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 a b c d e f
initial 3m q

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

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

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

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

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

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

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

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

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

19. 1--m a’ b’ c’ d’ e’ 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
P = U c>0 DTIME(n )
EXP = U c > 0 DTIME(2 )
EXPOLY = U c > 0 DTIME(2 )
PSPACE = U c > 0 DSPACE(n ) c 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