林兹第五版：计算机科学入门 - 形式语言与自动机

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《形式语言与自动机：第五版》是由彼得·林兹所著的一本经典教材，专为计算机科学和计算机工程专业的二年级和三年级学生设计，旨在介绍形式语言、自动机、计算理论等核心概念。这门课程在计算机科学的教学大纲中占据着重要地位，因为它为学生们提供了早期接触理论计算的基础。本书覆盖了形式语言理论的各个方面，包括语言的定义、构造、性质和分类，以及自动机（如确定型自动机、非确定型自动机、有限自动机、无限自动机等）的工作原理和应用。形式语言是描述符号序列的规则集合，而自动机则是用来识别这些语言的机器模型，它们在计算机科学中扮演着理解输入数据结构和执行决策的关键角色。书中探讨了如何通过构造正规表达式来描述特定的语言，以及如何使用图灵机等抽象模型来衡量一个系统的计算能力。此外，读者还将学习如何将实际问题转化为形式语言和自动机的问题，以便于设计算法和构建计算机程序。本书采用清晰易懂的方式阐述复杂的理论概念，配以丰富的例题和练习，帮助学生逐步掌握理论并将其应用到实践中。通过阅读和学习本书，学生不仅能够建立起坚实的理论基础，而且还能培养解决问题的逻辑思维和抽象建模能力，这对于后续的软件开发、系统分析和理论研究都至关重要。总结来说，《形式语言与自动机：第五版》是一本深入浅出的教材，适合对计算机科学有兴趣的学生深入理解形式化的抽象思维，以及那些希望在自动机理论和计算复杂性等领域进一步探索的专业人士。无论是在课堂学习还是自我研读中，它都是一个不可或缺的参考资料。

1. The subsets S

, S

,…S

are mutually disjoint;

2. S

∪ S

∪…∪ S

= S;

3. none of the S

is empty.

Then S

, S

,…S

is called a partition of S.

Functions and Relations

A function is a rule that assigns to elements of one set a unique element of another set. If f denotes a

function, then the first set is called the domain of f, and the second set is its range. We write

f : S

→ S

to indicate that the domain of f is a subset of S

and that the range of f is a subset of S

. If the domain

of f is all of S

, we say that f is a total function on S

; otherwise f is said to be a partial function.

In many applications, the domain and range of the functions involved are in the set of positive

integers. Furthermore, we are often interested only in the behavior of these functions as their

arguments become very large. In such cases an understanding of the growth rates may suffice and a

common order of magnitude notation can be used. Let f (n) and g (n) be functions whose domain is a

subset of the positive integers. If there exists a positive constant c such that for all sufficiently large n

we say that f has order at most g. We write this as

then f has order at least g, for which we use

Finally, if there exist constants c

and c

such that

f and g have the same order of magnitude, expressed as

In this order-of-magnitude notation, we ignore multiplicative constants and lower-order terms that

to υ

. The sequence of edges (υ

, υ

), (υ

, υ

), (υ

, υ

) is a cycle, but not a simple one. If the edges of a

graph are labeled, we can talk about the label of a walk. This label is the sequence of edge labels

encountered when the path is traversed. Finally, an edge from a vertex to itself is called a loop. In

Figure 1.1, there is a loop on vertex υ

Figure 1.1

On several occasions, we will refer to an algorithm for finding all simple paths between two

given vertices (or all simple cycles based on a vertex). If we do not concern ourselves with

efficiency, we can use the following obvious method. Starting from the given vertex, say υ

, list all

outgoing edges (υ

, υ

), (υ

, υ

),…At this point, we have all paths of length one starting at υ

. For all

vertices υ

, υ

,…so reached, we list all outgoing edges as long as they do not lead to any vertex

already used in the path we are constructing. After we do this, we will have all simple paths of length

two originating at υ

. We continue this until all possibilities are accounted for. Since there are only a

finite number of vertices, we will eventually list all simple paths beginning at υ

. From these we

select those ending at the desired vertex.

Trees are a particular type of graph. A tree is a directed graph that has no cycles, and that has one

distinct vertex, called the root, such that there is exactly one path from the root to every other vertex.

This definition implies that the root has no incoming edges and that there are some vertices without

outgoing edges. These are called the leaves of the tree. If there is an edge from υ

to υ

, then υ

is said

to be the parent of υ

, and υ

the child of υ

. The level associated with each vertex is the number of

edges in the path from the root to the vertex. The height of the tree is the largest level number of any

vertex. These terms are illustrated in Figure 1.2.

At times, we want to associate an ordering with the nodes at each level; in such cases we talk

about ordered trees.

Figure 1.2

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林兹第五版：计算机科学入门 - 形式语言与自动机

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