Geodesic
The complexity of the realization of symmetrical functions by formulae
Matematičeskie zametki, Tome 11 (1972) no. 1, pp. 109-120

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It is proved that every symmetric function in $k$-valued logic of $n$ arguments can be realized by a formula in any basis, the complexity of the formula not exceeding $n^C$, where $C$ is a constant depending on the basis. It is shown that in the case $k=2$, $C\leqslant 4,93$ for all bases. 
@article{MZM_1972_11_1_a12,
     author = {V. M. Khrapchenko},
     title = {The complexity of the realization of symmetrical functions by formulae},
     journal = {Matemati\v{c}eskie zametki},
     pages = {109--120},
     publisher = {mathdoc},
     volume = {11},
     number = {1},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_11_1_a12/}
}
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V. M. Khrapchenko. The complexity of the realization of symmetrical functions by formulae. Matematičeskie zametki, Tome 11 (1972) no. 1, pp. 109-120. http://geodesic.mathdoc.fr/item/MZM_1972_11_1_a12/