Finite Automata, Formal Logic, and Circuit Complexity

One of these, explored by McNaughton and Papert in their 1971 monograph Counter-free Automata, was the characterization of automata that admit first-order behavioral descriptions, in terms of the semigroup­ theoretic approach to automata that had recently been developed in the work of Krohn and Rhodes and of Schiitzenberger. 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 important 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.