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

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

Communication Pattern Logic: Epistemic and Topological Views

We propose communication pattern logic. A communication pattern describes how processes or agents inform each other, independently of the information content. The full-information protocol in distributed computing is the special case wherein all agents inform each other. We study this protocol in distributed computing models where communication might fail: an agent is certain about the messages it receives, but it may be uncertain about the messages other agents have received. In a dynamic epistemic logic with distributed knowledge and with modalities for communication patterns, the latter are interpreted by updating Kripke models. We propose an axiomatization of communication pattern logic, and we show that collective bisimilarity (comparing models on their distributed knowledge) is preserved when updating models with communication patterns. We can also interpret communication patterns by updating simplicial complexes, a well-known topological framework for distributed computing. We show that the different semantics correspond, and propose collective bisimulation between simplicial complexes

A dynamic epistemic logic analysis of equality negation and other epistemic covering tasks

In this paper we study the solvability of the equality negation task in a simple wait-free model where two processes communicate by reading and writing shared variables or exchanging messages. In this task, the two processes start with a private input value in the set {0,1,2}, and after communicating, each one must decide a binary output value, so that the outputs of the processes are the same if and only if the input values of the processes are different. This task is already known to be unsolvable; our goal here is to prove this result using the dynamic epistemic logic (DEL) approach introduced by Goubault et al. (2018) [18]. We show that in fact, there is no epistemic logic formula that explains why the task is unsolvable. Furthermore, we observe that this task is a particular case of an epistemic covering task. We thus establish a connection between the existing DEL framework and the theory of covering spaces in topology, and prove that the same result holds for any epistemic covering task: no epistemic formula explains the unsolvability

Wait-Free Solvability of Equality Negation Tasks

We introduce a family of tasks for n processes, as a generalization of the two process equality negation task of Lo and Hadzilacos (SICOMP 2000). Each process starts the computation with a private input value taken from a finite set of possible inputs. After communicating with the other processes using immediate snapshots, the process must decide on a binary output value, 0 or 1. The specification of the task is the following: in an execution, if the set of input values is large enough, the processes should agree on the same output; if the set of inputs is small enough, the processes should disagree; and in-between these two cases, any output is allowed. Formally, this specification depends on two threshold parameters k and l, with k<l, indicating when the cardinality of the set of inputs becomes "small" or "large", respectively. We study the solvability of this task depending on those two parameters. First, we show that the task is solvable whenever k+2 <= l. For the remaining cases (l = k+1), we use various combinatorial topology techniques to obtain two impossibility results: the task is unsolvable if either k <= n/2 or n-k is odd. The remaining cases are still open