Lower Bounds for DeMorgan Circuits of Bounded Negation Width

We consider Boolean circuits over {boolean OR, boolean AND, (sic)} with negations applied only to input variables. To measure the "amount of negation" in such circuits, we introduce the concept of their "negation width." In particular, a circuit computing a monotone Boolean function f(x(1), ... , x(n)) has negation width w if no nonzero term produced (purely syntactically) by the circuit contains more than w distinct negated variables. Circuits of negation width w = 0 are equivalent to monotone Boolean circuits, while those of negation width w = n have no restrictions. Our motivation is that already circuits of moderate negation width w = n(epsilon) for an arbitrarily small constant epsilon > 0 can be even exponentially stronger than monotone circuits. We show that the size of any circuit of negation width w computing f is roughly at least the minimum size of a monotone circuit computing f divided by K = min{w(m), m(w)}, where m is the maximum length of a prime implicant of f. We also show that the depth of any circuit of negation width w computing f is roughly at least the minimum depth of a monotone circuit computing f minus logK. Finally, we show that formulas of bounded negation width can be balanced to achieve a logarithmic (in their size) depth without increasing their negation width.