1. Lecture 1-2 Time and Space of DTM

2. Time of DTM

3. 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)}

4. 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

5. 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)}

6. Model
Multitape TM with write-only output.

7. Linear Speed Up

8. 1--m
Bee dance 3m

9. Model Independent Classes

10. 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.

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

12. 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)}.

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

14. 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)}

15. Tape Compression Theorem

16. 1--m 3m

17. 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

18. 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.

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

20. PSPACE EXPOLY

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

22. “Sufficiently large”

23. A, B P imply A U B P

24. A, B P imply AB P

25. L P implies L* P

26. All regular sets belong to P

27. Space Hierarchy Theorem

28. 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).

29. 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

30. Time Hierarchy

31. 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).

32. 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

33. P EXP
Could you prove

34. PSPACE≠EXP