On the completeness and decidability of duration calculus with iteration

AbstractThe extension of the duration calculus (DC) by iteration, which is also known as Kleene star, enables the straightforward specification of repetitive behaviour in DC and facilitates the translation of design descriptions between DC, timed regular expressions and timed automata. In this paper we present axioms and a proof rule about iteration in DC. We consider abstract-time DC and its extension by a state-variable binding existential quantifier known as higher-order DC (HDC). We show that the ω-complete proof systems for DC and HDC known from our earlier work can be extended by our axioms and rule in various ways in order to axiomatise iteration completely. The additions we propose include either the proof rule or an induction axiom. We also present results on the decidability of a subset of the extension DC* of DC by iteration

Probabilistic Interval Temporal Logic and Duration Calculus with Infinite Intervals: Complete Proof Systems

The paper presents probabilistic extensions of interval temporal logic (ITL) and duration calculus (DC) with infinite intervals and complete Hilbert-style proof systems for them. The completeness results are a strong completeness theorem for the system of probabilistic ITL with respect to an abstract semantics and a relative completeness theorem for the system of probabilistic DC with respect to real-time semantics. The proposed systems subsume probabilistic real-time DC as known from the literature. A correspondence between the proposed systems and a system of probabilistic interval temporal logic with finite intervals and expanding modalities is established too.Comment: 43 page

Axiomatisation and decidability of multi-dimensional Duration Calculus

AbstractThe Shape Calculus is a spatio-temporal logic based on an n-dimensional Duration Calculus tailored for the specification and verification of mobile real-time systems. After showing non-axiomatisability, we give a complete embedding in n-dimensional interval temporal logic and present two different decidable subsets, which are important for tool support and practical use

Duration Calculus of Weakly Monotonic Time

We extend Duration Calculus to a logic which allows description of Discrete Processes where several steps of computation can occur at the same time point. Moreover, the order of occurrence of these steps is relevant. The resulting logic is called Duration Calculus of Weakly Monotonic Time (WDC). It allows effects such as true synchrony and digitisation to be modelled. As an example, We formulate a new semantics of Timed CSP assuming that the communication and computation take no time. We also outline a semantics of shared variable concurrency under similar assumptions. We introduce a notion of deformation of time in WDC. We study the duration calculus properties which remain invariant under such deformation of time