LTL is closed under topological closure

Article history: We constructively prove that for every LTL formula ϕ, the smallest safety property containing the property expressed by ϕ is also expressible in LTL. It immediately follows that LTL admits the safety-liveness decomposition: any property expressed by an LTL formula is equivalent to the intersection of a safety property and a liveness property, both of them expressible in LTL. Our proof is based on constructing a minimal deterministic counter-free Buchi automaton that recognizes the smallest safety property containing the property expressed by ϕ.

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