The mistake in “A Solution of the P versus NP Problem”

The first step of the approach in [1] is a transformation which eliminates the negated variables in the monomials of the DNF representation and in the clauses of the CNF representation of the functions computed at the nodes of the given standard network resulting into the so-called reduced DNF and CNF representations of these functions. Theorem 5 characterizes the reduced DNF and CNF representations of the function computed at the output node of the standard network. Theorem 5 is correct. Given a standard network which computes a non-constant monotone Boolean Function at its output node and a CNF-DNF-approximator, the approximator was used for the approximation of the reduced formulas computed at the nodes of the network. I shall elaborate the mistake for the case that the reduced DNF formulas are approximated. With respect to the approximation of the reduced CNF formulas, an analogous reasoning can be given. If the reduced DNF representations are used for the evaluation of a standard network β with respect to a given input c, Method 1 which computes for each gate the value with respect to the input c cannot be used. This can be seen as follows. Consider an ∧-gate g with direct predecessors h1 and h2. Assume that each monomial in the reduced DNF-representation with respect to h1 which is fulfilled by c contains the variable xi. Furthermore assume that each