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- Languages and Machines An Introduction to the Theory of Computer Science
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## Languages and Machines An Introduction to the Theory of Computer Science

An imprint of Addison Wesley Longman, Inc. Includes bibliographical references and index. ISBN 1. Formal languages. Machine theory. Computational complexity. The programs and applications presented in this book have been included for their instructional value. They have been tested with care but are not guaranteed for any particular purpose.

The publisher does not offer any warranties or representations, nor does it accept any liabilities with respect to the programs or applications. Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks.

Where those designations appear in this book, and Addison-Wesley was aware of a trademark claim, the designations have been printed in initial caps or all caps. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.

Printed in the United States of America. The objective of the second edition remains the same as that of the first, to provide a mathematically sound presentation of the theory of computer science at a level suitable for junior- and senior-level computer science majors.

It is my hope that, like a fine wine, this book has improved with age. The improvement comes with the experience gained from eight years of use by students and faculty who generously shared their observations and suggestions with me. These comments helped to identify areas that would benefit from more comprehensive explanations and topics that would add depth and flexibility to the coverage. A brief description of some of the additions is given below.

An algorithm for mininimizing the number of states in a deterministic finite automa- ton has been added to Chapter 6. This completes the sequence of actions used to construct automata to accept complex languages: Initially use nondeterminism to assist in the de- sign process, algorithmically construct an equivalent deterministic machine, and, finally, minimize the number of states to obtain the optimal machine.

The correctness of the min- imization algorithm is proven using the Myhill-Nerode theorem, which has been included in Chapter 7. Rice's theorem has been added to the presentation of undecidability.

This further demonstrates the importance of reduction in establishing undecidability and provides a simple method for showing that many properties of recursively enumerable languages are undecidable. The coverage of computational complexity has been significantly expanded. Chap- ter 14 introduces the time complexity of Turing machines and languages. Properties of the. Chapter 15 is devoted solely to the issues of tractabil- ity and NP-completeness. Organization Since most students at this level have had little or no background in abstract mathematics, the presentation is designed not only to introduce the foundations of computer science but also to increase the student's mathematical sophistication.

This is accomplished by a rigorous presentation of concepts and theorems accompanied by a generous supply of examples.

Each chapter ends with a set of exercises that reinforces and augments the material covered in the chapter. The presentation of formal language and automata theory examines the relationships between the grammars and abstract machines of the Chomsky hierarchy.

Parsing context- free languages is presented via standard graph-searching algorithms immediately follow- ing the introduction of context-free grammars. Considering parsing at this point reinforces the need for a formal approach to language definition and motivates the development of normal forms.

Chapters on LL and LR grammars are included to permit a presen- tation of formal language theory that serves as a foundation to a course in compiler design. Finite-state automata and Turing machines provide the framework for the study of ef- fective computation. The equivalence of the languages generated by the grammars and recognized by the machines of the Chomsky hierarchy is established.

The coverage of computability includes decidability, the Church-Turing thesis, and the equivalence of Tur- ing computability and pi-recursive functions. The classes TPand NY of solvable decision problems and the theory of NP-completeness are introduced by analyzing the time com- plexity of Turing machines. To make these topics accessible to the undergraduate student, no special mathemati- cal prerequisites are assumed.

Instead, Chapter 1 introduces the mathematical tools of the theory of computing: naive set theory, recursive definitions, and proof by mathematical in- duction. With the exception of the specialized topics in Sections 1. Section 1. Equivalence rela- tions, introduced in Section 1. For students who have completed a course in discrete mathematics, most of the material in Chapter 1 can be treated as review. Prerequisite Chapter Chapters I 2 1 3 2 4 3 5 3 6 2 7 3,6 8 5,7 9 2 10 8,9 11 9 12 11 13 12 14 13 15 14 16 4,5 17 4,5,6.

The entire book can be comfortably covered in two semesters. Since courses on the foundations of computing may emphasize different topics, the text has been organized to allow the flexibility needed to design one-term courses that concentrate on one or two specific areas. Suggested outlines for such courses are. Notation The theory of computer science is a mathematical examination of the capabilities and lim- itations of effective computation.

The following notational conventions will be used through- out the book:. The use of roman letters for sets and mathematical structures is somewhat nonstandard but was chosen to make the components of a structure visually identifiable.

From the fonts alone it can be seen that G consists of three sets and a variable S. A three-part numbering system is used throughout the book; a reference is given by chapter, section, and item. One numbering sequence records definitions, lemmas, theo- rems, corollaries, and algorithms. A second sequence is used to identify examples. Exer- cises are referenced simply by the chapter and the number. The end of a proof is marked by 0 and the end of an example by E.

An index of symbols, including descriptions of their use and the numbers of the pages on which they are defined, is given in Appendix I. A number of people made significant contributions to this book, both in the original and the current edition.

I would like to extend my most sincere appreciation to the students and professors who have used this text over the past eight years. Your comments and suggestions provided the impetus for the inclusion of many of the new topics and more detailed explanations of material previously covered.

Lee Northwestern University , C. I would also like to acknowledge the assistance received from the staff of the Addison-Wesley Publishing Co. Thomas A.

The theory of computer science began with the questions that spur most scientific en- deavors: how and what. After these had been answered, the question that motivates many economic decisions, how much, came to the forefront.

The objective of this book is to explain the significance of these questions for the study of computer science and provide answers whenever possible. Formal language theory was initiated by the question, How are languages defined? In an attempt to capture the structure and nuances of natural language, linguist Noam Chomsky developed formal systems called grammars for generating syntactically correct sentences.

At approximately the same time, computer scientists were grappling with the problem of explicitly and unambiguously defining the syntax of programming languages. These two studies converged when the syntax of the programming language ALGOL was defined using a formalism equivalent to a context-free grammar.

The investigation of computability was motivated by two fundamental questions: What is an algorithm? An answer to the first question requires a formal model of computation.

It may seem that the combination of a computer and high-level programming language, which clearly constitute a computational system, would provide the ideal framework for the study of computability. Only a little consideration is needed to see difficulties with this approach.

What computer? How much memory should it have? What programming language? More- over, the selection of a particular computer and language may have inadvertent and un- wanted consequences for the answer to the second question. A problem that may be solved on one computer configuration may not be solvable on another. The question of whether a problem is algorithmically solvable should be independent of the model computation used: Either there is an algorithmic solution to a problem or there is no such solution.

Consequently, a system that is capable of performing all possible algorithmic computations is needed to appropriately address the question of computability. The characterization of general algorithmic computation has been a major area of research for mathematicians and logicians since the s. Many different systems have been pro- posed as models of computation, including recursive functions, the lambda calculus of Alonzo Church, Markov systems, and the abstract machines developed by Alan Turing.

All of these systems, and many others designed for this purpose, have been shown to be capable of solving the same set of problems. One interpretation of the Church-Turing the- sis, which will be discussed in Chapters 11 and 13, is that a problem has an algorithmic solution only if it can be solved in any and hence all of these computational systems.

Because of its simplicity and its similarity to the modern computer, we will use the Turing machine as our framework for the study of computation. The Turing machine has many features in common with a computer: It processes input, writes to memory, and produces output. Although Turing machine instructions are primitive compared with those of a computer, it is not difficult to see that the computation of a computer can be simulated by an appropriately defined sequence of Turing machine instructions.

The Turing machine model does, however, avoid the physical limitations of conventional computers; there is no upper bound on the amount of memory or time that may be used in a computation. Consequently, any problem that can be solved on a computer can be solved with a Turing machine, but the converse of this is not guaranteed. After accepting the Turing machine as a universal model of effective computation, we can address the question What are the capabilities and limitations of algorithmic compu- tation?

The Church-Turing thesis assures us that a problem is solvable only if there is a suitably designed Turing machine that solves it. To show that a problem has no solution reduces to demonstrating that no Turing machine can be designed to solve the problem. Chapter 1I follows this approach to show that several important questions concerning our ability to predict the outcome of a computation are unsolvable. Once a problem is known to be solvable, one can begin to consider the efficiency or optimality of a solution.

The question how much initiates the study of computational complexity. The time complexity of a Turing machine measures the number of transitions required by a computation.

## Languages and Machines: An Introduction to the Theory of Computer Science, 3rd Edition

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## Languages and Machines: An Introduction to the Theory of Computer Science

Enter your mobile number or email address below and we'll send you a link to download the free Kindle App. Then you can start reading Kindle books on your smartphone, tablet, or computer - no Kindle device required. The third edition of Languages and Machines: An Introduction to the Theory of Computer Science provides readers with a mathematically sound presentation of the theory of computer science at a level suitable for junior and senior level computer science majors.

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T. Sudkamp. Thomas Sudkamp. Ma Haoxiang Thus each machine has an associated language, the set of strings accepted by the machine. The computational.

Solutions Manual for. Languages and Machines: An Introduction to the Theory of. Computer Science. Third Edition. Thomas A. Sudkamp.

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