Bounded-depth, polynomial-size circuits for symmetric functions
Theoretical Computer Science
Almost optimal lower bounds for small depth circuits
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Log depth circuits for division and related problems
SIAM Journal on Computing
Journal of Computer and System Sciences - 3rd Annual Conference on Structure in Complexity Theory, June 14–17, 1988
Fast parallel arithmetic via modular representation
SIAM Journal on Computing
On threshold circuits and polynomial computation
SIAM Journal on Computing
The complexity of iterated multiplication
Information and Computation
A theorem on probabilistic constant depth Computations
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Uniform Circuits for Division: Consequences and Problems
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Parity, circuits, and the polynomial-time hierarchy
SFCS '81 Proceedings of the 22nd Annual Symposium on Foundations of Computer Science
Structural Complexity and Neural Networks
WIRN VIETRI 2002 Proceedings of the 13th Italian Workshop on Neural Nets-Revised Papers
The Complexity of Membership Problems for Circuits over Sets of Natural Numbers
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
Affine image matching is uniform TC0-complete
CPM'10 Proceedings of the 21st annual conference on Combinatorial pattern matching
On the complexity of szilard languages of regulated grammars
ICTAC'11 Proceedings of the 8th international conference on Theoretical aspects of computing
Determinism versus nondeterminism with arithmetic tests and computation: extended abstract
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Lower bounds for RAMs and quantifier elimination
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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Integer division has been known since 1986 [4, 13, 12] to be in slightly non-uniform TC°, i.e., computable by polynomial-size, constant depth threshold circuits. This has been perhaps the outstanding natural problem known to be in a standard circuit complexity class, but not known to be in its uniform version. We show that indeed division is in uniform TC°. A key step of our proof is the discovery of a first-order formula expressing exponentiation modulo any number of polynomial size.