## Section: New Results

### Highlights of the Year

#### Active Diagnosis for Probabilistic Systems

Diagnosis fits well with probabilistic systems since it is natural to model the uncertainty
about the behaviour of a partially observed system by distributions. We
had previously revisited the active diagnosis (which aims at controlling the system
to make it diagnosable) in discrete event systems designing optimal decision and
synthesis procedures [7] . This year, we have considered active
diagnosis for probabilistic discrete event systems, obtaining again optimal
procedures [26] . Furthermore we have refined the notion
of active diagnosis by introducing the *safe active diagnosis* which ensures
that after the control is applied, there is a positive probability that a fault
never occurs. Interestingly this problem is undecidable but for finite memory
controller we have shown that the problem becomes again decidable and we have designed
optimal decision and synthesis procedures. Our approach has raised an issue that
has not be observed by previous researchers: while in discrete event system, most variants
of diagnosis are in fact equivalent, this is no more the case for probabilistic systems.
So in [26] , we have undertaken the task of classifying the different versions
obtaining a complete landscape of the notions both in terms of relations and complexity.
Furthermore we have proposed a new notion of diagnosis, the *prediagnosis* that combines
the advantages of diagnosis and prediction.

#### Weighted automata and weighted logics

Weighted automata are a conservative quantitative extension of finite automata that enjoys applications, e.g., in language processing and speech recognition. Their expressive power, however, appears to be limited, especially when they are applied to more general structures than words, such as graphs. To address this drawback, we have introduced weighted pebble walking automata, which allow to navigate freely in the graph and may use pebbles to mark some positions.

In [20] , we have shown with examples from natural language modeling and quantitative model-checking that weighted expressions and automata with pebbles are more expressive and allow much more natural and intuitive specifications than classical ones. We have extended Kleene-Schu ̈tzenberger theorem showing that weighted expressions and automata with pebbles have the same expressive power. We focussed on an efficient translation from expressions to automata. We also proved that the evaluation problem for weighted automata can be done very efficiently if the number of reusable pebbles is low.

In [18] , we have studied the expressive power of these automata on words. We have proved that two-way pebble weighted automata, one-way pebble weighted automata, and our weighted logic with transitive closure are expressively equivalent. We also gave new logical characterizations of standard recognizable series.

In [30] , we addressed the more general case of graphs such as nested words, trees, pictures, Mazurkiewicz traces, ... We established that weighted pebble walking automata have the same expressive power as weighted first order logic with transitive closure logic, lifting a similar result by Engelfriet and Hoogeboom from the Boolean case to a quantitative setting.

#### Verification of concurrent recursive programs

Distributed systems form a crucially important but particularly challenging domain. Designing correct distributed systems is demanding, and verifying its correctness is even more so. The main cause of difficulty here is concurrency and interaction (or communication) between various distributed components. Hence it is important to provide a framework that makes easy the design of systems as well as their analysis. There are two schools of thought on reasoning about distributed systems: one following the interleaving based semantics, and one following the visual partial-order/graph based semantics. In [23] , we compare these two approaches and argue in favour of the latter. An introductory treatment of the split-width technique is also provided.

In [34] , we develop a general technique based on split-width for the verification of networks of multi-threaded recursive programs communicating via reliable FIFO channels. We extend the approach of [6] to this setting. Split-width offers an intuitive visual technique to decompose our behaviour graphs such as MSCs and nested words. The decomposition is mainly a divide-and-conquer technique which naturally results in a tree decomposition. Every behaviour can now be interpreted over its decomposition tree. Properties over the behaviour naturally transfer into properties over the decomposition tree. This allows us to use tree-automata techniques to obtain decision procedures for a range of problems such as reachability, model checking against logical formalisms etc. In this way, we obtain simple, uniform and optimal decision procedures for various verification problems parametrised by split-width. Furthermore, the simple visual mechanism of split-width is as powerful as yardstick graph measures such as tree-width or clique-width. Hence it captures any class of distributed behaviours with a decidable MSO theory.

Multi-threaded recursive programs communicating via channels are turing powerful, hence their verification has focussed on under-approximation techniques. Any error detected in the under-approximation implies an error in the system. However the successful verification of the under-approximation is not as useful if the system exhibits unverified behaviours. In [24] , we study controllers that observe/restrict the system so that it stays within the verified under-approximation. We identify some important properties that a good controller should satisfy. We consider an extensive under-approximation class, construct a distributed controller with the desired properties and also establish the decidability of verification problems for this class.

#### Regulation in Systems Biology

##### Rare events in Signalling Cascades

The visit in 2013 of Professor Monika Heiner from Cottbus University has led to a fruitful collaboration related to statistical model checking of rare events in signalling cascades (a regulatory biological system) [25] . This work has received one of the five top paper awards of the conference. In addition, we have improved the statistical methods used in our tool Cosmos.

##### Characterization of Reachable Attractors Using Petri Net Unfoldings

Attractors of network dynamics represent the long-term behaviours of the modelled system. Their characterization is therefore crucial for understanding the response and differentiation capabilities of a dynamical biological system. In the scope of qualitative models of interaction networks, the computation of attractors reachable from a given state of the network faces combinatorial issues due to the state space explosion.

In [33] , we have presented a new algorithm that exploits the concurrency between transitions of parallel acting components in order to reduce the search space. The algorithm relies on Petri net unfoldings that can be used to compute a compact representation of the dynamics. We have illustrated the applicability of the algorithm with Petri net models of cell signalling and regulation networks, boolean and multi-valued. The proposed approach aims at being complementary to existing methods for deriving the attractors of Boolean models, while being generic since it applies to any safe Petri net.