Finite Automata, Formal Logic, and Circuit Complexity

Howard Straubing, "Finite Automata, Formal Logic, and Circuit Complexity"
Birkhäuser Boston | 1994-05-03 | ISBN: 0817637192 | 240 pages

The material in this book is situated at the juncture of the automata theory, logic, computational complexity and semigroup theory. The first part of the book is devoted to the algebraic characterization of the regular languages definable in many different logical theories. This includes older results of Büchi on monadic second-order language, and of McNaughton and Papert on first-order logic and star-free languages, as well as many more recent developments that have never been treated in book form. The second part presents the recently-discovered connections between the algebraic theory of the automata and the complexity theory of small-depth circuits.
This self-contained work is suitable for use as a text for a course. Exercises appear at the end of each chapter, some to test knowledge, others to extend the investigation. Citations at the end of each chapter provide a useful guide to related topics mentioned in the text of the chapter. The book will be welcomed by both advanced students and researchers in theoretical computer science, algebra, and mathematical logic.


This work, intended for researchers and advanced students in theoretical computer science and mathematics, is situated at the juncture of automata theory, logic, semigroup theory and computational complexity. The first part focuses on the algebraic characterization of the regular languages definable in many different logical theories. The second part presents the recently-discovered connections between the algebraic theory of automata and the complexity theory of small-depth circuits. The first seven chapters of this text are devoted to the algebraic characterization of the regular languages definable in many different logical theories, obtained by varying both the kinds of quantification and the atomic formulas that are admitted. This includes the results of Buchi and of McNaughton and Papert, as well as more recent developments that are scattered throughout research journals and conference proceedings. The two tables at the end of Chapter 7 summarize most of the imprtant results of this first part of the book. Chapter 8 provides a brief account of the complexity theory of small-depth families of boolean circuits. Chapter 9 aims to tie all the threads together: it shows that questions about the structure of complexity classes of small-depth circuits are precisely equivalent to questions about the definability of regular languages in various versions of first-order logic.

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