In approaches to non-interleaving concurrency, more than one event may happen simultaneously. There is a plethora of formalisms for modeling and analyzing such concurrent systems, Petri nets , event structures , configuration structures [36, 35], asynchronous transition systems , or more recent variations such as dynamic event structures , Unravel nets  or ST-structures . They all share the idea of differentiating between concurrent and interleaving executions; in CCS notation , is not the same as .
In , van Glabbeek shows that (up to history-preserving bisimilarity) higher-dimensional automata (HDA), introduced by Pratt and van Glabbeek in [25, 32], encompass all other commonly used models for concurrency. However, their generality make HDA quite difficult to work with, and so the quest for useful and general models for concurrency continues.
In , Pratt introduces sculpting as a process to manage the complexity of HDA. Intuitively, sculpting takes one single hypercube, having enough concurrency ( enough events), and remove cells until the desired concurrent behavior is obtained. This is different from, and in a sense orthogonal to, composition, where a system is built by putting together smaller systems, which in HDA is done by gluing cubes. Pratt finishes the introduction of  saying that “sculpture on its own suffices […] for the abstract modeling of concurrent behavior.”
In this paper we make precise the intuition of Pratt  and give a definition of sculptures. We show that there is a close correspondence between sculptures, Chu spaces over  and ST-structures. We develop an algorithm to decide whether a HDA can be sculpted and show in Thm. 6.1 several natural examples of acyclic HDA which cannot be sculpted. We will carefully introduce these concepts later, but spend some time here to motivate our developments.
Combinatorial sculpting as described above is not to be confused with geometric sculpting, which consists of taking a geometric cube of some dimension and chiseling away hypercubes which one does not want to be part of the structure. Figure 1 shows a geometric sculpture; for a combinatorial sculpture see Fig. 2.
Geometric sculpting has been used by Fajstrup in [13, 12] and other papers to model and analyze so-called PV programs: processes which interact by locking and releasing shared resources. In the simplest case of linear processes without choice or iteration this defines a hypercube with forbidden hyperrectangles, which cannot be accessed due to resources’ access limits. See Fig. 5 for an example.
Technically, geometric sculptures are Euclidean cubical complexes; rewriting a proof in  we show that such complexes are precisely (combinatorial) sculptures. In other words, a HDA is Euclidean iff it can be sculpted, so that the geometric models for concurrency [13, 12] are closely related to the combinatorial ones [25, 32], through the notion of sculptures. Much work has been done in the geometric analysis of Euclidean HDA [14, 15, 12, 21, 30, 38]; through our equivalences these results are made available for the combinatorial models.
The notion of unfolding is commonly used to turn a complicated model into a simpler, but potentially infinite one. It may thus be expected that even if a HDA cannot be sculpted, its unfolding can, as illustrated by the two examples in Fig. 4 below.
However, this is not always the case, as witnessed by the example in Fig. 5 above which shows a HDA which cannot be sculpted and which is its own unfolding. This example features two agents, and , which compete to choose between two future events and . If the demon finishes his event first, then the choice between and is a demonic choice, already made when starting the event; if the angel finishes her event first, then we have an instance of an angelic choice between and . This concurrent system, introduced in , cannot be modeled as an ST-structure, but can be modeled as an ST-structure with cancellation [20, Sec. 5].
Even more concerning is the fact that there are HDA which can be sculpted, but their unfoldings cannot; in fact, Fig. 2 exposes one such example. This shows that for HDA, unfolding does not always return a simpler model, and seems to contradict Pratt’s claim that sculpting suffices for modeling.
In the geometric setting, this means that there are Euclidean cubical complexes whose unfoldings are not Euclidean. Since Goubault and Jensen’s seminal paper , directed topology has been developed in order to analyze concurrent systems as geometric objects [17, 13, 11]. Directed topology has been developed largely in analogy to algebraic topology, but the analogy has a tendency to break.
The mismatch we discover here, between Euclidean complexes and unfoldings, shows another such broken analogy. Unfoldings of HDA have been developed as a directed analogue to universal covering spaces in algebraic topology [32, 7, 8]. There are several other problems with this notion, and finding better definitions of directed coverings is active ongoing research, see for example .
Another motivation for Pratt’s  is that general HDA have no explicit notion of events. From the work in  on ST-structures, introduced as event-based counterparts of HDA, we know that it is not always possible to properly identify the events in a HDA. The example in Fig. 6 shows the (strong) asymmetric conflict from [36, 29, 20], having two events, and , with the provision that occurrence of disables . This can be modeled as a general event structure, but not as a pure event structure, hence also not as a configuration structure . It can also be modeled as an ST-structure, but when using HDA, one faces the problem that HDA transition labels do not carry events. The right part of Fig. 6 shows two different ways of sculpting the corresponding structure from a HDA, one in which the two -labeled transitions denote the same event and one in which they do not; à priori there is no way to tell which HDA is the “right” model. This also shows that the same HDA may be sculpted in several different ways.
Structure of the paper
We start in Sec. 2 by recalling the definitions of HDA, ST-structures, and Chu spaces. In Sec. 3 we introduce sculptures and show that they are isomorphic to regular ST-structures. The triple equivalence
|regular ST-structures — regular Chu spaces — sculptures|
embodies Pratt’s event-state duality . Regularity is a geometric closure condition introduced for ST-structures in  which ensures that for any ST-configuration, also all its faces are part of the structure, and they are all distinct. If regularity is dropped, then one has to pass to partial HDA  on the geometric side, and then the above equivalence becomes one between ST-structures and sculptures from partial HDA. For clarity of exposition we do not pursue this here, but also in that case, there will be acyclic partial HDA which cannot be sculpted.
Section 4 contains our main contribution, an algorithm to decide whether a given HDA can be sculpted. The algorithm essentially works by covering with the ST-structure which is built out of all paths in , and then trying to find a quotient of which is isomorphic to . We show that such a quotient exists iff can be sculpted.
Figure 7 shows a simple example: the empty square, a one-dimensional HDA with two interleaving transitions. The covering splits the upper-right corner, and the algorithm finds an equivalence on the four events which recovers (an ST-structure isomorphic to) : in this case we equate and , which corresponds to the standard way of identifying events in HDA as opposite sides of a filled-in square when it exists.
Another example is shown in Fig. 8. This one-dimensional acyclic HDA cannot be sculpted, and the algorithm detects this by noting that (1) all the -labeled transitions indeed need to be the same event, but then (2) the two states connected with a dashed line need to be identified, so that the ST-structure covering cannot be isomorphic to the original HDA model. This example also shows that no two-dimensional structure is needed for things to turn problematic: already in dimension there are acyclic HDA which cannot be sculpted.
In Sec. 5 we make the connection between the combinatorial and geometric models and show that HDA can be sculpted precisely if they are Euclidean. This necessitates a few notions from directed topology which can be found in appendix.
Figure 9 sums up the relations between the different models which we expose in this paper. (The dashed line indicates the common belief that Chu spaces over and acyclic HDA are equivalent, which we prove not to be the case.) We do not pay much attention to categorical notions or results here, beyond what is necessary for our developments; a precise categorical treatment is left for further work. Due to space constraints, all proofs have been confined to an appendix.
2 HDA, ST-Structures, and Chu Spaces
HDA are automata in which independence of events is indicated by higher-dimensional structure. They consist of states, transitions, and cubes of different dimensions which represent events running concurrently.
Technically, HDA are based on precubical sets as introduced below.
A precubical set is a graded set , with for , together with mappings , , satisfying the precubical identities, for ,
Elements of are called -cells (or simply cells), and for , is its dimension. The mappings and are called face maps, and we will usually omit the extra subscript and simply write and . Intuitively, each -cell has lower faces and upper faces , and the precubical identity expresses the fact that -faces of an -cell meet in common -faces; see Fig. 10 for an example.
Morphisms of precubical sets are graded functions which commute with the face maps: for all , , and . This defines a category of precubical sets. A precubical morphism is an embedding if it is injective; in that case we write . and are isomorphic, denoted , if there is a bijective morphism .
If two cells in a precubical set are in a face relation (for ), then this sequence can be rewritten in a unique way, using the precubical identities (1), so that the indices , see . is said to be non-selflinked if up to this rewriting, there is at most one face relation between any of its cells, that is, it holds for all that there exists at most one index sequence such that for .
In other words, is non-selflinked iff any is embedded in , hence iff all ’s iterated faces are genuinely different. This conveys a geometric intuition of regularity and is frequently assumed [10, 13], also in algebraic topology [2, Def. IV.21.1]. It means that for all cells in , each of their faces (and faces of faces etc.) are present in as distinct cells.
A precubical set is finite if is finite as a set. This means that is finite for each and that is finite-dimensional: there exists such that for all (equivalently, for all ). In that case, the smallest such is called the dimension of and denoted . A higher-dimensional automaton (HDA) is a finite non-selflinked precubical set with a designated initial cell . Morphisms of HDA are precubical morphisms which fix the initial cell, have .
A step in a HDA, with , , and , is either
A path is a sequence of steps , with . The first cell is denoted and the ending cell in a finite path is .
A cell in a HDA is reachable from another cell if there exists a path with and . is said to be connected if any cell is reachable from the initial state . is acyclic if there are no two different cells in such that is reachable from and is reachable from .
If a HDA is unconnected, then it contains cells which are not reachable during any computation, which are uninteresting from the point of view of computation. We will hence from now assume all HDA to be connected.
Note that the marking of the steps by can be deduced from the fact that the step goes from a lower cell to a higher cell for s-steps (and the opposite for t-steps). It is though useful in many of the proofs to have easily visible the exact map ( the index also) that the step uses, instead of explicitly assuming it every time. When the index is not important we only write or .
An ST-configuration over a finite set of events is a pair of sets . An ST-structure is a pair consisting of a finite set of events and a set of ST-configurations over .
Intuitively, in an ST-configuration the set contains events which have started and contains events which have terminated. Hence the condition : only events which have already started can terminate. The events in are running concurrently, and we call the concurrency degree of .
The notion of having events which are currently running, started but not terminated, is a key aspect captured by ST-structures and also by HDA through their higher dimensional cells. Other event-based formalisms such as configuration structures [35, 36] or event structures [23, 37] cannot express this.
A step between two ST-configurations is either
with , and , or
with , , and .
When the type is unimportant we write . A path of an ST-structure, denoted , is a sequence of steps, where the end of one is the beginning of the next,
A path is rooted if it starts in . An ST-structure is said to be
rooted if ;
connected if for any there exists a rooted path ending in ;
closed under single events if, for all and all , also and .
is regular if it satisfies all three conditions above.
ST-structures were introduced in  as an event-based counterpart of HDA that are also a natural extension of configuration structures and event structures. The notions of rootedness and connectedness for ST-structures are similar to connectedness for HDA. The notion of being closed under single events mirrors the fact that cells in HDA have all their faces, and (by non-selflinkedness) these are all distinct (see also [20, Prop. 3.40]). Thus regularity is assumed in some of the results below.
A morphism of ST-structures is a partial function of events which preserves ST-configurations and is locally total and injective, that is, for all , and for all , the restriction is a total and injective function. This defines a category of ST-structures. Two ST-structures are isomorphic, denoted , if there exists a bijective morphism between them.
For later use we record a notion of quotient of ST-structures under an equivalence relation on its events:
Let be an ST-structure and an equivalence relation. The quotient of under is the ST-structure , with .
It is clear that is again an ST-structure. To ease notation we will sometimes denote . The quotient map is generally not an ST-morphism, failing local injectivity.
An equivalence relation on an ST-structure is collapsing if there is and with and . Otherwise, is non-collapsing.
is non-collapsing iff the quotient map is an ST-morphism.
The model of Chu spaces has been developed by Gupta and Pratt [19, 27] in order to study the event-state duality . A Chu space over a finite set is a triple with and sets and a function called the matrix of the Chu space.
Chu spaces can be viewed in various equivalent ways [19, Chap. 5]. For our setting, we take the view of as the set of events and as the set of configurations. The structure is representing the possible values the events may take: is the classical case of an event being either not started () or terminated (), hence Chu spaces over correspond to configuration structures [35, 36] where an order of is used to define the steps in the system, steps between states must respect the increasing order when lifted pointwise from to .
ST-structures capture the “during” aspect in the event-based setting, extending configuration structures with this notion. Therefore we need another structure with the order , introducing the value to stand for during, or in transition. Note that  studies Chu spaces over , whereas Pratt proposed to study Chu spaces over and other structures in .
Using currying, we can view a Chu space over as a structure or . Consequently, we will often write or instead of below. The category of Chu spaces, , has morphisms between Chu spaces and defined to be pairs of maps and that satisfy the following two equations (called adjointness condition in [19, Ch. 4]) for all and for all : and .
Definition 3 (translations between ST and Chu)
For an ST-structure construct the associated Chu space over with the set of events from , and containing for each ST-configuration the state formed by assigning to each :
if and ;
if and ;
if and .111The case is dismissed by the requirement of ST-configurations.
Call this mapping when applied to an ST-configuration and when applied to an ST-structure. The other way, we translate a Chu space into an ST-structure over with one ST-configuration for each state using the inverse of the above mapping. We use for the ST-configuration obtained from the event listing .
For example, for an event listing make the ST-configuration where the last event does not appear neither in the first nor the second set of the ST-configuration.
Theorem 2.1 ([20, Sec. 3.4])
For any ST-structure , . For any Chu space over , .
Thus, an ST-configuration can be seen as a listing/tuple with values from ; which exact listing of the events is irrelevant once fixed. Therefore, when we later use ST-configurations to label cells of an , we can alternatively use the Chu spaces notation, interchangeably, to simplify arguments.
For any ST-structure the Chu space is extensional, meaning that no two states are identical ( .
In short, since ST-structures work with sets, in the set of ST-configurations there are no two ST-configurations that are the same, then the states produced by would also be different by the virtue of the assignment from Def. 3 which associates a unique valuation of the events of an ST-configuration.
In detail, for any they are created from some different , which implies the two cases:
When then pick some s.t. (or the other way around if needed) then the states generated by would have the valuations: and , thus making them different.
When but then pick some s.t. (or the other way around if needed) then the states would have (because ) and , thus making them different.
We call a precubical set a bulk if it is non-selflinked and generated by precisely one -cube, if . Any two -dimensional bulks are isomorphic, where the isomorphism is generated by a permutation of the directions of the two generating cells. Hence we may talk of the -dimensional bulk, and denote it by . We develop a naming scheme for bulks inspired by Chu spaces.
Fix and let . For , let be the set of tuples with precisely occurrences of . For and , define face maps as follows: for with , let and be the tuples with the -th occurrence of set to or , respectively. We call this the canonical naming for the bulk .
The above construction essentially labels the cells of the -bulk with lists of Chu-labels, with states from the full Chu space on events.
The structure as defined above is the -dimensional bulk.
It is trivial to check that is a precubical set. is also non-selflinked, and , thus by uniqueness up to isomorphism, is the -dimensional bulk, corresponding to the full Chu space over events.
The initial state of the bulk is the cell named in the canonical naming (any automorphism of fixes this cell). This turns bulks into HDA.
The -dimensional bulk can be embedded into any bulk of dimension by using the embedding which maps any cell to in the canonical naming. It can easily be shown that up to isomorphism, is the only HDA morphism from to , and also that there are no HDA morphisms for .
A sculpture is a HDA together with a bulk and a HDA embedding . A morphism of sculptures , is a pair of HDA morphisms , such that the square
By the above considerations, this entails that and is injective, hence also must be injective. Two sculptures are isomorphic, denoted , when and are isomorphisms (implying ).
For the special case of above, we see that any sculpture can be over-embedded into a sculpture for . Conversely, any sculpture admits a minimal bulk for which , such that there is no embedding of into for any . We call such a minimal embedding simplistic.
One precubical set can be seen as sculpted from two different-dimensional bulks, in both cases being a simplistic sculpture, it all depends on the embedding morphism ( Fig. 6). Because of this we cannot determine from a HDA alone in which sculpture it enters (if any). Working with unfoldings is not particularly good either. The interleaving square from Fig. 7 (left) can be sculpted from , but its unfolding may be sculpted from or ; we cannot decide which. All the sculptures in Figs. 6 and 7 are simplistic.
We show that sculptures and regular ST-structures are isomorphic while also respecting the computation steps. This result resolves the open problems noticed in [20, Sec. 3.3] that there is no adjoint between ST-structures and general HDA.
Recall that an ST-structure is regular if it is rooted, connected, and closed under single events. Through the observation from Sec. 2 the results in this section extend to (regular) Chu spaces over as well.
Definition 6 (from regular ST-structures to sculptures)
We define a mapping that for any regular ST-structure generates an HDA, as well as a bulk and an embedding, thus a sculpture, as follows. Consider with the events linearly ordered as a list . Then returns the HDA which
has cells ;
for any two cells and add the map entry where is the index of the event in the listing ;
for any two cells and add the map entry where is the index of the event in the listing .
is the listing restricted to the set . Build the bulk , with , using the canonical naming on the same listing of the events . The embedding is defined as returning the Chu-labeling as in Def. 3 on the same listing of events .
The mapping translates a regular ST-structure into a HDA, respecting all cubical laws. Moreover, it is immaterial which listing of the events is picked in the definition (these results are direct adaptations of results from ).
Definition 7 (from sculptures to regular ST-structures)
Define a mapping which to a sculpture associates the ST-structure as follows. Take a linearly ordered set (of events) of cardinality . The ST-configurations of are obtained from the cells of as .
Intuitively, since we have the bulk we can work with the canonical naming based on a fixed listing of events. Using Thm. 2.1 we can associate an ST-configuration to each cell of the HDA by going through the embedding to the corresponding cell in the bulk. It is clear that is rooted, connected and closed under single events, regular.
The following result shows a one-to-one correspondence between regular ST-structures and sculptures.
For any regular ST-structure , . For any sculpture , .
We can also understand as labeling every cell of the sculpture with an ST-configuration, or equivalently (because of Thm. 2.1) with a Chu state.
In a bulk every cell has a unique label (either as an ST-configuration or as a Chu-label representation). Thus, there are no two cells of the bulk with the same label.
The mapping is functorial, in the sense that an ST-morphism is translated into an HDA morphism , given by , up to an isomorphism on . If is injective, then is a sculpture morphism.
In the natural interpretation of HDA, the cubes represent events running concurrently. Thus, in a bulk the cell on the highest layer has different events running concurrently. When we look only at the transitions of the bulk, then there is a natural equivalence relation which identifies these events, given by equating opposite faces of squares. In terms of the canonical naming, the transitive closure then equates all labels that have the single value on the same position of the tuple, and everywhere else.
Inside a bulk define a relation on transitions of the bulk as
for some and . Consider the reflexive and transitive closure of , and denote it the same. This is now an equivalence relation on .
4 Decidability for the Class of Sculptures
We proceed to develop an algorithm to decide whether a given HDA can be sculpted. Note that simply searching for embeddings into bulks does not work, because the bulk can be of any dimension, so there are infinitely many embeddings to check. First we need an alternative way of translating HDA into ST-structures that works on paths starting from the shortest path in the initial cell.
Definition 9 (from HDA to ST-structures through paths)
Define a map which builds an ST-structure by associating to each rooted path in an ST-configuration as follows.
for the minimal rooted path, which ends in , associate ;
for any path that ends in a transition ,
add the ST-configuration ;
add the ST-configuration ;
for any path with , with , add the ST-configuration , with , , and .
In case 3 above the and always exist because we work with non- selflinked HDA. All cells are reachable through the paths considered in the above definition when applied inductively on the length of the distance from the initial cell.
For every transition adds one new event to the ST-structure. This adds too many events and does not capture the geometric intuition about concurrency, where transitions parallel in the sides of a filled square should represent the same event. Indeed, the construction is similar to an unfolding , except that no homotopy equivalence is applied. See [20, Def. 3.39] for a related construction.
If the HDA in question is a sculpture, then there is a natural equivalence relation on its cells which captures the notion of event:
In a sculpture the embedding generates the following equivalence on the cells of :
If, on the other hand, we are faced with an HDA which may or may not be a sculpture, then all we have is a minimal equivalence on its transitions which is generated by identifying opposite faces of squares:
For a HDA , define a relation on transitions of as
for some and . Consider the reflexive and transitive closure of , and denote it the same. This is now an equivalence relation on .
Note that this is a generalization of Def. 8 to HDA. Contrary to Def. 8, it may fail to identify events because of missing concurrency squares. It is clear that in any sculpture, because the bulk has more squares to build .
For a sculpture we have
We exhibit the bijective function between the events on the left, as an ordered list related to the bulk, and the events on the right, which are equivalence classes of transitions made only by the application of , which in turn comes from the bulk transitions via the embedding. To prove that preserves ST-configuration we use induction on the length of the rooted paths reaching some cell, making use of the canonical naming of the bulk and the translations between Chu and ST, through the embedding.
For a HDA , and using the notation of Def. 9, let be the relation . For an equivalence relation , let .
A HDA can be sculpted iff there exists an equivalence relation such that
for all there is precisely one element with , and
whenever , then .
For the forward direction we can use . For the other direction, the key of the proof is the diagram
(where the arrow is dashed because is a relation, not a function). The quotient map is an ST-morphism because is non-collapsing, and then is an isomorphism of HDA.
The number of equivalence relations on is finite, hence the above theorem translates into a decision procedure to determine whether is a sculpture. Below we give a more intuitive algorithm, using constructions which iteratively repair by constructing a finite increasing sequence of equivalence relations.
Let be a HDA and as above. First, the following lemma shows that we can restrict our attention to labelings of -cells:
If for some , , then also for some .
We can immediately rule out labelings in which a cell receives ST-configurations with different numbers of events:
If there is and with , then cannot be sculpted.
We will inductively construct equivalence relations , with the property that . This procedure will either lead to a relation as required in Thm. 4.2 or to an irreparable conflict as explained below.
Let , the minimal equivalence relation. If is a sculpture, then , hence we can safely start our procedure with .
Assume, inductively, that has been constructed for some . The next lemma shows that if there are two different cells which receive the same labeling under , then either is not a sculpture or we need to backtrack.
If there are with , and can be sculpted, then for any embedding .
We construct from by finding and repairing homotopy pairs, which consist of two paths of the form
with all intermediate states distinct and no other transitions or squares between them. The shortest homotopy pair is an interleaving, a pair of two transitions.
If is such that , then there is a homotopy pair with final state .
Now if the homotopy pair is an interleaving , , then we must repair by identifying with and with . If it is not, then there are several choices for identifying events, and some of them may lead into situations like in Lemma 7. Let be any permutation on with and , then we can identify with for all . The restriction on the permutation is imposed by the fact that we only identify transitions that can possibly be concurrent, which is not the case for two transitions starting from, or ending in, the same cell.
Let be the equivalence relation thus generated. As this inclusion is proper, it is clear that the described process either stops with a Lemma 7 situation which cannot be resolved using backtracking or with a relation which satisfies Thm. 4.2.
We give an example to illustrate why backtracking might be necessary when applying the algorithm. Figure 11 is a variation of the example in Fig. 8 which, as the labeling on the top right shows, can be sculpted. However, if we start our procedure by resolving the homotopy pair on the left in a “wrong” way, see the bottom of the figure, then we get into a contradiction in the top right corner and must backtrack.
5 Euclidean Cubical Complexes are Sculptures
This section provides a connection between the combinatorial intuition of sculptures and the geometric intuition of Euclidean HDA. We need to introduce some notions from directed topology first; see [17, 11] for motivation and background.
Directed topological spaces
A directed topological space, or d-space, is a pair consisting of a topological space and a set of directed paths in which contains all constant paths and is closed under concatenation, monotone reparametrization, and subpath.
On a d-space , we may define the reachability preorder by iff there is for which and . As contains all constant paths and is closed under concatenation, this is indeed a preorder. A d-space is said to be a partially ordered space, or po-space, if is a partial order, antisymmetric.
Prominent examples of po-spaces are the directed interval with the usual ordering and its cousins, the directed -cubes for . Similarly, we have the directed Euclidean spaces , with the usual ordering, for .
Morphisms of d-spaces are those continuous functions that are also directed, that is, satisfy for all . It can be shown that for an arbitrary d-space , .
The geometric realization of a precubical set is the d-space , where the equivalence relation is generated by and . (Technically, this requires us to define disjoint unions and quotients of d-spaces, but there is nothing surprising about these definitions, see .)
Geometric realization is naturally extended to morphisms of precubical sets: if is a morphisms of precubical sets, then is the directed map given by