Signed Digit Addition and Related Operations with Threshold Logic
IEEE Transactions on Computers
Practical low-cost CPL implementations threshold logic functions
GLSVLSI '01 Proceedings of the 11th Great Lakes symposium on VLSI
Uniform constant-depth threshold circuits for division and iterated multiplication
Journal of Computer and System Sciences - Complexity 2001
Number-theoretic constructions of efficient pseudo-random functions
Journal of the ACM (JACM)
Powering requires threshold depth 3
Information Processing Letters
Separating AC0 from depth-2 majority circuits
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Cryptographic hardness for learning intersections of halfspaces
Journal of Computer and System Sciences
Neural network architecture selection: can function complexity help?
Neural Processing Letters
Energy complexity and depth of threshold circuits
FCT'09 Proceedings of the 17th international conference on Fundamentals of computation theory
Counting, fanout and the complexity of quantum ACC
Quantum Information & Computation
Decomposition of threshold functions into bounded fan-in threshold functions
Information and Computation
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Let $\widehat{LT}_d$ denote the class of functions that can be computed by depth-$d$ threshold circuits with polynomial size and polynomially bounded integer weights. Using the results in [M. Goldman, J. Hastad, and A. Razborov, in Proc. 7th Annual Conference on Structure in Complexity Theory], [M. Goldman and M. Karpinski, Constructing depth d+1 majority circuits that simulate depth d threshold circuits, unpublished] we show that multiple sum is in $\widehat{LT}_2$, and multiplication and division are in $\widehat{LT}_3$. Moreover, it follows from the lower-bound results in [A. Hajnal et al., IEEE Sympos. Foundations of Comput. Sci., 28 (1987), pp. 99--110], [T. Hofmeister and P. Pudlak, Forschungbericht Nr. 477 Uni Dortmund, 1992] that these threshold circuits are optimal in circuit depth. The authors also indicate that these techniques can be applied to construct polynomial-size depth-3 threshold circuits for powering and depth-4 threshold circuits for multiple product.