Necessity of WPR-expressiveness for relative completeness in partial reverse Hoare logic

Ascertain whether the WPR-expressiveness assumption—that the assertion language can express weakest preconditions for all programs and postconditions—is necessary to prove the relative completeness of the ordinary proof system for partial reverse Hoare logic; specifically, prove or refute the conjecture that relative completeness holds without assuming WPR-expressiveness, analogous to Bergstra and Tucker’s result for partial Hoare logic.

Background

Throughout the paper, the authors assume an assertion language that is WPR-expressive in order to prove relative completeness: any valid partial reverse Hoare triple is provable under this expressiveness assumption.

They note an analogous result in partial Hoare logic by Bergstra and Tucker, where relative completeness does not require such expressiveness (for weakest liberal preconditions). Motivated by this precedent, the authors conjecture that relative completeness for partial reverse Hoare logic may similarly not depend on WPR-expressiveness, leaving a concrete question about the necessity of this assumption.