Behavioural equivalences for timed systems

Timed transition systems are behavioural models that include an explicit treatment of time flow and are used to formalise the semantics of several foundational process calculi and automata. Despite their relevance, a general mathematical characterisation of timed transition systems and their behavioural theory is still missing. We introduce the first uniform framework for timed behavioural models that encompasses known behavioural equivalences such as timed bisimulations, timed language equivalences as well as their weak and time-abstract counterparts. All these notions of equivalences are naturally organised by their discriminating power in a spectrum. We prove that this result does not depend on the type of the systems under scrutiny: it holds for any generalisation of timed transition system. We instantiate our framework to timed transition systems and their quantitative extensions such as timed probabilistic systems

Coalgebraic Semantics for Timed Processes

We give a coalgebraic formulation of timed processes and their operational semantics. We model time by a monoid called a “time domain”, and we model processes by “timed transition systems”, which amount to partial monoid actions of the time domain or, equivalently, coalgebras for an “evolution comonad ” generated by the time domain. All our examples of time domains satisfy a partial closure property, yielding a distributive law of a monad for total monoid actions over the evolution comonad, and hence a distributive law of the evolution comonad over a dual comonad for total monoid actions. We show that the induced coalgebras are exactly timed transition systems with delay operators. We then integrate our coalgebraic formulation of time qua timed transition systems into Turi and Plotkin’s formulation of structural operational semantics in terms of distributive laws. We combine timing with action via the more general study of the combination of two arbitrary sorts of behaviour whose operational semantics may interact. We give a modular account of the operational semantics for a combination induced by that of each of its components. Our study necessitates the investigation of products of comonads. In particular, we characterise when a monad lifts to the category of coalgebras for a product comonad, providing constructions with which one can readily calculate. Key words: time domains, timed transition systems, evolution comonads, delay operators, structural operational semantics, modularity, distributive laws

An Automatic Technique for Checking the Simulation of Timed Systems

International audienceIn this paper, we suggest an automatic technique for checking the timed weak simulation between timed transition systems. The technique is an observation-based method in which two timed transition systems are composed with a timed observer. A μ-calculus property that captures the timed weak simulation is then verified on the result of the composition. An interesting feature of the suggested technique is that it only relies on an untimed μ-calculus model-checker without any specific algorithm needed to analyze the result of the composition. We also show that our simulation relation supports interesting results concerning the trace inclusion and the preservation of linear properties. Finally, the technique is validated using the FIACRE/TINA toolset

The cones and foci proof techniques for timed transition systems

We propose an extension of the cones and foci proof technique that can be used to prove timed branching bisimilarity of states in timed transition systems. We prove the correctness of this technique and we give an example verification

On the compared expressiveness of arc, place and transition time Petri nets

International audienceIn this paper, we consider safe Time Petri Nets where time intervals (strict and large) are associated with places (TPPN), arcs (TAPN) or transitions (TTPN). We give the formal strong and weak semantics of these models in terms of Timed Transition Systems. We compare the expressiveness of the six models w.r.t. (weak) timed bisimilarity (behavioral semantics). The main results of the paper are : (i) with strong semantics, TAPN is strictly more expressive than TPPN and TTPN ; (ii) with strong semantics TPPN and TTPN are incomparable ; (iii) TTPN with strong semantics and TTPN with weak semantics are incomparable. Moreover, we give a complete classification by a set of 9 relations explained in a figure

A Calculus for Timed Automata (Extended Abstract)

A language for representing timed automata is introduced. Its semantics i defined in terms of timed automata. This language is complete in the sense that any timed automaton can be represented by a term in the language. We also define a direct operational semantics for the language in terms of (timed) transition systems. This is proven to be equivalent (or, more precisely, timed bisimilar) to the interpretation in terms of timed automata. In addition, a set of axioms is given that is shown to be sound for timed bisimulation. Finally, we introduce several features including the parallel composition and derived time operations like wait, time-out and urgency. We conclude with an example and show that we can eliminate non-reachable states using algebraic techniques

Extending nuXmv with Timed Transition Systems and Timed Temporal Properties

nuXmv is a well-known symbolic model checker, which implements various state-of-the-art algorithms for the analysis of finite- and infinite-state transition systems and temporal logics. In this paper, we present a new version that supports timed systems and logics over continuous super-dense semantics. The system specification was extended with clocks to constrain the timed evolution. The support for temporal properties has been expanded to include \textsc {MTL}_{0,\infty } formulas with parametric intervals. The analysis is performed via a reduction to verification problems in the discrete-time case. The internal representation of traces has been extended to go beyond the lasso-shaped form, to take into account the possible divergence of clocks. We evaluated the new features by comparing nuXmv with other verification tools for timed automata and \textsc {MTL}_{0,\infty }, considering different benchmarks from the literature. The results show that nuXmv is competitive with and in many cases performs better than state-of-the-art tools, especially on validity problems for \textsc {MTL}_{0,\infty }

Test Derivation from Timed Automata

A real-time system is a discrete system whose state changes occur in real-numbered time [AH97]. For testing real-time systems, specification languages must be extended with constructs for expressing real-time constraints, the implementation relation must be generalized to consider the temporal dimension, and the data structures and algorithms used to generate tests must be revised to operate on a potentially infinite set of states