On the complexity of finding falsifying assignments for Herbrand disjunctions

Suppose that $${{\it \Phi}}$$Φ is a consistent sentence. Then there is no Herbrand proof of $${\neg {\it \Phi}}$$¬Φ , which means that any Herbrand disjunction made from the prenex form of $${\neg {\it \Phi}}$$¬Φ is falsifiable. We show that the problem of finding such a falsifying assignment is hard in the following sense. For every total polynomial search problem R, there exists a consistent $${{\it \Phi}}$$Φ such that finding solutions to R can be reduced to finding a falsifying assignment to an Herbrand disjunction made from $${\neg {\it \Phi}}$$¬Φ . It has been conjectured that there are no complete total polynomial search problems. If this conjecture is true, then for every consistent sentence $${{\it \Phi}}$$Φ , there exists a consistent sentence $${\Psi}$$Ψ , such that the search problem associated with $${\Psi}$$Ψ cannot be reduced to the search problem associated with $${{\it \Phi}}$$Φ .