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

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