1. Lecture 4 Hierarchy Theorem

2. Space Hierarchy Theorem

3. 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, Space M (x) = s(n).

4. Space Hierarchy If s 2 (n) is a fully space-constructible function, s 1 (n)/s 2 (n) → 0 as n → infinity, s 1 (n) > log n, then DSPACE(s 2 (n)) DSPACE(s 1 (n)) ≠ Φ

7. Input tape (read only) Storage tapes Output tape (possibly, write only)

9. Input tape (read only) Storage tapes Output tape (possibly, write only)

10. Claim Proof

11. Claim Proof

13. Time Hierarchy

14. 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, Time M (x) = t(n).

15. Time Hierarchy If t 1 (n) > n+1, t 2 (n) is fully time-constructible, t 1 (n) log t 1 (n) /t 2 (n) → 0 as n → infinity, then DTIME(t 2 (n)) DTIME(t 1 (n)) ≠ Φ

16. 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 n c c

17. P c EXP

18. EXP ≠ PSAPACE

19. PSPACE≠EXP

20. PSPACE c EXPOLY

21. Problems in P Are they in P? Sorting minimum spanning tree shortest path maximum flow

22. Problems in P None of following is in P? Sorting minimum spanning tree shortest path maximum flow They are all polynomial-time computable functions

23. P contains only languages or decision problems A decision problem is a problem who has only two answers, YES and NO. A decision problem can be described by a language consisting of all inputs at which YES answer would be obtained.

24. Every optimization problem has a decision version Minimum spanning tree Decision version of minimum spanning tree

25. For optimization problem with integer value, the decision version is equivalent to it.

26. Problem in EXP Traveling Salesman Problem Minimum Vertex Cover Hamiltonian Cycle Satisfiability Partition

27. Edmonds Conjecture in 1965 Traveling Salesman Problem cannot be solved in polynomial time.