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Understanding the Theory of Computation: Foundations and Applications

This course, led by Assoc. Prof. Dr. Brahim Hnich at Izmir University of Economics, delves into the fundamental concepts of the Theory of Computation. It covers critical topics such as Automata Theory, Computability Theory, and Complexity Theory, illustrating what can be computed and the efficiency of algorithms. Students will engage with key textbooks and participate in hands-on learning experiences aimed at enhancing mathematical reasoning and analytical skills necessary for advanced studies in computer science. Office hours are available for personalized support.

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Understanding the Theory of Computation: Foundations and Applications

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  1. CS 208: Computing Theory Assoc. Prof. Dr. Brahim Hnich Faculty of Computer Sciences Izmir University of Economics

  2. Contact Info • Room 415 • Email: brahim.hnich@ieu.edu.tr • Office hours • Wednesdays: 14—16

  3. Textbooks • The course lecture are based on the following books • Harry R. Lewis and C.H. Papadimitriou. "Elements of the Theory of Computation", ISBN 0-13-272741-2, Prentice Hall. (Available in library) • Michael Sipser. "Introduction to the Theory of Computation", ISBN 053494728X, PWS Publishing Company.

  4. Textbooks • The course lecture are based on the following books • Michael R. Garey and David S. Johnson. “Computer and Intractability", ISBN 0-7167-1045-5, W.H. Freeman and Company. • C.H. Papadimitriou. “Computational Complexity", ISBN 0-201-53082-1, Addison-Wesley Publishing Company, Inc.

  5. Grading and exams • Midterm Exam 40% • Final Exam : 60%

  6. Programme • Part A: Preliminaries • Introduction • Mathematical Notations for Computations

  7. Programme • Part B: Automata Theory • Finite State Automata • Context Free Grammars

  8. Programme • Part C: Computability Theory • Turing Machines • Decidability and Undecidability

  9. Programme • Part D: Complexity Theory • Time Complexity • Intractability

  10. Programme • Part E: Summary and Concluding Remarks

  11. Good Luck!

  12. Outline • Introduction to the course • Goal • Organization • Outline • Mathematical Notations for Computations

  13. What is computation? • Definition: Processing of information based on a finite set of operations or rules

  14. What is computation? • Paper and pencil Arithmetic • 99 - 8 = 91 • Calculators • Bar code scanner • Digital camera • Computer programs in C and Java • …

  15. What do we want in a “theory” • Generality • Technology-independent • Abstraction: ignores inessential details • Precision • Mathematical, formal • Can prove theorems about computation • Positive: what can be computed • Negative: what cannot be computed

  16. So you want to be a good computer scientist/ software engineer? • Learn the rules • Algorithms, data structures, … • Learn basic principles • Data abstraction, … • Understand which problems you can solve with computers and which you simply cannot • … and why!

  17. So you want to be a good computer scientist/ software engineer? • Learn the rules • Algorithms, data structures, … • Learn basic principles • Data abstraction, … • Understand which problems you can solve efficiently with computers and which you cannot!

  18. Course Overall Objective • Understand the fundamental capabilities and limitations of computers • Make a theory out of the idea of computation

  19. Why study theory? • Pragmatic Reasons • Apply efficient algorithms to tractable problems • Avoid intractable or impossible problems • Learn to tell the difference

  20. Why study theory? • Mathematical education • Automata Theory • Computability Theory • Complexity Theory

  21. Why study theory? • Philosophy • What is computation? • What is provable?

  22. Course of two parts

  23. The Good News • You only have to take this course once • Except those of you who will not pay attention to the lectures • At the end, you can say that you understand what computers are really about • Good for postgraduate and professional life

  24. 2nd half of the course

  25. The Bad News • Theory course • Formal • Requires mathematics • Understanding Proofs • Analytical skills

  26. The Bad News • Theory course • Formal • Requires mathematics • Understanding Proofs • Analytical skills But, we will try to make it fun because it is really interesting!

  27. End of Introduction Next: Topics to be covered

  28. Three major topics • What is a computer? • Automata Theory • What can (cannot) be computed? • Computability Theory • What can (cannot) be computed efficiently? • Complexity Theory

  29. Three major topics • What is a computer? • Automata Theory • What can (cannot) be computed? • Computability Theory • What can (cannot) be computed efficiently? • Complexity Theory

  30. Complexity Theory

  31. Complexity Theory • Key notion: tractable vs. intractable problems

  32. Complexity Theory • Key notion: tractable vs. intractable problems EASY HARD

  33. Complexity Theory • A problem is a general question: • Description of parameters • Description of solution • E.g. • Input: Given a list of integers • Can you sort the list in ascending order?

  34. Complexity Theory • An algorithm is a step-by-step procedure: • A recipe • A computer program • A mathematical object

  35. Complexity Theory • We want the most efficient algorithm as a function of problem size: • Fastest (mostly) • Most economical with memory (sometimes)

  36. Example: Traveling Salesman Problem • Given • A set of m cities • A set of inter-city distances • Find a permutation of cities which makes a tour of minimum length

  37. Example: TSP instance a 9 10 5 c 6 3 b d 9 (not drawn to scale)

  38. Example: TSP instance a 9 10 5 c 6 3 b d 9 A solution: The tour a,b,d,c,a has length 27

  39. Problem Size • What is an appropriate measure of problem size? • m the number of nodes? • m(m+1)/2 distances?

  40. Problem Encoding • Use an encoding of the problem • Alphabet of symbols • Strings: abcd//10/5/9//6/9//3

  41. Measures • Problem size: length of the encoding • Time complexity: how long an algorithm takes, as a function of problem size

  42. Time Complexity • What is tractable? • A function f(n) is O(g(n)) whenever f(n) ≤ c. g(n) for n > some constant

  43. Time Complexity A polynomial-time algorithm is one whose time complexity is O(p(n)) for some polynomial p(n) e.g. O(n2)

  44. Time Complexity An exponential-time algorithm is one whose time complexity cannot be bounded by a polynomial e.g. O(nlog n)

  45. Tractability • Basic distinction • Polynomial time means tractable • Exponential time means intractable

  46. Tractability Execution time

  47. Effect of Speed-Ups Let us wait for faster hardware!

  48. Back to TSP • Your boss says: • “Get me an efficient traveling-salesman algorithm, or else!” • What are you going to do?

  49. Response • “Yes Ma’am, expect it this afternoon!”

  50. Response • “Yes Ma’am, expect it this afternoon!” • Problem is • All known algorithms (essentially) check all possible paths • Exhaustive checking is exponential! • … so best of luck!

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