# Tag Info

Accepted

### Oracle Construction for Grover's Algorithm

The oracle is basically just an implementation of the predicate you want to search for a satisfying solution to. For example, suppose you have a 3-sat problem: ...
• 1,458
Accepted

### Is there an oracle such that SAT is not infinitely often in sub-exponential time?

You can just take the oracle A s.t. NP$^A$=EXP$^A$ since EXP is not in i.o.-subexp. For SAT$^A$ it depends on the encoding, for example if the only valid SAT instances have even length then it is easy ...
• 8,546
Accepted

• 14.8k
Accepted

### Oracle comparing $EXP$ with $UP$

$\mathsf{UP} \neq \mathsf{EXP}$ is open. A UP-generic oracle* should make $\mathsf{P} \neq \mathsf{UP} = \mathsf{EXP}$, and since $\mathsf{UP} \subseteq \mathsf{\oplus P} \subseteq \mathsf{EXP}$ ...
• 35.7k
Accepted

### Is there a simplex-like algorithm that can be used with a separation oracle?

I'm not sure if you would consider the algorithms I'll discuss here "Simplex-like" (but see the comment about column generation at the end). If you have a weak separation oracle for a ...
• 8,143
Accepted

### Does there exist an oracle $A$ such that $(P^{\#P})^{A} \neq PSPACE^{A}$?

On popular request, here is my comment as an answer: There is an oracle separating $\mathrm{PP}$ from $\mathrm{PSPACE}$: Jacobo Toran, A combinatorial technique for separating counting complexity ...
• 2,828
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### Lower bounds for nonuniform circuits and oracles separating complexity classes

Yes, yes, and yes. The basic idea is to consider the characteristic function of a language $L$ (the oracle you're constructing) at length $n$ as a string of length $2^n$ that will be an input to a ...
• 35.7k

### Results comparing BQP and NEXP

The oracle you ask for has $P=NP\ne BQP=NEXP$, and therefore it has $BQP\ne PH$. Finding any oracle relative to which $BQP\ne PH$ was an open problem for twenty years until Raz and Tal [1] found such ...
• 1,704

### Lower bound on alternations needed in $BQP$ versus $PH$ result?

If you just want oracle separations with $\#P$, you don't need to use the new result of Raz and Tal. You can use the classic Parity/Majority not in $AC^0$ results from the 1980s. For example, the ...
• 13.3k

### P vs. NP in a logic with a random oracle

Yes on Question 1 (assuming ZFC is consistent). You don't need $f$ to be random exactly, any $f$ will do. And for the proof you need to also use the fact that there is an oracle $h$ with NP$^h=$P$^h$.
• 4,400
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### Is it possible to reduce an NP language to a NEXP language with exponentially smaller input length?

This is quite unlikely to hold, because $\mathrm{EXP_{poly}^{NEXP}}$ actually coincides with $\Theta^{\exp}_2$, the exponential analogue of the class $\Theta^P_2$, which is presumably a strict ...
• 14.8k
Accepted