Negation-limited formulas

Understanding the power of negation gates is crucial to bridge the exponential gap between monotone and non-monotone computation. We focus on the model of formulas over the De Morgan basis and study the connection of negation-limited formulas with negation-limited circuits and with monotone formulas. We show the following results: • We give an efficient transformation of formulas with t negation gates to circuits with log⁡t negation gates. This transformation provides a generic way to cast results for negation-limited circuits to the setting of negation-limited formulas. For example, using a result of Rossman (CCC '15), we obtain an average-case lower bound for formulas of polynomial-size on n variables with n^1/2−ε negations.• We prove that every formula that contains t negation gates can be shrunk using a random restriction to a formula of size O(t) with the shrinkage exponent of monotone formulas. As a result, the shrinkage exponent of formulas that contain a constant number of negation gates is equal to the shrinkage exponent of monotone formulas.The above results follow from an efficient structural decomposition theorem for formulas that depends on their negation complexity which may be of independent interest.