Lecture 1-2 Time and Space of DTM

1. Lecture 1-2 Time and Space of DTM

2. Model • Multitape TM with write-only output.

3. Time of DTM

4. Time Bound M is said to have a time bound t(n) if for every x with |x| < n, TimeM(x) < max {n+1, t(n)}

5. Theorem • For any multitape DTM M, there exists a one-tape DTM M’ to simulate M within time TimeM’(x) < c + (TimeM(x)) c is a constant. 2

6. Complexity Class • A language L has a (deterministic) time-complexity t(n) if there is a multitape DTM M accepting L, with time bound t(n). • DTIME(t(n)) = {L | L has a time bound t(n)}

7. Linear Speed Up

8. 1--m Bee dance 3m q

9. 1--m b c a e f d initial 3m q

10. 1--m b c a e f d 1st bee dance 3m c d q

11. 1--m b c a e d f 1st bee dance 3m e d c f q

12. 1--m b c a e d f 1st bee dance 3m e d c f q

13. 1--m b c a e d f 1st bee dance 3m a e d c b f q

14. 1--m b c a e d f 3m a’ e’ d’ c’ b’ f’ p

15. 1--m b c’ a e d’ f 2nd bee dance 3m a’ e’ b’ f’ p

16. 1--m b c’ a e’ d’ f’ 2nd bee dance 3m a’ b’ p

17. 1--m b c’ a e’ d’ f’ 2nd bee dance 3m a’ b’ p

18. 1--m b’ c’ a’ e’ d’ f’ 2nd bee dance 3m p

19. 1--m b’ c’ a’ e’ d’ f’ initial 3m p

20. Model Independent Classes

21. Space • SpaceM(x) = total # of cells that M visits on all working (storage) tapes during the computation on input x. • If M is a multitape DTM, then the working tapes do not include the input tape and the write-only output tape.

22. Multi-tape DTM Input tape (read only) working tapes Output tape (possibly, write only)

23. Space Bound • A DTM with k work tapes is said to have a space bound s(n) if for any input x with |x| < n, SpaceM(x) < max{k, s(n)}.

24. Time and Space • For any DTM with k work tapes, SpaceM(x) < k (TimeM(x) + 1)

25. Complexity Classes • A language L has a space complexity s(n) if it is accepted by a multitape with write-only output tape DTM with space bound s(n). • DSPACE(s(n)) = {L | L has space complexity s(n)}

26. Tape Compression Theorem

27. 1--m 3m

28. Model Independent Classes c • P = U c>0 DTIME(n ) • EXP = U c > 0 DTIME(2 ) • EXPOLY = U c > 0 DTIME(2 ) • PSPACE = U c > 0 DSPACE(n ) cn c n c

29. Extended Church-Turing Thesis • A function computable in polynomial time in any reasonable computational model using a reasonable time complexity measure is computable by a DTM in polynomial time.

30. P PSPACE • SpaceM(x) < k (TimeM(x) + 1)

31. PSPACE EXPOLY

32. Input tape (read only) working tapes Output tape (possibly, write only)

33. “Sufficiently large”

34. A, B P imply A U B P

35. A, B P imply AB P

36. L P implies L* P

37. All regular sets belong to P

38. Space Hierarchy Theorem

39. Space-constructible function • s(n) is fully space-constructible if there exists a DTM M such that for sufficiently large n and any input x with |x|=n, SpaceM(x) = s(n).

40. Space Hierarchy If • s2(n) is a fully space-constructible function, • s1(n)/s2(n) → 0 as n → infinity, • s1(n) > log n, then DSPACE(s2(n)) DSPACE(s1(n)) ≠ Φ

41. Time Hierarchy

42. Time-constructible function • t(n) is fully time-constructible if there exists a DTM such that for sufficiently large n and any input x with |x|=n, TimeM(x) = t(n).

43. Time Hierarchy If • t1(n) > n+1, • t2(n) is fully time-constructible, • t1(n) log t1(n) /t2(n) → 0 as n → infinity, then DTIME(t2(n)) DTIME(t1(n)) ≠ Φ

44. P EXP Could you prove

46. PSPACE≠EXP