Complexity and Randomness in Group Theory
Author: Frédérique Bassino
Publisher: Walter de Gruyter GmbH & Co KG
Published: 2020-06-08
Total Pages: 412
ISBN-13: 3110667525
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Complexity and Randomness in Group Theory
Author: Frédérique Bassino
Publisher: Walter de Gruyter GmbH & Co KG
Published: 2020-06-08
Total Pages: 386
ISBN-13: 3110667029
DOWNLOAD EBOOK →This book shows new directions in group theory motivated by computer science. It reflects the transition from geometric group theory to group theory of the 21st century that has strong connections to computer science. Now that geometric group theory is drifting further and further away from group theory to geometry, it is natural to look for new tools and new directions in group theory which are present.
Non-commutative Cryptography and Complexity of Group-theoretic Problems
Author: Alexei G. Myasnikov
Publisher: American Mathematical Soc.
Published: 2011
Total Pages: 402
ISBN-13: 0821853600
DOWNLOAD EBOOK →Examines the relationship between three different areas of mathematics and theoretical computer science: combinatorial group theory, cryptography, and complexity theory. It explores how non-commutative (infinite) groups can be used in public key cryptography. It also shows that there is remarkable feedback from cryptography to combinatorial group theory because some of the problems motivated by cryptography appear to be new to group theory.
Groups and Model Theory
Author: Olga Kharlampovich
Publisher: Walter de Gruyter GmbH & Co KG
Published: 2021-05-10
Total Pages: 250
ISBN-13: 3110719762
DOWNLOAD EBOOK →This monograph provides an overview of developments in group theory motivated by model theory by key international researchers in the field. Topics covered include: stable groups and generalizations, model theory of nonabelian free groups and of rigid solvable groups, pseudofinite groups, approximate groups, topological dynamics, groups interpreting the arithmetic. The book is intended for mathematicians and graduate students in group theory and model theory. The book follows the course of the GAGTA (Geometric and Asymptotic Group Theory with Applications) conference series. The first book, "Complexity and Randomness in Group Theory. GAGTA book 1," can be found here: http://www.degruyter.com/books/978-3-11-066491-1 .
Kolmogorov Complexity and Algorithmic Randomness
Author: A. Shen
Publisher: American Mathematical Society
Published: 2022-05-18
Total Pages: 511
ISBN-13: 1470470640
DOWNLOAD EBOOK →Looking at a sequence of zeros and ones, we often feel that it is not random, that is, it is not plausible as an outcome of fair coin tossing. Why? The answer is provided by algorithmic information theory: because the sequence is compressible, that is, it has small complexity or, equivalently, can be produced by a short program. This idea, going back to Solomonoff, Kolmogorov, Chaitin, Levin, and others, is now the starting point of algorithmic information theory. The first part of this book is a textbook-style exposition of the basic notions of complexity and randomness; the second part covers some recent work done by participants of the “Kolmogorov seminar” in Moscow (started by Kolmogorov himself in the 1980s) and their colleagues. This book contains numerous exercises (embedded in the text) that will help readers to grasp the material.
Topological and Asymptotic Aspects of Group Theory
Author: R. I. Grigorchuk
Publisher: American Mathematical Soc.
Published: 2006
Total Pages: 248
ISBN-13: 0821837567
DOWNLOAD EBOOK →The articles in this volume are based on the talks given at two special sessions at the AMS Sectional meetings held in 2004. The articles cover various topological and asymptotic aspects of group theory, such as hyperbolic and relatively hyperbolic groups, asymptotic cones, Thompson's group, Nielsen fixed point theory, homology, groups acting on trees, groups generated by finite automata, iterated monodromy groups, random walks on finitely generated groups, heat kernels, and currents on free groups.
Elementary Theory of Groups and Group Rings, and Related Topics
Author: Paul Baginski
Publisher: Walter de Gruyter GmbH & Co KG
Published: 2020-02-10
Total Pages: 272
ISBN-13: 311063838X
DOWNLOAD EBOOK →This proceedings volume documents the contributions presented at the conference held at Fairfield University and at the Graduate Center, CUNY in 2018 celebrating the New York Group Theory Seminar, in memoriam Gilbert Baumslag, and to honor Benjamin Fine and Anthony Gaglione. It includes several expert contributions by leading figures in the group theory community and provides a valuable source of information on recent research developments.
The Complexity Theory Companion
Author: Lane Hemaspaandra
Publisher: Springer Science & Business Media
Published: 2001-12-01
Total Pages: 396
ISBN-13: 9783540674191
DOWNLOAD EBOOK →Here is an accessible, algorithmically oriented guide to some of the most interesting techniques of complexity theory. The book shows that simple algorithms are at the heart of complexity theory. The book is organized by technique rather than by topic. Each chapter focuses on one technique: what it is, and what results and applications it yields.
Forcing with Random Variables and Proof Complexity
Author: Jan Krajíček
Publisher: Cambridge University Press
Published: 2010-12-23
Total Pages: 265
ISBN-13: 1139493922
DOWNLOAD EBOOK →This book introduces a new approach to building models of bounded arithmetic, with techniques drawn from recent results in computational complexity. Propositional proof systems and bounded arithmetics are closely related. In particular, proving lower bounds on the lengths of proofs in propositional proof systems is equivalent to constructing certain extensions of models of bounded arithmetic. This offers a clean and coherent framework for thinking about lower bounds for proof lengths, and it has proved quite successful in the past. This book outlines a brand new method for constructing models of bounded arithmetic, thus for proving independence results and establishing lower bounds for proof lengths. The models are built from random variables defined on a sample space which is a non-standard finite set and sampled by functions of some restricted computational complexity. It will appeal to anyone interested in logical approaches to fundamental problems in complexity theory.
Randomness and Complexity
Author: Cristian S. Calude
Publisher: World Scientific
Published: 2007
Total Pages: 466
ISBN-13: 9812770836
DOWNLOAD EBOOK →The book is a collection of papers written by a selection of eminent authors from around the world in honour of Gregory Chaitin''s 60th birthday. This is a unique volume including technical contributions, philosophical papers and essays. Sample Chapter(s). Chapter 1: On Random and Hard-to-Describe Numbers (902 KB). Contents: On Random and Hard-to-Describe Numbers (C H Bennett); The Implications of a Cosmological Information Bound for Complexity, Quantum Information and the Nature of Physical Law (P C W Davies); What is a Computation? (M Davis); A Berry-Type Paradox (G Lolli); The Secret Number. An Exposition of Chaitin''s Theory (G Rozenberg & A Salomaa); Omega and the Time Evolution of the n-Body Problem (K Svozil); God''s Number: Where Can We Find the Secret of the Universe? In a Single Number! (M Chown); Omega Numbers (J-P Delahaye); Some Modern Perspectives on the Quest for Ultimate Knowledge (S Wolfram); An Enquiry Concerning Human (and Computer!) [Mathematical] Understanding (D Zeilberger); and other papers. Readership: Computer scientists and philosophers, both in academia and industry.
