Decomposition-Based Checking and Local Revision for Propositional Circumscription via Minimal Reducts

Circumscription is a classical non-monotonic formalism in which selected atoms are minimized while other atoms are fixed or allowed to vary. For propositional clause theories, checking whether a candidate interpretation is a circumscription model amounts to a global minimality test. We study this checking problem through the minimal reduct of the candidate interpretation. The reduct turns the global test into a residual entailment problem; we then decompose that entailment problem along the collapsed negative dependency graph. The checker verifies source components over their ancestor scopes, contracts atoms whose obligations have been certified, and records certificate fragments that refer back to clauses of the original input theory. We give two exact local certification strategies: a direct SAT check and a MUS-based extraction procedure. Experiments on solved random 3CNF instances and industrial CNF instances show that the decomposition-based checker agrees with the global reduct baseline and that its certificates can be replayed. The MUS variant produces much smaller supports, but it also spends more time on extraction.