11 research outputs found
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 and a model
, we provide topological conditions for to be solvable in . 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 -resilient solution. In contrast, our
generalized computability theorem confirms its -resilient solvability in a
straightforward manner.Comment: 16 pages, 5 figure
Tight Bounds for Connectivity and Set Agreement in Byzantine Synchronous Systems
In this paper, we show that the protocol complex of a Byzantine synchronous
system can remain -connected for up to rounds,
where is the maximum number of Byzantine processes, and .
This topological property implies that rounds are
necessary to solve -set agreement in Byzantine synchronous systems, compared
to rounds in synchronous crash-failure systems. We
also show that our connectivity bound is tight as we indicate solutions to
Byzantine -set agreement in exactly synchronous
rounds, at least when is suitably large compared to . In conclusion, we
see how Byzantine failures can potentially require one extra round to solve
-set agreement, and, for suitably large compared to , at most that
Strong Equivalence Relations for Iterated Models
The Iterated Immediate Snapshot model (IIS), due to its elegant geometrical
representation, has become standard for applying topological reasoning to
distributed computing. Its modular structure makes it easier to analyze than
the more realistic (non-iterated) read-write Atomic-Snapshot memory model (AS).
It is known that AS and IIS are equivalent with respect to \emph{wait-free
task} computability: a distributed task is solvable in AS if and only if it
solvable in IIS. We observe, however, that this equivalence is not sufficient
in order to explore solvability of tasks in \emph{sub-models} of AS (i.e.
proper subsets of its runs) or computability of \emph{long-lived} objects, and
a stronger equivalence relation is needed. In this paper, we consider
\emph{adversarial} sub-models of AS and IIS specified by the sets of processes
that can be \emph{correct} in a model run. We show that AS and IIS are
equivalent in a strong way: a (possibly long-lived) object is implementable in
AS under a given adversary if and only if it is implementable in IIS under the
same adversary. %This holds whether the object is one-shot or long-lived.
Therefore, the computability of any object in shared memory under an
adversarial AS scheduler can be equivalently investigated in IIS
On the Bit Complexity of Iterated Memory
Computability, in the presence of asynchrony and failures, is one of the
central questions in distributed computing. The celebrated asynchronous
computability theorem (ACT) charaterizes the computing power of the read-write
shared-memory model through the geometric properties of its protocol complex: a
combinatorial structure describing the states the model can reach via its
finite executions. This characterization assumes that the memory is of
unbounded capacity, in particular, it is able to store the exponentially
growing states of the full-information protocol.
In this paper, we tackle an orthogonal question: what is the minimal memory
capacity that allows us to simulate a given number of rounds of the
full-information protocol? In the iterated immediate snapshot model (IIS), we
determine necessary and sufficient conditions on the number of bits an IIS
element should be able to store so that the resulting protocol is equivalent,
up to isomorphism, to the full-information protocol. Our characterization
implies that processes can simulate rounds of the
full-information IIS protocol as long as the bit complexity per process is
within and . Two processes, however, can simulate
any number of rounds of the full-information protocol using only bits per
process, which implies, in particular, that just bits per process are
sufficient to solve -agreement for arbitrarily small
.Comment: 21 pages, 4 figures. To be published in 31st International Colloquium
On Structural Information and Communication Complexity (SIROCCO 2024