We present global and local algorithms for solving systems of equations over Noetherian posets with a bottom element, a general setting underlying many verification problems. Our algorithms compute the solution of a selected variable by restricting exploration to those parts of the system required to determine its value. We achieve this by computing variable dependencies by means of dependency oracles. Oracles guide the exploration of the system and provide sound termination criteria for local fixed-point computation. A key advantage of our approach is its flexibility: oracles can be customized, composed, or over-approximated, offering a principled way to trade precision for performance without compromising correctness. We evaluate our solution against existing algorithms from the literature and show that our prototype implementation is competitive and often outperforms specialized solutions, while remaining simple and adaptable across diverse application domains.

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