Choosing What to Observe: Higher-Order Obstructions in Horn Abduction

Abstract

An explanation may exist in every complete context yet be impossible to select from the information observed. We study this distinction in finite propositional Horn abduction with integrity constraints. Observations are sufficient when each report admits one hypothesis set that is consistent and entails the manifestation in every compatible context. We characterize sufficiency by minimal context families with no common explanation. Their variable spans form a canonical hypergraph whose transversals are exactly the sufficient observation sets. The characterization recovers classical discernibility for unique explanations and supports composition across independent components. Minimal obstructions have unbounded cardinality even for acyclic rules, while rectangular explanation families admit a tight bound given by their largest choice domain. With circuit-defined contexts, sufficiency is $\Pi_3^P$-complete and selecting at most $k$ observable variables is $\Sigma_4^P$-complete, already for unrestricted contexts and acyclic rules. The selection reduction makes observation choices enforce globally fixed existential commitments. Explicit contexts reduce both tasks to NP-complete problems; tabulated explanation families make sufficiency polynomial. These results connect explanatory choice, information requirements, and representation cost.