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## Section: New Results

### Analysis and verification of quantitative systems

#### Quantitative verification of distributions of stochastic models

Participant : Blaise Genest.

In [24], we obtained conditions under which quantitative verification of distributions of stochastic systems is decidable. This is a challenging question as for general Markov Chains, verification of distribution is Skolem-complete, a problem on linear recurrence sequences whose decidability is a long-standing problem open for 40 years. In this paper, we approach this problem by studying the languages generated by Markov Chains, whose regularity would entail the decidability of quantitative verification. Given an initial distribution, we represent the trajectory of Markov Chain over time as an infinite word over a finite alphabet, where the ${n}^{th}$ letter represents a probability range after $n$ steps. We extend this to a language of trajectories (a set of words), one trajectory for each initial distribution from a (possibly infinite) set. We show that if the eigenvalues of the transition matrix associated with the Markov Chain are all distinct positive real numbers, then the language is effectively regular. Further, we show that this result is at the boundary of regularity, as non-regular languages can be generated when the restrictions are even slightly relaxed. The regular representation of the language allows us to reason about more general properties, e.g., robustness of a regular property in a neighbourhood around a given distribution.

#### Diagnosability of repairable faults

Participants : Éric Fabre, Loïc Hélouët, Hervé Marchand, Engel Lefaucheux.

#### Diagnosability of stochastic systems

Participants : Éric Fabre, Blaise Genest, Hugo Bazille, Ocan Sankur.

Diagnosis of partially observable stochastic systems prone to faults was introduced in the late nineties. Diagnosability, i.e. the existence of a diagnoser, may be specified in different ways: (1) exact diagnosability (called A-diagnosability) requires that almost surely a fault is detected and that no fault is erroneously claimed while (2) approximate diagnosability (called ε-diagnosability) allows a small probability of error when claiming a fault and (3) accurate approximate diagnosability (called AA-diagnosability) requires that this error threshold may be chosen arbitrarily small. In a recent work [27], we focused on approximate diagnoses. We first refined the almost sure requirement about finite delay introducing a uniform version and showing that while it does not discriminate between the two versions of exact diagnosability this is no more the case in approximate diagnosis. We then gave a complete picture of relations between the different diagnosability specifications for probabilistic systems and establish characterisations for most of them in the finite-state case. Based on these characterisations, we developped decision procedures, studied their complexity and proved their optimality. We also designed synthesis algorithms to construct diagnosers and we analysed their memory requirements. Finally we established undecidability of the diagnosability problems for which we provided no characterisation. Notably, we proved the AA-diagnosability problem to be undecidable, answering a longstanding open question.

In another work [28], we investigated semantical and computational issues for exact notions of diagnosability in the context of infinite-state probabilistic systems. We first showed established a characterisation of the so-called FF-diagnosability using a $G\delta$ set (instead of an open set for finite-state systems) and also for two other notions, IF- and IA-diagnosability, when models are finitely branching. We also proved that surprisingly the last notion, FA-diagnosability, cannot be characterised in this way even in the finitely branching case. Then we applied our characterisations for a partially observable probabilistic extension of visibly pushdown automata, yielding EXPSPACE procedures for solving diagnosability problems. In addition, we establish some computational lower bounds and show that slight extensions of these probabilistic visibly pushdown automata lead to undecidability.

#### Analysing decisive stochastic processes

Participant : Nathalie Bertrand.

In 2007, Abdulla et al. introduced the elegant concept of decisive Markov chain. Intuitively, decisiveness allows one to lift the good properties of finite Markov chains to infinite Markov chains. For instance, the approximate quantitative reachability problem can be solved for decisive Markov chains (enjoying reasonable effectiveness assumptions) including probabilistic lossy channel systems and probabilistic vector addition systems with states. In a recent work [26], we extended the concept of decisiveness to more general stochastic processes. This extension is non trivial as we consider stochastic processes with a potentially continuous set of states and uncountable branching (common features of real-time stochastic processes). This allowed us to obtain decidability results for both qualitative and quantitative verification problems on some classes of real-time stochastic processes, including generalized semi-Markov processes and stochastic timed automata.

#### Concurrent timed systems

Participants : Loïc Hélouët, Blaise Genest.

Adding real time information to Petri net models often leads to undecidability of classical verification problems such as reachability and boundedness. For instance, models such as Timed-Transition Petri nets (TPNs)  [47] are intractable except in a bounded setting. On the other hand, the model of Timed-Arc Petri nets  [50] enjoys decidability results for boundedness and control-state reachability problems at the cost of disallowing urgency (the ability to enforce actions within a time delay).

We have addressed semantics variants of time and timed Petri nets to obtain concurrent models with interesting expressive power, but yet allowing decidability of verification and robustness questions. Robustness of timed systems aims at studying whether infinitesimal perturbations in clock values can result in new discrete behaviors. A model is robust if the set of discrete behaviors is preserved under arbitrarily small (but positive) perturbations.

In [25] we have considered time in Petri nets under a strong semantics with multiple enabling of transitions. We focus on a structural subclass of unbounded TPNs, where the underlying untimed net is free-choice, and show that it enjoys nice properties under a multi-server semantics. In particular, we showed that the questions of fireability (whether a chosen transition can fire), and termination (whether the net has a non-terminating run) are decidable for this class. We then consider the problem of robustness under guard enlargement  [48], i.e., whether a given property is preserved even if the system is implemented on an architecture with imprecise time measurement. Unlike in [15], where decidability of several problems is obtained for bounded classes of nets, we showed that robustness of fireability is decidable for unbounded free choice TPNs with a multi-server semantics.

The robustness of time Petri nets was addressed in [15] by considering the model of parametric guard enlargement which allows time-intervals constraining the firing of transitions in TPNs to be enlarged by a (positive) parameter. We show that TPNs are not robust in general and checking if they are robust with respect to standard properties (such as boundedness, safety) is undecidable. We then extend the marking class timed automaton construction for TPNs to a parametric setting, and prove that it is compatible with guard enlargements. We apply this result to the (undecidable) class of TPNs which are robustly bounded (i.e., whose finite set of reachable markings remains finite under infinitesimal perturbations): we provide two decidable robustly bounded subclasses, and show that one can effectively build a timed automaton which is timed bisimilar even in presence of perturbations. This allows us to apply existing results for timed automata to these TPNs and show further robustness properties.

The goal of [23] is to investigate decidable classes of Petri nets with time that capture some urgency and still allow unbounded behaviors, which go beyond finite state systems. We have shown, up to our knowledge, the first decidability results on reachability and boundedness for Petri net variants that combine unbounded places, time, and urgency. For this, we have introduced the class of Timed-Arc Petri nets with restricted Urgency, where urgency can be used only on transitions consuming tokens from bounded places. We showed that control-state reachability and boundedness are decidable for this new class, by extending results from Timed-Arc Petri nets (without urgency) [43]. Our main result concerns (marking) reachability, which is undecidable for both TPNs (because of unrestricted urgency)  [46] and Timed-Arc Petri Nets (because of infinite number of “clocks”)  [49]. We obtained decidability of reachability for unbounded TPNs with restricted urgency under a new, yet natural, timed-arc semantics presenting them as Timed-Arc Petri Nets with restricted urgency. Decidability of reachability under the intermediate marking semantics is also obtained for a restricted subclass.

#### Petri nets realizability

Participants : Loïc Hélouët, Abd El Karim Kecir.

We considered in [30] the realizability of urban train schedules by stochastic concurrent timed systems. Schedules are high level views of desired timetables that a metro system should implement. They are represented as partial orders decorated with timing constraints. Train systems are represented as elementary stochastic time Petri nets. We have first considered logical realizability: a schedule is realizable by a net $𝒩$ if it embeds in a time process of $𝒩$ that satisfies all its constraints. However, with continuous time domains, the probability of a time process that realizes a schedule is null. We have extended the former notion of realizability to consider probabilistic realizability of schedules up to some imprecision $\alpha$. This probabilistic realizability holds if the probability that $𝒩$ logically realizes $S$ with constraints enlarged by $\alpha$ time units is strictly positive. We have shown that upon a sensible restriction guaranteeing time progress (systems can not perform an arbitrary number of actions within a single time unit), logical and probabilistic realizability of a schedule can be checked on the finite set of symbolic prefixes extracted from a bounded unfolding of the net. We have provided a construction technique for these prefixes and shown that they represent all time processes of a net occurring up to a given maximal date. We have then shown how to verify existence of an embedding and compute the probability of its realization.