Combinatorics, Automata and Number Theory
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Language and Automata Theory and Applications
LANGUAGE AND AUTOMATA THEORY AND APPLICATIONS Carlos Martín-Vide Characterization • It deals with the description of properties of sequences of symbols • Such an abstract characterization explains the interdisciplinary flavour of the field • The theory grew with the need of formalizing and describing the processes linked with the use of computers and communication devices, but its origins are within mathematical logic and linguistics A bit of history • Early roots in the work of logicians at the beginning of the XXth century: Emil Post, Alonzo Church, Alan Turing Developments motivated by the search for the foundations of the notion of proof in mathematics (Hilbert) • After the II World War: Claude Shannon, Stephen Kleene, John von Neumann Development of computers and telecommunications Interest in exploring the functions of the human brain • Late 50s XXth century: Noam Chomsky Formal methods to describe natural languages • Last decades Molecular biology considers the sequences of molecules formed by genomes as sequences of symbols on the alphabet of basic elements Interest in describing properties like repetitions of occurrences or similarity between sequences Chomsky hierarchy of languages • Finite-state or regular • Context-free • Context-sensitive • Recursively enumerable REG ⊂ CF ⊂ CS ⊂ RE Finite automata: origins • Warren McCulloch & Walter Pitts. A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biophysics, 5:115-133, 1943 • Stephen C. Kleene. Representation of events in nerve nets and -
2020 SIGACT REPORT SIGACT EC – Eric Allender, Shuchi Chawla, Nicole Immorlica, Samir Khuller (Chair), Bobby Kleinberg September 14Th, 2020
2020 SIGACT REPORT SIGACT EC – Eric Allender, Shuchi Chawla, Nicole Immorlica, Samir Khuller (chair), Bobby Kleinberg September 14th, 2020 SIGACT Mission Statement: The primary mission of ACM SIGACT (Association for Computing Machinery Special Interest Group on Algorithms and Computation Theory) is to foster and promote the discovery and dissemination of high quality research in the domain of theoretical computer science. The field of theoretical computer science is the rigorous study of all computational phenomena - natural, artificial or man-made. This includes the diverse areas of algorithms, data structures, complexity theory, distributed computation, parallel computation, VLSI, machine learning, computational biology, computational geometry, information theory, cryptography, quantum computation, computational number theory and algebra, program semantics and verification, automata theory, and the study of randomness. Work in this field is often distinguished by its emphasis on mathematical technique and rigor. 1. Awards ▪ 2020 Gödel Prize: This was awarded to Robin A. Moser and Gábor Tardos for their paper “A constructive proof of the general Lovász Local Lemma”, Journal of the ACM, Vol 57 (2), 2010. The Lovász Local Lemma (LLL) is a fundamental tool of the probabilistic method. It enables one to show the existence of certain objects even though they occur with exponentially small probability. The original proof was not algorithmic, and subsequent algorithmic versions had significant losses in parameters. This paper provides a simple, powerful algorithmic paradigm that converts almost all known applications of the LLL into randomized algorithms matching the bounds of the existence proof. The paper further gives a derandomized algorithm, a parallel algorithm, and an extension to the “lopsided” LLL. -
Numeration 2016
Numeration 2016 Prague, Czech Republic, May 23–27 2016 Department of Mathematics FNSPE Czech Technical University in Prague Numeration 2016 Prague, Czech Republic, May 23–27 2016 Department of Mathematics FNSPE Czech Technical University in Prague Print: Česká technika — nakladatelství ČVUT Editor: Petr Ambrož [email protected] Katedra matematiky Fakulta jaderná a fyzikálně inženýrská České vysoké učení technické v Praze Trojanova 13 120 00 Praha 2 List of Participants Rafael Alcaraz Barrera [email protected] IME, University of São Paulo Karam Aloui [email protected] Institut Elie Cartan de Nancy / Faculté des Sciences de Sfax Petr Ambrož petr.ambroz@fjfi.cvut.cz FNSPE, Czech Technical University in Prague Hamdi Ammar [email protected] Sfax University Myriam Amri [email protected] Sfax University Hamdi Aouinti [email protected] Université de Tunis Simon Baker [email protected] University of Reading Christoph Bandt [email protected] University of Greifswald Attila Bérczes [email protected] University of Debrecen Anne Bertrand-Mathis [email protected] University of Poitiers Dávid Bóka [email protected] Eötvös Loránd University Horst Brunotte [email protected] Marta Brzicová [email protected] FNSPE, Czech Technical University in Prague Péter Burcsi [email protected] Eötvös Loránd University Francesco Dolce [email protected] Université Paris-Est Art¯urasDubickas [email protected] Vilnius University Lubomira Dvořáková [email protected] FNSPE, Czech -
On the Behavior of Mahler's Measure Under Iteration 1
ON THE BEHAVIOR OF MAHLER'S MEASURE UNDER ITERATION PAUL FILI, LUKAS POTTMEYER, AND MINGMING ZHANG Abstract. For an algebraic number α we denote by M(α) the Mahler measure of α. As M(α) is again an algebraic number (indeed, an algebraic integer), M(·) is a self-map on Q, and therefore defines a dynamical system. The orbit size of α, denoted #OM (α), is the cardinality of the forward orbit of α under M. We prove that for every degree at least 3 and every non-unit norm, there exist algebraic numbers of every orbit size. We then prove that for algebraic units of degree 4, the orbit size must be 1, 2, or infinity. We also show that there exist algebraic units of larger degree with arbitrarily large but finite orbit size. 1. Introduction The Mahler measure of an algebraic number α with minimal polynomial f(x) = n anx + ··· + a0 2 Z[x] is defined as: n n Y Y M(α) = janj maxf1; jαijg = ±an αi: i=1 i=1 jαij>1 Qn where f(x) = an i=1(x − αi) 2 C[x]. It is clear that M(α) ≥ 1 is a real algebraic integer, and it follows from Kronecker's theorem that M(α) = 1 if and only if α is a root of unity. Moreover, we will freely use the facts that M(α) = M(β) whenever α and β have the same minimal polynomial, and that M(α) = M(α−1). D.H. Lehmer [10] famously asked in 1933 if the Mahler measure for an algebraic number which is not a root of unity can be arbitrarily close to 1. -
The Smallest Perron Numbers 1
MATHEMATICS OF COMPUTATION Volume 79, Number 272, October 2010, Pages 2387–2394 S 0025-5718(10)02345-8 Article electronically published on April 26, 2010 THE SMALLEST PERRON NUMBERS QIANG WU Abstract. A Perron number is a real algebraic integer α of degree d ≥ 2, whose conjugates are αi, such that α>max2≤i≤d |αi|. In this paper we com- pute the smallest Perron numbers of degree d ≤ 24 and verify that they all satisfy the Lind-Boyd conjecture. Moreover, the smallest Perron numbers of degree 17 and 23 give the smallest house for these degrees. The computa- tions use a family of explicit auxiliary functions. These functions depend on generalizations of the integer transfinite diameter of some compact sets in C 1. Introduction Let α be an algebraic integer of degree d, whose conjugates are α1 = α, α2,...,αd and d d−1 P = X + b1X + ···+ bd−1X + bd, its minimal polynomial. A Perron number, which was defined by Lind [LN1], is a real algebraic integer α of degree d ≥ 2 such that α > max2≤i≤d |αi|.Any Pisot number or Salem number is a Perron number. From the Perron-Frobenius theorem, if A is a nonnegative integral matrix which is aperiodic, i.e. some power of A has strictly positive entries, then its spectral radius α is a Perron number. Lind has proved the converse, that is to say, if α is a Perron number, then there is a nonnegative aperiodic integral matrix whose spectral radius is α.Lind[LN2] has investigated the arithmetic of the Perron numbers. -
COMPUTER MODELS and AUTOMATA THEORY in BIOLOGY and MEDICINE Ion C
Mathematical Modelling, Vol. 7, pp. 1513-1577,1986 Printed in the U,S.A. All rights reserved. Copyright C 1986 Pergamon Journals Ltd. REVIEW ARTICLE COMPUTER MODELS AND AUTOMATA THEORY IN BIOLOGY AND MEDICINE Ion C. Baianu University of Ulinois at Urbana Physical Chemistry and NMR Laboratories 567 Bevier Hall, 905 S. Goodwin Ave .Urbana, UIinois 61801 (Received 27 February 1985; revised 12 September 1985) 1. INTRODUCTION The applications of computers to biological and biomedical problem solving goes back to the very beginnings of computer science, automata theory [1], and mathematical biology [2]. With the advent of more versatile and powerful computers, biological and biomedical applications of computers have proliferated so rapidly that it would be virtually impossible to compile a comprehensive review of all developments in this field. Limitations of computer simulations in biology have also come under close scrutiny, and claims have been made that biological systems have limited information processing power [3]. Such general conjectures do not, however, deter biologists and biomedical researchers from developing new computer applications in biology and medicine. Microprocessors are being widely employed in biological laboratories both for automatic data acquisition/processing and modeling; one particular area, which is of great biomedical interest, involves fast digital image processing and is already established for routine clinical examinations in radiological and nuclear medicine centers, Powerful techniques for biological research are routinely employing dedicated, on-line microprocessors or array processors; among such techniques are: Fourier-transform nuclear magnetic resonance (NMR), NMR imaging (or tomography), x-ray tomography, x-ray diffraction, high performance liquid chromatography, differential scanning calorimetry and mass spectrometry. -
A Mathematical Theory of Computation?
A Mathematical Theory of Computation? Simone Martini Dipartimento di Informatica { Scienza e Ingegneria Alma mater studiorum • Universit`adi Bologna and INRIA FoCUS { Sophia / Bologna Lille, February 1, 2017 1 / 57 Reflect and trace the interaction of mathematical logic and programming (languages), identifying some of the driving forces of this process. Previous episodes: Types HaPOC 2015, Pisa: from 1955 to 1970 (circa) Cie 2016, Paris: from 1965 to 1975 (circa) 2 / 57 Why types? Modern programming languages: control flow specification: small fraction abstraction mechanisms to model application domains. • Types are a crucial building block of these abstractions • And they are a mathematical logic concept, aren't they? 3 / 57 Why types? Modern programming languages: control flow specification: small fraction abstraction mechanisms to model application domains. • Types are a crucial building block of these abstractions • And they are a mathematical logic concept, aren't they? 4 / 57 We today conflate: Types as an implementation (representation) issue Types as an abstraction mechanism Types as a classification mechanism (from mathematical logic) 5 / 57 The quest for a \Mathematical Theory of Computation" How does mathematical logic fit into this theory? And for what purposes? 6 / 57 The quest for a \Mathematical Theory of Computation" How does mathematical logic fit into this theory? And for what purposes? 7 / 57 Prehistory 1947 8 / 57 Goldstine and von Neumann [. ] coding [. ] has to be viewed as a logical problem and one that represents a new branch of formal logics. Hermann Goldstine and John von Neumann Planning and Coding of problems for an Electronic Computing Instrument Report on the mathematical and logical aspects of an electronic computing instrument, Part II, Volume 1-3, April 1947. -
Introduction to Theory of Computation
Introduction to Theory of Computation Anil Maheshwari Michiel Smid School of Computer Science Carleton University Ottawa Canada anil,michiel @scs.carleton.ca { } April 17, 2019 ii Contents Contents Preface vi 1 Introduction 1 1.1 Purposeandmotivation ..................... 1 1.1.1 Complexitytheory .................... 2 1.1.2 Computability theory . 2 1.1.3 Automatatheory ..................... 3 1.1.4 Thiscourse ........................ 3 1.2 Mathematical preliminaries . 4 1.3 Prooftechniques ......................... 7 1.3.1 Directproofs ....................... 8 1.3.2 Constructiveproofs. 9 1.3.3 Nonconstructiveproofs . 10 1.3.4 Proofsbycontradiction. 11 1.3.5 The pigeon hole principle . 12 1.3.6 Proofsbyinduction. 13 1.3.7 Moreexamplesofproofs . 15 Exercises................................. 18 2 Finite Automata and Regular Languages 21 2.1 An example: Controling a toll gate . 21 2.2 Deterministic finite automata . 23 2.2.1 Afirstexampleofafiniteautomaton . 26 2.2.2 Asecondexampleofafiniteautomaton . 28 2.2.3 A third example of a finite automaton . 29 2.3 Regularoperations ........................ 31 2.4 Nondeterministic finite automata . 35 2.4.1 Afirstexample ...................... 35 iv Contents 2.4.2 Asecondexample..................... 37 2.4.3 Athirdexample...................... 38 2.4.4 Definition of nondeterministic finite automaton . 39 2.5 EquivalenceofDFAsandNFAs . 41 2.5.1 Anexample ........................ 44 2.6 Closureundertheregularoperations . 48 2.7 Regularexpressions. .. .. 52 2.8 Equivalence of regular expressions and regular languages . 56 2.8.1 Every regular expression describes a regular language . 57 2.8.2 Converting a DFA to a regular expression . 60 2.9 The pumping lemma and nonregular languages . 67 2.9.1 Applications of the pumping lemma . 69 2.10Higman’sTheorem ........................ 76 2.10.1 Dickson’sTheorem . -
A Combinatorial Approach to the Theory of O~-Automata
INFORMATION AND CONTROL 48, 261-283 (1981) A Combinatorial Approach to the Theory of o~-Automata WOLFGANG THOMAS Mathematisches Institut der Universitdt, 7800 Freiburg, West Germany A combinatorial lemma is proved and used here to derive new results on ~o- automata and to give simpler proofs of known ones. In particular, we reprove McNaughton's fundamental theorem (characterizing the o~-regular sequence-sets), without having to construct a sophisticated w-automaton. The theorem is obtained by coding the behaviour of automata in a second-order language and a simple application of the lemma. In close analogy (now referring to a first-order language) a theory of counter-free co-automata is developed; it is shown that these automata are appropriate for characterizing the co-star-free sequence-sets. Finally, the lemma is applied in mathematical logic: Here new normal form theorems and also decidability results are proved for the first-order and the monadic second-order theory of certain structures over the ordering of natural numbers. 0. INTRODUCTION A central result in the theory of finite automata accepting co-sequences (short: co-automata) is the theorem of McNaughton (1966). It characterizes the sequence-sets accepted by co-automata as the co-regular sequence-sets. McNaughton's proof involves an intricate construction of an co-automaton; it shows "how a bounded memory (finite automaton) can be made to keep track of essential information, accruing in an ever growing past" (B/ichi (1977)). Modified and improved presentations of this construction have since been given in Rabin (1972), B/ichi (1973), Choueka (1974) and Trakhtenbrot and Barzdin (1973). -
New Families of Pseudo-Anosov Homeomorphisms with Vanishing Sah-Arnoux-Fathi Invariant
AN ABSTRACT OF THE THESIS OF Hieu Trung Do for the degree of Doctor of Philosophy in Mathematics presented on May 27, 2016. Title: New Families of pseudo-Anosov Homeomorphisms with Vanishing Sah-Arnoux-Fathi Invariant. Abstract approved: Thomas A. Schmidt Translation surfaces can be viewed as polygons with parallel and equal sides identified. An affine homeomorphism φ from a translation surface to itself is called pseudo-Anosov when its derivative is a constant matrix in SL2(R) whose trace is larger than 2 in absolute value. In this setting, the eigendirections of this matrix defines the stable and unstable flow on the translation surface. Taking a transversal to the stable flows, the first return map of the flow induces an interval exchange transformation T . The Sah-Arnoux-Fathi invariant of φ is the sum of the wedge product between the lengths of the subintervals of T and their translations. This wedge product does not depend on the choice of transversal. We apply Veech's construction of pseudo-Anosov homeomorphisms to produce infinite families of pseudo-Anosov maps in the stratum H(2; 2) with vanishing Sah-Arnoux-Fathi invariant, as well as sporadic examples in other strata. ©Copyright by Hieu Trung Do May 27, 2016 All Rights Reserved New Families of pseudo-Anosov Homeomorphisms with Vanishing Sah-Arnoux-Fathi Invariant by Hieu Trung Do A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Presented May 27, 2016 Commencement June 2016 Doctor of Philosophy thesis of Hieu Trung Do presented on May 27, 2016 APPROVED: Major Professor, representing Mathematics Chair of the Department of Mathematics Dean of the Graduate School I understand that my thesis will become part of the permanent collection of Oregon State University libraries. -
Ito-Sadahiro Numbers Vs. Parry Numbers
Acta Polytechnica Vol. 51 No. 4/2011 Ito-Sadahiro numbers vs. Parry numbers Z. Mas´akov´a, E. Pelantov´a Abstract We consider a positional numeration system with a negative base, as introduced by Ito and Sadahiro. In particular, we focus on the algebraic properties of negative bases −β for which the corresponding dynamical system is sofic, which β happens, according to Ito and Sadahiro, if and only if the (−β)-expansion of − is eventually periodic. We call β +1 such numbers β Ito-Sadahiro numbers, and we compare their properties with those of Parry numbers, which occur in thesamecontextfortheR´enyi positive base numeration system. Keywords: numeration systems, negative base, Pisot number, Parry number. ∈AN 1 Introduction According to Parry, the string x1x2x3 ... rep- resents the β-expansion of a number x ∈ [0, 1) if and The expansion of a real number in the positional only if number system with base β>1, as defined by R´enyi [12] is closely related to the transformation ∗ xixi+1xi+2 ...≺ d (1) (2) T :[0, 1) → [0, 1), given by the prescription T (x):= β βx −#βx$.Everyx ∈ [0, 1) is a sum of the infinite for every i =1, 2, 3,... series D { | ∞ Condition (2) ensures that the set β = dβ(x) xi − ∈ } D x = , where x = #βT i 1(x)$ (1) x [0, 1) is shift invariant, and so the closure of β βi i N i=1 in A , denoted by Sβ, is a subshift of the full shift N for i =1, 2, 3,... A . The notion of β-expansion can naturally be ex- Directly from the definition of the transformation T tended to all non-negative real numbers: The expres- we can derive that the ‘digits’ x take values in the set i sion of a positive real number y in the form {0, 1, 2,...,%β&−1} for i =1, 2, 3,... -
A Theory of Regular Queries
A Theory of Regular Queries Moshe Y. Vardi Rice University [email protected] ABSTRACT [22], he proposed using first-order logic as a declarative database A major theme in relational database theory is navigat- query language [23]. This led to the development of both ing the tradeoff between expressiveness and tractability for SEQUEL [16] and QUEL [54], as practical relational query query languages, where the query-containment problem is languages realizing Codd's idea, ultimately giving in 1986 considered a benchmark of tractability. The query class rise to the SQL standard [27], which has continued to evolve UCQ, consisting off unions of conjunctive queries, is a frag- over the past 30 years. ment of first-order logic that has a decidable query contain- Starting in the late 1970s, Codd's definition of first-order ment problem, but its expressiveness is limited. Extend- logic as \expressively complete" came under serious criti- ing UCQ with recursion yields Datalog, an expressive query cism [4], and various proposals emerged on how to extend language that has been studied extensively and has recently its expressive power [4, 6, 17], whose essence was to extend become popular in application areas such as declarative net- first-order logic with recursion, which was added to SQL in working. Unfortunately, Datalog has an undecidable query 1999 by way of common table expressions [29]. On the re- containment problem. Identifying a fragment of Datalog search side, the most popular relational language embodying that is expressive enough for applications but has a decid- recursion is Datalog [15], describe in more details below.