La puissance de désaccord d'un adversaire

International audienceUn des résultats fondamentaux de l'algorithmique distribuée est que le niveau d'accord qui peut être obtenu en présence de $t$ pannes est exactement $t+1$. Autrement dit un adversaire qui peut mettre en panne n'importe quel sous ensemble d'au plus $t$ processus peut empêcher les autres processus de se mettre d'accord sur $t$ valeurs. Mais quelle est la puissance des ($2^{2^n} -n$) autres adversaires qui ne peuvent mettre en panne que certaines combinaisons des processus? Cet article présente une caractérisation précise d'un adversaire. On y introduit la notion de "puissance de désaccord". La puissance de désaccord d'un adversaire est le plus grand entier $k$ tel que l'accord des processus sur $k$ valeurs soit possible avec cet adversaire. Puis on montre comment {\em calculer} automatiquement cet entier pour un adversaire donné

Relating L-Resilience and Wait-Freedom via Hitting Sets

The condition of t-resilience stipulates that an n-process program is only obliged to make progress when at least n-t processes are correct. Put another way, the live sets, the collection of process sets such that progress is required if all the processes in one of these sets are correct, are all sets with at least n-t processes. We show that the ability of arbitrary collection of live sets L to solve distributed tasks is tightly related to the minimum hitting set of L, a minimum cardinality subset of processes that has a non-empty intersection with every live set. Thus, finding the computing power of L is NP-complete. For the special case of colorless tasks that allow participating processes to adopt input or output values of each other, we use a simple simulation to show that a task can be solved L-resiliently if and only if it can be solved (h-1)-resiliently, where h is the size of the minimum hitting set of L. For general tasks, we characterize L-resilient solvability of tasks with respect to a limited notion of weak solvability: in every execution where all processes in some set in L are correct, outputs must be produced for every process in some (possibly different) participating set in L. Given a task T, we construct another task T_L such that T is solvable weakly L-resiliently if and only if T_L is solvable weakly wait-free

Locality and Checkability in Wait-free Computing

22 pagesThis paper studies several notions of locality that are inherent to the specification of distributed tasks and independent of the computing environment, and investigates the ability of a shared memory wait-free system to solve tasks satisfying various forms of locality. First, we define a task to be projection-closed if every partial output π(t) for a full input s is also a valid output for the partial input π(s) and prove that projection-closed tasks are precisely those tasks that are wait-free checkable. Our second main contribution is dealing with a stronger notion of lo- cality of topological nature. A task T = (I, O, ∆) is said to be locality- preserving if and only if O is a covering complex of I, that is, each simplex s of I is mapped by ∆ to a set of simplexes of O each isomorphic to s. This topological property yields obstacles for wait-free solvability different in nature from the classical agreement impossibility results. On the other hand, locality-preserving tasks are projection-closed and therefore always wait-free checkable. We provide a classification of locality-preserving tasks in term of their computational power, by establishing a correspondence between locality-preserving tasks and subgroups of the edgepath group of a complex. Using this correspondence, we prove the existence of hierarchies of locality-preserving tasks, each one containing a universal task (induced by the universal covering complex), and at the bottom the trivial identity task

A generalized asynchronous computability theorem

We consider the models of distributed computation defined as subsets of the runs of the iterated immediate snapshot model. Given a task $T$ and a model $M$, we provide topological conditions for $T$ to be solvable in $M$. When applied to the wait-free model, our conditions result in the celebrated Asynchronous Computability Theorem (ACT) of Herlihy and Shavit. To demonstrate the utility of our characterization, we consider a task that has been shown earlier to admit only a very complex $t$-resilient solution. In contrast, our generalized computability theorem confirms its $t$-resilient solvability in a straightforward manner.Comment: 16 pages, 5 figure