Bounded queries, approximations, and the Boolean hierarchy
Information and Computation
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COCOON '98 Proceedings of the 4th Annual International Conference on Computing and Combinatorics
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Some connections between bounded query classes and non-uniform complexity
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On the computational complexity of qualitative coalitional games
Artificial Intelligence
Proving SAT does not have small circuits with an application to the two queries problem
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A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
CSL '08 Proceedings of the 22nd international workshop on Computer Science Logic
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STACS'99 Proceedings of the 16th annual conference on Theoretical aspects of computer science
The 1-versus-2 queries problem revisited
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
A tight Karp-Lipton collapse result in bounded arithmetic
ACM Transactions on Computational Logic (TOCL)
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We show that if the Boolean hierarchy collapses to level $k$, then the polynomial hierarchy collapses to $BH_{3}(k)$, where $BH_{3}(k)$ is the $k$th level of the Boolean hierarchy over $\Sigma^{P}_{2}$. This is an improvement over the known results, which show that the polynomial hierarchy would collapse to $P^{NP^{NP}[O(\log n)]}$. This result is significant in two ways. First, the theorem says that a deeper collapse of the Boolean hierarchy implies a deeper collapse of the polynomial hierarchy. Also, this result points to some previously unexplored connections between the Boolean and query hierarchies of $Delta^{P}_{2}$ and $Delta^{P}_{3}$. Namely, \begin{eqnarray*} & BH(k) = {\rm co-}BH(k) \implies BH_{3}(k) = {\rm co-}BH_{3}(k), \\[\jot] & P^{{\rm NP}\|[k]} = P^{{\rm NP}\|[k+1]} \implies P^{{\rm NP}^{\rm NP}\|[k+1]} = P^{{\rm NP}^{\rm NP}\|[k+2]}. \end{eqnarray*}