Combinatorics Automata and Number Theory

Author: Valérie Berthé
Publisher: Cambridge University Press
ISBN: 0521515971
Format: PDF, Mobi
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This series is devoted to significant topics or themes that have wide application in mathematics or mathematical science and for which a detailed development of the abstract theory is less important than a thorough and concrete exploration of the implications and applications. Books in the Encyclopedia of Mathematics and its Applications cover their subjects comprehensively. Less important results may be summarised as exercises at the ends of chapters, For technicalities, readers can be referred to the bibliography, which is expected to be comprehensive. As a result, volumes are encyclopedic references or manageable guides to major subjects.

Sequences Groups and Number Theory

Author: Valérie Berthé
Publisher: Birkhäuser
ISBN: 331969152X
Format: PDF, Kindle
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This collaborative book presents recent trends on the study of sequences, including combinatorics on words and symbolic dynamics, and new interdisciplinary links to group theory and number theory. Other chapters branch out from those areas into subfields of theoretical computer science, such as complexity theory and theory of automata. The book is built around four general themes: number theory and sequences, word combinatorics, normal numbers, and group theory. Those topics are rounded out by investigations into automatic and regular sequences, tilings and theory of computation, discrete dynamical systems, ergodic theory, numeration systems, automaton semigroups, and amenable groups. This volume is intended for use by graduate students or research mathematicians, as well as computer scientists who are working in automata theory and formal language theory. With its organization around unified themes, it would also be appropriate as a supplemental text for graduate level courses.

Developments in Language Theory

Author: Marie-Pierre Beal
Publisher: Springer
ISBN: 3642387713
Format: PDF, Mobi
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This book constitutes the proceedings of the 17th International Conference on Developments in Language Theory, DLT 2013, held in Marne-la-Vallée, France, in June 2013. The 34 full papers presented in this volume were carefully reviewed and selected from 63 submissions. The scope of the conference includes, among others, the following topics and areas: combinatorial and algebraic properties of words and languages; grammars, acceptors and transducers for strings, trees, graphs, arrays; algebraic theories for automata and languages; codes; efficient text algorithms; symbolic dynamics; decision problems; relationships to complexity theory and logic; picture description and analysis; polyominoes and bidimensional patterns; cryptography; concurrency; cellular automata; bio-inspired computing; and quantum computing.

Algebraic Informatics

Author: Traian Muntean
Publisher: Springer
ISBN: 3642406637
Format: PDF
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This book constitutes the refereed proceedings of the 5th International Conference on Algebraic Informatics, CAI 2013, held in Porquerolles, France in September 2013. The 19 revised full papers presented together with 5 invited articles were carefully reviewed and selected from numerous submissions. The papers cover topics such as data models and coding theory; fundamental aspects of cryptography and security; algebraic and stochastic models of computing; logic and program modelling.

Formal Languages Automata and Numeration Systems

Author: Michel Rigo
Publisher: John Wiley & Sons
ISBN: 1119008220
Format: PDF, ePub
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Formal Languages, Automaton and Numeration Systems presents readers with a review of research related to formal language theory, combinatorics on words or numeration systems, such as Words, DLT (Developments in Language Theory), ICALP, MFCS (Mathematical Foundation of Computer Science), Mons Theoretical Computer Science Days, Numeration, CANT (Combinatorics, Automata and Number Theory). Combinatorics on words deals with problems that can be stated in a non-commutative monoid, such as subword complexity of finite or infinite words, construction and properties of infinite words, unavoidable regularities or patterns. When considering some numeration systems, any integer can be represented as a finite word over an alphabet of digits. This simple observation leads to the study of the relationship between the arithmetical properties of the integers and the syntactical properties of the corresponding representations. One of the most profound results in this direction is given by the celebrated theorem by Cobham. Surprisingly, a recent extension of this result to complex numbers led to the famous Four Exponentials Conjecture. This is just one example of the fruitful relationship between formal language theory (including the theory of automata) and number theory. Contents to include: • algebraic structures, homomorphisms, relations, free monoid • finite words, prefixes, suffixes, factors, palindromes • periodicity and Fine–Wilf theorem • infinite words are sequences over a finite alphabet • properties of an ultrametric distance, example of the p-adic norm • topology of the set of infinite words • converging sequences of infinite and finite words, compactness argument • iterated morphism, coding, substitutive or morphic words • the typical example of the Thue–Morse word • the Fibonacci word, the Mex operator, the n-bonacci words • wordscomingfromnumbertheory(baseexpansions,continuedfractions,...) • the taxonomy of Lindenmayer systems • S-adic sequences, Kolakoski word • repetition in words, avoiding repetition, repetition threshold • (complete) de Bruijn graphs • concepts from computability theory and decidability issues • Post correspondence problem and application to mortality of matrices • origins of combinatorics on words • bibliographic notes • languages of finite words, regular languages • factorial, prefix/suffix closed languages, trees and codes • unambiguous and deterministic automata, Kleene’s theorem • growth function of regular languages • non-deterministic automata and determinization • radix order, first word of each length and decimation of a regular language • the theory of the minimal automata • an introduction to algebraic automata theory, the syntactic monoid and the syntactic complexity • star-free languages and a theorem of Schu ̈tzenberger • rational formal series and weighted automata • context-free languages, pushdown automata and grammars • growth function of context-free languages, Parikh’s theorem • some decidable and undecidable problems in formal language theory • bibliographic notes • factor complexity, Morse–Hedlund theorem • arithmetic complexity, Van Der Waerden theorem, pattern complexity • recurrence, uniform recurrence, return words • Sturmian words, coding of rotations, Kronecker’s theorem • frequencies of letters, factors and primitive morphism • critical exponent • factor complexity of automatic sequences • factor complexity of purely morphic sequences • primitive words, conjugacy, Lyndon word • abelianisation and abelian complexity • bibliographic notes • automatic sequences, equivalent definitions • a theorem of Cobham, equivalence of automatic sequences with constant length morphic sequences • a few examples of well-known automatic sequences • about Derksen’s theorem • some morphic sequences are not automatic • abstract numeration system and S-automatic sequences • k − ∞-automatic sequences • bibliographic notes • numeration systems, greedy algorithm • positional numeration systems, recognizable sets of integers • divisibility criterion and recognizability of N • properties of k-recognizable sets of integers, ratio and difference of consec- utive elements: syndeticity • integer base and Cobham’s theorem on the base dependence of the recog- nizability • non-standard numeration systems based on sequence of integers • linear recurrent sequences, Loraud and Hollander results • Frougny’s normalization result and addition • morphic numeration systems/sets of integers whose characteristic sequence is morphic • towards a generalization of Cobham’s theorem • a few words on the representation of real numbers, β-integers, finiteness properties • automata associated with Parry numbers and numeration systems • bibliographic notes First order logic • Presburger arithmetic and decidable theory • Muchnik’s characterization of semi-linear sets • Bu ̈chi’s theorem: k-recognizable sets are k-definable • extension to Pisot numeration systems • extension to real numbers • decidability issues for numeration systems • applications in combinatorics on words

Combinatorics Words and Symbolic Dynamics

Author: Valérie Berthé
Publisher: Cambridge University Press
ISBN: 1316462528
Format: PDF, ePub
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Internationally recognised researchers look at developing trends in combinatorics with applications in the study of words and in symbolic dynamics. They explain the important concepts, providing a clear exposition of some recent results, and emphasise the emerging connections between these different fields. Topics include combinatorics on words, pattern avoidance, graph theory, tilings and theory of computation, multidimensional subshifts, discrete dynamical systems, ergodic theory, numeration systems, dynamical arithmetics, automata theory and synchronised words, analytic combinatorics, continued fractions and probabilistic models. Each topic is presented in a way that links it to the main themes, but then they are also extended to repetitions in words, similarity relations, cellular automata, friezes and Dynkin diagrams. The book will appeal to graduate students, research mathematicians and computer scientists working in combinatorics, theory of computation, number theory, symbolic dynamics, tilings and stringology. It will also interest biologists using text algorithms.

Advanced Graph Theory and Combinatorics

Author: Michel Rigo
Publisher: John Wiley & Sons
ISBN: 1119058643
Format: PDF, ePub, Mobi
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Advanced Graph Theory focuses on some of the main notions arising in graph theory with an emphasis from the very start of the book on the possible applications of the theory and the fruitful links existing with linear algebra. The second part of the book covers basic material related to linear recurrence relations with application to counting and the asymptotic estimate of the rate of growth of a sequence satisfying a recurrence relation.

Recurrence Sequences

Author: Graham Everest
Publisher: American Mathematical Soc.
ISBN: 1470423154
Format: PDF, Kindle
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Recurrence sequences are of great intrinsic interest and have been a central part of number theory for many years. Moreover, these sequences appear almost everywhere in mathematics and computer science. This book surveys the modern theory of linear recurrence sequences and their generalizations. Particular emphasis is placed on the dramatic impact that sophisticated methods from Diophantine analysis and transcendence theory have had on the subject. Related work on bilinear recurrences and an emerging connection between recurrences and graph theory are covered. Applications and links to other areas of mathematics are described, including combinatorics, dynamical systems and cryptography, and computer science. The book is suitable for researchers interested in number theory, combinatorics, and graph theory.

Oriented Matroids

Author: Anders Björner
Publisher: Cambridge University Press
ISBN: 9780521777506
Format: PDF, Kindle
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First comprehensive, accessible account; second edition has expanded bibliography and a new appendix surveying recent research.