Lecture 23 Space Complexity of DTM

1. Lecture 23 Space Complexity of DTM

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

3. Space Bound • A DTM is said to have a space bound s(n) if for any input x with |x| < n, SpaceM(x) < max{1, s(n)}.

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

5. 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)}

6. Tape Compression Theorem • For any function s(n) and any constant c > 0, DSPACE(s(n)) = DSPACE(c·s(n))

7. 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

8. 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.

9. P PSPACE

10. PSPACE EXPOLY

11. A, B ε P imply A U B ε P

12. A, B ε P imply AB ε P

13. L ε P implies L* ε P

14. All regular sets belong to P

15. Hierachy Theorem

16. 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).

17. 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

18. 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).

19. 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

20. P EXP

21. EXP ≠ PSAPACE