1. Part II Theory of Nondeterministic Computation

2. Cook introduced NP class in 1971
Edmonds Conjecture in 1965
Traveling Salesman Problem cannot be solved in polynomial time.
Cook introduced NP class in 1971
NP: Nondeterminastic Polynomial-time class.

3. 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.
Is Nondeterministic TM a reasonable computational model?

4. Lecture 2-1 Time and Space of NTM

5. Nondeterministic TM (NTM)
There are multiple choices for each transition.
For each input x, the NTM may have more than one computation paths.
An input x is accepted if at least one computation path leads to the final state.
L(M) is the set of all accepted inputs.

6. How many choices?
# of states
# of tape-symbols

7. A function f(x) is computed by an NTM M if for every x ε L(M), M gives output f(x) whenever it reaches the final state.
Theorem. A function can be computed by an NTM iff it is Turing-computable.
Proof. Idea: Enumerate all possible computation paths of certain length from small to large.
Implement: set a tape to do the following enumeration.

8. Suppose for each transition, there are at most c choices
Suppose for each transition, there are at most c choices. Then on the enumeration tape, the DTM enumerate all strings on alphabet {a1, a2, …, ac}: ε, a1, a2, …, ac, a1a2, a1a3, ….
When a string ai1ai2∙∙∙aim is written on the enumeration tape, the DTM simulates the NTM by making the ij –th choice at the j-th move.
DTM halts iff it found a computation path of NTM which halts.

9. Question: How many moves does a DMT needs to simulate t moves of a NTM?

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

11. 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,
TimeM(x) < max {n+1, t(n)} .

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

13. Relationship
P NP
NP ≠ EXP
NP EXPOLY

14. NP≠EXP

15. Theorem
Speed Up Theorem still holds.
Hierarch Theorem

16. Space For a NTM M and an input x,
SpaceM(x) = the minimum space, over all
computation paths, on input x
if x ε L(M)
= infinity
otherwise

17. 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,
SpaceM(x) ≤ max{k, s(n)}
k = # of work tapes

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

19. Relationship
NP NPSPACE
PSPACE = NPSPACE (why?)

20. Savich’s Theorem If s(n) ≥ log n, then NSPACE (s(n)) DSPACE(s(n) )
The proof will be given later! 2

21. Theorems
Compression Theorem holds.
Hierarchy Theorem holds.

22. Translation Lemma
Let s1(n), s2(n) and f(n) be fully space-constructible functions with s2(n) > n and f(n) > n. Then
NSPACE(s1(n)) NSPACE(s2(n))
implies
NSPACE(s1(f(n))) NSPACE(s2(f(n)))

23. Hierarchy NSPACE(n ) DSPACE(n ) DSPACE(n ) NSPACE(n )
For r > 1 and a > 0,
NSPACE(n ) ≠ NSPACE (n ) 4 8 9 ≠ 9
r+a r

25. Proof of Savitch’s Theorem

26. Savich’s Theorem If s(n) ≥ log n, then NSPACE (s(n)) DSPACE(s(n) )2
Proof.

27. How many configurations of M exist within space s(n)?
Input tape x1 x2 xn
Work tape

30. Depth-first Search

31. P NP
PSPACE P
EXP
EXPOLY
PSPACE
EXPOLY