The subject of this chapter is the study of formal languages (mostly languages recognizable by finite automata) in the framework of mathematical logic.

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Languages, automata, and logic

On uniformity within NC 1

Discoveries in finite semigroups have influenced several mathematical fields, including theoretical computer science, tropical algebra via matrix theory with coefficients in semirings, and other areas of modern algebra. This comprehensive, encyclopedic text will provide the reader - from the graduate student to the researcher/practitioner with a detailed understanding of modern finite semigroup theory, focusing in particular on advanced topics on the cutting edge of research. Key features: (1) Develops q-theory, a new theory that provides a unifying approach to finite semigroup theory via quantization; (2) Contains the only contemporary exposition of the complete theory of the complexity of finite semigroups; (3) Introduces spectral theory into finite semigroup theory; (4) Develops the theory of profinite semigroups from first principles, making connections with spectra of Boolean algebras of regular languages; (5) Presents over 70 research problems, most new, and hundreds of exercises. Additional features: (1) For newcomers, an appendix on elementary finite semigroup theory; (2) Extensive bibliography and index. The q-theory of Finite Semigroups presents important techniques and results, many for the first time in book form, and thereby updates and modernizes the literature of semigroup theory.

The q-theory of Finite Semigroups

1 Abstract In order to study circuit complexity classes within NC 1 in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC 1 uniformity Ru81,Co85], have appeared in recent research: Immerman's families of circuits deened by rst-order formulas Im87a,Im87b] and a uniformity corresponding to Buss' deterministic log-time reductions Bu87]. We show that these two notions are equivalent, leading to a natural notion of uniformity for low-level circuit complexity classes. We show that recent results on the structure of NC 1 Ba89] still hold true in this very uniform setting. Finally, we investigate a parallel notion of uniformity, still more restrictive, based on the regular languages. Here we give characterizations of sub-classes of the regular languages based on their logical expressibility, extending recent work of Straubing, Th erien, and Thomas STT88]. A preliminary version of this work appeared as BIS88]. 2 Introduction 2.1 Circuit Complexity Computer scientists have long tried to classify problems (deened as Boolean predicates or functions) by the size or depth of Boolean circuits needed to solve them. This eeort has

On uniformity within NC 1 .

Preface 1. Preliminaries 2. Codes 3. Prefix codes 4. Automata 5. Deciphering delay 6. Bifix codes 7. Circular codes 8. Factorizations of free monoids 9. Unambiguous monoids of relations 10. Synchronization 11. Groups of codes 12. Factorizations of cyclic groups 13. Densities 14. Polynomials of finite codes Solutions of exercises Appendix: Research problems References Index of notation Index.

Raymond E. Miller

Papers

Varieties of Formal Languages