Theory of Computation(TOC) Syllabus - Fourth Semester

Tribhuvan University

Institute of Science and Technology

Bachelor of Science in Computer Science and Information Technology

Course Title: Theory of Computation

Course no: CSC262

Semester: V

Nature of course: Theory + Lab

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 undecidability and intractability.

Course Objective : 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.

Course Contents:

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, Substring of a string, Concatenation of strings, Languages, Empty 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
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

Unit 3. Regular Expression 6 Hrs.

3.1 Regular Expressions, Regular Operators, 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
3.3 Properties of Regular Languages, Pumping Lemma, 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, Language of a grammar
4.3 Parse tree and its construction, Ambiguous grammar, Use of parse tree to show ambiguity in grammar
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 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,
5.4 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 Machine 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, Non-Deterministic 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, Cook’s 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 I: Basic Foundations (5 Hrs)

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

Unit II & III: Introduction to Finite Automata and Regular Expressions (14 Hrs)

Write programs for illustrating the concepts of
- Determinstic Finite Automata
- 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 IV & V: Context Free Grammar and Push Down Automata (14 Hrs)

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
- Acceptance by Final State
- Acceptance by Empty Stack

Unit VI: Turing Machines (12 Hrs)

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