East Carolina University
Department of Computer Science

CSCI 4602
Automata, Computability and Complexity
Standard Syllabus

Catalog entry

P: CSCI 2405, CSCI 2530. Finite automata and regular languages, Turing machines, Church-Turing thesis, Computability and Unsolvability, Complexity, Theory of NP-Completeness.

Course summary

Theory of computation comprises the fundamental mathematical properties of computer hardware, software and certain applications thereof. In studying this subject we seek to determine what can and cannot be computed, how quickly, with how much memory and on which type of computational model. Three central areas of theory of computation are Theory of Automata and Formal Languages, Computability Theory and Complexity Theory. In this course, we will take a brief look at each of these three areas.

Course topics

• Introduction. Mathematical background, strings and languages.
  
  
• Finite automata
• Deterministic finite automata. Definition, examples, analysis and design.
• Nondeterministic finite automata. Definition, examples, analysis, design. Equivalence of NFA and DFA models.
• NFA with ε-transitions. Definition, examples, analysis and design. Equivalence of NFAs with ε-transitions to ordinary NFAs.
• Closure properties of regular languages.
• Regular expressions. Definitions, examples, analysis and design.
• Kleene's theorem. Equivalence of regular expressions and finite automata.
• Non-regular languages. Prove languages to be non-regular using a chosen technique.
• Optional: The Pumping Lemma for regular languages and its application to prove non-regularity of languages.
• Optional: The Myhill-Nerode Theorem and its application to prove non-regularity of languages and to the existence of a unique minimal DFA for a regular language.
• Optional: Applying closure properties to prove non-regularity of languages.
• Optional: State minimization.
• Optional: Topics from applications of finite automata such as neural nets and markov chains.
  
  
• General computation
• Turing Machines. Definitions, examples, analysis and design.
• Church-Turing thesis.
• Universal Turing machines.
• Diagonalization and the halting problem.
• Reductions. Different kinds of reductions.
• More examples of undecidable problems.
• Optional: Proof of the undecidability of Post's Correspondence Problem.
• Optional: Rice's Theorem and applications of it.
• Optional: Recursion Theorems and applications of them.
  
  
• Basic Concepts of Complexity Theory
• Time and space complexity. Polynomial time.
• The classes P and NP.
• Polynomial time reductions and NP-Completness. The Cook-Levin Theorem.
• More examples of NP-Complete Problems.
• The P vs. NP problem. Different formulations of it. Significance of it.
• Optional: The class NP-Intermediate and Ladner's Theorem
• Optional: The class Co-NP, NP=Co-NP Problem, NP ∩ Co-NP = P Problem.
• Optional: Different methods of coping with NP-completeness such as approximation algorithms and Fixed-Parameter Tractability.
• Optional: Other topics from complexity theory such as space complexity, polynomial hierarchy, randomized complexity classes and intractabilty.

Student learning outcomes

• Analyze a given DFA and/or NFA and determine the language recognized by it.
• Design a DFA and/or NFA recognizing a specified language.
• Analyze and design a regular expression specifying a given language.
• Prove that a given language in not regular.
• Explain the significance of Church-Turing Thesis.
• Discuss the notion of universal Turing machine.
• Recognize properties of reductions.
• Use reductions to prove interesting results.
• Realize that there are problems not solvable by computers
• Describe the complexity classes P and NP.
• List several natural NP-complete problems.
• Demonstrate the NP-Completeness of some computational problems by reduction.
• Explain the significance of the P vs. NP problem.
• Students should both realize the importance of theory to the field of computer science and develop the abstract thinking needed to pursue it further as desired by them. This is more important than any of the specific outcomes listed above.

Textbook

Introduction to Theory of Computation by Michael Sipser, Cengage Learning, Third Edition, 2013.

Grading

Your grade in this course will be based on your performance in homework and examinations. There will be two regular examinations during the semester and a final comprehensive examination at the end of the semester. Homework will be based on "problem of the week" model, where you will be assigned one problem to solve each week.

Grade meanings