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CS221: Computational Complexity Prof. Salil Vadhan
Lecture 13: Does the PH collapse? and an application of
alternation, DTIME(n) 6 = DTIME(n)
10/21 Scribe: Grant Schoenebeck
Contents
1 Recap on Polynomial Hierarchy 1
2 Does PH Collapse? 1
3 An application of Alternation(Paul, Pippenser, Szemeredi, Trotter) 3
1 Recap on Polynomial Hierarchy
ΣkP: can have up to k alternations and starts with an exist alternation.
ΠkP: can have up to k alternations and starts with a for all alternation.
P H =
k ΣkP
QBFk = {ϕ(x 1 ,... , xk) : ∃x 1 , ∀x 2 ,... Qxk such that ϕ(x 1 ,... , xk) = 1}
Theorem 1 QBFk is complete for ΣkP
Theorem 2 ΣkP = NPΣk−^1 P(= NPΠk−^1 P)
ΠkP = co-NPΣk−^1 P
Definition 3 ∆kP = PΣk−^1 P
2 Does PH Collapse?
Theorem 4 ΣkP = ΠkP ⇒ ∀j ≥ k Σj P = Πj P = ΣkP. That is to say that the the PH would collapse to the kth level.
Remark 5 This is part of the general trend that equality translates upward and inequality translates downward.
Intuitively the classes are not closed under complement, and if they were the hierarchy would fall.
Proof:
It suffices to show that ΣkP = ΠkP ⇒ QBFk ∈ ΣkP. This would then imply that Σk+1P = ΣkP and it follows that that Σk+1P = Πk+1P. Then we can proceed by induction.
So we examine the problem ∃x 1 , ∀x 2 ,... , Qxk such that ϕ(x 1 ,... , xk) = 1. If we fix x 1 we get an instance of QBFk ∈ ΠkP. Call the formula with a fixed x 1 ψx 1. (It begins with a for all clause.) But by assumption ΠkP = ΣkP so QBFk ∈ ΣkP. So now we can guess x 1 and run the ΣkP algorithm on ψx 1 , but this is also a ΣkP algorithm.
Corollary 6 P = NP ⇒ PH collapses. PH = P = NP.
Corollary 7 NP = co-NP ⇒ PH collapses to NP.
Remark 8 Normally when we show such results we show them with a com- plete language and a reduction. For instance, we can reduce any language in NP to SAT so if SAT ∈ P ⇒ NP ⊆ P, but here we have shown that SAT ∈ P ⇒ PH = P without ever giving a reduction.
If NP ⊆ TIME(t(n)) ⇒ PH ⊂?? or ΣkP ⊂??. What if NTIME(n^2 ) = TIME(n^2 )?
Conjecture 9 PH does not collapse.
Evidence for it:
- Analogy with Arithmetic Hierarchy in Recursion Theory (known to work). But not everything seems analogous, for instance in recursion theory the recursive languages = RE languages ∩ coRE languages, but it seems unlikely that P = NP ∩ co-NP.
this means is that we divide the computations that M performs into a(n) = f (n) b(n) =^
f (n) segments of length
f (n).
How do we make such a TM? In some way we copy the information from the adjacent blocks to the current block. The specifics are messy.
Define a computational graph G [Valient]. The vertices of such a graph V = { 1 ,... , a(n)} correspond to the time segments. The edges are (i, i+1)∀i and (i, j) if for some tape of Mb the block of cells used in time j was last used in time i (this graph represents information flow through time in the computation).
Observe structural properties of graph. In degree ≤ k + 1 (k = number of tapes). Can decompose it into the union of 2k + 1 graphs of degree ≤ 1 (no crossing).
(Hard) Graph-theoretic lemma. Every such graph has o(a(n)) segregators of size o(a(n)). Can remove o(a(n)) ”predecessors” not necessarily immedi- ate.
Σ 4 simulation. ∃ Guess computations in removed time steps. ∀ Check that the computations link together well. (they have very few predecessors so there is relatively little to guess and to check).
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