Logical Approaches to Barriers in Computing and Complexity
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N° 92 Séminaire De Structures Algébriques Ordonnées 2016
N° 92 Séminaire de Structures Algébriques Ordonnées 2016 -- 2017 F. Delon, M. A. Dickmann, D. Gondard, T. Servi Février 2018 Equipe de Logique Mathématique Prépublications Institut de Mathématiques de Jussieu – Paris Rive Gauche Universités Denis Diderot (Paris 7) et Pierre et Marie Curie (Paris 6) Bâtiment Sophie Germain, 5 rue Thomas Mann – 75205 Paris Cedex 13 Tel. : (33 1) 57 27 91 71 – Fax : (33 1) 57 27 91 78 Les volumes des contributions au Séminaire de Structures Algébriques Ordonnées rendent compte des activités principales du séminaire de l'année indiquée sur chaque volume. Les contributions sont présentées par les auteurs, et publiées avec l'agrément des éditeurs, sans qu'il soit mis en place une procédure de comité de lecture. Ce séminaire est publié dans la série de prépublications de l'Equipe de Logique Mathématique, Institut de Mathématiques de Jussieu–Paris Rive Gauche (CNRS -- Universités Paris 6 et 7). Il s'agit donc d'une édition informelle, et les auteurs ont toute liberté de soumettre leurs articles à la revue de leur choix. Cette publication a pour but de diffuser rapidement des résultats ou leur synthèse, et ainsi de faciliter la communication entre chercheurs. The proceedings of the Séminaire de Structures Algébriques Ordonnées constitute a written report of the main activities of the seminar during the year of publication. Papers are presented by each author, and published with the agreement of the editors, but are not refereed. This seminar is published in the preprint series of the Equipe de Logique Mathématique, Institut de Mathématiques de Jussieu– Paris Rive Gauche (CNRS -- Universités Paris 6 et 7). -
Volume 16, Number 1
2 International Journal "Information Theories & Applications" Vol.16 / 2009 International Journal INFORMATION THEORIES & APPLICATIONS Volume 16 / 2009, Number 1 Editor in chief: Krassimir Markov (Bulgaria) International Editorial Staff Chairman: Victor Gladun (Ukraine) Adil Timofeev (Russia) Ilia Mitov (Bulgaria) Aleksey Voloshin (Ukraine) Juan Castellanos (Spain) Alexander Eremeev (Russia) Koen Vanhoof (Belgium) Alexander Kleshchev (Russia) Levon Aslanyan (Armenia) Alexander Palagin (Ukraine) Luis F. de Mingo (Spain) Alfredo Milani (Italy) Nikolay Zagoruiko (Russia) Anatoliy Krissilov (Ukraine) Peter Stanchev (Bulgaria) Anatoliy Shevchenko (Ukraine) Rumyana Kirkova (Bulgaria) Arkadij Zakrevskij (Belarus) Stefan Dodunekov (Bulgaria) Avram Eskenazi (Bulgaria) Tatyana Gavrilova (Russia) Boris Fedunov (Russia) Vasil Sgurev (Bulgaria) Constantine Gaindric (Moldavia) Vitaliy Lozovskiy (Ukraine) Eugenia Velikova-Bandova (Bulgaria) Vitaliy Velichko (Ukraine) Galina Rybina (Russia) Vladimir Donchenko (Ukraine) Gennady Lbov (Russia) Vladimir Jotsov (Bulgaria) Georgi Gluhchev (Bulgaria) Vladimir Lovitskii (GB) IJ ITA is official publisher of the scientific papers of the members of the ITHEA® International Scientific Society IJ ITA welcomes scientific papers connected with any information theory or its application. IJ ITA rules for preparing the manuscripts are compulsory. The rules for the papers for IJ ITA as well as the subscription fees are given on www.ithea.org . The camera-ready copy of the paper should be received by http://ij.ithea.org. Responsibility for papers published in IJ ITA belongs to authors. General Sponsor of IJ ITA is the Consortium FOI Bulgaria (www.foibg.com). International Journal “INFORMATION THEORIES & APPLICATIONS” Vol.16, Number 1, 2009 Printed in Bulgaria Edited by the Institute of Information Theories and Applications FOI ITHEA®, Bulgaria, in collaboration with the V.M.Glushkov Institute of Cybernetics of NAS, Ukraine, and the Institute of Mathematics and Informatics, BAS, Bulgaria. -
The Polynomial Hierarchy
ij 'I '""T', :J[_ ';(" THE POLYNOMIAL HIERARCHY Although the complexity classes we shall study now are in one sense byproducts of our definition of NP, they have a remarkable life of their own. 17.1 OPTIMIZATION PROBLEMS Optimization problems have not been classified in a satisfactory way within the theory of P and NP; it is these problems that motivate the immediate extensions of this theory beyond NP. Let us take the traveling salesman problem as our working example. In the problem TSP we are given the distance matrix of a set of cities; we want to find the shortest tour of the cities. We have studied the complexity of the TSP within the framework of P and NP only indirectly: We defined the decision version TSP (D), and proved it NP-complete (corollary to Theorem 9.7). For the purpose of understanding better the complexity of the traveling salesman problem, we now introduce two more variants. EXACT TSP: Given a distance matrix and an integer B, is the length of the shortest tour equal to B? Also, TSP COST: Given a distance matrix, compute the length of the shortest tour. The four variants can be ordered in "increasing complexity" as follows: TSP (D); EXACTTSP; TSP COST; TSP. Each problem in this progression can be reduced to the next. For the last three problems this is trivial; for the first two one has to notice that the reduction in 411 j ;1 17.1 Optimization Problems 413 I 412 Chapter 17: THE POLYNOMIALHIERARCHY the corollary to Theorem 9.7 proving that TSP (D) is NP-complete can be used with DP. -
Complexity Theory
Complexity Theory Course Notes Sebastiaan A. Terwijn Radboud University Nijmegen Department of Mathematics P.O. Box 9010 6500 GL Nijmegen the Netherlands [email protected] Copyright c 2010 by Sebastiaan A. Terwijn Version: December 2017 ii Contents 1 Introduction 1 1.1 Complexity theory . .1 1.2 Preliminaries . .1 1.3 Turing machines . .2 1.4 Big O and small o .........................3 1.5 Logic . .3 1.6 Number theory . .4 1.7 Exercises . .5 2 Basics 6 2.1 Time and space bounds . .6 2.2 Inclusions between classes . .7 2.3 Hierarchy theorems . .8 2.4 Central complexity classes . 10 2.5 Problems from logic, algebra, and graph theory . 11 2.6 The Immerman-Szelepcs´enyi Theorem . 12 2.7 Exercises . 14 3 Reductions and completeness 16 3.1 Many-one reductions . 16 3.2 NP-complete problems . 18 3.3 More decision problems from logic . 19 3.4 Completeness of Hamilton path and TSP . 22 3.5 Exercises . 24 4 Relativized computation and the polynomial hierarchy 27 4.1 Relativized computation . 27 4.2 The Polynomial Hierarchy . 28 4.3 Relativization . 31 4.4 Exercises . 32 iii 5 Diagonalization 34 5.1 The Halting Problem . 34 5.2 Intermediate sets . 34 5.3 Oracle separations . 36 5.4 Many-one versus Turing reductions . 38 5.5 Sparse sets . 38 5.6 The Gap Theorem . 40 5.7 The Speed-Up Theorem . 41 5.8 Exercises . 43 6 Randomized computation 45 6.1 Probabilistic classes . 45 6.2 More about BPP . 48 6.3 The classes RP and ZPP . -
Syllabus Computability Theory
Syllabus Computability Theory Sebastiaan A. Terwijn Institute for Discrete Mathematics and Geometry Technical University of Vienna Wiedner Hauptstrasse 8–10/E104 A-1040 Vienna, Austria [email protected] Copyright c 2004 by Sebastiaan A. Terwijn version: 2020 Cover picture and above close-up: Sunflower drawing by Alan Turing, c copyright by University of Southampton and King’s College Cambridge 2002, 2003. iii Emil Post (1897–1954) Alonzo Church (1903–1995) Kurt G¨odel (1906–1978) Stephen Cole Kleene Alan Turing (1912–1954) (1909–1994) Contents 1 Introduction 1 1.1 Preliminaries ................................ 2 2 Basic concepts 3 2.1 Algorithms ................................. 3 2.2 Recursion .................................. 4 2.2.1 Theprimitiverecursivefunctions . 4 2.2.2 Therecursivefunctions . 5 2.3 Turingmachines .............................. 6 2.4 Arithmetization............................... 10 2.4.1 Codingfunctions .......................... 10 2.4.2 Thenormalformtheorem . 11 2.4.3 The basic equivalence and Church’s thesis . 13 2.4.4 Canonicalcodingoffinitesets. 15 2.5 Exercises .................................. 15 3 Computable and computably enumerable sets 19 3.1 Diagonalization............................... 19 3.2 Computablyenumerablesets . 19 3.3 Undecidablesets .............................. 22 3.4 Uniformity ................................. 24 3.5 Many-onereducibility ........................... 25 3.6 Simplesets ................................. 26 3.7 Therecursiontheorem ........................... 28 3.8 Exercises -
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. -
Unclaimed Capital Credits If Your Name Appears on This List, You May Be Eligible for a Refund
Unclaimed Capital Credits If your name appears on this list, you may be eligible for a refund. Complete a Capital Credits Request Form and return it to TCEC. 0'GRADY JAMES ABERNATHY RENETHA A ADAIR ANDY 2 C INVESTMENT ABETE LILLIE ADAIR CURTIS 2 GIRLS AND A COF FEE ABEYTA EUSEBIO ADAIR KATHLEEN SHOP ABEYTA SAM ADAIR LEANARD 21 DIESEL SERVICE ABLA RANDALL ADAIR SHAYLIN 4 RED CATTLE CO ABODE CONSTRUCTIO N ADAME FERNAND 5 C FARMS ABRAHA EDEN M ADAMS & FIELD 54 DINER ABRAHAM RAMPHIS ADAMS ALFRED 54 TOWING & RECOV ERY ABREM TERRY ADAMS ALVIN A & C FEED STORE ABSOLUTE ENDEAVORS INC ADAMS ALVIN L A & I SKYLINE ROO FING ACEVEDO ARNALDO ADAMS ASHLEY A & M CONSTRUCTIO N CO ACEVEDO DELFINA A ADAMS BAPTIST CHURCH A & S FARMS ACEVEDO LEONARD ADAMS CHARLES G A C CROSSLEY MOTE L ACKER EARL ADAMS DARREL A W H ON CO ACKER LLOYD ADAMS DAVID A WILD HAIR ACKER NELDA ADAMS DELBERT AARON KENNETH ACKERMAN AMIRAH ADAMS DIANE M ABADI TEAME ACKERS MELVIN D ADAMS DORIS ABARE JASMINE ACOSTA ARMANDO ADAMS DOYLE G ABARE KALYN ACOSTA CESAR ADAMS FERTILIZER ABAYNEH SOLOMON ACOSTA FERNANDO ADAMS GARY ABB RANDALL CORPO RATION ACOSTA GARCIA R ADAMS HALEY D ABBOTT CLYDE ACOSTA GILBERTO E ADAMS HEATHER ABBOTT CLYDE V ACOSTA JOE ADAMS JEROD ABBOTT FLOYD K ACOSTA JOSE A ADAMS JERRY ABBOTT MAUREAN ACOSTA MARIA ADAMS JILL M ABBOTT ROBERT ACOSTA VICTOR ADAMS JOHN ABBOTT SCOTTY ACTION REALTY ADAMS JOHNNY ACTON PAM ADAMS LINDA Current as of February 2021 Page 1 of 190 Unclaimed Capital Credits If your name appears on this list, you may be eligible for a refund. -
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. -
Saint Michael's Church
! Saint Michael’s Church ! APRIL 28th 29th, 2018 First Holy Communion ! MASS SCHEDULE 7:30 AM & : Central Palisades Park, Saturday 5:30 Sunday 7:30, Korean, AM, Sunday Mass at : PM-Spanish OUR PARISH - Korean Reverend Minhyun Cho, Pastor Feast: AM, Ernest G. Rush, Parochial Vicar Vigil 7:30 Reverend Stanley Lobo, Priest In Residence Sister Clara Kim, Pastoral Associate Ms. Patricia Miller, Office Manager/Bookkeeper Mrs. Seung (Michaela) Kang, Secretary There will be a Prayer Group Each Sunday 3 !5pm in Religious Education Coordinator - the Chapel headed by Edna Kwon. Info inside bulletin. Mrs. Toni Fordyce (201-886-9034) Mrs. Chung Im Juliana Kim and RECONCILIATION (Confession) Each Saturday 5:00pm in the Church Notre Dame Interparochial Academy BAPTISM Principal, Grades Pre-K to . be the First St. Palisades Park, of at 10:30 AM Mass. And, all Korean Baptisms will be held on Third Sunday of the month at the 9:00am Mass. be on the of the at CONGRATULATIONS TO OUR 29 CANDIDATES ON ! 3:00 PM. MAKING THEIR SACRAMENT OF FIRST HOLY (201 233- COMMUNION ON SATURDAY, APRIL 28TH, 2018 ! 0782). come to the to WEDDING Contact the rectory year in advance of the wedding date OUR 29 CANDIDATES ! MAKING THEIR SACRAMENT OF FIRST HOLY COMMUNION ! Christopher Arreaga Espana, Irwing Camaja Vasquez, Brandon Cho, Melany Curruchich Sandoval, Julissa Cuyuch Mejia, David Despaigne, Matthew Guaillas, Brian Guerra Par, Jannelly Hidal- go Jimenez, Daniel Joseph Hummell, Hyunjun Lee, Emily Lopez, Mia Martinez, Edward Martinez, Brian Medina Duarte, Chloe McElkenny, Sofia Stela Morales, Anthony Morel, Evan Morin, Dnaly Morquecho, Bryant Morquecho, Jessica Ramos Castro, Alexander Rivera Pineda, Derek Sisneros, Damien Soler, Jaiden Soler, Elsa Elvira Teletor Taperia, Audrey Won, Justin Yoo. -
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 . -
New Computational Paradigmsnew Computational Paradigms
Lecture Notes in Computer Science 3526 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 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 New York University, NY, USA Doug Tygar University of California, Berkeley, CA, USA Moshe Y. Vardi Rice University, Houston, TX, USA Gerhard Weikum Max-Planck Institute of Computer Science, Saarbruecken, Germany S. Barry Cooper Benedikt Löwe Leen Torenvliet (Eds.) New Computational Paradigms First Conference on Computability in Europe, CiE 2005 Amsterdam, The Netherlands, June 8-12, 2005 Proceedings 13 Volume Editors S. Barry Cooper University of Leeds School of Mathematics Leeds, LS2 9JT, UK E-mail: [email protected] Benedikt Löwe Leen Torenvliet Universiteit van Amsterdam Institute for Logic, Language and Computation Plantage Muidergracht 24, 1018 TV Amsterdam, NL E-mail: {bloewe,leen}@science.uva.nl Library of Congress Control Number: 2005926498 CR Subject Classification (1998): F.1.1-2, F.2.1-2, F.4.1, G.1.0, I.2.6, J.3 ISSN 0302-9743 ISBN-10 3-540-26179-6 Springer Berlin Heidelberg New York ISBN-13 978-3-540-26179-7 Springer Berlin Heidelberg New York This work is subject to copyright.