THEORY OF COMPUTATION

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3 THEORY OF COMPUTATION

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5 THEORY OF COMPUTATION For B.E. (IT, CS), MCA, BSC, BCA, MSC and Other Computer Courses By Dr. Poonam Sinha Head (IT/MCA), BUIT, Barkatullah University Bhopal (MP) Sunita Gond Lecturer, (IT/MCA) Deptt. BUIT, Barkatullah University Bhopal (MP) Apurva Saxena Lecturer, (IT/MCA) Deptt. BUIT, Barkatullah University Bhopal (MP) UNIVERSITY SCIENCE PRESS (An Imprint of Laxmi Publications (P) Ltd.) BANGALORE CHENNAI COCHIN GUWAHATI HYDERABAD JALANDHAR KOLKATA LUCKNOW MUMBAI RANCHI NEW DELHI INDIA USA GHANA KENYA

6 THEORY OF COMPUTATION Copyright by Laxmi Publications (P) Ltd. All rights reserved including those of translation into other languages. In accordance with the Copyright (Amendment) Act, 2012, 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. Any such act or scanning, uploading, and or electronic sharing of any part of this book without the permission of the publisher constitutes unlawful piracy and theft of the copyright holder s intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained from the publishers. Printed and bound in India Typeset at Excellent Graphics, Delhi First Edition: 2015 UTC THEORY OF COMPUTATION-SIN ISBN Limits of Liability/Disclaimer of Warranty: The publisher and the author make no representation or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties. The advice, strategies, and activities contained herein may not be suitable for every situation. In performing activities adult supervision must be sought. Likewise, common sense and care are essential to the conduct of any and all activities, whether described in this book or otherwise. Neither the publisher nor the author shall be liable or assumes any responsibility for any injuries or damages arising herefrom. The fact that an organization or Website if referred to in this work as a citation and/or a potential source of further information does not mean that the author or the publisher endorses the information the organization or Website may provide or recommendations it may make. Further, readers must be aware that the Internet Websites listed in this work may have changed or disappeared between when this work was written and when it is read. All trademarks, logos or any other mark such as Vibgyor, USP, Amanda, Golden Bells, Firewall Media, Mercury, Trinity, Laxmi appearing in this work are trademarks and intellectual property owned by or licensed to Laxmi Publications, its subsidiaries or affiliates. Notwithstanding this disclaimer, all other names and marks mentioned in this work are the trade names, trademarks or service marks of their respective owners. Published in India by & Bangalore & Chennai , & Cochin , & Guwahati , & Hyderabad , & Jalandhar & Kolkata & Lucknow & Mumbai , & Ranchi , GOLDEN HOUSE, DARYAGANJ, NEW DELHI , INDIA Telephone : , Fax : , C info@laxmipublications.com Printed at: Branches

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9 CONTENT ONTENTS Preface Acknowledgements Syllabuses of Various Universities vii viii ix 1. Introduction of Automata Theory Formal Language Mathematics Preliminaries Relations and Functions Graphs Trees Alphabet Strings Principle of Induction Propositional Logic (Zero Order Logic) Predicate Logic (First Order Logic) Automata Automata and Definition Description Finite Automata/Finite State Machine (FSM) Equivalence of DFA and NFA Finite Automata with Null String or NFA with ε Moves to NFA without ε Minimisation of Finite State Machine or My-Hill-Nerode Algorithm FSM with Output Regular Expression Regular Grammar Pumping Lemma for Regular Grammar Formal Language Definition and Description Phase Structured Grammar or Chomsky Classification of Languages/ Chomsky Hierarchy Properties of Families of Languages ( vii )

10 ( viii ) 4. Regular Expression Finite Automata Equivalence of Finite Automata and Regular Expression Equivalence of Two-Way Finite Automata Context Free Grammar Grammar and Context Free Language Derivation Tree (Parse Tree) Unrestricted Grammar Ambiguous Context Free Language Simplification of Context Free Grammars Normal Form of CFG Pushdown Automata Pushdown Automata Nondeterministic Pushdown Automata Pushdown Automata and Context Free Grammar Two-way Pushdown Automaton Parsing Pumping Lemma for CFG Context Free Language Pumping Lemma for CFL (Context Free Language) Turing Machine Introduction Model of Turing Machine Languages Accepted by Turing Machine Universal Turing Machine Determinism in Turing Machine Non-determinism in Turing Machine Types of Turing Machine Introduction Halting Problem of Turing Machine Post Correspondence Problem... 68

11 ( ix ) 9. Computability Introduction to Complexity Theory Introduction to Undecidability Recursively Enumerable Sets Context Sensitive Language Context Sensitive Language Primitive Recursive Functions Linear Bounded Automata Types of Grammar and its Hierarchy Appendix Solved Examples Important University Solved Problems Objective Type Questions Question Papers Index

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13 PREF REFACE We are highly delighted to place the book title Theory of Computation in the hands of one esteemed readers. This book has been written for B.E. (IT & CS), MCA, BSc, BCA, MSc and other computer programmes students keeping in view the following special points : Conceptual clarity and practical orientation. To learn the basic concept used in the design and analysis of Theory of Computation. Introduces the principles of Theory of Computation. Provides various methods and techniques suitable for a variety of applications. Covers all aspects of automata. Covers abundance of examples and their solutions. We have put our sincere efforts and knowledge to make you understand the subject matter in simplest and easiest way. Valuable suggestions are always welcome. We have considered the syllabus of Barkatullah University, RGPV, Delhi University and Devi Ahilya University and tried to simplify language of other books. Authors ( xi )

14 ACKNO CKNOWLEDGEMENT WLEDGEMENTS We take this opportunity to expresses our gratitude and thanks to the collegues for their valuable technical suggestions and constant encouragement, without which this book would not come into existence. We are grateful to our family members for inspiring us for this project. We deeply express our heartful thanks to the publishers for publishing this book in such a beautiful get-up and well in time. Authors ( xii )

15 SYLLABUSES OF VARIOUS UNIVERSITIES BARKATULLAH UNIVERSITY (Theory of Computation) Unit I Introduction to Theory of Computation and Finite Automata. Mathematical Preliminaries and Notation : Sets, functions and relations, Graphs and Trees, Proof Techniques, Basic concepts, Languages, Grammars, automatas, deterministic finite accepters, Deterministic accepters and Transition Graphs, Languages, Non deterministic finite accepters, definition of a NDFA, Equivalence of DFA and NDFA, Reduction of the Number of states in finite automata. Unit II Grammars and Languages. Regular expression, Regular Grammar, Regular languages, closure properties of Regular languages, Context free grammars, Simplification of Context free grammars and Normal forms, Properties of context free languages. Unit III Pushdown Automata. Non deterministic pushdown automata : Definition of a pushdown automata, The language accepted by a pushdown automata, Pushdown automata and context free languages, Pushdown automata for context free languages, CFG s for PDA, Deterministic Pushdown automata and Deterministic Context free languages, Grammars and Deterministic context free languages. Unit IV Turing Machines. The Standard Turing Machine : Definition of a Turing Machine, Turing Machines as language accepters, and Turing Machines as Transducers. Combining Turing Machines for complicated tasks, Turing thesis, other models of Turing Machines. Unit V Computability. Limits of algorithmic computation, Some Problems that can not be solved by Turing Machines, Computability and Decidability, the Turing Machine Halting Problem, Reducing one Undecidable Problem to another, Undecidable Problems for Recursively Enumerable languages, The post correspondence problem: Indecidable problems for context free languages, Recursive function, Primitives recursive functions, Ackermanris functions, recursive functions, Post Systems : Rewriting systems : Matrix grammars, Markov Algorithms. ( xiii )

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