A Superpolynomial Lower Bound for a Circuit Computing the Clique Function with At Most (1/6) log log n Negation Gates

We investigate about a lower bound on the size of a Boolean circuit that computes the clique function with a limited number of negation gates. To derive strong lower bounds on the size of such a circuit we develop a new approach by combining the three approaches: the restriction applied for constant depth circuits[Has], the approximation method applied for monotone circuits[Raz2] and boundary covering developed in the present paper. Based on the approach the following statement is established: If a circuit C with at most [(1/6) log log m negation gates detects cliques of size (log m) 3(log m)1/2 in a graph with m vertices, then C contains at least 2 (1/5)(log m)(log m)1/2 gates. In addition, we present a general relationship between negation-limited circuit size and monotone circuit size of an arbitrary monotone function.