From the Publisher:
Although it originated as a method to study general dynamical systems, symbolic dynamics is useful in coding for data storage and transmission as well as in linear algebra. Requiring only a undergraduate knowledge of linear algebra, this first general textbook includes over 500 exercises.

An Introduction to Symbolic Dynamics and Coding

Publisher Summary This chapter discusses the several classes of sentence-generating devices that are closely related, in various ways, to the grammars of both natural languages and artificial languages of various kinds. By a language it simply mean a set of strings in some finite set V of symbols called the vocabulary of the language. By a grammar a set of rules that give a recursive enumeration of the strings belonging to the language. It can be said that the grammar generates these strings. The chapter discusses the aspect of the structural description of a sentence, namely, its subdivision into phrases belonging to various categories. A major concern of the general theory of natural languages is to define the class of possible strings; the class of possible grammars; the class of possible structural descriptions; a procedure for assigning structural descriptions to sentences, given a grammar; and to do all of this in such a way that the structural description assigned to a sentence by the grammar of a natural language will provide the basis for explaining how a speaker of this language would understand this sentence.

The Algebraic Theory of Context-Free Languages*

Publisher Summary
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

A famdy of efficient algorithms for referring certain subclasses of the regular languages from fmtte posttwe samples is presented These subclasses are the k-reversible languages, for k = 0, 1, 2, . . . . For each k there is an algorithm for finding the smallest k-reversible language containing any fimte posluve sample. It ts shown how to use this algorithm to do correct identification m the ILmlt of the kreversible languages from posmve data A reversible language is one that Is k-reverstble for some k __ 0. An efficient algonthrn is presented for mfernng reversible languages from posmve and negative examples, and it is shown that it leads to correct identification m the hmlt of the class of reversible languages. Numerous examples are gtven to dlustrate the algorithms and their behawor

Inference of Reversible Languages

A probabilistic or stochastic automaton (pa) is a device with a finite number of internal states that scans input words over a finite alphabet and responds by successively changing its state in a probabilistic way.

Probabilistic automata

A definite automaton is, roughly speaking, an automaton (sequential circuit) with the property that for some fixed integer k its action depends only on the last k inputs. The notion of a definite event introduced by Kleene, as well as the related concepts of definite automata and tables, are studied here in detail. Basic results relating to the minimum number of states required for synthesizing an automaton of a given degree of definiteness are proved. We give a characterization of all k-definite events definable by k+1 state automata. Various decision problems pertaining to definite automata are effectively solved. We also solve effectively the problem of synthesizing a minimal automaton defining a given definite event. The solutions of decision and synthesis problems given here are practical in the sense that if the problem is presented by n units of information, then the algorithm in question requires about n3 steps of a very elementary nature (rather than requiring about 2n steps as some algorithms for automata do, which puts them beyond the capacity of the largest computers even for relatively small values of n). A notion of equivalence of definite events is introduced and the uniqueness of the minimal automaton defining an event in an equivalence class is proved.

M. O. Rabin

Papers

The Theory of Definite Automata