Fourth Semester

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Theory of Computation

Course Title: Theory of Computation

Course No.: CSC257

Course Nature: Theory + Lab

Semester: Fourth

Year: Second

Full Marks: 60 + 20 + 20

Pass Marks: 24 + 8 + 8

Credit Hours: 3

Course Description: This course presents a study of Finite State Machines and their languages. It covers the details of finite state automata, regular expressions, context free grammars. More, the course includes design of the Push-down automata and Turing Machines. The course also includes basics of undecidabilty and intractability.

Course Objectives: The main objective of the course is to introduce concepts of the models of computation and formal language approach to computation. The general objectives are to, introduce concepts in automata theory and theory of computation, design different finite state machines and grammars and recognizers for different formal languages, identify different formal language classes and their relationships, and determine the decidability and intractability of computational problems.

Contents of Chapter

Unit 1: Basic Foundations (3 Hrs.)

1.1. Review of Set Theory, Logic, Functions, Proofs

1.2. Automata, Computability and Complexity: Complexity Theory, Computability Theory, Automata Theory

1.3. Basic concepts of Automata Theory: Alphabets, Power of Alphabet, Kleen Closure Alphabet, Positive Closure of Alphabet, Strings, Empty String, Suffix, Prefix and Substring of a string, Concatenation of strings, Languages, Empty Language, Membership in Language

Unit 2: Introduction to Finite Automata (8 Hrs.)

2.1. Introduction to Finite Automata, Introduction of Finite State Machine

2.2. Deterministic Finite Automata (DFA), Notations for DFA, Language of DFA, Extended Transition Function of DFA Non-Deterministic Finite Automaton (NFA), Notations for NFA, Language of NFA, Extended Transition

2.3. Equivalence of DFA and NFA, Subset- Construction

2.4. Method for reduction of NFA to DFA, Theorems for equivalence of Language accepted by DFA and NFA: For any NFA, N = (QN, ∑, N, q0, FN) accepting language L  ∑* there is a DFA D = (QD, ∑, D, q0’,FD) that also accepts L i.e. L (N) = L (D), A language L is accepted by some NFA if L is accepted by some DFA.

2.5. Finite Automaton with Epsilon Transition (ε - NFA), Notations for ε - NFA, Epsilon Closure of a State, Extended Transition Function of ε – NFA, Removing Epsilon Transition using the concept of Epsilon Closure, Equivalence of NFA and ε –NFA, Equivalence of DFA and ε – NFA

2.6. Finite State Machines with output: Moore Machine and Mealy Machines, Illustration of the Moore and Mealy Machines

Unit 3: Regular Expressions (6 Hrs.)

3.1. Regular Expressions, Operators of Regular Expressions (Union, Concatenation, Kleen), Regular Languages and their applications, Algebraic Rules for Regular Expressions

3.2. Equivalence of Regular Expression and Finite Automata, Reduction of Regular Expression to ε–NFA, Conversion of DFA to Regular Expression, Arden’s Theorem

3.3. Properties of Regular Languages, Pumping Lemma for regular expression, Application of Pumping Lemma, Closure Properties of Regular Languages over (Union, Intersection , Complement), Minimization of Finite State Machines: Table Filling Algorithm

Unit 4: Context Free Grammar (9 Hrs.)

4.1. Introduction to Context Free Grammar (CFG), Components of CFG, Use of CFG, Context Free Language (CFL)

4.2. Types of derivations: Bottomup and Topdown approach, Leftmost and Rightmost, Sentential Form (Left, Right), Language of a grammar

4.3. Parse tree and its construction, Ambiguous grammar, Use of parse tree to show ambiguity in grammar, Inherent Ambiguity

4.4. Regular Grammars: Right Linear and Left Linear, Equivalence of regular grammar and finite automata

4.5. Simplification of CFG: Removal of Useless symbols, Nullable Symbols, and Unit Productions, Chomsky Normal Form (CNF), Greibach Normal Form (GNF), Backus-Naur Form (BNF)

4.6. Context Sensitive Grammar, Chomsky Hierarchy(Type 0, 1, 2, 3) , Pumping Lemma for CFL, Application of Pumping Lemma, Closure Properties of CFL

Unit 5: Push Down Automata (7 Hrs.)

5.1. Introduction to Push Down Automata (PDA), Representation of PDA, Operations of PDA, Move of a PDA, Instantaneous Description for PDA

5.2. Deterministic PDA, Non Deterministic PDA, Acceptance of strings by PDA, Language of PDA

5.3. Construction of PDA by Final State , Construction of PDA by Empty Stack, Conversion of PDA by Final State to PDA accepting by Empty Stack and vice-versa, Conversion of CFG to PDA, Conversion of PDA to CFG

Unit 6: Turing Machines (10 Hrs.)

6.1. Introduction to Turing Machines (TM), Notations of Turing Machine, Language of a Turing Machine, Instantaneous Description for Turing Machine, Acceptance of a string by a Turing Machines

6.2. Turing Machine as a Language Recognizer, Turing Machine as a Computing Function, Turing Machine with Storage in its State, Turing Machine as a enumerator of stings of a language, Turing Machine as Subroutine

6.3. Turing Machine with Multiple Tracks, Turing

Machine with Multiple Tapes, Equivalence of Multitape-TM and Multitrack-TM, NonDeterministic Turing Machines, Restricted Turing Machines: With Semi-infinite Tape, Multistack Machines, Counter Machines

6.4. Curch Turing Thesis, Universal Turing Machine, Turing Machine and Computers, Encoding of Turing Machine, Enumerating Binary Strings, Codes of Turing Machine, Universal Turing Machine for encoding of Turing Machine

Unit 7: Undecidability and Intractability (5 Hrs.)

7.1. Computational Complexity, Time and Space complexity of a Turing Machine, Intractability

7.2. Complexity Classes, Problem and its types: Absract, Decision, Optimization

7.3. Reducibility, Turing Reducible, Circuit Satisfiability, Cooks Theorem

7.4. Undecidability, Undecidable Problems: Post’s Correspondence Problem, Halting Problem and its proof, Undecidable Problem about Turing Machines

Laboratory Works:

Student should write programs and prepare lab sheets for most of the units in the syllabus. Majorly, students should practice design and implementation of Finite State Machines viz. DFA, NFA, PDA, and Turing Machine. Students are highly recommended to construct Tokenizers/ Lexical analyzer over/for some language. The nature of programming can be decided by the instructor and students as per their comfort. The instructors have to prepare lab sheets for individual unit covering the concept of all units as per the requirement. The sample lab sessions can be as following descriptions;

Unit 1:

Write programs for illustrating the concepts of Strings, Prefix, Suffix and Substring of a String.

Unit 2, 3:

Write programs for illustrating the concepts of o Determinstic Finite Automata o Non-Deterministic Finite Automata

Write programs for implementing Tokenizers like for valid C-identifiers, Keywords, e-mail validators, phone number etc.

Write programs that implement NFA for text search.

Write programs for implementing regular expressions.

Unit-4, 5:

Write Program for simulation of Leftmost/Rightmost Derivations.

Write Program for Parse Tree Contruction.

Write programs for illustrating the concepts of context free grammar and its accptance using the concepts of Push Down Automata o Acceptance by Final State o Acceptance by Empty Stack

Unit-6:

Write programs for illustrating the concepts of Turing Machine as a Language Recognizer.

Text Books:

  1. John E. Hopcroft, Rajeev Motwani, Jeffrey D. Ullman, Introduction to Automata Theory, Languages, and Computation, 3rd Edition, Pearson - Addison-Wesley.

Reference Books:

  1. Harry R. Lewis and Christos H. Papadimitriou, Elements of the Theory of Computation, 2nd Edition, Prentice Hall.
  2. Michael Sipser, Introduction to the Theory of Computation, 3rd Edition, Thomson Course Technology
  3. Efim Kinber, Carl Smith, Theory of Computing: A Gentle introduction, Prentice- Hall.
  4. John Martin, Introduction to Languages and the Theory of Computation, 3rd Edition, Tata McGraw Hill.