1. Lecture 24 Time and Space of NTM

2. Time For a NDM M and an input x, Time M (x) = the minimum # of moves leading to accepting x if x ε L(M) = infinity if x not in L(M)

3. Time Bound A NTM M is said to have a time bound t(n) if for sufficiently large n and every x ε L(M) With |x|=n, Time M (x) < max {n+1, t(n)}.

4. Complexity Classes NTIME(t(n)) = {L(M) | M is a NTM with time bound t(n)} NP = U c > 0 NTIME(n ) c

5. Relationship P NP NP ≠ EXP NP EXPOLY

7. Theorem Speed Up Theorem still holds. Hierarchy Theorem may not.

8. Space For a NTM M and an input x, Space M (x) = the minimum, over all computation paths, of maximum space taken in each work tape on input x if x ε L(M) = infinity otherwise

9. Space bound A NTM M is said to have a space bound s(n) if sufficiently large n and every input x with |x|=n, Space M (x) ≤ max{1, s(n)}

10. Complexity Classes NSPACE(s(n)) = {L(M) | M is a NTM with space bound s(n)} NSPACE = U c>0 NSPACE(n ) c

11. Relationship NP NSPACE PSAPACE = NPSPACE (why?)

12. Savich’s Theorem If s(n) ≥ log n, then NSPACE (s(n)) DSPACE(s(n) ) The proof will be given in next lecture! 2

13. Theorems Compresion Theorem holds. Hierarchy Theorem may not.

14. Translation Lemma Let s 1 (n), s 2 (n) and f(n) be fully space- constructible functions with s 2 (n) > n and f(n) > n. Then NSPACE(s 1 (n)) NSPACE(s 2 (n)) implies NSPACE(s 1 (f(n))) NSPACE(s 2 (f(n)))

15. Hierarchy NSPACE(n ) DSPACE(n ) DSPACE(n ) NSPACE(n ) For r > 1 and a > 0, NSPACE(n ) ≠ NSPACE (n ) 48 ≠ 9 9 r r+a

16. Proof of Savitch’s Theorem