Automata-theoretic aspects of formal power series
Arto Salomaa, Matti Soittola (Author)
This book develops a theory of formal power series in noncommuting variables, the main emphasis being on results applicable to automata and formal language theory. This theory was initiated around 196O-apart from some scattered work done earlier in connection with free groups-by M. P. Schutzenberger to whom also belong some of the main results. So far there is no book in existence concerning this theory. This lack has had the unfortunate effect that formal power series have not been known and used by theoretical computer scientists to the extent they in our estimation should have been. As with most mathematical formalisms, the formalism of power series is capable of unifying and generalizing known results. However, it is also capable of establishing specific results which are difficult if not impossible to establish by other means. This is a point we hope to be able to make in this book. That formal power series constitute a powerful tool in automata and language theory depends on the fact that they in a sense lead to the arithmetization of automata and language theory. We invite the reader to prove, for instance, Theorem IV. 5. 3 or Corollaries III. 7. 8 and III. 7.- all specific results in language theory-by some other means. Although this book is mostly self-contained, the reader is assumed to have some background in algebra and analysis, as well as in automata and formal language theory
x, 171 pages ; 25 cm
9780387902821, 9783540902829, 0387902821, 3540902821
3706786
I. Introduction
I.1. Preliminaries from algebra and analysis
I.2. Preliminaries from automata and formal language theory
I.3. Formal power series in noncommuting variables
II. Rational series
II.1. Rational series and linear systems
II.2. Recognizable series
II.3. Hankel matrices
II.4. Operations preserving rationality
II.5. Regular languages and rational series
II.6. Fatou properties
II.7. On rational series with real coefficients
II.8. On positive series
II.9. Rational sequences
II.10. Positive sequences
II.11. On series in product monoids
II.12. Decidability questions
III. Applications of rational series
III.1. On rational transductions
III.2. Families of rational languages
III.3. Rational series and stochastic automata
III.4. On stochastic languages
III.5. On one-letter stochastic languages
III.6. Densities of regular languages
III.7. Growth functions of L systems: characterization results
III.8. Growth functions of L systems: decidability
IV. Algebraic series and context-free languages
IV.1. Proper algebraic systems of equations
IV.2. Reduction theorems
IV.3. Closure properties
IV.4. Theorems of Shamir and Chomsky-Schiitzenberger
IV.5. Commuting variables and decidability
IV.6. Generalizations of proper systems. Fatou extensions
IV.7. Algebraic transductions
Historical and bibliographical remarks
References