## Section: New Results

### Verification

#### Probabilistic $\omega $-automata

Participant : Nathalie Bertrand.

Probabilistic $\omega $-automata are a variant version of nondeterministic automata over infinite words where all choices are resolved by probabilistic distributions. Acceptance of a run for an infinite input word can be defined using traditional acceptance criteria for $\omega $-automata, such as Büchi, Rabin or Streett conditions. The accepted language of a probabilistic $\omega $-automata is then defined by imposing a constraint on the probability measure of the accepting runs. Together with Christel Baier and Marcus Grösser from TU Dresden, we studied a series of fundamental properties of probabilistic $\omega $-automata with three different language-semantics: (1) the probable semantics that requires positive acceptance probability, (2) the almost-sure semantics that requires acceptance with probability 1, and (3) the threshold semantics that relies on an additional parameter $\lambda $ in ]0,1[ that specifies a lower probability bound for the acceptance probability. We provided a comparison of probabilistic $\omega $-automata under these three semantics and nondeterministic $\omega $-automata concerning expressiveness and efficiency. Furthermore, we addressed closure properties under the Boolean operators union, intersection and complementation and algorithmic aspects, such as checking emptiness or language containment. This work was published in Journal of the ACM [6] .

#### Petri nets reachability graphs

Participant : Christophe Morvan.

In the article [10] , we investigate the decidability and complexity status of model-checking problems on unlabelled reachability graphs of Petri nets by considering first-order and modal languages without labels on transitions or atomic propositions on markings. We consider several parameters to separate decidable problems from undecidable ones. Not only are we able to provide precise borders and a systematic analysis, but we also demonstrate the robustness of our proof techniques.

#### Frequencies in timed automata

Participant : Amélie Stainer.

A quantitative semantics for infinite timed words in timed automata based on the frequency of a run was introduced earlier by Bertrand, Bouyer, Brihaye and Stainer. Unfortunately, most of the results are obtained only for one-clock timed automata because the techniques do not allow to deal with some phenomenon of convergence between clocks. On the other hand, the notion of forgetful cycle was introduced by Basset and Asarin, in the context of entropy of timed languages, and seems to detect exactly these convergences. In [20] , we investigate how the notion of forgetfulness can help to extend the computation of the set of frequencies to n-clock timed automata.

#### Bounded satisfiability for PCTL

Participant : Nathalie Bertrand.

While model checking PCTL for Markov chains is decidable in polynomial-time, the decidability of PCTL satisfiability, as well as its finite model property, are long standing open problems. While general satisfiability is an intriguing challenge from a purely theoretical point of view, we argue that general solutions would not be of interest to practitioners: such solutions could be too big to be implementable or even infinite. Inspired by bounded synthesis techniques, we turn to the more applied problem of seeking models of a bounded size: we restrict our search to implementable – and therefore reasonably simple – models. In [14] and together with John Fearnley and Sven Schewe from University of Liverpool, we propose a procedure to decide whether or not a given PCTL formula has an implementable model by reducing it to an SMT problem. We have implemented our techniques and found that they can be applied to the practical problem of sanity checking – a procedure that allows a system designer to check whether their formula has an unexpectedly small model.

#### Graph transformation systems

Participant : Nathalie Bertrand.

In [13] , we study decidability issues for reachability problems in graph transformation systems, a powerful infinite-state model. For a fixed initial configuration, we consider reachability of an entirely specified configuration and of a configuration that satisfies a given pattern (coverability). The former is a fundamental problem for any computational model, the latter is strictly related to verification of safety properties in which the pattern specifies an infinite set of bad configurations. In this paper we reformulate results obtained, e.g., for context-free graph grammars and concurrency models, such as Petri nets, in the more general setting of graph transformation systems and study new results for classes of models obtained by adding constraints on the form of reduction rules.