Finite automata, formal logic, and circuit complexity
著者
書誌事項
Finite automata, formal logic, and circuit complexity
（Progress in theoretical computer science）
Birkhäuser, 1994
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注記
Inclides bibliographical references and index
内容説明・目次
- 巻冊次
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: us ISBN 9780817637194
内容説明
The study of the connections between mathematical automata and for mal logic is as old as theoretical computer science itself. In the founding paper of the subject, published in 1936, Turing showed how to describe the behavior of a universal computing machine with a formula of first order predicate logic, and thereby concluded that there is no algorithm for deciding the validity of sentences in this logic. Research on the log ical aspects of the theory of finite-state automata, which is the subject of this book, began in the early 1960's with the work of J. Richard Biichi on monadic second-order logic. Biichi's investigations were extended in several directions. 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. In the more than twenty years that have passed since the appearance of McNaughton and Papert's book, the underlying semigroup theory has grown enor mously, permitting a considerable extension of their results. During the same period, however, fundamental investigations in the theory of finite automata by and large fell out of fashion in the theoretical com puter science community, which moved to other concerns.
目次
I Mathematical Preliminaries.- I.1 Words and Languages.- I.2 Automata and Regular Languages.- I.3 Semigroups and Homomorphisms.- II Formal Languages and Formal Logic.- II.1 Examples.- II.2 Definitions.- III Finite Automata.- III.1 Monadic Second-Order Sentences and Regular Languages.- III.2 Regular Numerical Predicates.- III.3 Infinite Words and Decidable Theories.- IV Model-Theoretic Games.- IV.1 The Ehrenfeucht-Fraisse Game.- IV.2 Application to FO[
- 巻冊次
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: sz ISBN 9783764337193
内容説明
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.
目次
- Part 1 Mathematical preliminaries: words and languages
- automata and regular languages
- semigroups and homomorphisms. Part 2 Formal languages and formal logic: examples
- definitions. Part 3 Finite automata: monadic second-order sentences and regular languages
- regular numerical predicates
- infinite words and decidable theories. Part 4 Model-theoretic games: the Ehrenfeucht-Fraisse game
- application to FO [decreasing]
- application to FO [+1]. Part 5 Finite semigroups: the syntactic monoid
- calculation of the syntactic monoid
- application to FO [decreasing]
- semidirect products
- categories and path conditions
- pseudovarieties. Part 6 First-order logic: characterization of FO [decreasing]
- a hierarchy in FO [decreasing]
- another characterization of FO [+1]
- sentences with regular numerical predicates. Part 7 Modular quantifiers: definition and examples
- languages in (FO + MOD(P))[decreasing]
- languages in (FO + MOD)[+1]
- languages in (FO + MOD)[Reg]
- summary. Part 8 Circuit complexity: examples of circuits
- circuits and circuit complexity classes
- lower bounds. Part 9 Regular languages and circuit complexity: regular languages in NC1
- formulas with arbitrary numerical predicates
- regular languages and non-regular numerical predicates
- special cases of the central conjecture. Appendices: proof of the Krohn-Rhodes theorem
- proofs of the category theorems.
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