Part II Theory of Nondeterministic Computation

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.

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?

Lecture 2-1 Time and Space of NTM

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.

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

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.

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.

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

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)

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)} .

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

Relationship P NP NP ≠ EXP NP EXPOLY

NP≠EXP

Theorem Speed Up Theorem still holds. Hierarch Theorem

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

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

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

Relationship NP NPSPACE PSPACE = NPSPACE (why?)

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

Theorems Compression Theorem holds. Hierarchy Theorem holds.

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)))

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

Proof of Savitch’s Theorem

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

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

Depth-first Search

P NP PSPACE P EXP EXPOLY PSPACE EXPOLY