形式语言与自动机导论：第六版英文原版

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"形式语言与自动机导论，第六版，英文原版计算机专业教材，作者PETER LINZ，由Jones&Bartlett Learning出版。" 在计算机科学领域，形式语言与自动机是理论计算机科学的基础部分，它研究的是如何用数学模型来描述和处理语言。这本书的第六版，"An Introduction to Formal Languages and Automata" 是一个深入浅出的教程，旨在帮助学生和专业人士理解这一领域的核心概念。形式语言是用特定的数学结构，如正规表达式、上下文无关文法或正则文法来表示的一类语言。它们在编译原理、数据压缩、模式识别、网络协议解析等领域有着广泛应用。书中会详细讲解如何定义和操作这些语言，以及它们的性质和分类。自动机，另一方面，是一种抽象计算模型，如确定性和非确定性有限状态自动机（DFA和NFA）、下推自动机（PDA）和图灵机（Turing Machine）。这些模型模拟了不同的计算过程，并且对应着不同级别的语言识别能力。例如，DFA可以识别正规集，而PDA可以识别上下文无关语言，图灵机则是通用计算模型，能模拟任何可计算的函数。此书可能涵盖以下主题： 1. 形式语言的基本概念：包括正规集、正规表达式、正规文法的构造和转换。 2. 有限状态自动机：详细阐述DFA和NFA的构造、等价性、最小化等问题。 3. 上下文无关文法和语言：介绍Chomsky层次，以及如何将文法转化为规范形式，如CNF（Chomsky Normal Form）。 4. 下推自动机（PDA）和上下文有关语言：探讨PDA的运行机制，以及与正规语言和上下文无关语言的关系。 5. 图灵机与计算理论：介绍图灵机的工作原理，以及停机问题、递归可枚举集合等概念。 6. 语言的包含关系和等价测试：如Myhill-Nerode定理和Parikh定理的应用。 7. 形式语言的复杂度分析：讨论识别语言的算法复杂度和计算复杂性理论。此外，该教材还可能包含丰富的例题、习题和实验，以帮助读者巩固理解并提升实践能力。对于学习者来说，这是一本理想的入门教材，它提供了从基础到高级的全面介绍，有助于建立坚实的理论基础，为进一步学习编译原理、算法分析、计算理论等高级课程打下坚实的基础。

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CHAPTER

INTRODUCTION TO THE

THEORY OF COMPUTATION

CHAPTER SUMMARY

This chapter prepares you for what is to come. In Section 1.1, we review

some of the main ideas from finite mathematics and establish the no-

tation used in the text. Particularly important are the proof techniques,

especially proof by induction and proof by contradiction.

Section 1.2 gives you a first look at the ideas that are at the core of

the book: automata, languages, grammars, their definitions, and their

relations. In subsequent chapters, we will expand these ideas and study

a number of different types of automata and grammars.

In Section 1.3, we encounter some simple examples of the role these

concepts play in computer science, particularly in programming lan-

guages, digital design, and text processing. We do not belabor this

issue here, as you will encounter applications of these concepts in a

number of other CS courses.

he subject matter of this book, the theory of computation, includes

several topics: automata theory, formal languages and grammars,

computability, and complexity. Together, this material constitutes

the theoretical foundation of computer science. Loosely speaking we can

think of automata, grammars, and computability as the study of what can

be done by computers in principle, while complexity addresses what can

be done in practice. In this book we focus almost entirely on the ﬁrst of

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2 Chapter 1 Introduction to the Theory of Computation

these concerns. We will study various automata, see how they are related

to languages and grammars, and investigate what can and cannot be done

by digital computers. Although this theory has many uses, it is inherently

abstract and mathematical.

Computer science is a practical discipline. Those who work in it often

have a marked preference for useful and tangible problems over theoretical

speculation. This is certainly true of computer science students who are

concerned mainly with diﬃcult applications from the real world. Theoreti-

cal questions interest them only if they help in ﬁnding good solutions. This

attitude is appropriate, since without applications there would be little in-

terest in computers. But given this practical orientation, one might well

ask “why study theory?”

The ﬁrst answer is that theory provides concepts and principles that

help us understand the general nature of the discipline. The ﬁeld of com-

puter science includes a wide range of special topics, from machine design

to programming. The use of computers in the real world involves a wealth

of speciﬁc detail that must be learned for a successful application. This

makes computer science a very diverse and broad discipline. But in spite

of this diversity, there are some common underlying principles. To study

these basic principles, we construct abstract models of computers and com-

putation. These models embody the important features that are common

to both hardware and software and that are essential to many of the special

and complex constructs we encounter while working with computers. Even

when such models are too simple to be applicable immediately to real-world

situations, the insights we gain from studying them provide the foundation

on which speciﬁc development is based. This approach is, of course, not

unique to computer science. The construction of models is one of the es-

sentials of any scientiﬁc discipline, and the usefulness of a discipline is often

dependent on the existence of simple, yet powerful, theories and laws.

A second, and perhaps not so obvious, answer is that the ideas we will

discuss have some immediate and important applications. The ﬁelds of

digital design, programming languages, and compilers are the most obvious

examples, but there are many others. The concepts we study here run

like a thread through much of computer science, from operating systems to

pattern recognition.

The third answer is one of which we hope to convince the reader. The

subject matter is intellectually stimulating and fun. It provides many chal-

lenging, puzzle-like problems that can lead to some sleepless nights. This is

problem solving in its pure essence.

In this book, we will look at models that represent features at the

core of all computers and their applications. To model the hardware of a

computer, we introduce the notion of an automaton (plural, automata).

An automaton is a construct that possesses all the indispensable features

of a digital computer. It accepts input, produces output, may have some

temporary storage, and can make decisions in transforming the input into

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1.1 Mathematical Preliminaries and Notation 3

the output. A formal language is an abstraction of the general charac-

teristics of programming languages. A formal language consists of a set

of symbols and some rules of formation by which these symbols can be

combined into entities called sentences. A formal language is the set of all

sentences permitted by the rules of formation. Although some of the for-

mal languages we study here are simpler than programming languages, they

have many of the same essential features. We can learn a great deal about

programming languages from formal languages. Finally, we will formalize

the concept of a mechanical computation by giving a precise deﬁnition of

the term algorithm and study the kinds of problems that are (and are

not) suitable for solution by such mechanical means. In the course of our

study, we will show the close connection between these abstractions and

investigate the conclusions we can derive from them.

In the ﬁrst chapter, we look at these basic ideas in a very broad way to

set the stage for later work. In Section 1.1, we review the main ideas from

mathematics that will be required. While intuition will frequently be our

guide in exploring ideas, the conclusions we draw will be based on rigorous

arguments. This will involve some mathematical machinery, although the

requirements are not extensive. The reader will need a reasonably good

grasp of the terminology and of the elementary results of set theory, func-

tions, and relations. Trees and graph structures will be used frequently,

although little is needed beyond the deﬁnition of a labeled, directed graph.

Perhaps the most stringent requirement is the ability to follow proofs and

an understanding of what constitutes proper mathematical reasoning. This

includes familiarity with the basic proof techniques of deduction, induc-

tion, and proof by contradiction. We will assume that the reader has this

necessary background. Section 1.1 is included to review some of the main

results that will be used and to establish a notational common ground for

subsequent discussion.

In Section 1.2, we take a ﬁrst look at the central concepts of languages,

grammars, and automata. These concepts occur in many speciﬁc forms

throughout the book. In Section 1.3, we give some simple applications of

these general ideas to illustrate that these concepts have widespread uses

in computer science. The discussion in these two sections will be intuitive

rather than rigorous. Later, we will make all of this much more precise; but

for the moment, the goal is to get a clear picture of the concepts with which

we are dealing.

1.1

MATHEMATICAL PRELIMINARIES AND NOTATION

Sets

A set is a collection of elements, without any structure other than mem-

bership. To indicate that x is an element of the set S, we write x ∈ S.

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