Continuous Tasks and the Chromatic Simplicial Approximation Theorem

09/03/2021
by   Hugo Rincon Galeana, et al.
0

The celebrated 1999 Asynchronous Computability Theorem (ACT) of Herlihy and Shavit characterized the distributed tasks that are wait-free solvable, and thus uncovered a deep connection with algebraic topology. We present a novel interpretation of this theorem, through the notion of continuous task, defined by an input/output specification that is a continuous function. To do so, we introduce a chromatic version of a foundational result for algebraic topology: the simplicial approximation theorem. In addition to providing a different proof of the ACT, the notion of continuous task seems interesting in itself. Indeed, besides the fact that certain distributed problems are naturally specified by continuous functions, continuous tasks have an expressive power that also allows to specify the density of desired outputs for each combination of possible inputs,for example.

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I Introduction

Given a finite set of values , a task is a problem where each process of a distributed system starts with a private input value from , communicates with the others, and halts with a private output value from . The input complex defines the set of possible assignments of input values to the processes, and the output complex defines the allowed decisions. The input/output relation specifies, for each input assignment , a set of valid output decisions. Processes are usually asynchronous, they can be halted or delayed without warning by cache misses, interrupts, or scheduler pre-emption. In asynchronous systems, it is desirable to design algorithms that are wait-free: any process that continues to run will produce an output value, regardless of delays or failures by other processes.

It is now almost 30 years since Herlihy and Shavit [HS99:ACT] presented the celebrated Asynchronous Computability Theorem (ACT). Roughly speaking, the theorem states that a task is wait-free solvable in a read/write shared memory distributed system if and only if can be subdivided times, and sent by a simplicial decision map to , respecting the input/output relation (carrier map) . Intuitively, is the number of rounds the processes need to communicate to solve the task, and hence yields a time complexity characterization as well [HoestS06].

Section I below states one direction of the ACT (restricted for reference purposes only); it actually holds in the other direction as well. It is the easier direction, and can be proved in several ways, see e.g. [AttiyaR02, BG93:STOC, HS99:ACT, SZ00:SIAM], essentially by considering an appropriate subset of all wait-free executions of a full-information protocol: in any wait-free read/write model, any protocol solving a given task induces a subdivision of the task’s input complex.

[ACT (if-direction) [HS99:ACT]] If is solvable, then there exists a chromatic subdivision and a chromatic simplicial map carried by .

The ACT reinterprets distributed computing geometrically, and provides an explanation of why some tasks are solvable, and not others. It is particularly useful for proving that some tasks are unsolvable, opening the doors to the very powerful machinery of combinatorial topology; notable examples are the set agreement [HS99:ACT] impossibility result, which generalizes the classic FLP consensus impossibility [FLP85], and the renaming [Castaeda2010NewCT] impossibility result. Furthermore, the theorem is the basis for task solvability characterizations in other distributed computing models: systems where at most processes may crash, synchronous and partially synchronous processes, Byzantine and dependent failures, stronger shared memory communication objects, message passing models and even robot coordination algorithms [AlcantaraCFR19], see Section VII. An overview of topological distributed computing theory, as started by the ACT, can be found in the book [HKR13].

The simplicial formulation of the input/output relation and the ACT in general both obfuscates certain properties and introduces technical difficulties, which considerably complicated the proofs, cp. [AttiyaR02, BG93:STOC, HS99:ACT, SZ00:SIAM]. For colorless tasks like set agreement, which can be defined without refering to process ids (discussed in more detail in Section VII), there is also a continuous version of the ACT. It roughly says that a colorless task is wait-free solvable if and only if there is a continuous map from to carried by , where denotes the geometric realization of the complex . Since all involved complexes are colorless here, the proofs are much simpler. However, we are not aware of any formulation of the ACT via continuous functions for general (chromatic) tasks.

Indeed, the main source of the technical difficulties in the proof of the ACT is that the objects involved are chromatic: each vertex of and is associated to one of the process ids in the system, and all the simplicial maps are required to preserve vertex ids, i.e., must be rigid in that they always preserves the dimension of the simplices. Quite some effort is needed to deal with the resulting requirement of a chromatic decision map in the case of a (non-colorless) task such as renaming, which can specify which values can be output by which process.

Figures/hourglass

This is illustrated by the Hourglass task shown to the right, where there are three processes denoted by green, red, and yellow. There is no input, hence only one input simplex, and the processes must output labels in as shown in the output complex. The carrier map defining this task requires (i) to map the corners (solo executions) of to the vertices labeled in , (ii) to map the boundary (executions where two processes participate) of to the boundary of , as shown for the yellow-green boundary in the figure; and (iii) the triangle of (executions where all participate), to any triangle in .

Interestingly, there is a continuous map from to carried by , i.e., the Hourglass task would satisfy the conditions of the colorless ACT. It fails to meet our conditions for a continuous task (see Definition 5), however. And indeed, it has already been proved in [HKR13] that it is not wait-free solvable. By contrast, as the output complex of the Hourglass trask is not link-connected, it is not possible to apply related characterization theorems like [GKM14:PODC, Thm. 8.4] to prove this unsolvability (see Section VII for a more detailed discussion).

A foundational result in algebraic topology is the simplicial approximation theorem first proved by Brouwer, which served to put homology theory on a rigorous basis [simpComp]. It guarantees that continuous maps can be (by a slight deformation) approximated by simplicial maps. Roughly, it says that given a continuous map from to , there is some such that has a corresponding simplicial map from to that is a simplicial approximation, where denotes the barycentric subdivision (or any other mesh-shrinking subdivision). The colorless version of the ACT [HKR13, Ch. 4] is essentially a distributed computing version of the simplicial approximation theorem. And indeed, in the case of the Hourglass task, the simplicial approximation theorem says that there is some such that there is a simplicial map from to respecting . However, this does not result in a wait-free algorithm, since there is no such that preserves colors, i.e., is rigid.

Whereas several constructions of a chromatic simplicial approximation have been used in the existing proofs of the ACT [HS99:ACT, AttiyaR02, BG93:STOC, SZ00:SIAM] and some generalizations [GKM14:PODC, SaraphHG18], they are complicated and tailored to the specific context, which ensures link connectivity. We are not aware of a simplicial approximation theorem that could guarantee a chromatic map under more general general conditions, like the ones for our continuous tasks (see Definition 5).

Contributions: Our paper has three main contributions: (1) we introduce the novel notion of a continuous task and show that it allows for more expressive task specifications, (2) we formulate and prove a chromatic version of the simplicial approximation theorem, and (3) we use these notions to formulate a continuous version the ACT, denoted CACT, which we prove to be equivalent to the ACT in the wait-free shared memory model. Rather than focusing on a specific model of computation, however, the only property we require from the model is that Section I holds, i.e., if some protocol solves a task, then the protocol determines a chromatic subdivision of the input complex. We refer to any such model by ASM, and give examples of such models (such as the read-write shared memory model of the original ACT [HS99:ACT] and the Iterated Immediate Snapshot (IIS) model) in Section VII.

In more detail, our paper contains the following contributions:

  1. We introduce a continuous task as a triple : The possible input and output configurations are determined by the chromatic input and output complexes , as in the usual task notion. The input/output specification is a continuous function from to , instead of an input/output relation, . We identify an additional property called chromatic for , which intuitively requires to satisfy a minimal color and local dimension preserving requirement, stated formally in Definition 4.

    Semantically, continuous tasks present interesting expressive facilities with respect to traditional input/output specifications in the form of a carrier map . Indeed, they open up various interesting research questions, which we can barely touch here: In Section V, we introduce refined versions of the well-known -approximate agreement task, which demonstrate that even constraints on the density of the outputs for a given input can be expressed by means of continuous task specifications.

  2. We state and prove Fig. 4, a chromatic version of the simplicial approximation theorem, for chromatic functions (see Definition 4), which may be of independent interest also. In a way, it off-loads part of the complexity of constructing geometric subdivisions that ensure rigid maps for an arbitrary continuous function to the definition of a chromatic function .

  3. Using our chromatic simplicial approximation Fig. 4, we prove that chromatic functions precisely capture the notion of solvability in an ASM model. This leads to our CACT Section IV, which states that a task is wait-free solvable if and only if there exists a corresponding continuous task . Finally, Section IV states that any continuous task is solvable, which implies the converse of Section I.

Overall, our results provide a refined explanation on the reasons of why a task may or may not be solvable in an ASM model, and provide an alternative perspective and proof of the ACT (Section VII discusses several related theorems).

Paper organization: In Section II, we define continuous tasks and the meaning of solving such a task, and present the chromatic simplicial approximation theorem. Section IV contains our CACT theorem, and Section V is devoted to the application showing the expressive power of continuous tasks. A discussion of additional related work and some conclusions and directions of future research are provided in Section VII and Section VI, respectively. In the appendix, we provide and all our proofs as well as a collection of background combinatorial topology and distributed computing definitions, primarily taken from [HKR13].

Ii Continuous Tasks and the Chromatic Simplicial Approximation Theorem

In this section, we extend the standard language (see e.g. [HKR13]) of combinatorial topology in distributed computing to be able to specify continuous tasks. More specifically, given arbitrary geometric realizations and , i.e., metric topological spaces formed as the union of geometric simplices corresponding to the respective abstract simplices in and , we introduce the concept of a chromatic function, which is a continuous function that plays the role of a minimal carrier map that determines task solvability.

Ii-a Chromatic Functions and Continuous Task Solvability

We first need to extend the notion of a coloring in order to assign a set of colors to a point in the geometric realization of a finite simplicial complex . We start with the carrier of a point in the geometric realization of a simplicial complex .

Definition 1.

For a point , let the carrier of , denoted , be the unique smallest simplex such that . We can also define the carrier of a set .

Definition 2 (Extended coloring).

Let be an -dimensional simplicial complex with a coloring . We define the extended coloring with respect to as for every .

In addition to defining an extended coloring, we also need a notion of closeness. In topology, a neighborhood of a point is a collection of open sets that include and defines what does it mean to be close to . We will use the following Definition 3, illustrated in Fig. 1.

Figures/neighborhood

Fig. 1: The inner (yellow) triangle, and the red segment are neighborhoods of of dimension and , respectively, in the outer (blue) triangle .
Definition 3 (Simplicial neighborhood).

Let be an abstract simplicial complex, a point in its geometric realization, and . We say that is a simplicial neighborhood of if , where each , and is a point in its interior. That is, is homeomorphic to a geometric simplex generated from points that belong to .

A chromatic function has the property that it does not allow to map a neighborhood of to a neighborhood of with smaller dimension, and which preserves colors:

Definition 4 (Chromatic Function).

We say that a continuous function is chromatic (with respect to and ) if, for every , and any simplicial neighborhood of , it guarantees .

We note that not every continuous function is chromatic. An example is shown in Fig. 2, where we assume that maps the entire red curve in the blue simplex on the left to the central red vertex in on the right. This collapses a 1-dimensional neighborhood of to a 0-dimensional neighborhood . Similarly, if we assume that also maps the whole area above this curve to the line connecting the central red vertex to the small yellow vertex on the boundary of the upper simplex in , it also collapses a 2-dimensional neighborhood to a 1-dimensional neighborhood. Finally, also violates the color preservation requirement, as a 1-dimensional neighborhood of lying in the face consisting of the (green, yellow) edge is mapped to , which lies on the the (red, green) edge in the upper simplex in .

Figures/nchromatic

Fig. 2: is non-chromatic, since it maps a 1-dimensional neighborhood of a point (the red curve) with to the 0-dimensional red central vertex. It also violates the color preservation requirement, as a 1-dimensional neighborhood of within the (green, yellow) edge in is mapped onto the (red, green) edge in the upper simplex in .

Section II-A and Section II-A show that it is possible to chromatically subdivide the input complex while mantaining the color preservation property of a given chromatic function . Indeed, even if some chromatic subdivision does not fit w.r.t. color preservation, we can make an arbitrarily small perturbation to the vertices in the subdivision fitting. We note that the proof of this lemma is substantially less involved than the perturbation argument in the proof of the ACT [HS99:ACT], [HKR13, Ch. 11]. In fact, rather than constructing subdivisions that ensure rigidity for arbitrary continuous functions , we only need subdivisions that allow to remain a chromatic function.

Let be a chromatic function with respect to and . For any 1-layer chromatic subdivision of , there exist a geometric realization such that is chromatic with respect to and .

Proof.

To prove the lemma, it suffices to show that is chromatic for any given face of dimension . For , this is trivial, so assume that we have shown this already for all faces of dimension , and consider a face with . Let be an internal point of . Since is chromatic with respect to , there exists a -dimensional simplicial neighborhood of such that , i.e., includes all colors. We will first show that there is a point such that : Assuming the contrary, would be contained in the skeleton of . Consequently, also does not include all colors, i.e., there exists a color . Since the skeleton of that includes color is a closed set that does not include , there is a -dimentional -ball around that does not intersect the skeleton of that includes as a color. Due to continuity of , there exists a -dimensional -ball around that is contained in , which is mapped to . Since every point in has all colors in its carrier, we also obtain , but , since it does not include . This contradicts that is chromatic.

Therefore, there exists a point such that . Since is hence an interior point in , there exists where every point has all colors in its carrier. Since and , there exists such that . Again from continuity of , it follows that there exists a -ball around that maps to , where every point in the latter has all colors in its carrier. Therefore, in the to-be-constructed subdivision , we can create the new central face of dimension inside , which will make chromatic with respect to .

According to the induction hypothesis, we have already subdivided the boundary of (with dimension ) in a way that makes chromatic. What still remains to be done, however, is to show that is also chromatic in the non-central faces of dimension generated by our subdivision. For this purpose, it suffices to show that every -dimensional face that includes at least one vertex from can be chosen such that it includes all its colors . This suffices, since lower-dimensional faces of inherently preserve the chromatic property of : (1) For lower-dimensional faces on the boundary of , this follows from the induction hypothesis. (2) For -dimensional faces in the interior of that originate from the intersection with another -dimensional face , the intersection of their color sets must contain colors, which guarantees that is chromatic also on . Finally, (3) for lower-dimensional faces lying in , the color set even comprises all colors and cannot hence make non-chromatic.

To finally justify that, for any -dimensional face of vertices in our subdivision, there must indeed be a simplicial neighborhood that connects these vertices in such that , we just repeat the argument in our first step above: Assuming the contrary, would be in the -skeleton of , which does not include and would hence lead to a contradiction to being chromatic with respect to and . Consequently, we can always find a suitable choice for the geometric realization for , namely . According to case (2) above, for neighboring faces and , we can choose the common border arbitrarily within .

This completes the proof, since we showed that we can insert interior faces and connect new vertices in the geometric subdivision without making non-chromatic. ∎

Let be a chromatic function with respect to and . For any chromatic subdivision of , there exists a geometric realization of such that is chromatic with respect to and .

Our first step towards establishing a discrete/continuous duality between chromatic functions and chromatic simplicial maps, i.e., simplicial maps that preserve the colors of vertices (and hence are rigid), is showing that the geometric realization of a chromatic simplicial map is a chromatic function.

Let be a chromatic simplicial map. Then is a chromatic function.

Proof.

Continuity follows from being an affine mapping from .

To show that it is a chromatic function, we start with the color preservation property. Notice that, since is simplicial and chromatic, for any simplex . This implies that, for any . Moreover, it follows that, for any point and any simplicial neighborhood of of dimension , . Since must be of dimension at least since has dimension , it follows that .

We further develop the discrete/continuous duality by defining continuous tasks. Since chromatic functions correspond to chromatic simplicial maps, and chromatic simplicial maps determine task solvability, we can also use chromatic functions to express solvable tasks.

Definition 5 (Continuous task).

We say that a triple is a continuous task if and are pure chromatic simplicial complexes of the same dimension, and is a chromatic function.

In order for our task definition to be complete, we also need to define a criterion for task solvability. A continuous task is solvable in ASM if there exists an algorithm with a decision map that “approximates” the chromatic function of the continuous task. Therefore we first need a formal definition for a chromatic approximation.

Definition 6 (Chromatic approximation).

Let be a chromatic function. We say that a chromatic simplicial map is a chromatic approximation to if, for all , .

Notice that while a chromatic approximation of is a more relaxed definition than a simplicial approximation of in the topological sense, as need not hold, it adds a color preservation constraint.

Definition 7 (ASM Continuous Task Solvability).

We say that an algorithm in ASM solves a continuous task , if induces a subdivision of , and a (simplicial) decision map that is a chromatic approximation, i.e., for each .

Another way to formulate this condition is by defining an induced task associated to .

Definition 8 (Induced Task).

Given a continuous task , we define the task induced by as , where is the carrier map induced by as given by .

Recall from the ACT Section I that is solvable in ASM if there exists a subdivision and a decision map carried by . According Definition 7, this indeed implies continuous task solvability for as well.

Ii-B The Chromatic Approximation Theorem

In the previous subsection, we provided the motivation and definitions for chromatic functions and continuous tasks as part of a discrete/continuous duality for ASM. We showed that chromatic simplicial maps generate chromatic functions. However, in order for this correspondence to be complete, we need to show that we can approximate any chromatic function with a simplicial chromatic map. In general algebraic topology, the simplicial approximation theorem allows us to discretize continuous functions. In the context of distributed computing, however, we cannot apply the simplicial approximation theorem directly, since it does not necessarily preserve the color structure.

Therefore, in this subsection, we will prove that any chromatic function from a geometric pure simplicial complex into a geometric pure simplicial complex of the same dimension admits a chromatic approximation. To this end, we introduce the notion of the chromatic projection for some color . It maps interior points of a geometric simplex to its border, by taking the ray from the vertex with color to , and mapping to the intersection of and the opposite border of .

Definition 9 (Chromatic projection).

Let be a pure chromatic simplicial complex, and . We define the chromatic projection with respect to as as follows: For , let be the carrier of in . If , we define . Now, assume that , in which case we must have . Writing , where each , and the ’s correspond to the affine coordinates of with respect to , we define .

An example for a chromatic projection can be found in Fig. 3, for : Both the points marked by the green and the red inner node on the boundary of are mapped to the respective border of their carriers that lies opposite of the yellow vertex.

Fig. 3 also illustrates the pivotal concept of star-covering introduced in Definition 10, which requires the image of the interior of a simplex to be contained in the open star of some vertex .

Definition 10 (Star-covered subdivision).

Let be a chromatic function. We say that a chromatic subdivision is star-covered with respect to if for any , for some .

Figures/retract

Fig. 3: The interior of a simplex is mapped to an open star of a vertex. All yellow vertices are mapped to the center of the star.

The following Section II-B shows that a sufficiently deep chromatic subdivision guarantees that a chromatic function will be star-covered with respect to .

Let be a chromatic function. Then, there exists such that is star-covered with respect to .

Proof.

Since is pure and of the same dimension as , the collection is an open covering of . We claim that , the collection of preimages of under is a finite open cover for . Indeed, for every , . Since is an open cover for , there exists such that . Since is open and is continuous, the preimage of a sufficiently small neighborhood is open and hence contained in .

Since is a finite simplicial complex, is a compact metric space. Therefore, there exists a Lebesgue number such that any set with a diameter less than is contained in an element of . Since the chromatic subdivision is a mesh shrinking operation on , there exists some such that any in has a diameter less than . Therefore, for any , for some . Consequently, for some , which confirms that is indeed star-covered with respect to . ∎

A star-covered subdivision induces a coloring for the facets . Definition 11 simply associates a facet with the colors of all star centers that cover in . For example, in Fig. 3, is assigned the yellow color here. This color assignment will be fundamental to the proof of Fig. 4.

Definition 11 (Star coloring).

Let be a star-covered subdivision with respect to . For , we define .

Definition 12.

Let be a star-covered subdivision with respect to and . We say that a facet is -covered if , and define the -subcomplex of as .

The following quite obvious Section II-B shows that independently subdivided -subcomplexes can be globally refined.

Let be subcomplexes of such that , and be a chromatic subdivision of each . There exists a chromatic subdivision that refines each .

Since a given facet may be -covered for several different colors , we pick one of those to obtain a color partition of , as provided by Section II-B.

Let be a star-covered subdivision with respect to , with its induced -colored subcomplexes , . There exists a partition , , of such that each , and for any pair , . We call the color partition of .

Proof.

Let , which is a star-covered subdivision with respect to . According to Section II-B, not all -subcomplexes of are empty. As we can find a permutation of the coloring that ensures , we can just define . Now consider . If , then . Otherwise, we can proceed inductively to define the remaining , . ∎

The following Section II-B is instrumental in the proof of our chromatic simplicial approximation Fig. 4. It shows that chromatic projections applied to chromatic functions provide chromatic functions, as illustrated in Fig. 4.

Figures/projection

Fig. 4: The chromatic projection , applied to the chromatic function shown in Fig. 3 with denoting the yellow (Y) vertex, induces a chromatic function in a lower dimension. The function maps the green vertex , the red-green border of , to represented by the green inner vertex, the carrier of which contains the yellow central vertex in Fig. 3. Applying retracts this point to the opposite red-green border.

Let be a chromatic function, and a star-covered subdivision with respect to . Assume that is a -colored subcomplex of , and let , . Then, defined by is a chromatic function.

Proof.

Let . Notice that from Definition 9, is continuous in . Since is continuous, must also be continuous in . Hence, is indeed continuous.

In order to show that is chromatic, let be an -dimensional simplicial neighborhood in . Note that is also an -dimensional neighborhood in . Section II-A implies that is also chromatic in , hence holds. Since as , it follows that . Consequently, by distributivity of set exclusion over intersection. Recalling and noticing that , it follows that . We conclude that . Therefore, is indeed a chromatic function. ∎

The following Section II-B is just the chromatic approximation theorem Fig. 4 written out for 1-dimensional simplicial complexes. It will serve as the induction basis for the proof of the latter.

Let be a continuous chromatic function such that and are pure simplicial complexes of dimension . There exists a chromatic subdivision of , and a chromatic simplicial map that is a chromatic approximation of .

Proof.

From Section II-B, it follows that there exists a chromatic subdivision which is star-covered with respect to . We define a simplicial vertex map as follows: For a vertex , we set if and . That is, maps vertex to the vertex of the same color in the carrier of , so obviously is a chromatic map.

Now consider a facet . Since is star-covered, for some . Assume w.l.o.g. that , and let . Notice that , as otherwise is impossible. Therefore, there exists a simplex that includes and . It follows that , which proves that is a chromatic simplicial map. It follows from the definition of that it is a chromatic approximation to . ∎

[Chromatic Approximation Theorem] Let be a chromatic function such that and are pure simplicial complexes of dimension , and with the same color set . There exists a chromatic subdivision of , and a chromatic simplicial map that is a chromatic approximation of .

Proof.

We induction over . The induction base is provided by Section II-B. For our induction hypothesis, we assume that Fig. 4 holds for arbitrary chromatic functions with respect to and with dimension less than . Let be a chromatic function with respect to and , where and .

It follows from Section II-B that there exists a chromatic subdivision of , which is star-covered with respect to . Section II-B thus ensures that there is a color partition of . Recall that each is an -covered subcomplex of . Therefore, for any , for some with color . Now, iteratively for , we will perform the following two steps:

As our first step for , we define a partial vertex map for vertices with color , as illustrated for the small yellow vertex on the boundary of in Fig. 5: Let be such that . Since is -covered, there exists some such that and . We choose any such , and set . Note carefully that this partial map defines uniquely a partial map for vertices of color also in further chromatic subdivisions of : Let be any such subdivision of and , . Taking as a point in the corresponding 0-dimensional geometric simplex in , consider . Since is chromatic also in by Section II-A, there exists a unique vertex such that . We can therefore consistently define .

Figures/subd

Fig. 5: Star-cover for the chromatic subdivision , which induces a trivial color partition consisting of a single , with color  representing yellow, containing all images of the subdivides simplices in . The small yellow vertex is mapped by to the yellow center of the star, the small read and green vertices will be retracted to the opposite border and appropriately mapped in the induction step in the proof of Fig. 4, and finally mapped to some border vertices as shown in Fig. 6.

Figures/fappr

Fig. 6: Mapping of the small red and green vertices in Fig. 5, which were retracted to the outer border of the star, to appropriate border vertices via and (denoted as in the figure for brevity) in the induction step in the proof of Fig. 4.

As our second step for , we define a partial vertex map for vertices with a color , as illustrated for the small red and green vertices on the boundary of in Fig. 5. Let and . Since is chromatic, Section II-B implies that each defined by is also chromatic. Notice that each and has dimension less than . Therefore, the induction hypothesis holds, and there exist that are chromatic approximations to , as illustrated in Fig. 6. As before, each defines a vertex map for any further chromatic subdivision of in the same way as did for further subdivisions of .

Since each defines a vertex map for vertices of color different to , and defines a vertex map for vertices of color , the join defines a vertex map for for all colors.

To combine all these partial functions into a global function , we make use of Section II-B. It ensures that there is a chromatic subdivision of that refines every . Recall that every is well defined for any further chromatic subdivision . Therefore, we can globally define as , where is the smallest color such that in a sufficiently deep chromatic subdivision of .

In only remains to prove that the so constructed is a chromatic approximation to . It suffices to show that for every facet . Let . Since it generally holds for facets in a pure simplicial complex that , we only need to prove that for each . This follows from the inductive construction of each and and the properties of the elements of color partition , however, which are based on star-covered subdivisions. ∎

Iii Combinatorial Topology Preliminaries

For ease of reference, we provide a collection of basic combinatorial topology definitions, which have primarily been taken from [HKR13]. Section III-A is dedicated to basic combinatorial topology, Section III-B adds definitions required for the modeling of a distributed systems using combinatorial topology.

Iii-a Combinatorial Topology

The first definition that we need is the definition of an abstract simplicial complex, which can can be thought of as a “high-dimensional” graph. Abstract simplicial complexes have the advantage of having a discrete combinatorial construction, while at the same time, they are able to express triangulated manifolds.

Definition 13 (Abstract simplicial complex).

An abstract simplicial complex is a pair , where is a set, , and for any such that and , then . is called the vertex set, and is the face set of . The elements of are called the vertices, and the elements of are called the faces or simplices. We say that an abstract simplicial complex is finite if its vertex set is finite.

We say that a simplex is a facet if it is maximal with respect to containment.

Traditionally, dimension is a number that represents how many axes we need to describe a point in space. The dimension of a simplex of an abstract simplicial complex is just the number of its vertices minus 1.

Definition 14 (Dimension).

Let be an abstract simplicial complex, and be a simplex. We say that has dimension , denoted by , if it has a cardinality of . We say that is of dimension if it has a simplex of maximum dimension .

In the context of distributed computing, local processes’ states correspond to vertices, and global configurations correspond to faces. Consequently, we only consider simplicial complexes where all simplices are of the same dimension. We call this particular type of simplicial complexes pure abstract simplicial complexes.

Now that we have defined the basic objects in combinatorial topology, we can define the morphisms that preserve the structure.

Definition 15 (Simplicial maps).

Let and be abstract simplicial complexes. We say that a vertex map is a simplicial map if for any .

We say that a simplicial map is rigid if for any of dimension , is also of dimension

It should be noted that simplicial maps are the discrete analogon to continuous functions. This equivalence is formally stated through the simplicial approximation theorem.

In the context of distributed systems, it is sometimes inevitable to enrich simplicial complexes with a coloring. Such a coloring corresponds to extracting the processes’ ids from the local states.

Definition 16 (Coloring).

Let be a finite abstract simplicial complex of dimension . We say that a function is a proper coloring if for any simplex , is injective at .

In order to simplify notation, whenever we have two disjoint abstract simplicial complexes and , both of dimension , with colorings and , we will instead implicitly consider a "global" coloring , since it is simpler and usually free from ambiguity.

Since we are interested that morphisms preserve the color structure, we must also add a coloring restriction.

Definition 17 (Chromatic simplicial maps).

Let resp.  be abstract simplicial complexes of dimension , and resp.  proper colorings for resp. . We say that a simplicial map is chromatic if for any .

It should be noted that all chromatic simplicial maps are rigid.

We need to define a substructure relation. Notice that since an abstract simplicial complex is a pair of two sets, it is natural to define the substructure relation based on set containment.

Definition 18 (Subcomplex).

Let and be abstract simplicial complexes, we say that is a subcomplex of if and .

Since we have a definition for the subcomplex relation, we can now define the -dimensional skeleton of an abstract simplicial complex.

Definition 19 (r-Skeleton).

Let be a -dimensional abstract simplicial complex, and . We define the -skeleton of , as the subcomplex induced by the -dimensional simplices of . More precisely:

Definition 20 (Carrier Map).

Let and be abstract simplicial complexes and . We say that is a carrier map if is a subcomplex of for any , and for any , .

We say that a carrier map is rigid if for every simplex of dimension , the subcomplex is pure of dimension . It is is strict if for every simplices , . Finally, it carries a simplicial vertex map if, for any , .

Definition 21 (Star of a vertex).

Let . We define the star of as the subcomplex of of all simplices that contain .

Subdivisions are used to create refined simplicial complexes out of a given simplicial complex. In the barycentric subdivision, the original faces become vertices, and simplex chains become the faces of the new subdivision.

Definition 22 (Barycentric subdivision).

Let be an abstract simplicial complex. We define the barycentric subdivision, denoted by through its vertex set and faces as follows.

Intuitively, the barycentric subdivision adds a vertex in the center of each simplex, and joins each of the original vertices to the new vertex. Since it does not preserve colors, the more complex chromatic subdivision is typically used instead.

Definition 23 (Chromatic subdivision).

Let be a -dimensional abstract simplicial complex with a proper coloring . We define the chromatic subdivision through its vertex set and faces as follows.