Community detection in graphs
, 2009
"... The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of th ..."
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Cited by 801 (1 self)
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The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such
Results 1  10
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