Codes and Automata (Encyclopedia of Mathematics and its Applications, Series Number 129)

This major revision of Berstel and Perrin's classic Theory of Codes has been rewritten with a more modern focus and a much broader coverage of the subject. The concept of unambiguous automata, which is intimately linked with that of codes, now plays a significant role throughout the book, reflecting developments of the last 20 years. This is complemented by a discussion of the connection between codes and automata, and new material from the field of symbolic dynamics. The authors have also explored links with more practical applications, including data compression and cryptography. The treatment remains self-contained: there is background material on discrete mathematics, algebra and theoretical computer science. The wealth of exercises and examples make it ideal for self-study or courses. In summary, this is a comprehensive reference on the theory of variable-length codes and their relation to automata. This Major Revision Of Berstel And Perrin's Classic Theory Of Codes Has Been Rewritten With A More Modern Focus And A Much Broader Coverage Of The Subject. The Concept Of Unambiguous Automata, Which Is Intimately Linked With That Of Codes, Now Plays A Significant Role Throughout The Book, Reflecting Developments Of The Last 20 Years. This Is Complemented By A Discussion Of The Connection Between Codes And Automata, And New Material From The Field Of Symbolic Dynamics. The Authors Have Also Explored Links With More Practical Applications, Including Data Compression And Cryptography. The Treatment Remains Self-contained: There Is Background Material On Discrete Mathematics, Algebra And Theoretical Computer Science. The Wealth Of Exercises And Examples Make It Ideal For Self-study Or Courses. In Sum This Is A Comprehensive Reference On The Theory Of Variable-length Codes And Their Relation To Automata. Preliminaries -- Codes -- Prefix Codes -- Automata -- Deciphering Delay -- Bifix Codes -- Circular Codes -- Factorizations Of Free Monoids -- Unambiguous Monoids Of Relations -- Synchronization -- Groups Of Codes -- Factorizations Of Cyclic Groups -- Densities -- Polynomials Of Finite Codes. Jean Berstel, Dominique Perrin, Christophe Reutenauer. Includes Bibliographical References And Indexes. Cover; Title Page; Copyright; Contents; Preface; 1 Preliminaries; 1.1 Notation; 1.2 Monoids; 1.3 Words; 1.4 Automata; 1.5 Transducers; 1.6 Semirings and matrices; 1.7 Formal series; 1.8 Power series; 1.9 Nonnegative matrices; 1.10 Weighted automata; 1.11 Probability distributions; 1.12 Ideals in a monoid; 1.13 Permutation groups; 1.14 Notes; 2 Codes; 2.1 Definitions; 2.2 Codes and free submonoids; 2.3 A test for codes; 2.4 Codes and Bernoulli distributions; 2.5 Complete sets; 2.6 Composition; 2.7 Prefix graph of a code; 2.8 Exercises; 2.9 Notes; 3 Prefix codes; 3.1 Prefix codes; 3.2 Automata 3.3 Maximal prefix codes3.4 Operations on prefix codes; 3.5 Semaphore codes; 3.6 Synchronized codes; 3.7 Recurrent events; 3.8 Length distributions; 3.9 Optimal prefix codes; 3.10 Exercises; 3.11 Notes; 4 Automata; 4.1 Unambiguous automata; 4.2 Flower automaton; 4.3 Decoders; 4.4 Exercises; 4.5 Notes; 5 Deciphering delay; 5.1 Deciphering delay; 5.2 Maximal codes; 5.3 Weakly prefix codes; 5.4 Exercises; 5.5 Notes; 6 Bifix codes; 6.1 Basic properties; 6.2 Maximal bifix codes; 6.3 Degree; 6.4 Kernel; 6.5 Finite maximal bifix codes; 6.6 Completion; 6.7 Exercises; 6.8 Notes; 7 Circular codes 10.5 Exercises10.6 Notes; 11 Groups of codes; 11.1 Groups and composition; 11.2 Synchronization of semaphore codes; 11.3 Group codes; 11.4 Automata of bifix codes; 11.5 Depth; 11.6 Groups of finite bifix codes; 11.7 Examples; 11.8 Exercises; 11.9 Notes; 12 Factorizations of cyclic groups; 12.1 Factorizations of cyclic groups; 12.2 Bayonets; 12.3 Hooks; 12.4 Exercises; 12.5 Notes; 13 Densities; 13.1 Probability; 13.2 Densities; 13.3 Entropy; 13.4 Probabilities over a monoid; 13.5 Strict contexts; 13.6 Exercises; 13.7 Notes; 14 Polynomials of finite codes; 14.1 Positive factorizations 7.1 Circular codes7.2 Limited codes; 7.3 Length distributions; 7.4 Exercises; 7.5 Notes; 8 Factorizations of free monoids; 8.1 Factorizations; 8.2 Finite factorizations; 8.3 Exercises; 8.4 Notes; 9 Unambiguous monoids of relations; 9.1 Unambiguous monoids of relations; 9.2 The Schützenberger representations; 9.3 Rank and minimal ideal; 9.4 Very thin codes; 9.5 Group and degree of a code; 9.6 Interpretations; 9.7 Exercises; 9.8 Notes; 10 Synchronization; 10.1 Synchronizing pairs; 10.2 Uniformly synchronized codes; 10.3 Locally parsable codes and local automata; 10.4 Road coloring 14.2 The factorization theorem14.3 Noncommutative polynomials; 14.4 Proof of the factorization theorem; 14.5 Applications; 14.6 Commutative equivalence; 14.7 Complete reducibility; 14.8 Exercises; 14.9 Notes; Solutions of exercises; Appendix: Research problems; References; Index of notation; Index A major revision of Berstel and Perrin's classic Theory of Codes, this book has been written with a more modern focus and a much broader coverage of the subject. It is a comprehensive reference on the theory of variable-length codes and their relation to automata.