Fifty Years of the Spectrum Problem: Survey and New Results
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On Sum Coloring of Graphs
Mohammadreza Salavatipour A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Computer Science University of Toronto Copyright @ 2000 by Moharnrnadreza Salavatipour The author has gmted a non- L'auteur a accordé une licence non exclusive licence aiiowing the exclusive permettant il la National Library of Canada to Bibliothèque nationale du Canada de nproduce, loan, distribute or sell ~eproduire,prk, distribuer ou copies of this thesis in microform, vendre des copies de cette thése sous paper or electronic formats. la forme de mierofiche/6im, de reproduction sur papier ou sur format The author ntains omership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui proage cette thèse. Ni la thèse ni des extraits substantiels de celle-ci ne doivent être imprimes reproduced without the author's ou autrement reproduits sans son autorisation. Abstract On Sum Coloring of Grsphs Mohammadreza Salavatipour Master of Science Graduate Department of Computer Science University of Toronto 2000 The sum coloring problem asks to find a vertex coloring of a given graph G, using natural numbers, such that the total sum of the colors of vertices is rninimized amongst al1 proper vertex colorings of G. This minimum total sum is the chromatic sum of the graph, C(G), and a coloring which achieves this totd sum is called an optimum coloring. There are some graphs for which the optimum coloring needs more colors thaa indicated by the chromatic number. The minimum number of colore needed in any optimum colo~g of a graph ie dedthe strength of the graph, which we denote by @). -
Order-Preserving Graph Grammars
Order-Preserving Graph Grammars Petter Ericson DOCTORAL THESIS,FEBRUARY 2019 DEPARTMENT OF COMPUTING SCIENCE UMEA˚ UNIVERSITY SWEDEN Department of Computing Science Umea˚ University SE-901 87 Umea,˚ Sweden [email protected] Copyright c 2019 by Petter Ericson Except for Paper I, c Springer-Verlag, 2016 Paper II, c Springer-Verlag, 2017 ISBN 978-91-7855-017-3 ISSN 0348-0542 UMINF 19.01 Front cover by Petter Ericson Printed by UmU Print Service, Umea˚ University, 2019. It is good to have an end to journey toward; but it is the journey that matters, in the end. URSULA K. LE GUIN iv Abstract The field of semantic modelling concerns formal models for semantics, that is, formal structures for the computational and algorithmic processing of meaning. This thesis concerns formal graph languages motivated by this field. In particular, we investigate two formalisms: Order-Preserving DAG Grammars (OPDG) and Order-Preserving Hyperedge Replacement Grammars (OPHG), where OPHG generalise OPDG. Graph parsing is the practise of, given a graph grammar and a graph, to determine if, and in which way, the grammar could have generated the graph. If the grammar is considered fixed, it is the non-uniform graph parsing problem, while if the gram- mars is considered part of the input, it is named the uniform graph parsing problem. Most graph grammars have parsing problems known to be NP-complete, or even ex- ponential, even in the non-uniform case. We show both OPDG and OPHG to have polynomial uniform parsing problems, under certain assumptions. We also show these parsing algorithms to be suitable, not just for determining membership in graph languages, but for computing weights of graphs in graph series. -
Algorithms for Graphs of Small Treewidth
Algorithms for Graphs of Small Treewidth Algoritmen voor grafen met kleine boombreedte (met een samenvatting in het Nederlands) PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de Rector Magnificus, Prof. Dr. J.A. van Ginkel, ingevolge het besluit van het College van Decanen in het openbaar te verdedigen op woensdag 19 maart 1997 des middags te 2:30 uur door Babette Lucie Elisabeth de Fluiter geboren op 6 mei 1970 te Leende Promotor: Prof. Dr. J. van Leeuwen Co-Promotor: Dr. H.L. Bodlaender Faculteit Wiskunde & Informatica ISBN 90-393-1528-0 These investigations were supported by the Netherlands Computer Science Research Founda- tion (SION) with financial support from the Netherlands Organization for Scientific Research (NWO). They have been carried out under the auspices of the research school IPA (Institute for Programming research and Algorithmics). Contents Contents i 1 Introduction 1 2 Preliminaries 9 2.1GraphsandAlgorithms.............................. 9 2.1.1Graphs................................... 9 2.1.2GraphProblemsandAlgorithms..................... 11 2.2TreewidthandPathwidth............................. 13 2.2.1PropertiesofTreeandPathDecompositions............... 15 2.2.2ComplexityIssuesofTreewidthandPathwidth............. 19 2.2.3DynamicProgrammingonTreeDecompositions............. 20 2.2.4FiniteStateProblemsandMonadicSecondOrderLogic......... 24 2.2.5ForbiddenMinorsCharacterization.................... 28 2.3RelatedGraphClasses.............................. 29 2.3.1ChordalGraphsandIntervalGraphs.................. -
Logic in Computer Science, 2003. Proceedings. 18Th Annual IEEE Symposium On
Spectrum Hierarchies and Subdiagonal Functions Aaron Hunter Department of Computing Science Simon Fraser University Burnaby, Canada [email protected] Abstract been done on the properties of the classes in Fagin’s pro- posed spectrum hierarchy. The spectrum of a first-order sentence is the set of cardi- Similarly, very little is known about the classes of spec- nalities of its finite models. Relatively little is known about tra obtained by varying the number of binary relation sym- the subclasses of spectra that are obtained by looking only bols, or the number of unary relation symbols allowed in at sentences with a specific signature. In this paper, we sentences. In this paper, we will use model-theoretic tech- study natural subclasses of spectra and their closure prop- niques to explore the properties of these restricted classes erties under simple subdiagonal functions. We show that of spectra. many natural closure properties turn out to be equivalent to the collapse of potential spectrum hierarchies. We prove all 1.2. Closure Properties of our results using explicit transformations on first-order structures. Given a set of natural numbers S and a function f over the natural numbers, we write f(S) as a shorthand notation for {f(n)|n ∈ S}. We say that a class of spectra S is closed 1. Introduction under f if f(S) ∈Swhenever S ∈S. Using explicit transformations on graphs, More proves 1.1. Historical Motivation that F2 is closed under every polynomial with rational co- efficients that is assymptotically greater than the identity The spectrum of a first-order sentence φ is the set of car- function [12]. -
Partitioning a Graph Into Disjoint Cliques and a Triangle-Free Graph
This is a repository copy of Partitioning a graph into disjoint cliques and a triangle-free graph. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/85292/ Version: Accepted Version Article: Abu-Khzam, FN, Feghali, C and Muller, H (2015) Partitioning a graph into disjoint cliques and a triangle-free graph. Discrete Applied Mathematics, 190-19. 1 - 12. ISSN 0166-218X https://doi.org/10.1016/j.dam.2015.03.015 © 2015, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/ Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Partitioning a Graph into Disjoint Cliques and a Triangle-free Graph Faisal N. Abu-Khzam, Carl Feghali, Haiko M¨uller Abstract A graph G =(V, E) is partitionable if there exists a partition {A,B} of V such that A induces a disjoint union of cliques and B induces a triangle- free graph. -
Hyperoperations and Nopt Structures
Hyperoperations and Nopt Structures Alister Wilson Abstract (Beta version) The concept of formal power towers by analogy to formal power series is introduced. Bracketing patterns for combining hyperoperations are pictured. Nopt structures are introduced by reference to Nept structures. Briefly speaking, Nept structures are a notation that help picturing the seed(m)-Ackermann number sequence by reference to exponential function and multitudinous nestings thereof. A systematic structure is observed and described. Keywords: Large numbers, formal power towers, Nopt structures. 1 Contents i Acknowledgements 3 ii List of Figures and Tables 3 I Introduction 4 II Philosophical Considerations 5 III Bracketing patterns and hyperoperations 8 3.1 Some Examples 8 3.2 Top-down versus bottom-up 9 3.3 Bracketing patterns and binary operations 10 3.4 Bracketing patterns with exponentiation and tetration 12 3.5 Bracketing and 4 consecutive hyperoperations 15 3.6 A quick look at the start of the Grzegorczyk hierarchy 17 3.7 Reconsidering top-down and bottom-up 18 IV Nopt Structures 20 4.1 Introduction to Nept and Nopt structures 20 4.2 Defining Nopts from Nepts 21 4.3 Seed Values: “n” and “theta ) n” 24 4.4 A method for generating Nopt structures 25 4.5 Magnitude inequalities inside Nopt structures 32 V Applying Nopt Structures 33 5.1 The gi-sequence and g-subscript towers 33 5.2 Nopt structures and Conway chained arrows 35 VI Glossary 39 VII Further Reading and Weblinks 42 2 i Acknowledgements I’d like to express my gratitude to Wikipedia for supplying an enormous range of high quality mathematics articles. -
A Model-Theoretic Characterisation of Clique Width Achim Blumensath
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Annals of Pure and Applied Logic 142 (2006) 321–350 www.elsevier.com/locate/apal A model-theoretic characterisation of clique width Achim Blumensath Universit¨at Darmstadt, Fachbereich Mathematik, AG 1, Schloßgartenstraße 7, 64289 Darmstadt, Germany Received 3 June 2004; received in revised form 15 March 2005; accepted 20 February 2006 Available online 19 April 2006 Communicated by A.J. Wilkie Abstract We generalise the concept of clique width to structures of arbitrary signature and cardinality. We present characterisations of clique width in terms of decompositions of a structure and via interpretations in trees. Several model-theoretic properties of clique width are investigated including VC-dimension and preservation of finite clique width under elementary extensions and compactness. c 2006 Elsevier B.V. All rights reserved. Keywords: Clique width; Monadic second-order logic; Model theory 1. Introduction In the last few decades, several measures for the complexity of graphs have been defined and investigated. The most prominent one is the tree width, which appears in the work of Robertson and Seymour [12] on graph minors and which also plays an important role in recent developments of graph algorithms. When studying non-sparse graphs and their monadic second-order properties, the measure of choice seems to be the clique width defined by Courcelle and Olariu [7]. Although no hard evidence has been obtained so far, various partial results suggest that the property of having a finite clique width constitutes the dividing line between simple and complicated monadic theories. -
An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics
STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters 10.1090/stml/087 An Introduction to Ramsey Theory STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters Editorial Board Satyan L. Devadoss John Stillwell (Chair) Rosa Orellana Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 05D10, 03-01, 03E10, 03B10, 03B25, 03D20, 03H15. Jan Reimann was partially supported by NSF Grant DMS-1201263. For additional information and updates on this book, visit www.ams.org/bookpages/stml-87 Library of Congress Cataloging-in-Publication Data Names: Katz, Matthew, 1986– author. | Reimann, Jan, 1971– author. | Pennsylvania State University. Mathematics Advanced Study Semesters. Title: An introduction to Ramsey theory: Fast functions, infinity, and metamathemat- ics / Matthew Katz, Jan Reimann. Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Student mathematical library; 87 | “Mathematics Advanced Study Semesters.” | Includes bibliographical references and index. Identifiers: LCCN 2018024651 | ISBN 9781470442903 (alk. paper) Subjects: LCSH: Ramsey theory. | Combinatorial analysis. | AMS: Combinatorics – Extremal combinatorics – Ramsey theory. msc | Mathematical logic and foundations – Instructional exposition (textbooks, tutorial papers, etc.). msc | Mathematical -
A Survey of Recursive Analysis and Moore's Notion of Real Computation
A Survey of Recursive Analysis and Moore’s Notion of Real Computation Walid Gomaa INRIA Nancy Grand-Est Research Centre, France, Faculty of Engineering, Alexandria University, Egypt [email protected] Abstract The theory of analog computation aims at modeling computational systems that evolve in a continuous manner. Unlike the situation with the discrete setting there is no unified theory of analog computation. There are several proposed theories, some of them seem quite orthogonal. Some theories can be considered as generalizations of the Turing machine theory and classical recursion theory. Among such are recursive analysis and Moore’s class of recursive real functions. Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. Real computation in this context is viewed as effective (in the sense of Turing machine theory) convergence of sequences of rational numbers. In 1996 Moore introduced a function algebra that captures his notion of real computation; it consists of some basic functions and their closure under composition, integration and zero- finding. Though this class is inherently unphysical, much work have been directed at stratifying, restricting, and comparing it with other theories of real computation such as recursive analysis and the GPAC. In this article we give a detailed exposition of recursive analysis and Moore’s class and the relationships between them. 1 Introduction Analog computation is a computational paradigm that attempts to model systems whose internal states are continuous rather than discrete. Unlike the case of discrete computation, which has enjoyed a kind of uniformity and conceptual universality, there have been several approaches to analog computations some of which are not even comparable. -
0.1 Axt, Loop Program, and Grzegorczyk Hier- Archies
1 0.1 Axt, Loop Program, and Grzegorczyk Hier- archies Computable functions can have some quite complex definitions. For example, a loop programmable function might be given via a loop program that has depth of nesting of the loop-end pair, say, equal to 200. Now this is complex! Or a function might be given via an arbitrarily complex sequence of primitive recursions, with the restriction that the computed function is majorized by some known function, for all values of the input (for the concept of majorization see Subsection on the Ackermann function.). But does such definitional|and therefore, \static"|complexity have any bearing on the computational|dynamic|complexity of the function? We will see that it does, and we will connect definitional and computational complexities quantitatively. Our study will be restricted to the class PR that we will subdivide into an infinite sequence of increasingly more inclusive subclasses, Si. A so-called hierarchy of classes of functions. 0.1.0.1 Definition. A sequence (Si)i≥0 of subsets of PR is a primitive recur- sive hierarchy provided all of the following hold: (1) Si ⊆ Si+1, for all i ≥ 0 S (2) PR = i≥0 Si. The hierarchy is proper or nontrivial iff Si 6= Si+1, for all but finitely many i. If f 2 Si then we say that its level in the hierarchy is ≤ i. If f 2 Si+1 − Si, then its level is equal to i + 1. The first hierarchy that we will define is due to Axt and Heinermann [[5] and [1]]. -
Mathematical Logic
Copyright c 1998–2005 by Stephen G. Simpson Mathematical Logic Stephen G. Simpson December 15, 2005 Department of Mathematics The Pennsylvania State University University Park, State College PA 16802 http://www.math.psu.edu/simpson/ This is a set of lecture notes for introductory courses in mathematical logic offered at the Pennsylvania State University. Contents Contents 1 1 Propositional Calculus 3 1.1 Formulas ............................... 3 1.2 Assignments and Satisfiability . 6 1.3 LogicalEquivalence. 10 1.4 TheTableauMethod......................... 12 1.5 TheCompletenessTheorem . 18 1.6 TreesandK¨onig’sLemma . 20 1.7 TheCompactnessTheorem . 21 1.8 CombinatorialApplications . 22 2 Predicate Calculus 24 2.1 FormulasandSentences . 24 2.2 StructuresandSatisfiability . 26 2.3 TheTableauMethod......................... 31 2.4 LogicalEquivalence. 37 2.5 TheCompletenessTheorem . 40 2.6 TheCompactnessTheorem . 46 2.7 SatisfiabilityinaDomain . 47 3 Proof Systems for Predicate Calculus 50 3.1 IntroductiontoProofSystems. 50 3.2 TheCompanionTheorem . 51 3.3 Hilbert-StyleProofSystems . 56 3.4 Gentzen-StyleProofSystems . 61 3.5 TheInterpolationTheorem . 66 4 Extensions of Predicate Calculus 71 4.1 PredicateCalculuswithIdentity . 71 4.2 TheSpectrumProblem . .. .. .. .. .. .. .. .. .. 75 4.3 PredicateCalculusWithOperations . 78 4.4 Predicate Calculus with Identity and Operations . ... 82 4.5 Many-SortedPredicateCalculus . 84 1 5 Theories, Models, Definability 87 5.1 TheoriesandModels ......................... 87 5.2 MathematicalTheories. 89 5.3 DefinabilityoveraModel . 97 5.4 DefinitionalExtensionsofTheories . 100 5.5 FoundationalTheories . 103 5.6 AxiomaticSetTheory . 106 5.7 Interpretability . 111 5.8 Beth’sDefinabilityTheorem. 112 6 Arithmetization of Predicate Calculus 114 6.1 Primitive Recursive Arithmetic . 114 6.2 Interpretability of PRA in Z1 ....................114 6.3 G¨odelNumbers ............................ 114 6.4 UndefinabilityofTruth. 117 6.5 TheProvabilityPredicate . -
Graph Structure and Monadic Second-Order Logic: Language Theoretical Aspects?
Graph Structure and Monadic Second-order Logic: Language Theoretical Aspects? Bruno Courcelle UniversitéBordeaux-1,LaBRI,CNRS Institut Universitaire de France 351, Cours de la Libération 33405, Talence cedex, France [email protected] Abstract. Graph structure is a flexible concept covering many differ- ent types of graph properties. Hierarchical decompositions yielding the notions of tree-width and clique-width, expressed by terms written with appropriate graph operations and associated with Monadic Second-order Logic are important tools for the construction of Fixed-Parameter Tracta- ble algorithms and also for the extension of methods and results of Formal Language Theory to the description of sets of finite graphs. This informal overview presents the main definitions, results and open problems and tries to answer some frequently asked questions. Tree-width and monadic second-order (MS) logic are well-known tools for constructing fixed-parameter tractable (FPT) algorithms taking tree-width as parameter. Clique-width is, like tree-width, a complexity measure of graphs from which FPT algorithms can be built, in particular for problems specified in MS logic. These notions are thus essential for constructing (at least theoretically) tractable algorithms but also in the following three research fields: - the study of the structure of graphs excluding induced subgraphs, minors or vertex-minors (a notion related to clique-width, see [48] or [18]); - the extension of language theoretical notions in order to describe and to transform sets of finite and even countable graphs; - the investigation of classes of finite and countable graphs on which MS logic is decidable. Although these four research fields have been initially developed indepen- dently, they are now more and more related.