A Sheaf-Theoretic Characterization of Tasks in Distributed Systems

Assessing the solvability or impossibility of a given task in a distributed system is at the center of distributed systems research. A variety of theoretical tools have been developed by sourcing from different fields such as combinatorics, topology and epistemic logic. One of the earliest results is the two generals problem impossibility [2, 19], which roughly states that consensus is impossible in networks with message loss and unbounded delays. Numerous results have followed since: the celebrated FLP [11] impossibility shows that consensus cannot be achieved in asynchronous and crash-prone systems, the lossy-link impossibility [26] (the authors referred to it as mobile failures) that shows that consensus is impossible even in a synchronous two-process model if just one message may be lost in each round, [13] characterized consensus in distributed systems via epistemic logic and [23] provides a topological characterization of consensus under general message adversaries.

Although the aforementioned results are mostly related to problem solvability, they are specifically focused on consensus. Nevertheless, these results undoubtedly contributed to the development of frameworks that would later address general task solvability. Some such noteworthy results are Herlihy and Shavit’s characterization of wait-free t-resilient shared memory distributed computation, via combinatorial topology [17], Attiya, Nowak and Castañeda’s characterization of terminating tasks [4], and Alcántara, Castañeda, Flores and Rajsbaum’s characterization of look-compute-move wait-free robot tasks [3].

Sheaf theory originated from studying how local constraints give rise to global solutions, paired with cohomology theory as a measure of obstructions to such global constructions. Cellular sheaves were introduced by Curry [7] as a combinatorial counterpart to the common notion of sheaves found in the literature [20], with the interest of having a theory that is computable. Most importantly to us, the theory of cellular sheaves presents the concept of sheaves in an approachable manner, whence it becomes clear its usefulness to depict global properties of local information, as is the case for distributed computing. Sheaves have proven valuable in studying similar local-to-global phenomena, from opinion dynamics [14] to contextuality in quantum mechanics [1] and sensor integration [25, 18]. We now extend this framework to distributed systems.

In this paper, we extend the brief announcement [9] introducing a generic sheaf-theoretic framework for analyzing task solvability parametrized by indistinguishability relations on global states of the system, forming a graph-like structure. These indistinguishability graphs are closely related to Kripke frames in epistemic logic, as explored in [8]. Furthermore, our framework enables us to leverage the computational methods in cohomology theory so to analyze task solvability, and even derive solving algorithms in systems where the indistinguishability graph admits a cohomology group construction.

Contributions We introduce a novel sheaf-theoretic framework for analyzing distributed task solvability. Our approach complements the existing frameworks by connecting category theory with distributed computing. This novel bridge allows us to directly apply computational methods for decision function synthesis using linear algebra. We expect this first connection to provide fertile ground for more applications in distributed computing.

More precisely, we provide the following.

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  A task sheaf construction (Definition˜4.1) that captures both the local knowledge of each process (local constraints) and relates it to valid solutions, assessing global consistency;

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  A finite substructure of the executions that preserves the task solvability as system slices (Definition˜3.6);

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  An equivalence between the existence of solutions to a task and the existence of sections over the corresponding system slice (Theorem˜4.4);

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  A way to use cohomology to transform impossibility arguments into computational problems (Theorem˜5.5).

Paper Organization We start by defining the model of a distributed computing system we use in Section˜2, giving special attention to the discrete structures used to depict the system executions (Section˜2.1) and its tasks (Section˜2.2). Section˜3 introduces system slices, the smallest sufficient structure needed to assess task solvability, and how it brought to the categorical machinery. Section˜4 defines the task sheaf, first by its explicit construction (Section˜4.1), then by its equivalent categorical formulation (Section˜4.2), so to state the main theorem relating the solvability of tasks to the existence of sections (Section˜4.3). Section˜5 uses the categorical machinery of task sheaves to define its cohomology construction, so used to computationally assess task solvability. Section˜6 provides some concluding words.

We now define how processes evolve, communicate and make decisions, in order to define the runs of that system. Our main contribution on task solvability solely depends on the runs and the task specification and consequently our approach is applicable to any system providing runs consisting of global states consisting of local states. As there are multiple approaches to modeling distributed systems, we do not formally specify how to obtain said runs, but exemplify it via a synchronous message passing system, which is one common choice to represent a distributed system (e.g., [24]).

We call local protocols processes $\Pi=\{p\}_{i\in{[n]}}$ which are deterministic state machines $\mathcal{P}_{p}$ containing the possible local states, together with distinguished input states. The local protocol is defined by transition function, a communication function and a decision function $\delta_{p}$ mapping local states to task-specific outputs. Processes evolve depending on their previous local state and some partial information about other processes local states, usually represented via messages they received. The possibilities of (partial) information reception of the processes about the whole system state is captured as the adversary, also called the message adversary [26, 27, 28], heard-of predicates [6], time-varying-graphs [5], scheduler [22] etc., which influence the behavior of the system. Since there is a pallette of different adversaries, each different in their way they impact the system, and the precise capabilities of the adversary are not important in stating our results, we do not formalize it further. The only object we need is a run consisting of a sequence of configurations, defined in what follows.

Termination on an execution graph can be represented as a set of configurations that cuts the execution graph in half. Indeed, termination by definition is just a set of configurations that intersects any run, together with some extra conditions that we will define in the following. Unfortunately, not every such cut through the execution graph need be finite and therefore easy to find. Thus we do not concern ourselves with how we look for one, instead describe its shape.

An execution cut represents a set of “unavoidable” configurations within the system. Furthermore, we say that an execution cut $A\subset V(\mathcal{E})$ is terminal iff any $U\in A$ is terminal, i.e., at configuration $U$, all processes must have decided.

A causal closure $ccl(\mathcal{N}(A))$ of a terminal execution cut $A$ extends to the partially terminated configurations in $\mathcal{N}(A)\setminus A$, where at least one process $p$ has already terminated, but not necessarily all of them. Any partially terminated configuration $\left\langle g\right\rangle$eventually results into a terminated configuration $\left\langle h\right\rangle$in $A$. The causal closure contains all successors of partially terminated configurations up until they result in a fully terminated configuration in $A$. As some processes have already decided in a partially terminated configuration $\left\langle g\right\rangle$, we ensure that they keep their decided values in any successor configurations $\left\langle h\right\rangle$(reflecting that decisions are final) by constructing the causal consistency $\{\simeq_{p}\}_{p\in\Pi}$ relation such that $\left\langle g\right\rangle\simeq_{p}\left\langle h\right\rangle$.

Intuitively, given a causal closure of a terminal execution cut $ccl(\mathcal{N}(A))$, the causal consistency relation relates all configurations where a process terminated in a preceding or indistinguishable configuration. As $\leqslant$ and $\{\sim_{p}\}_{p\in\Pi}$ are reflexive, $\{\simeq_{p}\}_{p\in\Pi}$ is reflexive, and contains both relations $\leqslant$ and $\{\sim_{p}\}_{p\in\Pi}$. Observe that any causally dependent configurations in $\left\langle g\right\rangle,\left\langle h\right\rangle\in A$ such that $\left\langle g\right\rangle\leqslant\left\langle h\right\rangle$ are related $\left\langle g\right\rangle\simeq\left\langle h\right\rangle$ (and also $\left\langle h\right\rangle\simeq\left\langle g\right\rangle$) for all processes. Configurations $\left\langle g\right\rangle$in $ccl(\mathcal{N}(A))$ that are not in $A$, are related to a configuration $\left\langle h\right\rangle$in $A$ via the indistinguishability relation of $p$ and any successor of $\left\langle g\right\rangle$is also related to $\left\langle h\right\rangle$for $p$. This ensures that the partially terminated process $p$ keeps its terminated value.

In general system slices need not be finite, see Example˜3.7 for an explicit construction of an execution graph with a necessarily infinite system slice. An finite system slice is depicted in Example˜4.6

System slices naturally form a graph structure where we can define sheaves, where the set of global configurations are its vertices and the causal consistency relations its edges. This suffices for our purposes, but note that this consists of the more general structure of a cellular complex, where vertices and edges are $0$ and $1$ cells, and the execution structure naturally satisfies its local finite requirement. Cellular complexes are fundamental objects in algebraic topology, and a full account can be found in [15]. Most importantly, a cellular complex $X$ has an underlying partial ordering of its cells (vertices and edges), denoted $P_{X}$, which we exemplify now.

Finally, in order to access the categorical machinery needed to define the task sheaves, we need need to introduce the concept of a cellular category [21], which categorifies the cellular complex $X$. by viewing its associated poset $P_{X}$ as a category, which preserves the topological structure while enabling a categorical perspective better suited for defining sheaves.

Observe that any system slice is an undirected graph and therefore induces a cellular category. We will make ample use of this in the the following.

In this section, we introduce the task sheaf as a mathematical framework for encoding the solvability constraints of a distributed task. Sheaves provide a formalism that captures both the global structure of a task definition and the local constraints which protocols must respect. We present two equivalent formulations. First, an explicit construction, used to build the intuition over the running examples. Then, we present its categorical foundations, which enables the use of cohomology later in Section˜5. We then state the main theorem relating the solvability of a task to the existence of sections in the appropriate task sheaf.

Given a communication adversary $K$ representing some adversarial entity together with a task $T$, we search for a protocol together with a decision map. As the full-information protocol provides the finest possible execution tree, its natural to start the search process there. Slices that are finite in size can be recursively enumerated and, although that’s computationally inefficient, tested. If we find a finite slice $A$ that allows for a terminal decision map we can try to optimize the protocol inducing the execution tree. If we do not find a slice, but keep looking forever, then a finite slice does not exist and the task is not wait-free solvable, although it might be solvable given an infinite slice.

In this section we focus on wait-free solvability, so assume we have found a finite terminating slice $ccl(\mathcal{N}(A))$, we computationally determine the space of all sections and therefore the decision map $\delta$. We do this via cohomology, i.e., we turn our structure into an abelian group and extract from it all possible solutions.

Given a task sheaf $\mathcal{F}$, the process of obtaining the $n$th sheaf cohomology can be understood as an iteration of the following steps.

Where $\mathbf{sAb}$ is the category of simplicial abelian groups, $\mathbf{Ch_{\mbox{$\mathbb{Z}$}}}$ is the category of (co)chain complexes with integer coefficients and $\mathbf{Ab}$ the category of Abelian groups, where our cohomology lives.

We provide a brief explanation, and illustrate it below in Example˜5.4. Given a cset $\mathcal{O}$ containing the possible system output states, we can obtain a simplicial abelian group through the left adjoint of the forgetful functor that sends it to the underlying simplicial set²²2Here chromatic semi-simplicial sets are treated as simplicial sets for simplicity, as they only add the process labeling that would require extra bookkeeping.. The third map gets us a cochain complex with differentials defined from the alternating sum of the (co)face maps. Finally, we obtain the $n$-th cohomology group, which corresponds to a space of sections of our sheaf. This construction is well known in the literature of algebraic topology [15] and we adapt it as a tool for understanding distributed tasks.

The two cochain groups are connected via a linear coboundary map $\delta$. This maps a specific choice of output vectors to the individual choices along the indistinguishability edges defined by the restriction maps. To define $\delta$ we chose an arbitrary direction on each indistinguishability edge $e=\left\langle g\right\rangle\rightarrow\left\langle h\right\rangle$ just to facilitate an algebraic representation.

We can represent the coboundary map $d$ as a coboundary matrix $D$ where rows are indexed by edges and columns are indexed by configurations:

One can think of $D$ simply as computing the difference between two indistinguishable configurations. A section on our sheaf is a $0$-cochain that is mapped to $0$ by $d$, so any assignment to configurations such that any process that cannot distinguish two configurations, decides the same thing. The set of all sections is then the kernel $ker(D)$.