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Realization of Boolean functions by formulas in continuous bases containing a continuum of constants. (English. Russian original) Zbl 1283.06025
Math. Notes 92, No. 2, 166-175 (2012); translation from Mat. Zametki 92, No. 2, 181-191 (2012).
Summary: For an arbitrary Boolean function of \( n \) variables, we show how to construct formulas of complexity \( O (2^{n/2})\) in the bases \[ \left\{ {x - y,xy,\left| x \right|} \right\}\cup {\left[ {0,1} \right], \quad } \left\{ {x - y,x*y,2x,\left| x \right|} \right\}\cup {\left[ {0,1} \right],} \] where \( x * y = \max(-1, \min(1, x ))\max(-1, \min(1, y ))\). The obtained estimates are, in general, order-sharp.
MSC:
06E30 Boolean functions
94C10 Switching theory, application of Boolean algebra; Boolean functions (MSC2010)
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