Edsger W. Dijkstra proposed the -model  as a way to study properties of concurrent programs. One way to interpret this model when the program does not contain loops is to consider the states of such a program as a subset of that consist of unions of unit cubes called a Euclidean cubical complex. Each axis represents a sequence of actions a process completes in the program execution. A directed path (non-decreasing in all coordinates) represents a (partial) program execution. Such executions are equivalent if the corresponding directed paths are directed homotopic, i.e., if the space of such paths is connected. We give a well-known example of a Euclidean cubical complex with directed paths in Figure 1. This example is commonly known as the Swiss Flag. Using Euclidean cubical complexes as topological models for concurrent programming has proven to be beneficial. One example is that verifying properties of one execution in each connected component of the space of such paths in a Euclidean cubical complex verifies those properties for the entire concurrent program. A more in depth introduction to this view of concurrency, corresponding geometry, and benefits can be found in [5, 6].
A non-trivial Euclidean cubical complex contains uncountably many directed paths and more information than we need for understanding the topology of the spaces of directed paths. The main question we ask is, How can we simplify a directed Euclidean cubical complexes while still preserving spaces of directed paths? In the realm of concurrent computing, answering this question can help simplify concurrent programs and speed up the program verification process.
Past links are local representations of a Euclidean cubical complex at vertices. They were introduced in  as a means to show that any finite homotopy type can be realized as a connected component of the space of execution paths for some -model. In , we found conditions for when the local information of past links preserve the global information on the homotopy type of spaces of directed paths. Because of these relationships between past links and directed path spaces, we define collapsing in terms of past links. We call this type of collapsing link-preserving directed collapse (LPDC). We aim to compress a Euclidean cubical complex by LPDCs before attempting to answer questions about directed path spaces.
The main result of this paper is Theorem 4 and is a very simple criterion for such a collapsing to be allowed: A pair of cubes, , is an LPDC pair if and only if it is a collapsing pair in the non-directed sense and does not contain the minimum vertex of . This condition greatly simplifies the definition of LPDC and is easy to add to a collapsing algorithm for Euclidean cubical complexes in the undirected setting. Algorithms and implementations in this setting already exist such as in . Furthermore, we provide conditions for when LPDCs preserve the contractability and connectedness of directed path spaces (Section 4) along with a discussion of some of the limitations (Section 5). This work provides a start at the mathematical foundations for developing polynomial time algorithms that collapse Euclidean cubical complexes and preserve directed path spaces.
This paper builds on our prior work , as well as work by others [18, 14, 6, 8, 9]. In this section, we recall the definitions of directed Euclidean cubical complexes, which are the objects that we study in this paper. Then, we discuss the relationship between spaces of directed paths and past links in directed Euclidean cubical complexes. For additional background on directed topology (including generalizations of the definitions below), we refer the reader to . We also assume the reader is familiar with the notion of homotopy equivalence of topological spaces (denoted using in this paper) and homotopy between paths as presented in .
2.1 Directed Spaces and Euclidean Cubical Complexes
Let be a positive integer. A (closed) elementary cube in is a product of closed intervals of the following form:
where and . We often refer to elementary cubes simply as cubes
. The dimension of the cube is the number of unit entries in the vector; specifically, the dimension of the cube in Eq. (1) is the sum: . In particular, when , the elementary cube is a single point and often denoted using just . If and are elementary cubes such that , we say that is a face of and that is a coface of . Cubical sets were first introduced in the 1950s by Serre  in a more general setting; see also [2, 11, 7].
Elementary cubes stratify , where two points are in the same stratum if and only if they are members of the same set of elementary cubes; we call this the cubical stratification of . Each stratum in the stratification is either an open cube or a single point. A Euclidean cubical complex is a subspace that is equal to the union of a set of elementary cubes, together with the stratification induced by the cubical stratification of ; see Fig. 2. We topologize using the subspace topology with the standard topology on . Note that if , then, since is a union of elementary cubes, all of its faces are necessarily in as well. If with no proper cofaces, then we say that is a maximal cube in . We denote the set of closed cubes in by ; the set of closed cubes in is in one-to-one correspondence with the open cubes in . Specifically, vertices in correspond to vertices in and all other elementary cubes in correspond to their interiors in . Throughout this paper, we denote the set of zero-cubes in by and note that , since all cubes in are elementary cubes.
The product order on , denoted , is the partial order such that for two points and in , we have if and only if for each coordinate . Using this partial order, we define the interval of points in between and as
The point is the minimum vertex of the interval and is the maximum vertex of the interval, with respect to . Notationally, we write this as and . When and , for some , the interval is an elementary cube as defined in Eq. (1). If, in addition, is not the zero vector, then we say that is a lower coface of .
Using the fact that the partial order induces a partial order on the points in , we define directed paths in as the set of non-decreasing paths in : A path in is a continuous map from the unit interval to . We say that a path goes from to . Letting denote the set of all paths in , the set of directed paths is
We call the set of directed paths (dipaths), and we topologize it with the compact-open topology. For , we denote the subspace of paths from to by . We refer to as a directed Euclidean cubical complex.888Directed Eulcidean cubical complexes are an example of a more general concept known as directed space (d-spaces). To define a d-space, we have a topological space and we define a set of dipaths that contains all constant paths, and is closed under taking non-decreasing reparameterizations, concatenations, and subpaths. Indeed, satisfies these properties. The connected components of are exactly the equivalence classes of directed paths, up to dihomotopy. If two dipaths, and are homotopic through a continuous family of dipaths, then and are called dihomotopic.
Given a directed Euclidean cubical complex, certain subcomplexes will be of interest:
Definition 1 (Special Complexes).
Let be a directed Euclidean cubical complex in . Let and let be an elementary cube (that need not be in ).
The complex above is .
The complex below is .
The reachable complex from is .
The complex restricted to is
If , then we call the standard unit cubical complex and often denote it by . If for some , then is a full-dimensional unit cubical complex.
2.2 Past Links of Directed Cubical Compelxes
An abstract simplicial complex is a finite collection of sets that is closed under the subset relation, i.e., if and is a set such that , then . The sets in are called simplices. If the simplex has elements, then we say that the dimension of is , and we say is a -simplex. For example, the zero-simplices are the singleton sets and are often referred to as vertices. Since every element of a set gives rise to a singleton set in the finite set , must be finite.
In the cubical stratification of , the link of a point is the intersection of an arbitrarily small sphere around with the space ; that is, the link of a point is an -sphere. If , the link inherits the stratification as a subcomplex of , and is a simplicial complex whose -simplices are in one-to-one correspondence with the -dimensional cofaces of . The past link of is the restriction of the link using the set of lower cofaces of instead of all cofaces. Thus, we can represent each simplex in the past link as a vector in , where the vector represents the cube in the simplex-cube correspondence.999The point is the zero vector, and will never be in the past link. We differentiate this from , which denotes the origin of . As a simplicial complex, the past link of in has vertices , and represents the simplex . We are now ready to define the past link of a vertex in a Euclidean cubical complex:
Definition 2 (Past Link).
Let be a directed Euclidean cubical complex in . Let . The past link of is the following simplicial complex:
As a set, the past link represents all elementary cubes in for which is the maximum vertex. As a simplicial complex, it describes (locally) the different types of dipaths to or through in ; see Fig. 3.
We conclude this section with a lemma summarizing properties of the past link, most of which follow directly from definitions:
Lemma 1 (Properties of Past Links).
Let be a directed Euclidean cubical complex in . Then, the following statements hold for all :
If is a subcomplex of , then .
If is a full-dimensional unit cubical complex and , then is the complete simplicial complex on vertices.
Statement 1: If , then all past links are empty and the equality trivially holds. If , then is a finite non empty set. Thus, there exists such that, for all , . Let . Then, and so . Hence, , which means that . The reverse inclusion follows from the fact that each of these statements holds if and only if.
Statement 2: Observe that if , then, by definition of the past link, . Since , we have . Therefore, we can conclude that .
Statement 4: Let . Then, since is full-dimensional, for all , . Thus, by definition of past link, we have that the past link of is: , which is the complete simplicial complex on vertices. ∎
2.3 Relationship Between Past Links and Path Spaces
The topology of the past links is intrinsically related to the one of the spaces of directed paths. Specifically, in  we prove that the contractability and/or connectedness of past links of vertices in directed Euclidean cubical complexes with a minimum vertex101010In , the minimum (initial) vertex was often assumed to be for ease of exposition. We restate the lemmas and theorems here using more general notation, where has a minimum vertex . implies that all spaces of directed paths with as initial point are also contractible and/or connected.
Theorem 1 (Contractability [1, Theorem 1]).
Let be a directed Euclidean cubical complex in that has a minimum vertex . If, for all vertices , the past link is contractible, then the space is contractible for all .
An analogous theorem for connectedness also holds.
Theorem 2 (Connectedness [1, Theorem 2]).
Let be a directed Euclidean cubical complex in that has a minimum vertex . Suppose that, for all , the past link is connected. Then, for all , the space is connected.
Furthermore, we proved a partial converse to Theorem 2. Specifically, the converse holds only if is a reachable directed Euclidean cubical complex as defined in Statement 3 of Definition 1. This is expected: Properties of a part of the directed Euclidean complex which are not reachable from , will not influence the path spaces from .
Theorem 3 (Realizing Obstructions [1, Theorem 3]).
Let be a directed Euclidean cubical complex in . Let , and let . Let . If the past link is disconnected, then the space is disconnected.
3 Directed Collapsing Pairs
Although simplicial collapses preserve the homotopy type of the underlying space [12, Proposition 6.14] and hence of all path spaces, this type of collapsing in directed Euclidean cubical complexes may not preserve topological properties of spaces of directed paths. In this section, we study a specific type of collapsing called a link-preserving directed collapse. We define link-preserving directed collapses in Section 3.1 and give properties of link-preserving directed collapses in Section 3.2.
3.1 Link-Preserving Directed Collapses
Since we are interested in preserving the directed path spaces through collapses, the results from Section 2.3 motivate us to study a type of directed collapse (DC) via past links, introduced in . However, we will call it link-preserving directed collapse (LPDC) (as opposed to directed collapse) since we show in the last sections of this paper that when the spaces of directed paths starting from the minimum vertex are not connected, the following definition of collapse does not preserve the number of components.
Definition 3 (Link Preserving Directed Collapse).
Let be a directed Euclidean cubical complex in . Let be a maximal cube, and let be a proper face of such that no other maximal cube contains (in this case, we say that is a free face of ). Then, we define the -collapse of as the subcomplex obtained by removing everything in between and :
and let denote the stratification of the set induced by the cubical stratification of (thus, ).
We call the directed Euclidean cubical complex a link-preserving directed collapse (LPDC) of if, for all , the past link is homotopy equivalent to (denoted ). The pair is then called an LPDC pair.
Remark 1 (Simplicial Collapses).
The study of simplicial collapses is known as simple homotopy theory [17, 3], and traces back to the work of Whitehead in the 1930s . The idea is very similar: If is an abstract simplicial complex and such that is a proper face of exactly one maximal simplex , then the following complex is the -collapse of in the simplicial setting:
Note that we use only the free face () when defining a simplicial collapse, as doing so helps to distinguish between discussing a simplicial collapse and a directed Euclidean cubical collapse. In addition, we always explicitly state “in the simplicial setting” when talking about a simplicial collapse.
Applying a sequence of LPDCs to a directed Euclidean cubical complex can reduce the number of cubes, and hence can more clearly illustrate the number of dihomotopy classes of directed paths within the directed Euclidean cubical complex. For an example, see Fig. 4. However, it is not necessarily true that LPDCs preserve directed path spaces. We discuss the relationship between directed path spaces and LPDCs in Section 4.
3.2 Properties of LPDCs
We give a combinatorial condition for a collapsing pair to be an LPDC pair; namely, the condition is that does not contain the vertex . From the definition of an LPDC, we see that finding an LPDC pair requires computing the past link of all vertices in . In , we discussed how we can reduce the check down to only the vertices in since no other vertices have their past links affected. In this paper, we prove we need to only check one condition to determine if we have an LPDC pair. The one simple condition dramatically reduces the number of computations we need to perform in order to verify we have an LPDC. This result given in Theorem 4 depends on the following lemmas about the properties of past links on vertices.
Lemma 2 (Properties of Past Links in a Vertex Collapse).
Let be a directed Euclidean cubical complex in . Let and such that . If is a free face of and is the -collapse, then the following two statements hold:
To ease notation, we define the following two sets:
First, we prove Statement 1 (that ). We start with the forward inclusion. Let . By the definition of past links (see Definition 2), we know that . By the definition of (see Definition 1), . This implies . Therefore, . For the backward inclusion, let . Then, since and is an elementary cube by assumption, and by definition of , we have . Since , all faces must be in ; hence, . Therefore, , and so . Since we have both inclusions, then Statement 1 holds.
Now, we prove Statement 2 (that ). Again, we prove the inclusions in both directions. For the forward inclusion, let . By Statement 2 of Lemma 1, we have , and so . Next, we must show that . Assume, for a contradiction, that . Then, by definition of , . Since , we obtain the partial order . This implies that . Since is an elementary cube in , then its face must also be an elementary cube in . Setting and observing , we observe that is not an elementary cube in by Eq. (2). This gives us a contradiction and so . Therefore, .
Using Lemma 2, we see why cannot be the vertex when performing an LPDC. If , then
If is the maximum vertex of , then we obtain . This computation gives us the following corollary, which we illustrate in Fig. 5 when is a single closed three-cube.
Corollary 1 (Caution for a -Collapse).
Let be a directed Euclidean cubical complex in . Let , , and . If is a free face and is the -collapse. Then, the past link of in is:
In particular, if and , then the past link is the complete complex on elements before the collapse, and, after the collapse, it is homeomorphic to . Thus, is not an LPDC pair.
The following lemma shows under which condition a directed Euclidean cubical collapse induces a simplicial collapse in the past link.
Lemma 3 (Vertex Collapse Induces Simplicial Collapse).
Let be a directed Euclidean cubical complex in . Let and such that and . If is a free face of and is the -collapse, then is the -collapse of in the simplicial setting.
Consider . Since and is maximal in , we know and are elementary cubes in . Since is a free face of , we further know that is the only maximal proper coface of in . By definition of past link (Definition 2), we then have that and are simplices in , and is the only maximal proper coface of in . Hence, is free in . Moreover, is the -collapse of . One can see this by using Statement 2 of Lemma 2 by which can be characterized as the -collapse of .
Next, we prove two lemmas concerning relationships of the past link of a vertex in the original directed Euclidean cubical complex and in the collapsed directed Euclidean cubical complex. These relationships depend on where is located with respect to . In the first lemma, we consider the case where , and we present a sufficient condition for past links in and the -collapse to be equal. See Fig. 6 for an example that illustrates the result of this lemma.
Lemma 4 (Condition for Past Links in and to be Equal).
Let be a directed Euclidean cubical complex in . Let such that is a face of . If is a free face of and is the -collapse, then, for all such that , we have .
In the following lemma, we consider the case where , and we present a sufficient condition for past links in the -collapse and the -collapse to be equal. See Fig. 7 for an example that illustrates this result.
Lemma 5 (Condition for Past Links in and to be Equal).
Let be a directed Euclidean cubical complex in such that there exists cubes with a free face of . Let be the -collapse and let be the -collapse. If and , then and .
We first show . If is a zero-cube (and hence in ), then , which means that . On the other hand, if is not a zero-cube, then we have . In particular, . Thus, . And so, by definition of as a -collapse and since , we conclude that .
Case 1 (): Since , we know that . Thus, by Eq. (2), we have , which is a contradiction. So, Case 1 cannot happen.
Case 2 (): If , then, by the definition of a -collapse in Definition 3, we know and thus .
Hence, . Since we have both subset inclusions, we conclude . ∎
The next result states that vertex collapses result in homotopy equivalent past links as long as we are not collapsing the minimum vertex of the directed Euclidean cubical complex.
Lemma 6 (Past Links in a Vertex Collapse).
Let be a directed Euclidean cubical complex in . Let and let such that . Let with If is a free face of and is the -collapse, then .
We consider three cases:
Case 1 (): By definition of past link (Definition 2), if , then the past links and are equal.
Case 2 (): By Lemma 4, if , again we have equality of the past links and .
Lastly, we are ready to prove the main result.
Theorem 4 (Main Theorem).
Let be a directed Euclidean cubical complex in such that there exist cubes with a free face of . Then, is an LPDC pair if and only if .
Let be the -collapse. Let be the cubical complex such that . Since , we know (i.e., is a unit cube). Since is a single unit cube and is the maximal elementary cube, all proper faces of , including and , are free faces. Thus, let be the -collapse of , and let be the -collapse of .
We first prove the forward direction by contrapositive (if , then is not an LPDC pair. Assume . Since is a face of , we know . Since and since , we know that . Thus, and so . Applying Lemma 5 using , we obtain . Let . By Corollary 1, we obtain is homeomorphic to and is homeomorphic to . Thus,
and so . Since and are sets of faces of , the past link of remains the same outside of in both and .Thus, and so we conclude that is not an LPDC pair, as was to be shown.
Next, we show the backwards direction. Suppose . Let , and consider two cases: and .
Case 1 (): By Lemma 4, we have . Hence, . Since was arbitrarily chosen, we conclude that is an LPDC pair.
4 Preservation of Spaces of Directed Paths
We focus on LPDCs because of the relationships between past links and spaces of directed paths that were proved in . We explain how those relationships extend to the LPDC setting in this section.
One result from (Theorem 1) states that for a directed Euclidean cubical complex with a minimum vertex, if all past links are contractible, then all spaces of directed paths starting at that minimum vertex are also contractible. If we start with a directed Euclidean cubical complex with a minimum vertex that has all contractible past links, then all spaces of directed paths from the minimum vertex are contractible by this theorem. Applying an LPDC will preserve the homotopy type of past links by definition. Hence, applying the theorem again, we see that any LPDC will also have contractible directed path spaces from the minimum vertex. Notice that the minimum vertex is not removed in an LDPC, since it is a vertex and minimal in all cubes containing it (including the maximal cube). We give an example of this in Example 1.
Example 1 ( filled grid).
Let be the filled grid. For all , is contractible. By Theorem 1, this implies that all spaces of directed paths starting at are contractible. Applying an LPDC such as the edge results in contractible past links in and so all spaces of directed paths in are also contractible. See Fig. 8. We can generalize this example to any filled grid where .
An analogous result holds for connectedness (Theorem 2). If we start with a directed Euclidean cubical complex that has all connected past links, then all directed path spaces are connected. Any LPDC will result in a directed Euclidean cubical complex that also has connected directed path spaces. See Example 2.
Example 2 (Outer Cubes of the Grid).
Let , which, as an undirected complex, is homeomorphic to a thickened two-sphere. For all , is connected. By Theorem 2, this implies that for all , the space of directed paths is connected. Applying an LPDC such as with the vertex in the cube results in connected past links in and so all spaces of directed paths are connected. We can generalize this example to any grid where and the inner cubes of dimension are removed.
Both Theorem 1 and Theorem 2 have assumptions on the topology of past links and results on the topology of spaces of directed paths from the minimum vertex. We may ask if the converse statements are true. Does knowing the topology of spaces of directed paths from the minimum vertex tell us anything about the topology of past links? The converse to Theorem 1 holds. To prove this, we first need a lemma whose proof appears in .
Lemma 7 (Homotopy Equivalence [18, Prop. 5.3]).
Let be a directed Euclidean cubical complex in