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

Positive loop-closed automata: a decidable class of hybrid systems

AbstractThe model-checking problem for real-time and hybrid systems is very difficult, even for a well-formed class of hybrid systems—the class of linear hybrid automata—the problem is still undecidable in general. So an important question for the analysis and design of real-time and hybrid systems is the identification of subclasses of such systems and corresponding restricted classes of analysis problems that can be settled algorithmically. In this paper, we show that for a class of linear hybrid automata called positive loop-closed automata, the satisfaction problem for linear duration properties can be solved by linear programming. We extend the traditional regular expressions with duration constraints and use them as a language to describe the behaviour of this class of linear hybrid automata. The extended notation is called duration-constrained regular expressions. Based on this formalism, we show that the model-checking problem can be reduced formally to linear programs

Checking Hybrid Automata for Linear Duration Invariants ⋆

Abstract. In this paper, we consider the problem of checking hybrid systems modelled by hybrid automata for a class of real-time properties represented by linear duration invariants, which are constructed from linear inequalities of integrated durations of system states. Based on linear programming, an algorithm is developed for solving the problem for a class of hybrid automata