## Section: New Results

### New results on Axis 3: Population Models

#### Verification

**Participants :** Nathalie Bertrand, Anirban Majumdar

##### Networks of many identical agents communicating by broadcast.

Broadcast networks allow one to model networks of identical nodes communicating through message broadcasts [17]. Their parameterized verification aims at proving a property holds for any number of nodes, under any communication topology, and on all possible executions. We focus on the coverability problem which dually asks whether there exists an execution that visits a configuration exhibiting some given state of the broadcast protocol. Coverability is known to be undecidable for static networks, i.e. when the number of nodes and communication topology is fixed along executions. In contrast, it is decidable in PTIME when the communication topology may change arbitrarily along executions, that is for reconfigurable networks. Surprisingly, no lower nor upper bounds on the minimal number of nodes, or the minimal length of covering execution in reconfigurable networks, appear in the literature. We showed tight bounds for cutoff and length, which happen to be linear and quadratic, respectively, in the number of states of the protocol. We also introduced an intermediary model with static communication topology and non-deterministic message losses upon sending. We showed that the same tight bounds apply to lossy networks, although, reconfigurable executions may be linearly more succinct than lossy executions. Finally, we showed NP-completeness for the natural optimisation problem associated with the cutoff.

##### Randomized distributed algorithms for consensus.

Randomized fault-tolerant distributed algorithms pose a number of challenges for automated verification: (i) parameterization in the number of processes and faults, (ii) randomized choices and probabilistic properties, and (iii) an unbounded number of asynchronous rounds. This combination makes verification hard. Challenge (i) was recently addressed in the framework of threshold automata. We extended threshold automata to model randomized consensus algorithms that perform an unbounded number of asynchronous rounds. For non-probabilistic properties, we showed [18] that it is necessary and sufficient to verify these properties under round-rigid schedules, that is, schedules where processes enter round $r$ only after all processes finished round $r-1$. For almost-sure termination, we analyzed these algorithms under round-rigid adversaries, that is, fair adversaries that only generate round-rigid schedules. This allowed us to do compositional and inductive reasoning that reduces verification of the asynchronous multi-round algorithms to model checking of a one-round threshold automaton. We applied this framework and automatically verified the following classic algorithms: Ben-Or's and Bracha's seminal consensus algorithms for crashes and Byzantine faults, 2-set agreement for crash faults, and RS-Bosco for the Byzantine case.

#### Control

**Participants :** Nathalie Bertrand, Blaise Genest, Anirban Majumdar

##### Controlling a population

We introduced a new setting where a population of agents [6], each modelled by a finite-state system, are controlled uniformly: the controller applies the same action to every agent. The framework is largely inspired by the control of a biological system, namely a population of yeasts, where the controller may only change the environment common to all cells. We studied a synchronisation problem for such populations: no matter how individual agents react to the actions of the controller, the controller aims at driving all agents synchronously to a target state. The agents are naturally represented by a non-deterministic finite state automaton (NFA), the same for every agent, and the whole system is encoded as a 2-player game. The first player (Controller) chooses actions, and the second player (Agents) resolves non-determinism for each agent. The game with m agents is called the m-population game. This gives rise to a parameterized control problem (where control refers to 2 player games), namely the population control problem: can Controller control the m-population game for all m $\in \mathbb{N}$ whatever Agents does? In this work, we proved that the population control problem is decidable, and it is a EXPTIME-complete problem. As far as we know, this is one of the first results on the control of parameterized systems. Our algorithm, which is not based on cut-off techniques, produces winning strategies which are symbolic, that is, they do not need to count precisely how the population is spread between states. The winning strategies produced by our algorithm are optimal with respect to the synchronisation time: the maximal number of steps before synchronisation of all agents in the target state is at most polynomial in the number of agents m, and exponential in the size of the NFA. We also showed that if there is no winning strategy, then there is a population size M such that Controller wins the m-population game if and only if m $\le $ M. Surprisingly, M can be doubly exponential in the number of states of the NFA, with tight upper and lower bounds.

##### Concurrent multiplayer games with arbitrary many players

Traditional concurrent games on graphs involve a fixed number of
players, who take decisions simultaneously, determining the next state
of the game. In [16], we introduced a parameterized variant of
concurrent games on graphs, where the parameter is precisely the
number of players. Parameterized concurrent games are described by
finite graphs, in which the transitions bear regular languages to
describe the possible move combinations that lead from one vertex to
another. We considered the problem of determining whether the first
player, say Eve, has a strategy to ensure a reachability objective
against any strategy profile of her opponents as a coalition. In
particular Eve's strategy should be independent of the number of
opponents she actually has. Technically, we focused on an *a
priori* simpler setting where the languages labeling transitions only
constrain the number of opponents (but not their precise action
choices). These constraints are described as semilinear sets, finite
unions of intervals, or intervals. We established the precise
complexities of the parameterized reachability game problem, ranging
from PTIME-complete to PSPACE-complete, in a variety of situations
depending on the contraints (semilinear predicates, unions of
intervals, or intervals) and on the presence or not of
non-determinism.