A dynamic epistemic logic analysis of equality negation and other epistemic covering tasks☆,☆☆
Introduction
Functions are the basic objects of study in computability theory. A function is computable if there exists a Turing machine which, given an input of the function domain, returns the corresponding output. If instead of one Turing machine, we have many, and each one gets only one part of the input, and should compute one part of the output, we are in the setting of distributed computability, e.g. [1], [28]. The sequential machines are called processes,1 and are allowed to be infinite state machines, to concentrate on the interaction aspects of computability, disregarding sequential computability issues. The notion corresponding to a function is a task, roughly, the domain is a set of input vectors, the range is a set of output vectors, and the task specification Δ is an input/output relation between them. An input vector I specifies in its i-th entry the (private) input to the i-th process, and an output vector states that it is valid for each process i to produce as output the i-th entry of O, whenever the input vector is I. An important example of a task is consensus, where each process is given an input from a set of possible input values, and the participating processes have to agree on one of their inputs.
A distributed computing model has to specify various details related to how the processes communicate with each other and what type of failures may occur. It turns out that different models may have different power; they can solve different sets of tasks. In this paper we consider the layered message-passing model [19], both because of its relevance to real systems, and because it is the basis to study task computability. This simple, wait-free round-based model where messages can be lost, is described in Section 2.
The theory of distributed computability has been well-developed since the early 1990's [22], with origins even before [5], [13], and surveyed in a book [19]. It was discovered that the reason for why a task may or may not be computable is of a topological nature. The input and output sets of vectors are best described as simplicial complexes, and a task can be specified by a relation Δ from the input complex to the output complex . The main result is that a task is solvable in the layered message-passing model if and only if there is a certain subdivision of the input complex and a certain simplicial map δ to the output complex that respects the specification Δ. This is why the layered message-passing model is fundamental; models that can solve more tasks than the layered message-passing model preserve the topology of the input complex less precisely (they introduce “holes”).
We are interested in understanding distributed computability from the epistemic point of view. What is the knowledge that the processes should gain, to be able to solve a task? This question began to be addressed in [18], using dynamic epistemic logic (DEL). Here is a brief overview of the approach taken in [18]. A new simplicial model for a multi-agent system was introduced, instead of the usual Kripke epistemic S5 model based on graphs. Then, the initial knowledge of the processes is represented by a simplicial model, denoted as , based on the input complex of the task to be solved. The distributed computing model is represented by an action model , and the knowledge at the end of the executions of a protocol is represented by the product update , another simplicial model.2 Remarkably, the task specification is also represented by an action model , and the product update gives a simplicial model representing the knowledge that should be acquired, by a protocol solving the task. The task is solvable in whenever there exists a morphism such that the diagram of simplicial models below commutes (where π denotes the projection of a product update model onto the initial model, making sure that initial values are preserved).
Thus, to prove that a task is unsolvable, one needs to show that no such δ exists. But one would want to produce a specific formula that concretely represents knowledge that exists in , but has not been acquired after running the protocol, i.e. in . Indeed, it was shown in [18] that two of the main impossibilities in distributed computability, consensus [13], [27] and approximate agreement [19], can be expressed by such a formula. However, for other unsolvable tasks (e.g. set agreement), no such formula has been found, despite the fact that no morphism δ exists.
In this paper we show that actually, there are unsolvable tasks for which no such formula exists, namely, the equality negation task, defined by Lo and Hadzilacos [26]. This task was introduced as the central idea to prove that the consensus hierarchy [20], [23] is not robust.
Consider two processes and , each of which has a private input value, drawn from the set of possible input values . After communicating, each process must irrevocably decide a binary output value, either 0 or 1, so that the outputs of the processes are the same if and only if the input values of the processes are different.
It is interesting to study the solvability of the equality negation task from the epistemic point of view. It is well known that there is no wait-free algorithm for consensus in our model [6], [27]. The same is true for equality negation, as shown in [26], [17]. This is intriguing because there is a formula that shows the impossibility of consensus (essentially reaching common knowledge on input values) [18], while, as we show here, there is no such formula for equality negation. In more detail, it is well known that consensus is intimately related to connectivity, and hence to common knowledge, while its specification requires deciding values that are in disconnected components of the output complex. The equality negation task is unsolvable for a different reason, since its output complex is connected. Moreover, equality negation is strictly weaker than consensus: consensus can implement equality negation, but not vice versa (the latter is actually a difficult proof in [26]). So it is interesting to understand the difference between the knowledge required to solve each of these tasks.
The binary consensus task and the equality negation task are depicted next. For each one, the input complex and output complex are represented as graphs (one-dimensional simplicial complexes), with vertices colored red or blue, associated to and respectively. Each vertex also has a number, an input value in the case of vertices of or an output value in the case of vertices of . An edge means that a possible initial configuration of the system is when starts with input value and starts with input value . The relation Δ is a carrier map (see Section 4.2). If an edge , where , it means that according to the task specification, it is valid for to decide and for to decide , in any execution starting on input configuration e. For instance, the carrier map Δ specifying consensus sends the input edge to the set of two output edges , meaning that if the processes start with inputs 0 and 1, they can either both decide 0, or both decide 1 (we write xy as shorthand for the edge ). Similarly, the carrier map for equality negation sends the edges 00, 11 and 22 to the set ; and the edges 01, 02, 10, 12, 20 and 21 to the set .
We show that the reason why there is no formula showing the impossibility of equality negation is that the simplicial models and are bisimilar. The simplicial model is depicted in Section 5.2, notice that it is different from the equality negation output complex above. For this, we work out in detail a bisimulation notion for simplicial models.
Furthermore, we show that the reason why and are bisimilar is that the equality negation task is an epistemic covering task. The k-generalized equality negation task (Section 5.4.1) is similar, but defined in terms of a set of output values of cardinality 2k. For it coincides with the equality negation task. For every , the task is unsolvable in the wait-free model. However, they are all epistemic covering tasks, and hence the output model is bisimilar to the input model. Thus, no formula can show the impossibility for any of these tasks.
Epistemic covering tasks are a rich family of tasks, with a hierarchical structure. For any input simplicial model, a family of epistemic covering tasks can be defined in terms of covering complexes [14], [30]. None of them are wait-free solvable (except for the trivial ones), yet no formula exists to show that.
Finally, we propose a version of our framework based on an extension of DEL allowing factual change of atomic propositions [8], [4], [7]. In DEL with factual change we can express whether the epistemic covering task is solvable, as this is now represented as reachability of a given goal formula after action model execution. This approach also applies to other tasks involving the setting or resetting of decision variables. Intuitively, the reason why we cannot find a formula witnessing the unsolvability of the task in DEL without factual change is because the update expressivity of DEL without factual change is less than that of DEL with factual change. So, our solution is to enrich the language by adding the facility to change the value of atomic propositions, allowing us to express the required formula in an elegant intuitive way.
Section 2 recalls the DEL framework introduced in [18]. We define the layered message-passing model in this context in Section 3. Then, we develop a notion of bisimulation on simplicial models, which is both interesting in its own, but more fundamentally here, will be instrumental in our proof that some tasks cannot be proven unsolvable using DEL. We define the equality negation task in Section 5, and end up this section by generalizing this task to what we call epistemic covering tasks. The important remark is that these tasks are such that they produce product updates in the DEL sense that are similar to topological coverings of the input. In Section 6 we put the pieces together and prove that there is no DEL formula that shows that the equality negation task as well as general covering tasks are not solvable. We end the paper by discussing possible DEL extensions that would give us an explanatory formula for every unsolvable epistemic covering task. Additionally, in Appendix B we prove that the unsolvability of these tasks can be proven in an extension of DEL, with factual change.
Section snippets
Topological models for epistemic logic
We recap here the new kind of model for epistemic logic based on chromatic simplicial complexes, introduced in [18]. The geometric nature of simplicial complexes allows us to consider higher-dimensional topological properties of our models, and investigate their meaning in terms of knowledge. The idea of using simplicial complexes comes from distributed computability [19], [24]. After describing simplicial models, we explain how to use them in DEL.
Syntax. Let At be a countable set of atomic
Layered message-passing
In this section, we describe the computational model that we are interested in. We first explain distributed computing intuition, and then formalize it using action models.
We start with an overview of the layered message-passing model for two agents, or processes as they are called in distributed computing. More details about this model can be found in [19]. This model is known to be equivalent to the well-studied read/write wait-free model, in the sense that it solves the same set of tasks.
Bisimulation for simplicial models
The notion of bisimulation for simplicial models is fundamental in itself, and indeed will play a crucial role in our study. We start with a definition directly derived from the usual definition for Kripke models. Then we reformulate it into the language of combinatorial topology, using carrier maps.
Equality negation and epistemic covering tasks
We first formalize the notion of task from distributed computing using the action model formalism in Section 5.1. We define one important example of such a task, the equality negation task, in Section 5.2. Then, in Section 5.4 we observe that, from our perspective, it is nothing more than an instance of a large class of epistemic tasks called covering tasks. In order to define this family we recall some basic notions about covering spaces in Section 5.3.
A DEL analysis of equality negation and other covering tasks
Our goal is to use the DEL framework of Section 3 to prove the unsolvability of equality negation. In [18], we devised a proof method to establish such impossibility results, and successfully applied it to two classic distributed-computing examples: consensus and approximate-agreement. The proof method of [18] relies on an epistemic logic formula φ witnessing the impossibility of solving the task. For the so-called set-agreement task, we were not able to find a suitable formula φ.
We will see in
Discussion
The main result of this paper, Theorem 14, shows a limit of the DEL approach for studying distributed tasks. Indeed, we exhibited an example of a task, equality negation, which is known to be unsolvable using traditional topological methods, but for which there is no epistemic logic formula witnessing this impossibility. This is disappointing because, as shown in [18], the DEL approach has the major benefit that the formula φ used in the impossibility proof pinpoints the amount of knowledge
Conclusion
The equality negation task is known to be unsolvable in the wait-free read/write model. In fact, equality negation it is an instance of a rich family of tasks, that we call epistemic covering tasks, defined in terms of covering spaces, well-known in topology. All covering tasks are unsolvable in the wait-free model.
In this paper, we study these tasks using the simplicial complex semantics of DEL that we proposed in [18]. There are several purposes of doing this. First, the logical formula
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Eric Goubault and Jérémy Ledent were partially supported by DGA project “Validation of Autonomous Drones and Swarms of Drones” and the academic chair “Complex Systems Engineering” of Ecole Polytechnique-ENSTA-Télécom-Thalès-Dassault-Naval Group-DGA-FX-FDO-Fondation ParisTech. Sergio Rajsbaum was partially supported by the UNAM-PAPIIT project IN106520, by the France-Mexico Binational SEP-CONACYT-ANUIES-ECOS grant M12M01, and benefited from the invited professor programme from Ecole Polytechnique. Marijana Lazić was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 787367 (PaVeS). Hans van Ditmarsch is also affiliated to IMSc, Chennai, India, as research associate.