Another glance at the Alpern-Schneider characterization of safety and liveness in concurrent executions

Abstract

In order to derive a result such as the Alpern-Schneider theorem characterizing safety and liveness properties of concurrent program executions, it is shown that all that is needed is a ∨-preservingmap ϕ between complete Boolean algebras. Every property becomes a conjunction of a safety and a liveness property and safety properties can be characterized by sets of configurations that are to be “avoided”. Aside from the original result of B. Alpern and F.B. Schneider we also provide a new application by considering transition systems with a UNITY-style logic. Safety properties are characterized by a set of forbidden pairs of successive states and progress properties are those allowing all possible state-successor pairs. Every property of a transition system is shown to be a conjunction of a safety and a progress property.