## Section: New Results

### Theory of Distributed Systems

#### $t$-Resilient Immediate Snapshot is Impossible

Participant : Michel Raynal.

Immediate snapshot is the basic communication object on which relies the read/write distributed computing model made up of n crash-prone asynchronous processes, called iterated distributed model. Each iteration step (usually called a round) uses a new immediate snapshot object, which allows the processes to communicate and cooperate. More precisely, the $x$-th immediate snapshot object can be used by a process only when it executes the $x$-th round. An immediate snapshot object can be implemented by an $\left(n-1\right)$-resilient algorithm, i.e. an algorithm that tolerates up to $\left(n-1\right)$ process crashes (also called wait-free algorithm). Considering a $t$-crash system model (i.e. a model in which up to t processes are allowed to crash), this work [46] is on the construction of an extension of immediate snapshot objects to t-resiliency. In the $t$-crash system model, at each round each process may be ensured to get values from at least $n-t$ processes, and $t$-immediate snapshot has the properties of classical immediate snapshot (1-immediate snapshot) but ensures that each process will get values form at least $n-t$ processes. Its main result is the following. While there is a (deterministic) $t$-resilient read/write-based algorithm implementing $t$-immediate snapshot in a $t$-crash system when $t=n-1$, there is no $t$-resilient algorithm in a $t$-crash model when $t\in [1..(n-2\left)\right]$. This means that the notion of $t$-resilience is inoperative when one has to implement immediate snapshot for these values of $t$: the model assumption “at most $t<n-1$ processes may crash” does not provide us with additional computational power allowing for the design of genuine $t$-resilient algorithms (genuine meaning that such a $t$-resilient algorithm would work in the $t$-crash model, but not in the $(t+1)$-crash model). To show these results, the paper relies on well-known distributed computing agreement problems such as consensus and $k$-set agreement.

This work was done in collaboration with Carole Delporte, Hugues Fauconnier, and Sergio Rajsbaum, and appeared at SIROCCO 2016.

#### Two-Bit Messages are Sufficient to Implement Atomic Read/Write Registers in Crash-Prone Systems

Participant : Michel Raynal.

Atomic registers are certainly the most basic objects of computing science. Their implementation on top of an $n$-process asynchronous message-passing system has received a lot of attention. It has been shown that $t<n/2$ (where t is the maximal number of processes that may crash) is a necessary and sufficient requirement to build an atomic register on top of a crash-prone asynchronous message-passing system. Considering such a context, this work [49] presents an algorithm which implements a single-writer multi-reader atomic register with four message types only, and where no message needs to carry control information in addition to its type. Hence, two bits are sufficient to capture all the control information carried by all the implementation messages. Moreover, the messages of two types need to carry a data value while the messages of the two other types carry no value at all. As far as we know, this algorithm is the first with such an optimality property on the size of control information carried by messages. It is also particularly efficient from a time complexity point of view.

This work was done in collaboration with Achour Mostefaoui, and appeared at PODC 2016.