 Equivalence of weighted formula utility representations

Given two weighted formula utility representations with binary rational numbers as weights, decide whether their values agree on every Boolean valuation. This problem is co-NP-complete.

For membership, a valuation witnessing unequal values certifies non-equivalence. Evaluate all propositional formulas and add their weights exactly in polynomial time. For hardness, compare the one-rule base $\{(\theta ,1)\}$ with the empty base. They are equivalent exactly when $\theta$ is an unsatisfiable Boolean formula. Thus even very small weighted rule bases express a universal equivalence question.