Rapport D'activité 2007-2012
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Logics with Invariantly Used Relations Eickmeyer, Kord (2020)
Logics with Invariantly Used Relations Eickmeyer, Kord (2020) DOI (TUprints): https://doi.org/10.25534/tuprints-00013503 Lizenz: CC-BY 4.0 International - Creative Commons, Attribution Publikationstyp: Habilitation Fachbereich: 04 Department of Mathematics Quelle des Originals: https://tuprints.ulb.tu-darmstadt.de/13503 Fachbereich Mathematik Logics with Invariantly Used Relations vom Fachbereich Mathematik der Technischen Universität Darmstadt genehmigte Habilitationsschrift von Dr. Kord Eickmeyer aus Lage Datum des Habilitationsvortrags: 13. Dezember 2019 1. Gutachter: Prof. Dr. Martin Otto 2. Gutachter: Prof. Dr. Marc Pfetsch 3. Gutachter: Prof. Dr. Rodney Downey 4. Gutachter: Prof. Dr. Mikołaj Bojańczyk D 17 Logics with Invariantly Used Relations Habilitationsschrift Dr. Kord Eickmeyer Contents I Background 11 1 Preliminaries and Notation 13 1.1 Logic Preliminaries . 14 2 Logics with Invariantly Used Relations 25 2.1 Invariantly Used Relations . 25 2.2 Undecidability of the Syntax . 30 3 Structural Properties of Graphs 35 3.1 Trees and Tree Decompositions . 37 3.2 Treedepth . 40 3.3 Graphs of Bounded Genus . 44 3.4 Minors and Topological Subgraphs . 46 II Expressivity and Succinctness 51 4 Variants of First-Order Logic 53 4.1 Separation Results . 54 4.2 Trees . 55 4.3 Locality . 59 4.4 Depth-Bounded Structures . 62 5 Order-Invariant Monadic Second-Order Logic 75 5.1 Depth-Bounded Structures . 75 5.2 Decomposable Structures . 80 III Algorithmic Meta-Theorems 83 6 The Complexity of Model Checking 85 3 7 Successor-Invariant First-Order Logic 91 7.1 Interpreting a Successor Relation . 92 7.2 k-Walks in Graphs with Excluded Minors . 96 7.3 k-Walks in Graphs with Excluded Topological Subgraphs . -
Successor-Invariant First-Order Logic on Graphs with Excluded Topological Subgraphs
SUCCESSOR-INVARIANT FIRST-ORDER LOGIC ON GRAPHS WITH EXCLUDED TOPOLOGICAL SUBGRAPHS KORD EICKMEYER AND KEN-ICHI KAWARABAYASHI Abstract. We show that the model-checking problem for successor- invariant first-order logic is fixed-parameter tractable on graphs with excluded topological subgraphs when parameterised by both the size of the input formula and the size of the exluded topo- logical subgraph. Furthermore, we show that model-checking for order-invariant first-order logic is tractable on coloured posets of bounded width, parameterised by both the size of the input for- mula and the width of the poset. Results of this form, i.e. showing that model-checking for a cer- tain logic is tractable on a certain class of structures, are often referred to as algorithmic meta-theorems since they give a unified proof for the tractability of a whole range of problems. First-order logic is arguably one of the most important logics in this context since it is powerful enough to express many computational prob- lems (e.g. the existence of cliques, dominating sets etc.) and yet its model-checking problem is tractable on rich classes of graphs. In fact, Grohe et al. [21] have shown that model-checking for FO is tractable on all nowhere dense classes of graphs. Successor-invariant FO is a semantic extension of FO by allow- ing the use of an additional binary relation which is interpreted as a directed Hamiltonian cycle, restricted to formulae whose truth value does not depend on the specific choice of a Hamiltonian cy- cle. While this is very natural in the context of model-checking (after all, storing a structure in computer memory usually brings with it a linear order on the structure), the question of how the computational complexity of the model-checking problem for this arXiv:1605.01198v1 [cs.LO] 4 May 2016 richer logic compares to that of plain FO is still open. -
Proposal for a Joint Special Session at the AMS-ASL Meeting of 2009, January 5-8, 2009, Washington, D.C
Proposal for a Joint Special Session at the AMS-ASL Meeting of 2009, January 5-8, 2009, Washington, D.C. Approved by the ASL, still needs approval of the AMS. Model Theoretic Methods in Finite Combinatorics Organizers: • Martin Grohe Institut fur¨ Informatik, Humboldt-Universit¨at zu Berlin Berlin, Germany homepage: www2.informatik.hu-berlin.de/∼grohe/ e-mail: [email protected] • Johann A. Makowsky (corresponding organizer) Department of Computer Science, Technion - Israel Institute of Technology, Haifa, Israel homepage: www.cs.technion.ac.il/∼janos e-mail: [email protected] Purpose of the special session We want to bring the various aspects of the interaction between Model Theory and Finite Combinatorics to a wider audience. Altough the work on 0-1 laws is by now widely known, the other applications on counting functions, graph polynomials, extremal combinatorics, graph minors and regularity lemmas, have not yet received their deserved attention. Background In the last twenty years several applications of Logic, in particular Model theory, to problems in Combinatorics emerged. Among them we have • 0-1 laws and their variations. This is well summarized in the book by J. Spencer [Spe01]. • Modular linear recurrence relations for combinatorial counting functions (The Specker-Blatter Theorem) This theorem has remained widely unnoticed, and deserves wider attention and further study [Spe88, Fis03, FM03] • Recent progress in undertanding the ultimate periodicity of finite spectra by Y. Gurevich and S. Shelah, and E. Fischer and J.A. Makowsky [FM04]. • Applications of meta-theorems which apply to whole families of combina- torial problems, instead of just specific problems. -
Lecture Notes in Computer Science 4800 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan Van Leeuwen
Lecture Notes in Computer Science 4800 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen University of Dortmund, Germany Madhu Sudan Massachusetts Institute of Technology, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max-Planck Institute of Computer Science, Saarbruecken, Germany Arnon Avron Nachum Dershowitz Alexander Rabinovich (Eds.) Pillars of Computer Science Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday 13 Volume Editors Arnon Avron Nachum Dershowitz Alexander Rabinovich Tel Aviv University School of Computer Science Ramat Aviv, Tel Aviv, 69978 Israel E-mail: {aa; nachumd; rabinoa}@post.tau.ac.il Library of Congress Control Number: 2008920893 CR Subject Classification (1998): F.1, F.2.1-2, F.4.1, F.3, D.2.4, D.2-3, I.2.2 LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues ISSN 0302-9743 ISBN-10 3-540-78126-9 Springer Berlin Heidelberg New York ISBN-13 978-3-540-78126-4 Springer Berlin Heidelberg New York This work is subject to copyright. -
Lecture Notes in Computer Science 4800 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan Van Leeuwen
Lecture Notes in Computer Science 4800 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen University of Dortmund, Germany Madhu Sudan Massachusetts Institute of Technology, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max-Planck Institute of Computer Science, Saarbruecken, Germany Arnon Avron Nachum Dershowitz Alexander Rabinovich (Eds.) Pillars of Computer Science Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday 13 Volume Editors Arnon Avron Nachum Dershowitz Alexander Rabinovich Tel Aviv University School of Computer Science Ramat Aviv, Tel Aviv, 69978 Israel E-mail: {aa; nachumd; rabinoa}@post.tau.ac.il Library of Congress Control Number: 2008920893 CR Subject Classification (1998): F.1, F.2.1-2, F.4.1, F.3, D.2.4, D.2-3, I.2.2 LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues ISSN 0302-9743 ISBN-10 3-540-78126-9 Springer Berlin Heidelberg New York ISBN-13 978-3-540-78126-4 Springer Berlin Heidelberg New York This work is subject to copyright. -
Logical Approaches to Barriers in Computing and Complexity
Arnold Beckmann Christine Gaßner Benedikt L¨owe (Eds.) Logical Approaches to Barriers in Computing and Complexity International Workshop, Greifswald, Germany, February 2010 Organized by the Alfried Krupp Wissenschaftskolleg Greifswald under the auspices of the Deutsche Vereinigung f¨urMathematische Logik und f¨ur Grundlagen der Exakten Wissenschaften (DVMLG), the Polskie Towarzystwo Logiki i Filozofii Nauki (PTLiFN), the Association for Computability in Europe (ACiE) and the European Association for Computer Science Logic (EACSL) Abstract Booklet Funded by the Alfried Krupp von Bohlen und Halbach-Stiftung, Essen and the Deutsche Forschungsgemeinschaft, Bonn Preface This booklet contains the extended abstracts of talks presented at the workshop on Logical Approaches to Barriers in Computing and Complexity, held on February 17{20, 2010, in Greifswald, Germany. The workshop was initiated by the Deutsche Vereinigung f¨urMathematische Logik und f¨urGrundlagen der Exakten Wissenschaften (DVMLG), the Polskie Towarzystwo Logiki i Filozofii Nauki (PTLiFN), the Association for Computability in Europe (ACiE) and the European Association for Computer Science Logic (EACSL) and organized under the auspices of these four learn`edsocieties. The workshop was funded by the Alfried Krupp von Bohlen und Halbach-Stiftung, Essen and the Deutsche Forschungsgemeinschaft, Bonn, and it was run by and held at the Alfried Krupp Wissenschaftskolleg in Greifswald, Germany. Scientific Scope. Computability theory and complexity theory have their origins in logic, with connec- tions to foundations of mathematics. The fundamental goal in these areas is to understand the limits of computability (that is analysing which problems can be solved on nowadays and future computers in principle) and effective computability (that is understanding the class of problems which can be solved quickly and with restricted resources.) Logic provides a multifarious toolbox of techniques to analyse questions like this, some of which promise to provide a deep insight in the structure of limit of compu- tation.