1. Recall last lecture and Nondeterministic TMs Ola Svensson

2. Computational Complexity

3. TURING MACHINES

4. The goals A model that can compute anything computable A model so simple and clean so that we can prove interesting results

5. start symbolblank symbol

6. IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft StayLeft Stay

7. Example IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft StayLeft Stay Input: Work/output: … …

8. Example IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft StayLeft Stay Input: Work/output: … …

9. Example IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft StayLeft Stay Input: Work/output: … …

10. Example IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft StayLeft Stay Input: Work/output: … …

11. Example IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft StayLeft Stay Input: Work/output: … …

12. Example IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft StayLeft Stay Input: Work/output: … …

13. Example IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft StayLeft Stay Input: Work/output: … …

14. Example IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft StayLeft Stay Input: Work/output: … …  Hey, don’t miss this!

15. Example IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft StayLeft Stay Input: Work/output: … …

16. Example IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft StayLeft Stay Input: Work/output: … …

17. Example IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft StayLeft Stay Input: Work/output: … … What do we compute?

18. Efficiency and Running Time

22. Robustness of definition of Turing Machine The size of the alphabet doesn’t really matter The number of tapes do not really matter (up to a square in the running time) It is able to simulate all other realistic models with polynomial slow down (except potentially quantum computers that still need to be built) Church-Turing Thesis: Any computable function can be computed by a Turing machine

24. Universal Turing Machine

25. DIAGONALIZATION

26. Cantor’s argument The set of infinite binary strings are uncountable … s is different from any string in the enumeration so it is not part of the enumeration

27. Halting Problem …

28. …

29. Deterministic Time Hierarchy …

30. …

31. Diagonalization The proof technique used two things about TMs 1.They can be encoded efficiently 2.There is a TM that can emulate any other without increasing running time or space too much Otherwise, they treated TMs as ”black boxes” We will show today that such a proof cannot alone resolve P vs NP

32. NONDETERMINISTIC TURING MACHINES

33. Nondeterministic Turing machines (NDTMs)

34. Example IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft Input: Work/output: … …

35. Example Input: Work/output: … … IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft

36. Example Input: Work/output: … … IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft

37. Example Input: Work/output: … … IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft

38. Example Input: Work/output: … … IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft

39. Example Input: Work/output: … … IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft

40. Example Input: Work/output: … … IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft

41. Example Input: Work/output: … …  Now normal deterministic execution… IFTHEN Input symbol read Work (output) tape read Current state Move input head New work (output) symbol Move work head New state Right StayLeft What language do we accept?