Monotone Real Circuits are More Powerful than Monotone Boolean Circuits

Recently Pavel Pudlak (1995) and independently Armin Haken and Steve Cook (1995) gave exponential lower bounds for the size of monotone real circuits computing clique type functions. In both cases, the lower bound was established in the monotone boolean case and then extended to the real case. This left open the question of whether monotone boolean circuits are in general polynomially equivalent to monotone real circuits for boolean functions. By a simple construction, we show that monotone real circuits are exponentially more powerful than general boolean circuits by proving that linear-size log-depth fan in 2 monotone real circuits can compute any slice function.