A Hierarchy of Partial Order Temporal Properties

We propose a classification of partial order temporal properties into a hierarchy, which is a generalization of the safety-progress hierarchy of Chang, Manna and Pnueli. The classes of the hierarchy are characterized through three views: language-theoretic, topological and temporal. Instead of the domain of strings, we take the domain of Mazurkiewicz traces as a basis for our considerations. For the language-theoretic view, we propose operations on trace languages which define the four main classes of properties: safety, guarantee, persistence and response. These four classes are shown to correspond precisely to the two lower levels of the Borel hierarchy of the Scott topology of the domain of traces relativized to the infinite traces. In addition, a syntactic characterization of the classes is provided in terms of a sublogic of the Generalized Interleaving Set Temporal Logic GISTL (an extension of ISTL).