1. Mathematical Preliminaries 1
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1 Preface List of Important Symbols and Notations vii xix 1. Mathematical Preliminaries 1 Chapter Objective 1 Introduction SET Theory Alphabets Strings and Languages Relations Functions Graphs and Trees Proof Techniques 22 Summary 24 Review Questions 25 Graded Exercises 26 Objective Questions 27 Reference for Extra Reading 27 Online Sources Finite Automata 29 Chapter Objective 29 Introduction Finite State Machines and its Model Deterministic Finite Automata Simplified Notation FA with and without Epsilon Transitions Language of Deterministic Finite Automata Acceptability of a String by a DFA Processing of Strings by DFA 40
2 xiv q Contents 2.8 Nondeterministic Finite Automata Language of NFA Equivalence between DFA and NFA NFA W ith and W ithout Epsilon Transitions Two Way Finite Automata FA with Output:Moore and Mealy Machines From Finite Automata to Moore Machine Interconversion between the Machines Equivalence between Moore and Mealy Machines Minimisation of FA Properties of Transition Function (d) Extending Transition Function to Strings Applications of Finite Automata Limitations of Finite State Machines 83 Summary 83 Review Questions 84 Graded Exercises 89 Objective Questions 91 Reference for Extra Reading 91 Online Sources Formal Languages 97 Chapter Objective 97 Introduction Theory of Formal Languages Kleene and Positive Closure Defining Languages Recursive Definition of Languages Arithmetic Expressions Grammars Classification of Grammars and Languages Languages and their Relations Operations on Languages Chomsky Hierarchy 124 Summary 127 Review Questions 127 Graded Exercises 130 Objective Questions 132 Reference for Extra Reading 137 Online Sources R egular Language and R egular G rammar 138 Chapter Objective 138 Introduction 138
3 Contents q xv 4.1 Regular Language Regular Expressions Operators of Regular Expressions Identity Rules Algebraic Laws for RE Finite Automata and Regular Expressions Equivalence of Two Regular Expressions Myhill Nerode Theorem Regular Sets Closure Properties of Regular Sets Regular Grammar and FA Regular Expressions and Regular Grammar Left Linear and Right Linear Regular Grammar Applications of Regular Expressions Non-Regular Languages 167 Summary 167 Review Questions 168 Graded Exercises 173 Objective Questions 174 Reference for Extra Reading 177 Online Sources Properties of Regular Languages 178 Chapter Objective 178 Introduction Closure Properties of Regular Languages Decision Properties of Regular Languages Pumping Lemma for Regular Languages Proving Languages not to be Regular Languages Regular Language and Right Linear Grammar 189 Summary 189 Review Questions 190 Graded Exercises 190 Objective Questions 191 Reference for Extra Reading 191 Online Sources Context Free G rammar and Context Free Language 192 Chapter Objective 192 Introduction Definition of Context Free Grammar Context Free Language Deterministic Context Free Language (DCFL) Derivations 196
4 xvi q Contents 6.5 Parse Trees From Inference to Tree Derivation Tree and New Notation of Arithmetic Expressions Sentential Forms Rightmost and Leftmost Derivation of Strings Ambiguity in Grammar and Language Removal of Ambiguity Ambiguous to Unambiguous Context-Free Grammar Useless Symbols in CFG Elimination of Null and Unit Productions Chomsky and Greibach Normal Form CYK Algorithm Applications of CFG 237 Summary 290 Review Questions 241 Graded Exercises 243 Objective Questions 244 Reference for Extra Reading 248 Online Sources Push D ow n Automata 249 Chapter Objective 249 Introduction Description and Definition Definition and Model of PDA Language of PDA Graphical Notations for PDA Acceptance by Final State and Empty Stack From Empty Stackto Final State and Vice Versa Deterministic Push Down Automata Nondeterministic Push Down Automata Equivalence of PDA and Context Free Language PDA and Regular Languages Equivalence of PDA and Context Free Grammar Two StackPDA Auxiliary Push Down Automata Parsing and PDA (Top Down and Bottom Up) Deterministic PDA and Deterministic CFL 286 Summary 286 Review Questions 287 Graded Exercises 288 Objective Questions 290 Reference for Extra Reading 293 Online Sources 293
5 Contents q xvii 8. Properties of Regular and Context Free Languages 294 Chapter Objective 294 Introduction Pumping Lemma for Context Free Languages Decision Properties and Algorithm Closure Properties of CFLs Mixing of CFLs and RLs 309 Summary 313 Review Questions 313 Graded Exercises 314 Objective Questions 314 Reference for Extra Reading 316 Online Sources Turing Machines 317 Chapter Objective 317 Introduction Model of Turing Machines Definition of Turing Machine Halt and Crash Conditions Equivalence of Two Turing Machines Representation of Turing Machines Designs for Turing Machines Programming Techniques Turing Machine and Computation Types of Turing Machines Universal Turing Machine Church-Turing Hypothesis Language Accepted by Turing Machine Recursive and Recursively Enumerable Language Turing Machine and Type-0Grammar Undecidable Problems about Turing Machines Turing Machine as Language Acceptor and Generator Turing Transducer 354 Summary 356 Review Questions 356 Graded Exercises 359 Objective Questions 361 Reference for Extra Reading 363 Online Sources U ndecidability and C omputability 365 Chapter Objective 365 Introduction 365
6 xviii q Contents 10.1 Unsolvable Problems Involving CFLs Undecidable Problems that are Recursively Enumerable Post Correspondence Problem Modified Post Correspondence Problem Languages that are not Recursively Enumerable Context Sensitive Languages Computability Recursive Function Theory Ackermann s Function Reducing One Undecidable Problem to Another Rice s Theorem Computational Complexity Rewriting Systems MatrixGrammar MarkovAlgorithm 381 Summary 383 Review Questions 384 Graded Exercises 385 Objective Questions 385 Reference for Extra Reading 388 Online Sources N P-C ompleteness 390 Chapter Objective 390 Introduction Time Complexity Growth Rate of Functions Polynomial Time Polynomial Time Reduction P and NP Classes NP-Completeness NP-Hard Cook s Theorem Some NP-Complete Problems 404 Summary 405 Review Questions 406 Graded Exercises 406 Objective Questions 407 Reference for Extra Reading 409 Online Sources 409 Appendix: Answers to Objective Questions 411 Index 415
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