The Duration Calculus can be used as a high-level specification language for properties of real-time systems. The question arises whether reasoning about such specifications can be automated. To this end, we first discuss the decidability of the realisability problem of the Duration Calculus: is there an algorithm that for a given Duration Calculus formula decides whether this formula can be realised. By using proof techniques of Zhou Chaochen, M.R. Hansen, and P. Sestoft, we show that for a subset of the Duration Calculus and the discrete-time domain this problem is indeed decidable. However, for the general case of continuous time it is not. The proofs of these results shed light on the difference between these two time domains.

Next we introduce the subset of implementables due to A.P. Ravn. This subset provides certain patterns of formulas formalising concepts like stability and progress that are convenient for specifying the behaviour of controllers. Finally, we introduce Constraint Diagrams due to C. Kleuker as a graphical representation of a subset of Duration Calculus. These diagrams specify timed behaviours in an assumption/commitment style. We show that the implementables all have lucid representations as Constraint Diagrams. In general, Constraint Diagrams are more expressive than implementables.

Decidability results

Zhou Chaochen, M.R. Hansen, and P. Sestoft showed that the problem whether a given DC formula is satisfiable is decidable for a subset of DC when discrete time is assumed [ZHS93]. This result has been exploited by P.K. Pandya in a tool called DCVALID for automatically checking satisfiability and validity of formulas in this subset [Pan01].