Modal logics as introduced in Chapter 2 can express local capabilities and necessities of processes such as “tick is a possible next action” or “tick must happen next.” However, they cannot express enduring capabilities such as “tick is always a possible next action” or long term inevitabilities such as “tick eventually happens.” These features, especially in the guise of safety or liveness properties, have been found to be very useful when analysing the behaviour of concurrent systems. Another abstraction from behaviour is a run of a process that is a finite or infinite length sequence of transitions. Runs provide a basis for understanding longer term capabilities. Logics where properties are primarily ascribed to runs of systems are called “temporal” logics. An alternative foundation for temporal logic is to view enduring features as extremal solutions to recursive modal equations.
KeywordsModal Logic Temporal Property Modal Equation Safety Property Extremal Solution
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