Finite representation and computation for weakest hypotheses in the full goal language

Determine whether, for the full goal language of Rowe, Guttman, and Liskov—whose hypotheses may contain role-position and parameter atoms—the weakest hypothesis is always expressible as a finite disjunction and whether that weakest hypothesis can be computed.

Background

The paper solves the weakest-trust problem only when the variable portion of a goal hypothesis is a conjunction of trust atoms drawn from a finite set. The full goal language additionally permits atoms referring to role positions, parameters, and ordering relations, so the candidate hypothesis space may be infinite.

The authors note that the Galois connection still exists over the complete lattice of up-sets of hypotheses, but the weakest hypothesis need not have a finite representation in the extended goal language. They identify two unresolved issues: whether the relevant conjunctions form a well-quasi-order, which would guarantee finite generation of up-sets, and whether the weakest hypothesis can be computed in this general setting.

References

The main open question is whether this extends to the full goal language of , where hypotheses may also contain role-position and parameter atoms. The Galois connection exists there in the lattice of up-sets of hypotheses. What is not known is whether the weakest hypothesis is always a finite disjunction, and so a formula of the extended goal language, and whether it can be computed (Section~\ref{sec:lim-a2}).

— Trust a Few: The Weakest Assumptions a Protocol Needs  (2610.09730 - Mittal et al., 7 Oct 2026) in Conclusion; Section 7.1 (Section \ref{sec:lim-a2})

We do not know whether the conjunctions of atoms of under their order have this property.

— Trust a Few: The Weakest Assumptions a Protocol Needs  (2610.09730 - Mittal et al., 7 Oct 2026) in Section 7.1, “Finitely Many Candidates and Trust Hypotheses Only (A1, A2)” (Section \ref{sec:lim-a2})

A tight core that discards unneeded assumptions would need conclusions free of trust atoms, for instance $S(\tau)$ with its origination atoms removed. Whether the correspondence, or some replacement for it, holds for such conclusions is open.

— Trust a Few: The Weakest Assumptions a Protocol Needs  (2610.09730 - Mittal et al., 7 Oct 2026) in Section 4.3 “Tight Trusts and Attainable Goals” (Section \ref{sec:lim-tight}); Conclusion