Duration Calculus (or DC in short) presents a formal notation to specify properties of real-time systems and a calculus to formally prove such properties. Decidability is the underlying foundation to automated reasoning. But, excepting some of its simple fragments, DC has been shown to be undecidable. DC takes the set of real numbers to represent time. The main reason of undecidability comes from the assumption that, in a real-time system, state changes can occur at any time point. But an implementation of a specification is ultimately executed on a computer, and there states change according to a system clock. Under such an assumption, it has been shown that the decidability results can be extended to cover relatively richer subsets of DC. In this report, we extend such decidability results to still richer subsets of DC. Under the assumption that states change at real-numbered points, it has been shown that a propositional fragment of DC is undecidable. We also show that the same set still remains undecidable, even if we restrict that states change at points as designated by rational numbers.
