10 research outputs found
Extension-Based Proofs for Synchronous Message Passing
There is no wait-free algorithm that solves k-set agreement among n ? k+1 processes in asynchronous systems where processes communicate using only registers. However, proofs of this result for k ? 2 are complicated and involve topological reasoning. To explain why such sophisticated arguments are necessary, Alistarh, Aspnes, Ellen, Gelashvili, and Zhu recently introduced extension-based proofs, which generalize valency arguments, and proved that there are no extension-based proofs of this result.
In the synchronous message passing model, k-set agreement is solvable, but there is a lower bound of t rounds for any k-set agreement algorithm among n > kt processes when at most k processes can crash each round. The proof of this result for k ? 2 is also a complicated topological argument. We define a notion of extension-based proofs for this model and we show there are no extension-based proofs that t rounds are necessary for any k-set agreement algorithm among n = kt+1 processes, for k ? 2 and t > 2, when at most k processes can crash each round. In particular, our result shows that no valency argument can prove this lower bound
Partial Product Updates for Agents of Detectable Failure and Logical Obstruction to Task Solvability
The logical method proposed by Goubault, Ledent, and Rajsbaum provides a
novel way to show the unsolvability of distributed tasks by means of a logical
obstruction, which is an epistemic logic formula describing the reason of
unsolvability. In this paper, we introduce the notion of partial product
update, which refines that of product update in the original logical method, to
encompass distributed tasks and protocols modeled by impure simplicial
complexes. With this extended notion of partial product update, the original
logical method is generalized so that it allows the application of logical
obstruction to show unsolvability results in a distributed environment where
the failure of agents is detectable. We demonstrate the use of the logical
method by giving a concrete logical obstruction and showing that the consensus
task is unsolvable by the single-round synchronous message-passing protocol
A Simplicial Model for : Epistemic Logic with Agents that May Die
The standard semantics of multi-agent epistemic logic S5 is based on Kripke
models whose accessibility relations are reflexive, symmetric and transitive.
This one dimensional structure contains implicit higher-dimensional information
beyond pairwise interactions, that we formalized as pure simplicial models in a
previous work (Information and Computation, 2021). Here we extend the theory to
encompass simplicial models that are not necessarily pure. The corresponding
class of Kripke models are those where the accessibility relation is symmetric
and transitive, but might not be reflexive. Such models correspond to the
epistemic logic KB4 . Impure simplicial models arise in situations where two
possible worlds may not have the same set of agents. We illustrate it with
distributed computing examples of synchronous systems where processes may
crash
Simplicial Models for the Epistemic Logic of Faulty Agents
In recent years, several authors have been investigating simplicial models, a
model of epistemic logic based on higher-dimensional structures called
simplicial complexes. In the original formulation, simplicial models were
always assumed to be pure, meaning that all worlds have the same dimension.
This is equivalent to the standard S5n semantics of epistemic logic, based on
Kripke models. By removing the assumption that models must be pure, we can go
beyond the usual Kripke semantics and study epistemic logics where the number
of agents participating in a world can vary. This approach has been developed
in a number of papers, with applications in fault-tolerant distributed
computing where processes may crash during the execution of a system. A
difficulty that arises is that subtle design choices in the definition of
impure simplicial models can result in different axioms of the resulting logic.
In this paper, we classify those design choices systematically, and axiomatize
the corresponding logics. We illustrate them via distributed computing examples
of synchronous systems where processes may crash
A simplicial model for KB4n : epistemic logic with agents that may die
The standard semantics of multi-agent epistemic logic S5n is based on Kripke models whose accessibility relations are reflexive, symmetric and transitive. This one dimensional structure contains implicit higher-dimensional information beyond pairwise interactions, that we formalized as pure simplicial models in a previous work in Information and Computation 2021 [10]. Here we extend the theory to encompass simplicial models that are not necessarily pure. The corresponding class of Kripke models are those where the accessibility relation is symmetric and transitive, but might not be reflexive. Such models correspond to the epistemic logic KB4n. Impure simplicial models arise in situations where two possible worlds may not have the same set of agents. We illustrate it with distributed computing examples of synchronous systems where processes may crash
Impure Simplicial Complexes: Complete Axiomatization
Combinatorial topology is used in distributed computing to model concurrency
and asynchrony. The basic structure in combinatorial topology is the simplicial
complex, a collection of subsets called simplices of a set of vertices, closed
under containment. Pure simplicial complexes describe message passing in
asynchronous systems where all processes (agents) are alive, whereas impure
simplicial complexes describe message passing in synchronous systems where
processes may be dead (have crashed). Properties of impure simplicial complexes
can be described in a three-valued multi-agent epistemic logic where the third
value represents formulae that are undefined, e.g., the knowledge and local
propositions of dead agents. In this work we present an axiomatization for the
logic of the class of impure complexes and show soundness and completeness. The
completeness proof involves the novel construction of the canonical simplicial
model and requires a careful manipulation of undefined formulae