Welcome to MTH401A!! Theory of Computation

Size: px

Start display at page:

Download "Welcome to MTH401A!! Theory of Computation"

Horace Walker
5 years ago
Views:

1 Welcome to MTH401A!! Theory of Computation

2 Contact Information Instructor : Office : 327, Faculty Building Phone : 7880 or akasha@iitk.ac.in Web : Office Hours : After class Any other times : By (preferred) By appointment

3 Question! Can computers solve all computational problems?

4 Question! Can computers solve all computational problems? No!

5 Question! Can computers solve all computational problems? No! One of the primary goals of this course is to formally show that no is indeed the correct answer.

6 Question! Can computers solve all computational problems? No! One of the primary goals of this course is to formally show that no is indeed the correct answer. we need to make sense of the question itself

7 Question! Can computers solve all computational problems? No! One of the primary goals of this course is to formally show that no is indeed the correct answer. we need to make sense of the question itself

8 Computation

9 Computation Problem: Given two numbers, find their sum.

10 Computation Problem: Given two numbers, find their sum. Given numbers a and b add multistep procedure involving symbolic representation, reading, writing, remembering Produce a number c such that c = a + b

11 Computation Problem: Given two numbers, find their sum. Given numbers a and b add multistep procedure involving symbolic representation, reading, writing, remembering Produce a number c such that c = a + b Problem Instance: What is SEVEN plus THREE?

12 Computation Problem: Given two numbers, find their sum. Given numbers a and b add multistep procedure involving symbolic representation, reading, writing, remembering Produce a number c such that c = a + b Problem Instance: What is SEVEN plus THREE? symbol 7 3 symbol

13 Computation Problem: Given two numbers, find their sum. Given numbers a and b add multistep procedure involving symbolic representation, reading, writing, remembering Produce a number c such that c = a + b Problem Instance: What is SEVEN plus THREE?

14 Computation Problem: Given two numbers, find their sum. Given numbers a and b add multistep procedure involving symbolic representation, reading, writing, remembering Produce a number c such that c = a + b Problem Instance: What is SEVEN plus THREE? Computation: =

15 Computation Problem: Given two numbers, find their sum. Problem Instance: What is One million five hundred eighty eight thousand two hundred and seventy eight plus three million two hundred ninety two thousand eight hundred fifty nine?

16 Computation Problem: Given two numbers, find their sum. Problem Instance: What is One million five hundred eighty eight thousand two hundred and seventy eight plus three million two hundred ninety two thousand eight hundred fifty nine?

17 Computation Problem: Given two numbers, find their sum. Problem Instance: What is One million five hundred eighty eight thousand two hundred and seventy eight plus three million two hundred ninety two thousand eight hundred fifty nine? Computation: =

18 Computation Both computations produce unique answer to a given problem.

19 Computation Both computations produce unique answer to a given problem. Computational problem can be viewed as a function: Every valid input has a desired output.

20 Computation Both computations produce same answer to a given problem. Computational problem can be viewed as a function: Every valid input has a desired output. Computation: the process of deriving the desired output from given input(s).

21 Description Languages, Machines, etc.

22 Description Languages, Machines, etc. Computational Problem: Given A and B, compute C, their sum.

23 Description Languages, Machines, etc. Computational Problem: Given A and B, compute C, their sum. Simplified Computational Problem: Given A, B and C, check if C is the sum of A and B.

24 Description Languages, Machines, etc. Computational Problem: Given A and B, compute C, their sum. Simplified Computational Problem: Given A, B and C, check if C is the sum of A and B. Output is YES or NO.

25 Description Languages, Machines, etc. Computational Problem: Compute all primes p less than N.

26 Description Languages, Machines, etc. Computational Problem: Compute all primes p less than N. Simplified Computational Problem: Given p, check if p is a prime. Output is YES or NO.

27 Description Languages, Machines, etc. Computational Problem: Compute all primes p less than N. Simplified Computational Problem: Given p, check if p is a prime. Output is YES or NO. Solution of this problem is equivalent to checking the membership of the set { p : p is a prime }

28 Description Languages, Machines, etc. Computational Problem: Compute all primes p less than N. Simplified Computational Problem: Given p, check if p is a prime. Output is YES or NO. Solution of this problem is equivalent to checking the membership of the set { p : p is a prime } = { 2, 3, 5, 7, 11, 13, }

29 Description Languages, Machines, etc. Computational Problem: Compute all primes p less than N. Simplified Computational Problem: Given p, check if p is a prime. Output is YES or NO. members are strings of symbols coming from {0,1, 2, 3, 4, 5, 6, 7, 8, 9} Solution of this problem is equivalent to checking the membership of the set { p : p is a prime } = { 2, 3, 5, 7, 11, 13, }

30 Description Languages, Machines, etc. Computational Problem: Compute all primes p less than N. Simplified Computational Problem: Given p, check if p is a prime. Output is YES or NO. Solution of this problem is equivalent to checking the membership of the set { p : p is a prime } = { 2, 3, 5, 7, 11, 13, } = { 10, 11, 101, 111, 1011, 1101, }

31 Description Languages, Machines, etc. Computational Problem: Compute all primes p less than N. Simplified Computational Problem: Given p, check if p is a prime. Output is YES or NO. members are strings of symbols coming from {0,1} Solution of this problem is equivalent to checking the membership of the set { p : p is a prime } = { 2, 3, 5, 7, 11, 13, } = { 10, 11, 101, 111, 1011, 1101, }

members are strings of symbols coming from {0,1} Solution of this problem is equivalent to checking

32 Description Languages, Machines, etc. Computational Problem: Compute all primes p less than N. Language! Simplified Computational Problem: Given p, check if p is a prime. Output is YES or NO. members are strings of symbols coming from {0,1} Solution of this problem is equivalent to checking the membership of the set { p : p is a prime } = { 2, 3, 5, 7, 11, 13, } = { 10, 11, 101, 111, 1011, 1101, }

33 Description Languages, Machines, etc. Formalizing and understanding computing utilizes the idea of languages.

34 Description Languages, Machines, etc. Formalizing and understanding computing utilizes the idea of languages. Two ways of describing a language Machine description. Grammar description.

35 Course Outline

36 Course Outline Regular languages Finite state machines. Non-deterministic finite state machines. Regular expressions. Algorithms to decide about regular languages.

37 Course Outline Context free languages Context free grammars. Pushdown automata. Algorithms for context free grammars.

38 Course Outline Recursive and recursively enumerable languages Turing machines. Decidability of problems. What can be computed? NP-completeness and beyond.

39 Evaluation Policy Quizzes (Short, Unannounced) 20% Mid Semester Exam 30% End Semester Exam 50%

40 Books Elements of the Theory of Computation Christos H. Papadimitriou Introduction to Automata Theory, Languages, and Computation John E. Hopcroft, Rajeev Motwani, Jeffrey D. Ullman

Source of Slides: Introduction to Automata Theory, Languages, and Computation By John E. Hopcroft, Rajeev Motwani and Jeffrey D.

Source of Slides: Introduction to Automata Theory, Languages, and Computation By John E. Hopcroft, Rajeev Motwani and Jeffrey D. Ullman And Introduction to Languages and The by J. C. Martin Basic Mathematical