Non-cancellative Boolean Circuits: A Generalization of Monotone Boolean Circuits

Cancellations are known to be helpful in e cient algebraic computation of polynomials over elds. We de ne a notion of cancellation in Boolean circuits and de ne Boolean circuits that do not use cancellation to be non-cancellative. Non-cancellative Boolean circuits are a natural generalization of monotone Boolean circuits. We show that in the absence of cancellation, Boolean circuits require super-polynomial size to compute the determinant interpreted over GF(2). This non-monotone Boolean function is known to be in P . In the spirit of monotone complexity classes, we de ne complexity classes based on non-cancellative Boolean circuits. We show that when the Boolean circuit model is restricted by withholding cancellation, P and popular classes within P are restricted as well, but NP and circuit de nable classes above it remain unchanged.