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Full Description
This book is a systematic exposition of the theory of ratio- nal formal power series and the associated formal languages. It is the first to presentan algebraic approach. It con- tains all fundamental results, starting with the Kleene- Sch}tzenberger theorem, up to the latest developments. Re- lations with number theory and the theory of codes are em- phasized.
Contents
I. Rational Series.- 1 Semirings.- 2 Formal Series.- 3 The Topology of K"X".- 4 Rational Series.- 5 Recognizable Series.- 6 The Fundamental Theorem.- Exercises for Chapter I.- Notes to Chapter I.- II. Minimization.- 1 Syntactic Ideals.- 2 Reduced Linear Representations.- 3 The Reduction Algorithm.- Exercises for Chapter II.- Notes to Chapter II.- III. Series and Languages.- 1 The Theorem of Kleene.- 2 Series and Rational Languages.- 3 Supports.- 4 Iteration.- 5 Complementation.- Exercises for Chapter III.- Notes to Chapter III.- IV. Rational Series in One Variable.- 1 Rational Functions.- 2 The Exponential Polynomial.- 3 A Theorem of Polya.- 4 A Theorem of Skolem, Mahler and Lech.- Notes to Chapter IV.- V. Changing the Semiring.- 1 Rational Series over a Principal Ring.- 2 Positive Rational Series.- 3 Fatou Extensions.- Exercises for Chapter V.- Notes to Chapter V.- VI. Decidability.- 1 Problems of Supports.- 2 Growth.- Exercises for Chapter VI.- Notes to Chapter VI.- VII. Noncommutative Polynomials.- 1 The Weak Algorithm.- 2 Continuant Polynomials.- 3 Inertia.- 4 Gauss's Lemma.- Exercises for Chapter VII.- Notes to Chapter VII.- VIII. Codes and Formal Series.- 1 Codes.- 2 Completeness.- 3 The Degree of a Code.- 4 Factorization.- Exercises for Chapter VIII.- Notes to Chapter VIII.- References.
