Geodesic
Realization of Boolean Functions by Formulas in Continuous Bases Containing a Continuum of Constants
Matematičeskie zametki, Tome 92 (2012) no. 2, pp. 181-191.

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For an arbitrary Boolean function of $n$ variables, we show how to construct formulas of complexity $O(2^{n/2})$ in the bases $$ \{x-y,xy,|x|\} \cup [0,1],\qquad \{x-y,x*y,2x,|x|\} \cup [0,1], $$ where ${x*y=\max(-1,\min(1,x))\max(-1,\min(1,y))}$. The obtained estimates are, in general, order-sharp.
Keywords: Boolean function, complexity of the realization of Boolean functions, Shannon function, Lipschitz condition, continuous basis, almost-finite basis. 
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Ya. V. Vegner; S. B. Gashkov. Realization of Boolean Functions by Formulas in Continuous Bases Containing a Continuum of Constants. Matematičeskie zametki, Tome 92 (2012) no. 2, pp. 181-191. http://geodesic.mathdoc.fr/item/MZM_2012_92_2_a1/

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