Equivalence of definitive-set characterisation with Alpern–Schneider safety/liveness

Establish that the characterisation of safety, co-safety, liveness, and co-liveness properties given via definitive sets—namely, liveness if all finite traces are prefixes of traces in the definitive set, co-liveness if all finite traces are prefixes of traces in the definitive complement, co-safety if the set of definitive prefixes equals the set of its infinite traces, and safety as the complement of co-safety—coincides with the Alpern and Schneider metric-topological characterisation in which safety properties are limit-closed and liveness properties are dense in the space of infinite traces.

Background

Alpern and Schneider provide a topological characterisation over the metric space of infinite traces: safety properties are those closed under limits (limit-closed), while liveness properties are dense. Subsequent work also considers co-safety (guarantee) and co-liveness (morbidity) as complements.

The paper introduces definitive sets that include finite prefixes sufficient to confirm or refute properties. Using this notion, it proposes alternative conditions for liveness (no finite refutation), co-liveness (no finite confirmation), co-safety (finite confirmation always possible), and safety (via complement of co-safety), and conjectures the equivalence of these conditions to the Alpern–Schneider characterisation.