  Feasible interpolation is a general technique for proving proof complexity
lower bounds. The monotone version of the technique converts, in its basic
variant, lower bounds for monotone Boolean circuits separating two NP-sets to
proof complexity lower bounds. In a generalized version of the technique,
dag-like communication protocols are used instead of monotone Boolean circuits.
We study three kinds of protocols and compare their strength.
  Our results establish the following relationships in the sense of polynomial
reducibility: Protocols with equality are at least as strong as protocols with
inequality and protocols with equality have the same strength as protocols with
a conjunction of two inequalities. Exponential lower bounds for protocols with
inequality are known. Obtaining lower bounds for protocols with equality would
immediately imply lower bounds for resolution with parities (R(LIN)).
