Extended Tropical Circuits

In this chapter, we first show that just one division $$(/)$$ gate as the output gate can exponentially decrease the size of monotone arithmetic $$(+,\times )$$ circuits. As a consequence, just one subtraction $$(-)$$ gate at the “very end,” as the output gate, can exponentially decrease the size of tropical $$(\min ,+)$$ circuits. We then show that even $$(\min ,\max ,+)$$ circuits (with monotone $$\max $$ operation instead of non-monotone subtraction) can be exponentially smaller than tropical $$(\min ,+)$$ circuits. Finally, we show that, in contrast, subtractions at the “very beginning,” at input gates, are of almost no help. Namely, the size of $$(\min ,+)$$ circuits cannot be substantially reduced by allowing the circuits, besides input variables $$x_1,\ldots ,x_n$$ , to also use their tropical reciprocals $$-x_1,\ldots ,-x_n$$ as inputs. The question of whether reciprocal inputs can substantially decrease the size of $$(\min ,\max ,+)$$ circuits remains open.