Language Equations,9780387986265

Language Equations

by
ISBN13: 9780387986265
ISBN10: 038798626X
Format: Hardcover
Pub. Date: 1999-03-01
Publisher(s): Springer-Verlag New York Inc
List Price: $106.96

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Summary

Beginning with an informal introduction to language equations, this book presents a framework for a general theory for solving systems of equations and relations between languages. Classical language equations, generalized derivatives, Boolean language equations, and implicit equations are presented systematically. An exploration of mixed systems and open problems rounds out the presentation.

Table of Contents

Preface vii
1 An Informal Introduction to Language Equations
1(9)
1.1 Arithmetic Equations
2(1)
1.2 Language Equations
3(1)
1.3 An Example
4(1)
1.4 Nonlinear Equations
5(1)
1.5 Invertibility of Operations
6(1)
1.6 Implicit and Explicit Equations
6(1)
1.7 Overview of the Book
7(2)
1.8 Assumed Background of the Reader
9(1)
2 Basic Definitions
10(11)
2.1 The Alphabet
10(1)
2.2 The Constant Languages
11(1)
2.3 The Language Operations
11(2)
2.4 The Variables
13(1)
2.5 The Expressions
13(2)
2.6 The Equations
15(2)
2.6.1 Two-Sided Equations
16(1)
2.6.2 Explicit and Implicit Equations
16(1)
2.7 The Relations
17(1)
2.8 Examples
17(2)
2.9 Bibliographical Notes
19(2)
3 Classical Language Equations and the Substitution Property
21(17)
3.1 Introduction
21(2)
3.2 The Syntactic Solution Approach
23(5)
3.3 The Substitution Property
28(3)
3.4 Constructing Nondeterministic Finite Automata for the Solution Languages
31(3)
3.5 A Note on Context-Free Grammars
34(1)
3.6 Bibliographical Notes
35(1)
3.7 Problems
36(2)
4 Boolean Language Equations
38(26)
4.1 Boolean Automata
39(3)
4.2 The Lambda-Property and the Normal Form Theorem
42(4)
4.3 The Uniqueness Criterion for Boolean Equations
46(1)
4.4 Generalized Derivatives of Expressions
47(3)
4.5 Constructing a Boolean Automation for the Solution of Boolean Equations
50(10)
4.6 Boolean Equations with Multiple Solution
60(1)
4.7 Bibliographical Notes
61(1)
4.8 Problems
62(2)
5 More on Generalized Derivatives
64(15)
5.1 Derivatives and the Lambda-Property
64(5)
5.2 Existence and Uniqueness of Solutions
69(2)
5.3 Examples
71(4)
5.4 Techniques for Determining Whether a Word Is in a Solution
75(2)
5.5 Bibliographical Notes
77(1)
5.6 Problems
77(2)
6 Star Equations
79(7)
6.1 Bibliographical Notes
84(1)
6.2 Problems
84(2)
7 Explicit Equations over a One-Letter Alphabet
86(23)
7.1 Properties of Languages over a One-Letter Alphabet
87(2)
7.2 Normal Forms of Expressions Without Complementation
89(2)
7.3 Two Key Theorems
91(8)
7.4 Solving General Equations in One Variable
99(2)
7.5 Solving Systems of Equations
101(1)
7.6 Uniqueness and Regularity of Solutions
102(3)
7.6.1 Uniqueness
102(1)
7.6.2 Regularity
103(1)
7.6.3 CFL over {a}
104(1)
7.7 Explicit Equations over {a} with Complementation
105(2)
7.8 Conclusion and Open Problems
107(1)
7.9 Bibliographical Notes
107(1)
7.10 Problems
108(1)
8 Implicit Equations with Union and Left-Concatenation
109(20)
8.1 Basic Properties of Implicit Language Equations
111(3)
8.1.1 Existence of Solutions
111(1)
8.1.2 Nonclosure of CFL Under Implicit Language Equations
112(2)
8.2 A Procedure for Determining That No Solution Exists
114(2)
8.3 Solving Implicit Language Equations
116(5)
8.4 Characterizing Uniqueness in Implicit Language Equations
121(5)
8.5 Conclusion
126(1)
8.6 Bibliographical Notes
127(1)
8.7 Problems
127(2)
9 Regular Implicit Equations over a One-Letter Alphabet with Union, Concatenation, and Star
129(17)
9.1 Properties of Expressions over a One-Letter Alphabet
130(3)
9.2 Solving One Equation in One Variable
133(3)
9.3 Solving Systems of Equations
136(4)
9.4 Uniqueness of Solutions
140(1)
9.5 Examples
141(3)
9.6 Conclusion
144(1)
9.7 Bibliographical Note
144(1)
9.8 Problems
145(1)
10 Explicit Relations with Union and Left-Concatenation
146(25)
10.1 Introduction
146(3)
10.2 Properties of Single Explicit Relations in One Variable
149(2)
10.3 Solving Systems with Several Explicit Equations for One Variable
151(8)
10.4 Solving Strictly Decoupled Systems
159(3)
10.5 Solving General Decoupled Systems
162(6)
10.6 Conclusion
168(1)
10.7 Bibliographical Note
168(1)
10.8 Problems
169(2)
11 Implicit Relations with Union and Left-Concatenation
171(13)
11.1 Existence of Solutions
174(3)
11.2 Uniqueness of Solutions
177(2)
11.3 Examples
179(2)
11.4 Conclusion
181(1)
11.5 Bibliographical Note
182(1)
11.6 Problems
182(2)
12 Two-Sided Language Equations
184(14)
12.1 Bibliographical Note
196(1)
12.2 Problems
196(2)
13 Mixed Systems
198(9)
13.1 Mixed Systems of Equations with Union and Left-Concatenation
199(3)
13.2 Mixed Systems of Equations over a One-Letter Alphabet with Regular Constants and Union, Concatenation, and Star
202(3)
13.3 Bibliographical Note
205(1)
13.4 Problems
205(2)
14 Open Problems
207(6)
14.1 Classification
207(1)
14.2 The Cardinality of the Alphabet
208(1)
14.3 The Class of Constant Languages
209(1)
14.4 The Set of Operations Involved
209(1)
14.5 Two-Sided, Implicit, and Explicit Equations
210(1)
14.6 Effectiveness of Constructions
211(2)
References 213(2)
Index 215

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