Necessary and sufficient cross-sort condition for localization

Determine a necessary and sufficient condition on a pair of many-sorted patterns $(\Gamma,\varphi)$ under which the localization argument for global completeness survives across sorts.

Background

Global completeness holds for the one-sorted fixpoint-free system but fails in the many-sorted setting, even for a satisfiable theory. The counterexample exploits the absence of a chain of symbol argument positions connecting the sort of the hypotheses to the sort of the conclusion.

The paper suggests as a candidate condition that every sort occurring in the conclusion be reachable along chains of argument positions from a sort occurring in the theory, but does not establish whether this condition is sufficient or necessary.