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The Communication Complexity of Interleaved Group Products
The communication complexity of interleaved group products W. T. Gowers∗ Emanuele Violay April 2, 2015 Abstract Alice receives a tuple (a1; : : : ; at) of t elements from the group G = SL(2; q). Bob similarly receives a tuple (b1; : : : ; bt). They are promised that the interleaved product Q i≤t aibi equals to either g and h, for two fixed elements g; h 2 G. Their task is to decide which is the case. We show that for every t ≥ 2 communication Ω(t log jGj) is required, even for randomized protocols achieving only an advantage = jGj−Ω(t) over random guessing. This bound is tight, improves on the previous lower bound of Ω(t), and answers a question of Miles and Viola (STOC 2013). An extension of our result to 8-party number-on-forehead protocols would suffice for their intended application to leakage- resilient circuits. Our communication bound is equivalent to the assertion that if (a1; : : : ; at) and t (b1; : : : ; bt) are sampled uniformly from large subsets A and B of G then their inter- leaved product is nearly uniform over G = SL(2; q). This extends results by Gowers (Combinatorics, Probability & Computing, 2008) and by Babai, Nikolov, and Pyber (SODA 2008) corresponding to the independent case where A and B are product sets. We also obtain an alternative proof of their result that the product of three indepen- dent, high-entropy elements of G is nearly uniform. Unlike the previous proofs, ours does not rely on representation theory. ∗Royal Society 2010 Anniversary Research Professor. ySupported by NSF grant CCF-1319206. -
DIMACS Series in Discrete Mathematics and Theoretical Computer Science
DIMACS Series in Discrete Mathematics and Theoretical Computer Science Volume 65 The Random Projection Method Santosh S. Vempala American Mathematical Society The Random Projection Method https://doi.org/10.1090/dimacs/065 DIMACS Series in Discrete Mathematics and Theoretical Computer Science Volume 65 The Random Projection Method Santosh S. Vempala Center for Discrete Mathematics and Theoretical Computer Science A consortium of Rutgers University, Princeton University, AT&T Labs–Research, Bell Labs (Lucent Technologies), NEC Laboratories America, and Telcordia Technologies (with partners at Avaya Labs, IBM Research, and Microsoft Research) American Mathematical Society 2000 Mathematics Subject Classification. Primary 68Q25, 68W20, 90C27, 68Q32, 68P20. For additional information and updates on this book, visit www.ams.org/bookpages/dimacs-65 Library of Congress Cataloging-in-Publication Data Vempala, Santosh S. (Santosh Srinivas), 1971– The random projection method/Santosh S. Vempala. p.cm. – (DIMACS series in discrete mathematics and theoretical computer science, ISSN 1052- 1798; v. 65) Includes bibliographical references. ISBN 0-8218-2018-4 (alk. paper) 1. Random projection method. 2. Algorithms. I. Title. II. Series. QA501 .V45 2004 518.1–dc22 2004046181 0-8218-3793-1 (softcover) Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
A Short History of Computational Complexity
The Computational Complexity Column by Lance FORTNOW NEC Laboratories America 4 Independence Way, Princeton, NJ 08540, USA [email protected] http://www.neci.nj.nec.com/homepages/fortnow/beatcs Every third year the Conference on Computational Complexity is held in Europe and this summer the University of Aarhus (Denmark) will host the meeting July 7-10. More details at the conference web page http://www.computationalcomplexity.org This month we present a historical view of computational complexity written by Steve Homer and myself. This is a preliminary version of a chapter to be included in an upcoming North-Holland Handbook of the History of Mathematical Logic edited by Dirk van Dalen, John Dawson and Aki Kanamori. A Short History of Computational Complexity Lance Fortnow1 Steve Homer2 NEC Research Institute Computer Science Department 4 Independence Way Boston University Princeton, NJ 08540 111 Cummington Street Boston, MA 02215 1 Introduction It all started with a machine. In 1936, Turing developed his theoretical com- putational model. He based his model on how he perceived mathematicians think. As digital computers were developed in the 40's and 50's, the Turing machine proved itself as the right theoretical model for computation. Quickly though we discovered that the basic Turing machine model fails to account for the amount of time or memory needed by a computer, a critical issue today but even more so in those early days of computing. The key idea to measure time and space as a function of the length of the input came in the early 1960's by Hartmanis and Stearns. -
A Variation of Levin Search for All Well-Defined Problems
A Variation of Levin Search for All Well-Defined Problems Fouad B. Chedid A’Sharqiyah University, Ibra, Oman [email protected] July 10, 2021 Abstract In 1973, L.A. Levin published an algorithm that solves any inversion problem π as quickly as the fastest algorithm p∗ computing a solution for ∗ ∗ ∗ π in time bounded by 2l(p ).t , where l(p ) is the length of the binary ∗ ∗ ∗ encoding of p , and t is the runtime of p plus the time to verify its correctness. In 2002, M. Hutter published an algorithm that solves any ∗ well-defined problem π as quickly as the fastest algorithm p computing a solution for π in time bounded by 5.tp(x)+ dp.timetp (x)+ cp, where l(p)+l(tp) l(f)+1 2 dp = 40.2 and cp = 40.2 .O(l(f) ), where l(f) is the length of the binary encoding of a proof f that produces a pair (p,tp), where tp(x) is a provable time bound on the runtime of the fastest program p ∗ provably equivalent to p . In this paper, we rewrite Levin Search using the ideas of Hutter so that we have a new simple algorithm that solves any ∗ well-defined problem π as quickly as the fastest algorithm p computing 2 a solution for π in time bounded by O(l(f) ).tp(x). keywords: Computational Complexity; Algorithmic Information Theory; Levin Search. arXiv:1702.03152v1 [cs.CC] 10 Feb 2017 1 Introduction We recall that the class NP is the set of all decision problems that can be solved efficiently on a nondeterministic Turing Machine. -
The Book Review Column1 by William Gasarch Department of Computer Science University of Maryland at College Park College Park, MD, 20742 Email: [email protected]
The Book Review Column1 by William Gasarch Department of Computer Science University of Maryland at College Park College Park, MD, 20742 email: [email protected] Welcome to the Book Reviews Column. We hope to bring you at least two reviews of books every month. In this column six books are reviewed. 1. The following three books are all reviewed together by William Gasarch. Descriptive Com- plexity Theory by Neal Immerman, Finite Model Theory by Heinz-Dieter Ebbinhaus and Jorg Flum, and Descriptive Complexity and Finite Models (Proceedings from a DIMACS workshop) edited by Neil Immerman and Phokion Kolaitis. These books deal with how complicated it is to describe a set in terms of how many quantifiers you need and what symbols are needed in the language. There are many connections to complexity theory in that virtually all descriptive classes are equivalent to the more standard complexity classes. 2. Theory of Computing: A Gentle Introduction by Efim Kinber and Carl Smith, reviewed by Judy Goldsmith. This book is a textbook aimed at undergraduates who would not be happy with the mathematical rigour of the automata books of my youth, such as Hopcroft and Ullman's book. 3. Microsurveys in Discrete Probability (Proceedings from a DIMACS workshop) edited by David Aldous and James Propp is reviewed by Hassan Masum. This is a collection of articles (not all surveys) in the area of probability. 4. Term Rewriting and all that by Franz Baader and Tobias Nipkow, reviewed by Paliath Narendran. This is intended as both a text and a reference book on term rewriting. -
1. Introduction
Logical Methods in Computer Science Vol. 7 (3:14) 2011, pp. 1–24 Submitted Nov. 17, 2010 www.lmcs-online.org Published Sep. 21, 2011 RANDOMISATION AND DERANDOMISATION IN DESCRIPTIVE COMPLEXITY THEORY KORD EICKMEYER AND MARTIN GROHE Humboldt-Universit¨at zu Berlin, Institut f¨ur Informatik, Logik in der Informatik Unter den Linden 6, 10099 Berlin, Germany e-mail address: [email protected], [email protected] Abstract. We study probabilistic complexity classes and questions of derandomisation from a logical point of view. For each logic L we introduce a new logic BPL, bounded error probabilistic L, which is defined from L in a similar way as the complexity class BPP, bounded error probabilistic polynomial time, is defined from P. Our main focus lies on questions of derandomisation, and we prove that there is a query which is definable in BPFO, the probabilistic version of first-order logic, but not ω in C∞ω, finite variable infinitary logic with counting. This implies that many of the standard logics of finite model theory, like transitive closure logic and fixed-point logic, both with and without counting, cannot be derandomised. Similarly, we present a query on ordered structures which is definable in BPFO but not in monadic second-order logic, and a query on additive structures which is definable in BPFO but not in FO. The latter of these queries shows that certain uniform variants of AC0 (bounded-depth polynomial sized circuits) cannot be derandomised. These results are in contrast to the general belief that most standard complexity classes can be derandomised. -
BU CS 332 – Theory of Computation
BU CS 332 –Theory of Computation Lecture 21: • NP‐Completeness Reading: • Cook‐Levin Theorem Sipser Ch 7.3‐7.5 • Reductions Mark Bun April 15, 2020 Last time: Two equivalent definitions of 1) is the class of languages decidable in polynomial time on a nondeterministic TM 2) A polynomial‐time verifier for a language is a deterministic ‐time algorithm such that iff there exists a string such that accepts Theorem: A language iff there is a polynomial‐time verifier for 4/15/2020 CS332 ‐ Theory of Computation 2 Examples of languages: SAT “Is there an assignment to the variables in a logical formula that make it evaluate to ?” • Boolean variable: Variable that can take on the value / (encoded as 0/1) • Boolean operations: • Boolean formula: Expression made of Boolean variables and operations. Ex: • An assignment of 0s and 1s to the variables satisfies a formula if it makes the formula evaluate to 1 • A formula is satisfiable if there exists an assignment that satisfies it 4/15/2020 CS332 ‐ Theory of Computation 3 Examples of languages: SAT Ex: Satisfiable? Ex: Satisfiable? Claim: 4/15/2020 CS332 ‐ Theory of Computation 4 Examples of languages: TSP “Given a list of cities and distances between them, is there a ‘short’ tour of all of the cities?” More precisely: Given • A number of cities • A function giving the distance between each pair of cities • A distance bound 4/15/2020 CS332 ‐ Theory of Computation 5 vs. Question: Does ? Philosophically: Can every problem with an efficiently verifiable solution also be solved efficiently? -
Chicago Journal of Theoretical Computer Science the MIT Press
Chicago Journal of Theoretical Computer Science The MIT Press Volume 1997, Article 2 3 June 1997 ISSN 1073–0486. MIT Press Journals, Five Cambridge Center, Cambridge, MA 02142-1493 USA; (617)253-2889; [email protected], [email protected]. Published one article at a time in LATEX source form on the Internet. Pag- ination varies from copy to copy. For more information and other articles see: http://www-mitpress.mit.edu/jrnls-catalog/chicago.html • http://www.cs.uchicago.edu/publications/cjtcs/ • ftp://mitpress.mit.edu/pub/CJTCS • ftp://cs.uchicago.edu/pub/publications/cjtcs • Kann et al. Hardness of Approximating Max k-Cut (Info) The Chicago Journal of Theoretical Computer Science is abstracted or in- R R R dexed in Research Alert, SciSearch, Current Contents /Engineering Com- R puting & Technology, and CompuMath Citation Index. c 1997 The Massachusetts Institute of Technology. Subscribers are licensed to use journal articles in a variety of ways, limited only as required to insure fair attribution to authors and the journal, and to prohibit use in a competing commercial product. See the journal’s World Wide Web site for further details. Address inquiries to the Subsidiary Rights Manager, MIT Press Journals; (617)253-2864; [email protected]. The Chicago Journal of Theoretical Computer Science is a peer-reviewed scholarly journal in theoretical computer science. The journal is committed to providing a forum for significant results on theoretical aspects of all topics in computer science. Editor in chief: Janos Simon Consulting -
László Lovász Avi Wigderson of Eötvös Loránd University of the Institute for Advanced Study, in Budapest, Hungary and Princeton, USA
2021 The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2021 to László Lovász Avi Wigderson of Eötvös Loránd University of the Institute for Advanced Study, in Budapest, Hungary and Princeton, USA, “for their foundational contributions to theoretical computer science and discrete mathematics, and their leading role in shaping them into central fields of modern mathematics.” Theoretical Computer Science (TCS) is the study of computational lens”. Discrete structures such as the power and limitations of computing. Its roots go graphs, strings, permutations are central to TCS, back to the foundational works of Kurt Gödel, Alonzo and naturally discrete mathematics and TCS Church, Alan Turing, and John von Neumann, leading have been closely allied fields. While both these to the development of real physical computers. fields have benefited immensely from more TCS contains two complementary sub-disciplines: traditional areas of mathematics, there has been algorithm design which develops efficient methods growing influence in the reverse direction as well. for a multitude of computational problems; and Applications, concepts, and techniques from TCS computational complexity, which shows inherent have motivated new challenges, opened new limitations on the efficiency of algorithms. The notion directions of research, and solved important open of polynomial-time algorithms put forward in the problems in pure and applied mathematics. 1960s by Alan Cobham, Jack Edmonds, and others, and the famous P≠NP conjecture of Stephen Cook, László Lovász and Avi Wigderson have been leading Leonid Levin, and Richard Karp had strong impact on forces in these developments over the last decades. the field and on the work of Lovász and Wigderson. -
Algorithms: a Quest for Absolute Definitions∗
Algorithms: A Quest for Absolute De¯nitions¤ Andreas Blassy Yuri Gurevichz Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include speci¯cation, validation and veri¯ca- tion of software and hardware systems. We describe the quest to understand and de¯ne the notion of algorithm. We start with the Church-Turing thesis and contrast Church's and Turing's approaches, and we ¯nish with some recent investigations. Contents 1 Introduction 2 2 The Church-Turing thesis 3 2.1 Church + Turing . 3 2.2 Turing ¡ Church . 4 2.3 Remarks on Turing's analysis . 6 3 Kolmogorov machines and pointer machines 9 4 Related issues 13 4.1 Physics and computations . 13 4.2 Polynomial time Turing's thesis . 14 4.3 Recursion . 15 ¤Bulletin of European Association for Theoretical Computer Science 81, 2003. yPartially supported by NSF grant DMS{0070723 and by a grant from Microsoft Research. Address: Mathematics Department, University of Michigan, Ann Arbor, MI 48109{1109. zMicrosoft Research, One Microsoft Way, Redmond, WA 98052. 1 5 Formalization of sequential algorithms 15 5.1 Sequential Time Postulate . 16 5.2 Small-step algorithms . 17 5.3 Abstract State Postulate . 17 5.4 Bounded Exploration Postulate and the de¯nition of sequential al- gorithms . 19 5.5 Sequential ASMs and the characterization theorem . 20 6 Formalization of parallel algorithms 21 6.1 What parallel algorithms? . 22 6.2 A few words on the axioms for wide-step algorithms . 22 6.3 Wide-step abstract state machines . 23 6.4 The wide-step characterization theorem . -
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T G¨ P 2012 C N Deadline: December 31, 2011 The Gödel Prize for outstanding papers in the area of theoretical computer sci- ence is sponsored jointly by the European Association for Theoretical Computer Science (EATCS) and the Association for Computing Machinery, Special Inter- est Group on Algorithms and Computation Theory (ACM-SIGACT). The award is presented annually, with the presentation taking place alternately at the Inter- national Colloquium on Automata, Languages, and Programming (ICALP) and the ACM Symposium on Theory of Computing (STOC). The 20th prize will be awarded at the 39th International Colloquium on Automata, Languages, and Pro- gramming to be held at the University of Warwick, UK, in July 2012. The Prize is named in honor of Kurt Gödel in recognition of his major contribu- tions to mathematical logic and of his interest, discovered in a letter he wrote to John von Neumann shortly before von Neumann’s death, in what has become the famous P versus NP question. The Prize includes an award of USD 5000. AWARD COMMITTEE: The winner of the Prize is selected by a committee of six members. The EATCS President and the SIGACT Chair each appoint three members to the committee, to serve staggered three-year terms. The committee is chaired alternately by representatives of EATCS and SIGACT. The 2012 Award Committee consists of Sanjeev Arora (Princeton University), Josep Díaz (Uni- versitat Politècnica de Catalunya), Giuseppe Italiano (Università a˘ di Roma Tor Vergata), Mogens Nielsen (University of Aarhus), Daniel Spielman (Yale Univer- sity), and Eli Upfal (Brown University). ELIGIBILITY: The rule for the 2011 Prize is given below and supersedes any di fferent interpretation of the parametric rule to be found on websites on both SIGACT and EATCS. -
HISTORY of UKRAINE and UKRAINIAN CULTURE Scientific and Methodical Complex for Foreign Students
Ministry of Education and Science of Ukraine Flight Academy of National Aviation University IRYNA ROMANKO HISTORY OF UKRAINE AND UKRAINIAN CULTURE Scientific and Methodical Complex for foreign students Part 3 GUIDELINES FOR SELF-STUDY Kropyvnytskyi 2019 ɍȾɄ 94(477):811.111 R e v i e w e r s: Chornyi Olexandr Vasylovych – the Head of the Department of History of Ukraine of Volodymyr Vynnychenko Central Ukrainian State Pedagogical University, Candidate of Historical Sciences, Associate professor. Herasymenko Liudmyla Serhiivna – associate professor of the Department of Foreign Languages of Flight Academy of National Aviation University, Candidate of Pedagogical Sciences, Associate professor. ɇɚɜɱɚɥɶɧɨɦɟɬɨɞɢɱɧɢɣɤɨɦɩɥɟɤɫɩɿɞɝɨɬɨɜɥɟɧɨɡɝɿɞɧɨɪɨɛɨɱɨʀɩɪɨɝɪɚɦɢɧɚɜɱɚɥɶɧɨʀɞɢɫɰɢɩɥɿɧɢ "ȱɫɬɨɪɿɹ ɍɤɪɚʀɧɢ ɬɚ ɭɤɪɚʀɧɫɶɤɨʀ ɤɭɥɶɬɭɪɢ" ɞɥɹ ɿɧɨɡɟɦɧɢɯ ɫɬɭɞɟɧɬɿɜ, ɡɚɬɜɟɪɞɠɟɧɨʀ ɧɚ ɡɚɫɿɞɚɧɧɿ ɤɚɮɟɞɪɢ ɩɪɨɮɟɫɿɣɧɨʀ ɩɟɞɚɝɨɝɿɤɢɬɚɫɨɰɿɚɥɶɧɨɝɭɦɚɧɿɬɚɪɧɢɯɧɚɭɤ (ɩɪɨɬɨɤɨɥʋ1 ɜɿɞ 31 ɫɟɪɩɧɹ 2018 ɪɨɤɭ) ɬɚɫɯɜɚɥɟɧɨʀɆɟɬɨɞɢɱɧɢɦɢ ɪɚɞɚɦɢɮɚɤɭɥɶɬɟɬɿɜɦɟɧɟɞɠɦɟɧɬɭ, ɥɶɨɬɧɨʀɟɤɫɩɥɭɚɬɚɰɿʀɬɚɨɛɫɥɭɝɨɜɭɜɚɧɧɹɩɨɜɿɬɪɹɧɨɝɨɪɭɯɭ. ɇɚɜɱɚɥɶɧɢɣ ɩɨɫɿɛɧɢɤ ɡɧɚɣɨɦɢɬɶ ɿɧɨɡɟɦɧɢɯ ɫɬɭɞɟɧɬɿɜ ɡ ɿɫɬɨɪɿɽɸ ɍɤɪɚʀɧɢ, ʀʀ ɛɚɝɚɬɨɸ ɤɭɥɶɬɭɪɨɸ, ɨɯɨɩɥɸɽ ɧɚɣɜɚɠɥɢɜɿɲɿɚɫɩɟɤɬɢ ɭɤɪɚʀɧɫɶɤɨʀɞɟɪɠɚɜɧɨɫɬɿ. ɋɜɿɬɭɤɪɚʀɧɫɶɤɢɯɧɚɰɿɨɧɚɥɶɧɢɯɬɪɚɞɢɰɿɣ ɭɧɿɤɚɥɶɧɢɣ. ɋɬɨɥɿɬɬɹɦɢ ɪɨɡɜɢɜɚɥɚɫɹ ɫɢɫɬɟɦɚ ɪɢɬɭɚɥɿɜ ɿ ɜɿɪɭɜɚɧɶ, ɹɤɿ ɧɚ ɫɭɱɚɫɧɨɦɭ ɟɬɚɩɿ ɧɚɛɭɜɚɸɬɶ ɧɨɜɨʀ ɩɨɩɭɥɹɪɧɨɫɬɿ. Ʉɧɢɝɚ ɪɨɡɩɨɜɿɞɚɽ ɩɪɨ ɤɚɥɟɧɞɚɪɧɿ ɫɜɹɬɚ ɜ ɍɤɪɚʀɧɿ: ɞɟɪɠɚɜɧɿ, ɪɟɥɿɝɿɣɧɿ, ɩɪɨɮɟɫɿɣɧɿ, ɧɚɪɨɞɧɿ, ɚ ɬɚɤɨɠ ɪɿɡɧɿ ɩɚɦ ɹɬɧɿ ɞɚɬɢ. ɍ ɩɨɫɿɛɧɢɤɭ ɩɪɟɞɫɬɚɜɥɟɧɿ ɪɿɡɧɨɦɚɧɿɬɧɿ ɞɚɧɿ ɩɪɨ ɮɥɨɪɭ ɿ ɮɚɭɧɭ ɤɥɿɦɚɬɢɱɧɢɯ