A Structural Classification of Resilience for aRc | ba

We study the resilience of Boolean regular path queries of the form aRc | ba, where a, b, c are distinct and R is a fixed regular language over an alphabet B containing b but excluding a and c. Writing D = B ∖ {b}, we establish a complexity classification: set resilience is polynomial-time computable when R is local and either R ⊆ D∗ or R ⊆ D∗b; in all other cases, the decision problem is NP-complete. In particular, we prove NP-completeness under set semantics for ab∗c | ba, a previously unclassified example. For the tractable cases, minimum-cut constructions account for each shared database fact exactly once and support nonnegative rational deletion costs. Three base reductions and path-replacement arguments establish the hardness cases. We also give a polynomial-time classification procedure for explicitly represented DFAs and show that the same complexity boundary holds for bag resilience with binary-encoded multiplicities. A companion software artifact provides separate Lean 4.35 formalizations of the main classification theorem and six exact-value reduction identities.