Safety and Liveness, Weakness and Strength, and the Underlying Topological Relations

We present a characterization that shows what it means for a formula to be a weak or strong version of another formula. We show that the weak version of a formula is not the same as Alpern and Schneider's safety component, but can be achieved by taking the closure in the Cantor topology over an augmented alphabet in which every formula is satisfiable. The resulting characterization allows us to show that the set of semantically weak formulas is exactly the set of nonpathological safety formulas Furthermore, we use the characterization to show that the original versions of the IEEE standard temporal logics PSL and SVA are broken, and we show that the source of the problem lies in the semantics of the SERE intersection and fusion operators. Finally, we use the topological characterization to show the internal consistency of the alternative semantics adopted by the latest version of the PSL standard.