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This chapter presents the field of finite automata, which is a branch of mathematics connected with the algebraic theory of semigroups and associative algebras. From another viewpoint, finite automata are a branch of algorithm design concerned with string manipulation and sequence processing. The chapter presents the basic definitions and a proof of the fundamental result of S. C. Kleene. The chapter discusses the notion of star-height, which is one of the fast developing areas of automata theory. The chapter discusses star-free sets, which is an important notion especially because of its connection with logic. The chapter presents a complete proof of P. Schutzenberger's theorem, stating the equality between star-free sets and aperiodic sets. It also presents the syntactic characterization of two important subfamilies of star-free sets, namely, locally testable and piecewise testable. The chapter describes the applications of finite automata, such as string matching or file indexing. The chapter introduces the field of automata recognizing numbers expanded at some basis. This is an aspect of finite automata regarding several fields in classical mathematics, such as number theory and ergodic theory.

CHAPTER 1 – Finite Automata

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Computability and Unsolvability.

Formal models of language learning.

Structural properties, local and asymptotic, of members of a class of simple, realtime generable sequences are analyzed.

Uniform tag sequences

Preface to the Second Edition. Preface to the First Edition. The State of American Linguistics in the Mid 1950s. The Chomskyan Revolution. From Syntactic Structures to Aspects of the Theory of Syntax. The Rise of Generative Semantics. The Linguistic Wars. The Extended Standard Theory. The New Consensus and the New Rift in Generative Syntax. Current Approaches to Syntax. References. Name Index. Subject Index.

Linguistic theory in America

On the generative power of transformational grammars

A note on the universal base hypothesis

The class of sets acceptable by finite automata has been studied ext.ensively in the recent literature. A characterization of this class in terms of weak second order arithmetic [1, 2] together with undeeidability results in [4] shows that the set of squares of non-negative integers is not, an acceptable set (see [1, p. 81]). Another characterization of the acceptable sets given, for example, in [5] is purely combinatorial and avoids the use of mathematical logic. However, it is not easy to see how to draw conclusions about common number-theoretic sets from this alternate characterization. This note proceeds directly from the definitions of finite automata and of acceptable sets to show that the set of squares is not acceptable. (The reader may note that the same argument in fact shows tha t the set of odd squares is also not acceptable, tha t a similar argument will show tha t the set of even squares is not acceptable, and tha t these results can be combined to show that such sets as the set of numbers of the form z (z -a) for aay fixed a are not acceptable.) The main result hinges on the lemma in Section 3 which relates squares and powers of 2. This lemma and the simplicity of its proof may be of independent interest since elosely related problems seem to require much more powerful methods for their solution (see, for example, [3] and [6, Theorem 5]).

Finite Automata and the Set of Squares

The ability of context-sensitive grammars togenerate non-context-free languages is well known. However, phrase-structure rules are often used in both natural and artificial languages, not to generate sentences, but rather toanalyze or parse given putative sentences. Linguistic arguments have been advanced that this is the more fruitful use of CS rules for natural languages, and that, further, it is the purported phrasestructure tree which is presented and analyzed, rather than merely the terminal string itself. In this interpretation, for example the rules {S → AB, A → a/__b, B → b/ a__} “parse” the sentenceab for they “analyze” the tree represented by the labeled bracketing [S[Aa]A[Bb]B]S, even though these rules do not generateab when interpreted as a CS grammar. We shall say that a languageL is parsed by the finite setR of context-sensitive rules ifL consists of the terminal strings of trees analyzed byR. An actual parsing of an artificial language might be obtained, for example, by using a standard “parsing algorithm” for the associated CF grammar (obtained by deleting all contexts from rules) to generate putative trees and then analyzing these trees with the given CS grammar. In this paper, a language is shown to becontext-free if and only if there is a finite set ofcontext-sensitive rules which parse this language; i.e., if and only if there is a collection of trees whose terminal strings are the sentences of this language and a finite set of CS rules which analyze exactly these trees.

Context-sensitive immediate constituent analysis: Context-free languages revisited

Strong representability of partial functions in arithmetic theories

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