Computational limitations of small-depth circuits
Computational limitations of small-depth circuits
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
Nondeterministic circuits, space complexity and quasigroups
Theoretical Computer Science
ACM SIGACT News
On limited nondeterminism and the complexity of the V-C dimension
Journal of Computer and System Sciences
On the Complexity of Some Problems on Groups Input as Multiplication Tables
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
On the nlog n isomorphism technique (A Preliminary Report)
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
On the Hardness of Graph Isomorphism
SIAM Journal on Computing
Parity, circuits, and the polynomial-time hierarchy
SFCS '81 Proceedings of the 22nd Annual Symposium on Foundations of Computer Science
Separating the polynomial-time hierarchy by oracles
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
Reductions to Graph Isomorphism
Theory of Computing Systems - Special Section: Algorithmic Game Theory; Guest Editors: Burkhard Monien and Ulf-Peter Schroeder
The complexity of quasigroup isomorphism and the minimum generating set problem
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
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We give a new upper bound for the Group and Quasigroup Isomorphism problems when the input structures are given explicitly by multiplication tables. We show that these problems can be computed by polynomial size nondeterministic circuits of unbounded fan-in with O(log log n) depth and O(log2 n) nondeterministic bits, where n is the number of group elements. This improves the existing upper bound for the problems. In the previous bound the circuits have bounded fan-in but depth O(log2 n) and also O(log2 n) nondeterministic bits. We then prove that the kind of circuits from our upper bound cannot compute the Parity function. Since Parity is AC0-reducible to Graph Isomorphism, this implies that Graph Isomorphism is strictly harder than Group or Quasigroup Isomorphism under the ordering defined by AC0 reductions. We extend this result to the stronger ACC0[p] reduction and its randomized version.