Undecidability of the Word Problem for Groups: the Point of View of Rewriting Theory
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Elements of Programming Elements of Programming Alexander Stepanov Paul McJones (ab)c = a(bc) Semigroup Press Palo Alto • Mountain View Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed with initial capital letters or in all capitals. The authors and publisher have taken care in the preparation of this book, but make no expressed or implied warranty of any kind and assume no responsibility for errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of the use of the information or programs contained herein. Copyright c 2009 Pearson Education, Inc. Portions Copyright c 2019 Alexander Stepanov and Paul McJones All rights reserved. Printed in the United States of America. This publication is protected by copyright, and permission must be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permissions, request forms and the appropriate contacts within the Pearson Education Global Rights & Permissions Department, please visit www.pearsoned.com/permissions/. ISBN-13: 978-0-578-22214-1 First printing, June 2019 Contents Preface to Authors' Edition ix Preface xi 1 Foundations 1 1.1 Categories of Ideas: Entity, Species, -
CLASS DESCRIPTIONS—WEEK 2, MATHCAMP 2016 Contents 9:10
CLASS DESCRIPTIONS|WEEK 2, MATHCAMP 2016 Contents 9:10 Classes 2 Dynamical Systems 2 Field Extensions and Galois Theory (Week 1 of 2) 2 Model Theory 2 Neural Networks 3 Problem Solving: Induction 3 10:10 Classes 4 Almost Planar 4 Extending Inclusion-Exclusion 4 Multilinear Algebra 5 The Word Problem for Groups 5 Why Are We Learning This? A seminar on the history of math education in the U.S. 6 11:10 Classes 6 Building Mathematical Structures 6 Graph Minors 7 History of Math 7 K-Theory 7 Linear Algebra (Week 2 of 2) 7 The Banach{Tarski Paradox 8 1:10 Classes 8 Divergent Series 8 Geometric Group Theory 8 Group Actions 8 The Chip-Firing Game 9 Colloquia 9 There Are Eight Flavors of Three-Dimensional Geometry 9 Tropical Geometry 9 Oops! I Ran Out of Axioms 10 Visitor Bios 10 Moon Duchin 10 Sam Payne 10 Steve Schweber 10 1 MC2016 ◦ W2 ◦ Classes 2 9:10 Classes Dynamical Systems. ( , Jane, Tuesday{Saturday) Dynamical systems are spaces that evolve over time. Some examples are the movement of the planets, the bouncing of a ball around a billiard table, or the change in the population of rabbits from year to year. The study of dynamical systems is the study of how these spaces evolve, their long-term behavior, and how to predict the future of these systems. It is a subject that has many applications in the real world and also in other branches of mathematics. In this class, we'll survey a variety of topics in dynamical systems, exploring both what is known and what is still open. -
CS411-2015F-14 Counter Machines & Recursive Functions
Automata Theory CS411-2015F-14 Counter Machines & Recursive Functions David Galles Department of Computer Science University of San Francisco 14-0: Counter Machines Give a Non-Deterministic Finite Automata a counter Increment the counter Decrement the counter Check to see if the counter is zero 14-1: Counter Machines A Counter Machine M = (K, Σ, ∆,s,F ) K is a set of states Σ is the input alphabet s ∈ K is the start state F ⊂ K are Final states ∆ ⊆ ((K × (Σ ∪ ǫ) ×{zero, ¬zero}) × (K × {−1, 0, +1})) Accept if you reach the end of the string, end in an accept state, and have an empty counter. 14-2: Counter Machines Give a Non-Deterministic Finite Automata a counter Increment the counter Decrement the counter Check to see if the counter is zero Do we have more power than a standard NFA? 14-3: Counter Machines Give a counter machine for the language anbn 14-4: Counter Machines Give a counter machine for the language anbn (a,zero,+1) (a,~zero,+1) (b,~zero,−1) (b,~zero,−1) 0 1 14-5: Counter Machines Give a 2-counter machine for the language anbncn Straightforward extension – examine (and change) two counters instead of one. 14-6: Counter Machines Give a 2-counter machine for the language anbncn (a,zero,zero,+1,0) (a,~zero,zero,+1,0) (b,~zero,~zero,-1,+1) (b,~zero,zero,−1,+1) 0 1 (c,zero,~zero,0,-1) 2 (c,zero,~zero,0,-1) 14-7: Counter Machines Our counter machines only accept if the counter is zero Does this give us any more power than a counter machine that accepts whenever the end of the string is reached in an accept state? That is, given -
BRANCHING PROGRAM UNIFORMIZATION, REWRITING LOWER BOUNDS, and GEOMETRIC GROUP THEORY IZAAK MECKLER Mathematics Department U.C. B
BRANCHING PROGRAM UNIFORMIZATION, REWRITING LOWER BOUNDS, AND GEOMETRIC GROUP THEORY IZAAK MECKLER Mathematics Department U.C. Berkeley Berkeley, CA Abstract. Geometric group theory is the study of the relationship between the algebraic, geo- metric, and combinatorial properties of finitely generated groups. Here, we add to the dictionary of correspondences between geometric group theory and computational complexity. We then use these correspondences to establish limitations on certain models of computation. In particular, we establish a connection between read-once oblivious branching programs and growth of groups. We then use Gromov’s theorem on groups of polynomial growth to give a simple argument that if the word problem of a group G is computed by a non-uniform family of read-once, oblivious, polynomial-width branching programs, then it is computed by an O(n)-time uniform algorithm. That is, efficient non-uniform read-once, oblivious branching programs confer essentially no advantage over uniform algorithms for word problems of groups. We also construct a group EffCirc which faithfully encodes reversible circuits and note the cor- respondence between certain proof systems for proving equations of circuits and presentations of groups containing EffCirc. We use this correspondence to establish a quadratic lower bound on the proof complexity of such systems, using geometric techniques which to our knowledge are new to complexity theory. The technical heart of this argument is a strengthening of the now classical the- orem of geometric group theory that groups with linear Dehn function are hyperbolic. The proof also illuminates a relationship between the notion of quasi-isometry and models of computation that efficiently simulate each other. -
A Microcontroller Based Fan Speed Control Using PID Controller Theory
A Microcontroller Based Fan Speed Control Using PID Controller Theory Thu Thu Zue Zin and Hlaing Thida Oo University of Computer Studies, Yangon thuthuzuezin @gmail.com Abstract PIC control application is widely used and User define popular in modern elections. The aim of this paper Microcontroller speed PIC 16F877 is design and construct the fan speed control system Duty cycle with microcontroller. PIC 16F877 and photo transistor is used for the system. User define speed can be selected by keypads for various speed. Motor is controlled by the pulse width modulation. Driver The system can be operate with normal mode and circuit timer mode. Delay time for timer mode is 10 seconds. The feedback signal from sensor is controlled by the proportional integral derivative equations to reproduce the desire duty cycle. Sensor Motor 1. Introduction Figure 1. Block Diagram of the System Motors are widely used in many applications, such as air conditioners , slide doors, washing machines and control areas. Motors are 2. Background Theory derivate it’s output performance due to the PID Controller tolerances, operating conditions, process error and Proportional Integral Derivative controller is measurement error. Pulse Width Modulation a generic control loop feedback mechanism widely control technique is better for control the DC motor used in industrial control systems. A PID controller than others techniques. PWM control the speed of attempts to correct the error between a measured motor without changing the voltage supply to process variable and a desired set point by calculating motor. Series of pulse define the speed of the and then outputting a corrective action that can adjust motor. -
Copyright © 1998, by the Author(S). All Rights Reserved
Copyright © 1998, by the author(s). All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission. WHATS DECIDABLE ABOUT HYBRID AUTOMATA by Thomas A. Henzinger, Peter W. Kopke, Anuj Puri, and Pravin Varaiya Memorandum No. UCB/ERL M98/22 15 April 1998 WHAT'S DECIDABLE ABOUT HYBRID AUTOMATA by Thomas A. Henzinger,Peter W. Kopke, Anuj Puri, and Pravin Varaiya Memorandum No. UCB/ERL M98/22 15 April 1998 ELECTRONICS RESEARCH LABORATORY College ofEngineering University of California, Berkeley 94720 What's Decidable About Hybrid Automata?*^ Thomas A. Henzinger^ Peter W. Kopke^ Anuj Puri^ Pravin Varaiya^ Abstract. Hybrid automata model systems with both digital and analog compo nents, such as embedded control programs. Many verification tasks for such programs can be expressed as reachabibty problems for hybrid automata. By improving on pre vious decidability and undecidability results, we identify a precise boundary between decidability and undecidability for the reachabibty problem of hybrid automata. On the positive side, we give an (optimal) PSPACE reachabibty algorithmfor the case of initiabzed rectangular automata, where ab analog variables foUow independent tra jectories within piecewise-bnear envelopes and are reinitiabzed whenever the envelope changes. Our algorithm is based on the construction ofa timedautomaton that contains ab reachabibty information about a given initiabzed rectangular automaton. -
Turing Machines [Fa’16]
Models of Computation Lecture 6: Turing Machines [Fa’16] Think globally, act locally. — Attributed to Patrick Geddes (c.1915), among many others. We can only see a short distance ahead, but we can see plenty there that needs to be done. — Alan Turing, “Computing Machinery and Intelligence” (1950) Never worry about theory as long as the machinery does what it’s supposed to do. — Robert Anson Heinlein, Waldo & Magic, Inc. (1950) It is a sobering thought that when Mozart was my age, he had been dead for two years. — Tom Lehrer, introduction to “Alma”, That Was the Year That Was (1965) 6 Turing Machines In 1936, a few months before his 24th birthday, Alan Turing launched computer science as a modern intellectual discipline. In a single remarkable paper, Turing provided the following results: • A simple formal model of mechanical computation now known as Turing machines. • A description of a single universal machine that can be used to compute any function computable by any other Turing machine. • A proof that no Turing machine can solve the halting problem—Given the formal description of an arbitrary Turing machine M, does M halt or run forever? • A proof that no Turing machine can determine whether an arbitrary given proposition is provable from the axioms of first-order logic. This is Hilbert and Ackermann’s famous Entscheidungsproblem (“decision problem”). • Compelling arguments1 that his machines can execute arbitrary “calculation by finite means”. Although Turing did not know it at the time, he was not the first to prove that the Entschei- dungsproblem had no algorithmic solution. -
Complexity: Automata DICE 2015
An in-between “implicit” and “explicit” complexity: Automata DICE 2015 Clément Aubert Queen Mary University of London 12 April 2015 Introduction: Motivation (Dal Lago, 2011, p. 90) Question Is there an in-between? Answer One can also restrict the quality of the resources. ICC is machine-independent and without explicit bounds. 2 (Ibarra, 1971, p. 88) (Girard et al., 1992, p. 18) Machine-independant Bounded recursion on notation (Cobham, 1965), Bounded linear logic (Girard et al., 1992), Bounded arithmetic (Buss, 1986), . Explicit bounds Automaton, Descriptive complexity (Fagin, 1973), Auxiliary pushdown machine, Recursion on notation (Bellantoni and Cook, 1992), (Boolean circuit,) . Tiered recurrence (Leivant, 1993), . Implicit bounds Introduction: What is ICC? Machine-dependant Turing machine, Random access machine, Counter machine, . 3 (Ibarra, 1971, p. 88) (Girard et al., 1992, p. 18) Explicit bounds Automaton, Descriptive complexity (Fagin, 1973), Auxiliary pushdown machine, Recursion on notation (Bellantoni and Cook, 1992), (Boolean circuit,) . Tiered recurrence (Leivant, 1993), . Implicit bounds Introduction: What is ICC? Machine-dependant Machine-independant Turing machine, Bounded recursion on notation (Cobham, 1965), Random access machine, Bounded linear logic (Girard et al., 1992), Counter machine, . Bounded arithmetic (Buss, 1986), . 3 (Ibarra, 1971, p. 88) Explicit bounds Automaton, Descriptive complexity (Fagin, 1973), Auxiliary pushdown machine, Recursion on notation (Bellantoni and Cook, 1992), (Boolean circuit,) . Tiered recurrence (Leivant, 1993), . Implicit bounds Introduction: What is ICC? Machine-dependant Machine-independant Turing machine, Bounded recursion on notation (Cobham, 1965), Random access machine, Bounded linear logic (Girard et al., 1992), Counter machine, . Bounded arithmetic (Buss, 1986), . (Girard et al., 1992, p. 18) 3 (Ibarra, 1971, p. 88) (Girard et al., 1992, p. -
Bounded Counter Languages
Bounded Counter Languages⋆ Holger Petersen Reinsburgstr. 75 D-70197 Stuttgart Abstract. We show that deterministic finite automata equipped with k two-way heads are equivalent to deterministic machines with a single two-way input head and k − 1 linearly bounded counters if the accepted ∗ ∗ ∗ language is strictly bounded, i.e., a subset of a1a2 · · · am for a fixed se- quence of symbols a1,a2,...,am. Then we investigate linear speed-up for counter machines. Lower and upper time bounds for concrete recognition problems are shown, implying that in general linear speed-up does not hold for counter machines. For bounded languages we develop a technique for speeding up computations by any constant factor at the expense of adding a fixed number of counters. 1 Introduction The computational model investigated in the present work is the two-way counter machine as defined in [4]. Recently, the power of this model has been compared to quantum automata and probabilistic automata [17,14]. We will show that bounded counters and heads are equally powerful for finite deterministic devices, provided the languages under consideration are strictly bounded. By equally powerful we mean that up to a single head each two- way input head of a deterministic finite machine can be simulated by a counter bounded by the input length and vice versa. The condition that the input is bounded cannot be removed in general, since it is known that deterministic finite two-way two-head automata are more powerful than deterministic two- way one counter machines if the input is not bounded [2]. The special case of equivalence between deterministic one counter machines and two-head automata over a single letter alphabet has been shown with the help of a two-dimensional automata model as an intermediate step in [10], see also [11]. -
Programs = Data = First-Class Citizens in a Computational World
PROGRAMS = DATA = FIRST-CLASS CITIZENS IN A COMPUTATIONAL WORLD Lars Hartmann Neil D. Jones Jakob Grue Simonsen + Visualization by Søren Bjerregaard Vrist (All now or recently at the University of Copenhagen) IMDEA and UCM Madrid (March 2012) Sources: I Conference CS2BIO Computer Science to Biology (ENTCS proceedings June 2010) I Journal Scientific Annals of Computer Science (2011, Vol. XXI) I Festschrift for Carolyn Talcott (November 2011, Springer Festschrift Series, LNCS vol. 7000) I Article accepted to appear in Philosophical Transactions A of the Royal Society | 0 | ALAN TURING STARTED THE BALL ROLLING (IN 1936) 1. A convincing analysis of the nature of computation 2. A very early model of computation (MOC) 3. The first programmers' manual 4. Undecidability of the halting problem 5. Universal Turing machine (a self-interpreter) 6. Contributor to the “Confluence of ideas'": that all sensible models of computation are equivalent, e.g., I Turing machine I Lambda calculus I Recursive function definitions I String rewrite systems | 1 | 75 YEARS OF MODELS OF COMPUTATION (just a few) Lambda calculus Church 1936 Turing machine 1936 von Neumann architecture 1945 Finite automata Rabin and Scott Counter machine Lambek and Minsky Random access machine (RAM) Cook and Reckhow Random access stored program (RASP) Elgot and Robinson Cellular automaton, LIFE,.. von Neumann, Conway, Wolfram Abstract state machine Gurevich et al Text register machine Moss Blob model 2010 | 2 | ABOUT MOCS (MODELS OF COMPUTATION) Criteria I How to compare I How to improve I Lacks, failings, inconveniences I What are they suitable for? Programming? Theorem proving ? Modeling ? . New direction: biological computing I Enormous potential (price, concurrency, automation, . -
The Compressed Word Problem for Groups
Markus Lohrey The Compressed Word Problem for Groups – Monograph – April 3, 2014 Springer Dedicated to Lynda, Andrew, and Ewan. Preface The study of computational problems in combinatorial group theory goes back more than 100 years. In a seminal paper from 1911, Max Dehn posed three decision prob- lems [46]: The word problem (called Identitatsproblem¨ by Dehn), the conjugacy problem (called Transformationsproblem by Dehn), and the isomorphism problem. The first two problems assume a finitely generated group G (although Dehn in his paper requires a finitely presented group). For the word problem, the input con- sists of a finite sequence w of generators of G (also known as a finite word), and the goal is to check whether w represents the identity element of G. For the con- jugacy problem, the input consists of two finite words u and v over the generators and the question is whether the group elements represented by u and v are conju- gated. Finally, the isomorphism problem asks, whether two given finitely presented groups are isomorphic.1 Dehns motivation for studying these abstract group theo- retical problems came from topology. In a previous paper from 1910, Dehn studied the problem of deciding whether two knots are equivalent [45], and he realized that this problem is a special case of the isomorphism problem (whereas the question of whether a given knot can be unknotted is a special case of the word problem). In his paper from 1912 [47], Dehn gave an algorithm that solves the word problem for fundamental groups of orientable closed 2-dimensional manifolds. -
Séminaire BOURBAKI Octobre 2018 71E Année, 2018–2019, N 1154 ESPACES ET GROUPES NON EXACTS ADMETTANT UN PLONGEMENT GROSSIER
S´eminaireBOURBAKI Octobre 2018 71e ann´ee,2018{2019, no 1154 ESPACES ET GROUPES NON EXACTS ADMETTANT UN PLONGEMENT GROSSIER DANS UN ESPACE DE HILBERT [d'apr`esArzhantseva, Guentner, Osajda, Spakula]ˇ by Ana KHUKHRO 1. INTRODUCTION The landscape of modern group theory has been shaped by the use of geometry as a tool for studying groups in various ways. The concept of group actions has always been at the heart of the theory, and the idea that one can link the geometry of the space on which the group acts to properties of the group, or that one can view groups as geometric objects themselves, has opened up many possibilities of interaction between algebra and geometry. Given a finitely generated discrete group and a finite generating set of this group, one can construct an associated graph called a Cayley graph. This graph has the set of elements of the group as its vertex set, and two vertices are connected by an edge if one can obtain one from the other by multiplying on the right by an element of the generating set. This gives us a graph on which the group acts by isometries, viewing the graph as a metric space with the shortest path metric. While a different choice of generating set will result in a non-isomorphic graph, the two Cayley graphs will be the same up to quasi-isometry, a coarse notion of equivalence for metric spaces. The study of groups from a geometric viewpoint, geometric group theory, often makes use of large-scale geometric information.