Non-cancellative boolean circuits: A generalization of monotone boolean circuits

Abstract

Cancellations are known to be helpful in efficient algebraic computation of polynomials over fields. We define a notion of cancellation in Boolean circuits and define 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 define 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 classes NP and above remain unchanged.