1. 1 Time Complexity We use a multitape Turing machine We count the number of steps until a string is accepted We use the O(k) notation

2. 2 Example: Algorithm to accept a string : Use a two-tape Turing machine Copy the on the second tape Compare the and

3. 3 Copy the on the second tape Compare the and Time needed: Total time:

4. 4 For string of length time needed for acceptance:

5. 5 Language class: A Deterministic Turing Machine accepts each string of length in time

6. 6

7. 7 In a similar way we define the class for any time function: Examples:

8. 8 Example: The membership problem for context free languages (CYK - algorithm) Polynomial time

9. 9 Theorem:

10. 10 Polynomial time algorithms: Represent tractable algorithms: For small we can compute the result fast

11. 11 for all The class All tractable problems Polynomial time

12. 12 CYK-algorithm

13. 13 Exponential time algorithms: Represent intractable algorithms: Some problem instances may take centuries to solve

14. 14 Example: the Hamiltonian Problem Question: is there a Hamiltonian path from s to t? s t

15. 15 s t YES!

16. 16 Exponential time Intractable problem A solution: search exhaustively all paths L = { : there is a Hamiltonian path in G from s to t}

17. 17 Example: The Satisfiability Problem Boolean expressions in Conjunctive Normal Form: Variables Question: is expression satisfiable?

18. 18 Satisfiable: Example:

19. 19 Not satisfiable Example:

20. 20 For variables: Algorithm: search exhaustively all the possible binary values of the variables exponential

21. 21 Non-Determinism Language class: A Non-Deterministic Turing Machine accepts each string of length in time

22. 22 Example: Non-Deterministic Algorithm to accept a string : Use a two-tape Turing machine Guess the middle of the string and copy on the second tape Compare the two tapes

23. 23 Time needed: Total time: Use a two-tape Turing machine Guess the middle of the string and copy on the second tape Compare the two tapes

24. 24

25. 25 In a similar way we define the class for any time function: Examples:

26. 26 Non-Deterministic Polynomial time algorithms:

27. 27 for all The class Non-Deterministic Polynomial time

28. 28 Example: The satisfiability problem Non-Deterministic algorithm: Guess an assignment of the variables Check if this is a satisfying assignment

29. 29 Time for variables: Total time: Guess an assignment of the variables Check if this is a satisfying assignment

30. 30 The satisfiability problem is an - Problem

31. 31 Observation: Deterministic Polynomial Non-Deterministic Polynomial

32. 32 Open Problem: WE DO NOT KNOW THE ANSWER

33. 33 Example: Does the Satisfiability problem have a polynomial time deterministic algorithm? WE DO NOT KNOW THE ANSWER Open Problem: