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

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

4. For string of length time needed for acceptance:

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

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

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

9. Theorem:

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

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

12. CYK-algorithm

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

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

15. s t YES!

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

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

18. Example: Satisfiable:

19. Example: Not satisfiable

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

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

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

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

26. Non-Deterministic Polynomial time algorithms:

27. The class for all Non-Deterministic Polynomial time

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

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

30. The satisfiability problem is an - Problem

31. Observation: Deterministic Polynomial Non-Deterministic Polynomial

32. Open Problem: WE DO NOT KNOW THE ANSWER

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