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10
votes
3answers
219 views

Examples of $\Sigma_2^p$ complete problems?

I need a list of $\Sigma_2^p$ complete languages. There are two such problems listed in the Complexity Zoo, namely: Minimum equivalent DNF. Given a DNF formula F and integer k, is there a DNF ...
6
votes
1answer
145 views

Coding theory and complete problems

Coding theory is an useful topic in theoretical computer science. There are known examples of problems coming from coding theory which turn out to be NP complete. My questions are the following: ...
4
votes
0answers
78 views

Variant of Toda's theorem for intermediate levels of the polynomial hierarchy

Is there a version of Toda's theorem for intermediate levels of the polynomial hierarchy ? More precisely, is there any variant of the Toda's theorem that states: Let $\# wSAT$ be the number of ...
3
votes
0answers
83 views

Gradual increase in hardness from P to PH of #SAT

We know that counting the number of solutions to $3-SAT$ is the canonical $\#P-Complete$ problem. Equivalently, it is the canonical $PH-Hard$ problem. However, counting the number of solutions to ...
2
votes
0answers
57 views

Weight enumerator and levels of polynomial hierarchy

Let $A_i$ be the number of codewords in a binary linear code $\mathcal{C}$ of weight $i$. It is known that: $A_k$ is in $P$, where $k = \mathcal{O}(\log_2 n)$. $A_{n}$ is in $\#P-Complete$, ...
3
votes
0answers
101 views

Computing the permanent with polylog size matrices

The complexity of computing the permanent of a $l\times l$ binary matrix is known to be $\#\mathsf{P}$-complete, from the famous result of Valiant, where $l = \Theta(n)$. We know that the problem is ...
11
votes
1answer
219 views

Is the collapse of $PH$ known to extend downward to classes in-between its levels?

Contained in-between each level of the polynomial hierarchy are various complexity classes, including $\Delta_i^{\text{P}}$, $\text{DP}$, $\text{BH}_k$, and $\Sigma_i^\text{P} \cap \Pi_i^\text{P}$. ...
2
votes
0answers
194 views

An analog of DP for the second level of the polynomial hierarchy

The complexity class DP can be defined as the set of all languages that are the intersection of an NP language with a coNP language. I have a language for which I wish to determine the exact ...
4
votes
1answer
323 views

On the proof of Meyer's Theorem

Meyer's theorem is one of the classical results about collapse of the polynomial hierarchy such as famous Karp Lipton's theorem, and states that $EXP \subseteq P/poly \Rightarrow EXP = \Sigma_{2}^{p} ...
8
votes
1answer
556 views

Interesting SUBSET-SUM problems

I know the following variants of SUBSETSUM problems: $ \mathtt{UNARY\mbox{-}SUBSETSUM} \in \mathsf{L} $ (Elberfeld at. al., 2010), NP-complete $ \mathtt{SUBSETSUM} $, and NEXP-complete $ ...
0
votes
1answer
185 views

PH and Optimization Problems

If we have a machine which can solve any problem in the second level of PH, can this machine solve optimization problems which is generalized version of NP-complete problems such as MAX-CLIQUE or ...
8
votes
2answers
361 views

Do good PCPs for NP give us good PCPs for the entire polynomial hierarchy?

The PCP Theorem states that every decision problem in NP has probabilistically checkable proofs (or equivalently, that there exists a complete and quasi-sound proof system for theorems in NP using ...
12
votes
1answer
352 views

Oracle relative to which $\mathsf{BPP}$ is not contained in $Δ_2 \mathsf{P}$

Complexity Zoology by Greg Kuperberg states that there is a language $X$ such that $\mathsf{BPP}^X \nsubseteq \mathsf{\Delta_2 \mathsf{P}}^X$ — in other words, $\mathsf{BPP}^X \nsubseteq ...
6
votes
1answer
542 views

Complexity of Exactly $A$-SAT

Let define the "Exactly $A$-SAT" problem : Given a CNF formula $F$ and a set $A$ of positive integers, is it true that the number of models of $F$ is in $A$? What is the complexity of Exactly ...
8
votes
4answers
1k views

Consequences of $NP=coNP$ and $P\ne NP$?

We know that if $P=NP$ then the whole PH collapses. What if the polynomial hierarchy collapses partially ? (Or how to understand that PH could collapse above a certain point and not below ?) In ...
3
votes
2answers
389 views

Is there a PSPACE-intermediate language?

Suppose PH is strictly contained in PSPACE. Is there a problem in PSPACE that is not in PH and not PSPACE-complete? I encountered a language that is in PSPACE. The question is whether it's in PH. ...
40
votes
0answers
1k views

Can one amplify P=NP beyond P=PH?

In Descriptive Complexity, Immerman has Corollary 7.23. The following conditions are equivalent: 1. P = NP. 2. Over finite, ordered structures, FO(LFP) = SO. This can be thought of as ...
12
votes
5answers
871 views

Why doesn't P=NP imply P=AP (i.e. P=PSPACE)?

It is well known that if $\mathbf{P}=\mathbf{NP}$ then the polynomial hierarchy collapses and $\mathbf{P}=\mathbf{PH}$. This can easily be understood inductively using oracle machines. The question ...
14
votes
2answers
538 views

Is there a Time Hierarchy theorem for PH?

Is it true that there are problems in the polynomial hierarchy solvable in time $O(n^k)$ (by an alternating Turing machine in some level of the polynomial hierarchy) that are not solvable in ...
1
vote
1answer
214 views

Complexity of a certain leaf language with Prime & Composite number of accepting paths.

Given a non-deterministic Turing Machine that runs in polynomial time, it accepts if the number of accepting paths are composite, it rejects if the number of accepting paths are prime and it outputs I ...
7
votes
1answer
357 views

canonical complete problems for $\Delta^P_n$

Finding whether or not a QBF can be satisfied is a canonical complete problem for both $\Sigma^P_n$ (start from $\exists$) and $\Pi^P_n$ (start from $\forall$). What is the canonical complete problem ...
30
votes
4answers
2k views

Is $PH \subseteq PP$?

We know that the first level of the polynomial hierarchy (i.e. NP and co-NP) is in PP, and that $PP \subseteq PSPACE$. We also know from Toda's Theorem that $PH \subseteq P^{PP}$. Do we know whether ...
26
votes
3answers
1k views

A decision problem which is not known to be in PH but will be in P if P=NP

Edit: As Ravi Boppana correctly pointed out in his answer and Scott Aaronson also added another example in his answer, the answer to this question turned out to be “yes” in a way which I had not ...