2 The algebra of contextualized ontologies

Contextualized ontologies are composed of three parts: an entity and a context, which are both ontologies, and a link between them. When referring to ubiquitous computing, we can think about the entity, for example, as being a mobile device. The context can be thought of as the environment where the entity operates, which, in this case, could be a physical environment where this mobile device is located. The link is the immersion of the entity into its context. Under this model, information concerning either the physical or computational environment is treated as a relevant part of the application, deserving its own representation. This representation is self-contained, what gives a high degree of modularity. The following definition is adapted from Maedche and Staab (Reference Maedche and Staab2001):

Usually, an ontology is a set of DL (description logic) sentences representing a theory—the knowledge base. From the semantic point of view, when this set of sentences is consistent we have a model, which we call an ontology structure. In an ontology structure O = (C, R, H ^C, rel, A) for a given ontology Ω, C and R are the set of non logical symbols that appear in the sentences of Ω (in C, the unnary symbols and in R, the n-ary symbols). H ^C embodies the is–a relation of Ω, and rel is an extensional description of the non-taxonomical relations. A is the set of sentences of Ω, as long as it is consistent. For any ontology structure O, there exists a set of corresponding valid sentences: the ontology Ω induced by the given ontology structure (of course, it is provided modulled ontology isomorphism). However, for some ontologies Ω there are no corresponding ontology structures. For example, for $$\[--><$> {\rm{\rOmega }}\, = \,{\rm{\{ }}\neg p\, \sqcup \,p{\rm{\} }} <$><!--$$ . In this paper we focus on ontology structures, that is, the relationships between ontologies will be given at the semantic level. It is clear that any semantic mapping induces a mapping between theories.

We define a link between ontology structures (semantic mapping) as

By condition (i), links preserve hierarchy of concepts. By (ii), links preserve relations. By (iii), axioms of the domain ontology structure, when properly translated to the vocabulary of the codomain ontology structure, hold for the codomain. The process of translation is a canonical process over the structure of the sentence. We will denote by trans_fg(a) the translation of an axiom a by f and g. By Th(A′) we mean the set of sentences that are provable from axioms A′. From (iii) of Definition 2, we see that a link between ontology structures induces a mapping between theories:

Let O = (C, R, H ^C, rel, A) be an ontology structure for Ω, and $$\[--><$> {\rm{(}}f,g,h{\rm{)}}\,:\,O\, \rightarrow \,O^{\prime} <$><!--$$ be a link between ontology structures, where $$\[--><$> O^{\prime}\, = \,{\rm{(}}C^{\prime},R^{\prime},{{H}^{C^{\prime}}}, rel^{\prime},A^{\prime}{\rm{)}} <$><!--$$ . Then, by (iii) of Definition 2, if a ∈ A then $$\[--><$> A^{\prime} \vdash \,tran{{s}_{fg}}{\rm{(}}a{\rm{)}} <$><!--$$ . O′ induces an ontology Ω′ whose axioms are A′ and the translation of the axiomatization of H ^C. Then there is a syntactical morphism between Ω and Ω′: for all $$\[--><$> a\, \in \,{\rm{\rOmega }},{\rm{\rOmega ^{\prime}}}\, \vdash \,tran{{s}_{fg}}{\rm{(}}a{\rm{)}} <$><!--$$ .

Since we adopt a semantic-oriented approach, in the remainder of this paper we use the term ontology as a placeholder for ontology structure.

Example 1. Figure 1 shows a simplification of the representation of Prof. Silva and its contextualization in PUC-Rio. The ontology at left describes Silva as a concept Prof. Silva and a subconcept SMP_Silva, the smart phone of Prof. Silva. In the ontology at right there is a simplified representation of PUC-Rio. In this situation, the ontology representing the professor plays the role of entity and the ontology representing the institution PUC-Rio plays the role of context. The contextualized entity shows how Prof. Silva is considered as a member of PUC-Rio. In this figure, one can notice that the hierarchy of concepts is preserved.

Figure 1

An entity: Silva, a contex: PUC-Rio, and a contextualized entity: Silva at PUC-Rio

2.1 Operations on contextualized ontologies

The Algebra of Contextualized Ontologies is formed by two sets of operations devoted to compose and decompose contextualized ontologies in several ways. There is a duality between the actions of composing and decomposing which, in the formal sense, takes the form of reversing arrows, and is clearly noted by abstracting the role of an ontology to entity or context. It is sufficient to define one of the sets of operations, and then, by duality, to deduce the other. This homogeneity in the definition is also reflected in the algorithms that compute the algebra operations, what contributes to the simplification of the framework as a whole.

The operations have the general purpose of manipulating information of ontologies in order to produce a coherent and concise body of information. For that sake, to play the role of establishing binary relations between contexts or between entities, we define alignment and coalignment operations. We define also context integration, collapsed union, entity integration and relative intersection operations, which are devoted to produce new entities, contexts or contextualized entities according to the related information. The composition operations are alignment, context integration and collapsed union. Decomposition operations are coalignment, entity integration and relative intersection.

Alignment. It is a situation where an entity has more than one context: C ₁←E_Med←C ₂, as shown in Figure 2(a). By defining a binary relation, the alignment makes possible the partial mapping between contexts. This feature makes possible to deal with situations where a concept of a context does not make sense in the other context. On the other hand, the entity must be totally mapped on both contexts: all concepts of the entity must be understood in both contexts.

Figure 2

The algebra of contextualized ontologies operations

Context integration. In an alignment C ₁←E_Med←C ₂, the context integration produces a new context C, to which C ₁ and C ₂ are linked: C ₁→C←C ₂, as shown in Figure 2(b). This new context combines information of C ₁ and C ₂ preserving the coherence with the entity E_Med. The integration performs the amalgamated union of contexts, collapsing components that are related by the alignment C ₁←E_Med←C ₂. The result contains all information of the original contexts, but identifies parts related by the mediator entity.

Collapsed union. It is the amalgamated union of two contextualized ontologies mediated by a third contextualized ontology, as shown in Figure 2(c). It is the combined composition of entities and contexts, where the ontology links ensure the preservation of structure, relations and axioms of each ontology and coherence of each entity with respect to contexts. It produces a new contextualized ontology with all components of the original ones, but collapsing components that have the same source in the mediator.

Coalignment. It is a mechanism of establishing a correspondence between vocabularies of two ontologies by the use of an intermediate target ontology: E ₁←C_Med←E ₂, as shown in Figure 2(f). It constitutes a binary relation between the two ontologies, where related components are those that are mapped in the same component of the intermediate target. By defining a binary relation, the coalignment makes possible the partial mapping between entities, that is, not all concepts of one entity make sense for the other entity. Both entities, however, must be totally mapped in the context.

Entity integration. Given a coalignment E ₁←C_Med←E ₂, the entity integration produces a new entity E contextualized by the original ones (and by transitivity, by the original context C_Med), as shown in Figure 2(d). The entity integration performs the semantic intersection of the entities under the mediation of the context, that is, the new entity will embody all information of the original entities that are related by the coalignment E ₁←C_Med←E ₂.

Relative intersection. It is the intersection of two contextualized ontologies mediated by a third contextualized ontology, as shown in Figure 2(e). It is the combined intersection of entities and contexts, where the ontology links ensure the preservation of structure, relations and axioms of each ontology and coherence of each entity with respect to its context. It produces a new contextualized ontology having just the components of the originals that are mapped in the mediator.

The reader who is familiar with Category Theory can find in Cafezeiro et al. (Reference Cafezeiro, Haeusler and Rademaker2008a), a detailed explanation about the categorical foundation of each operation. In this paper, we briefly remark that the contextualization of an entity is a morphism in the category of ontology structures. Alignment is a pushout in the same category, and the dual coalignment is a pullback. Context integration is a pushout in the category of ontology structures and the dual entity integration is a pullback in the same category. Collapsed union is a pushout in the category of links between ontology structures, and the dual relative integration is the pullback in the same catogory.

3 Specifying context-aware applications

Considering that it would be very difficult to specify a system by working directly with ontologies and ontology links defined by the algebra, we adopt in this approach three kinds of diagrams: diagram of reactive elements, diagram of entities and diagram of ontologies. These diagrams give structure to the process of specification by supporting different levels of expression. The diagram of reactive elements gives a general view of the system by representing the exchange of information between its components. The diagram of entities, sketches the meaning of a single entity, making possible to describing an entity without considering all the formal details of a specification through ontologies and ontology links. Finally, the diagram of ontologies introduces all formal details of a system. These three diagrams do not have the purpose of establishing a step-by-step process of specification, but to offer a simple way of expressing each level of abstraction.

Diagram of reactive elements. It specifies all computational elements (pieces of software/hardware) that compose the ubiquitous system and the kind of information that is exchanged among them. Each element of a ubiquitous system is a node of the diagram, which is linked to other nodes by a directed arrow labelled with the information that is to be transmited from an element to another.

Diagram of entities. An entity is any element of the system that produces a reaction caused by the interaction with another element of the system. For example, a Professor, that must be identified when arriving at work. Each element of the previous diagram is related to a diagram of entities. Through the Algebra of Contextualized Ontologies, the entities of this diagram can be operated generating new entities wherein the element being described can extract the necessary information to work on.

Diagram of ontologies. It provides a detailed view of the diagram of entities, where each entity is described by an ontology and the links are consistently defined in the Algebra of Contextualized Ontologies.

These three diagrams are related in the following way: each element of the diagram of reactive elements correspond to a diagram of entities. In the diagram of entities, details of entities and their connection are hidden. These details appear in the diagram of ontologies, where entities’ specifications and their connections are sufficiently detailed in order to make possible consistency tests.

3.1 Formalizing the scenario

Following, we give more details on the scenario presented in Subsection 1.2, highlighting the main elements involved.

Silva carries with him his smart phone (SMP_Silva), which hosts some context-aware applications, such as the Configuration Management Service (CMS) and the Personal Agenda (PA). While CMS is responsible for dynamically setting the configurations of SMP_Silva (e.g. switching it to silent mode), PA is responsible for managing Silva's personal agenda, notifying him about important events. The ambient infrastructure in each university also hosts context-aware applications, such as the Ambient Management Service (AMS) and the UbiForm Project Agenda (UPA). AMS is responsible for identifying the devices (and users) connected to the local wireless network and setting up users’ workspaces, that is, providing access to the files they will need. UPA is responsible for managing the project agenda and tracking the project's participants.

When Silva arrives at PUC-Rio, the AMS of PUC-Rio (AMS_PUC) detects SMP_Silva and identifies that it belongs to him. The UbiForm Project Agenda (UPA_PUC) informs the members of UbiForm Project about Silva's arrival. The PA application running on SMP_Silva contacts UPA_PUC with a request to be notified about the beginning of events. The CMS, requests to be notified whenever Silva is in a room in which an activity (e.g. a technical presentation, a brainstorming session) has started, so that it may switch SMP_Silva to silent mode.

When Silva arrives at UFF, SMP_Silva gets a connection to the local wireless network, and based on its GPS coordinates finds out that Silva is at UFF. The the AMS of UFF (AMS_UFF) registers SMP_Silva and identifies that the device belongs to Silva, a member of UbiForm project who works at a partner institution. AMS_UFF contacts AMS_PUC, authenticating the user and recovering his preferences. It then configures a workspace for him. Although Silva is identified as a visitor at UFF, he will also be perceived as being a professor in his institution. According to UFF's UbiForm resource access policy any professor (local or visitor) can access the meeting room's printers at UFF. Then, AMS_UFF will also recognize this access permission.

3.1.1 Diagrams of reactive elements

Figure 3 composes the diagram of reactive elements of the scenario described above. It shows AMS_PUC, UPA_PUC, CMS, PA, AMS_UFF and UPA_UFF that exchange information as a result of the perception of the presence of Silva. Two of these elements are applications executing on Silva's smart phone: the CMS, which configures the phone ring mode according to the location or activity of Prof. Silva, and PA, a personal agenda that must be synchronized with the project agenda. The other elements, AMS_PUC, UPA_PUC, AMS_UFF and UPA_UFF are applications executing at each university's ambient infrastructure and play the role of monitoring information about the environment and providing specific functionalities for the smart phone's applications (e.g. notifying the arrival of a project participant). Figure 3(a) pictures the exchange of information when Silva arrives at PUC. In Figure 3(b), we show the diagram of reactive elements that describes the arrival of Silva at UFF. In this case, AMS_UFF interacts with AMS_PUC, so that Silva is authenticated at UFF and the resources at UFF can be allocated for him according to his preferences and permissions.

Figure 3

Diagrams of reactive elements

3.1.2 Diagrams of entities

Each element of the diagram of reactive elements should then be detailed in terms of a diagram of entities. For example, for AMS_UFF it is necessary to combine personal and professional information of Silva in order to use personal preferences to configure a workspace for him, providing quick access to the files he will be manipulating. The Algebra of Contextualized Ontologies is used at this specification level to guide the construction of a coherent and consistent body of the information over which each element reacts. Details of the algebraic operation will appear only in the diagrams of ontologies. In order to illustrate the use of diagrams of entities in the following, we describe the diagram for the AMS.

Figure 4 represents the situation where AMS informs other participants of UbiForm project about his arrival. Prof_i at PUC results from the context integration of each i and the ontology describing information of PUC, under the mediation of professional information of each i (Prof_i). Figure 4(a) shows this context integration for i = Silva. The entity integration of each Prof_i and Prof Silva under the context of PUC will make the connection among the i professors of PUC and Professor Silva. The resulting entity will be composed by each professor and is shown in Figure 4(b).

Figure 4

Diagrams of entities

Later, Professor Silva is visiting UFF, where he is registered as a visitor researcher. Within the context SilvaAtUFF that results from integration $$\[--><$> Silva\,{\mathop{ \leftarrow }\limits^{{AMS}}} \,Prof.\,Silva\,{\mathop{ \rightarrow }\limits^{{AMS}}} \,UFF <$><!--$$ (lateral square at left in Figure 5), AMS_UFF can properly set the professor's workspace. But some of Silva's permissions for the use of resources come from the fact that he is a Professor at PUC. Thus, information about Silva's status at PUC must also be taken into account. The context integration $$\[--><$> UFF\,{\mathop{ \leftarrow }\limits^{{AMS}}} \,Prof.\,Silva\,{\mathop{ \rightarrow }\limits^{{AMS}}} \,PUC <$><!--$$ generates a context where AMS_UFF can find information about Silva as a PUC professor and as a UFF visitor researcher in the joint project UFF/PUC (base square of Figure 5). The context integration $$\[--><$> SilvaAtUFF\,{\mathop{ \leftarrow }\limits^{{AMS}}} \,Silva\,{\mathop{ \rightarrow }\limits^{{AMS}}} \,SilvaAtPUC <$><!--$$ generates a context where AMS_UFF can find not only information about Silva as a PUC professor or as a UFF visitor researcher, but also personal information about Silva (top square of Figure 5). Note that Figure 5 also pictures a combined integration: the collapsed union of the contextualized entities UFF → SilvaAtUFF, PUC → SilvaAtPUC mediated by Prof. Silva → Silva.

Figure 5

Diagram of entities of AMS: each face of the cube shows a context integration. The complete cube is the collapsed union of the contextualized entities UFF → SilvaAtUFF, PUC → SilvaAtPUC mediated by Prof. Silva → Silva. AMS=Ambient Management Service

3.1.3 Diagrams of ontologies

The diagrams of ontologies are a refinement of the diagrams of entities, were each entity appears described as an onlology and the connections of the entity diagrams are mapped to links between ontologies. We selected an illustrative diagram of entities to show how this framework provides the required information to adapt services or behaviours according to the context changes. We consider a situation in which information coming from one context enables decisions about an entity in a different context. For instance, as any professor (local or visitor) can access the meeting room's printers at UFF, Silva is allowed to use that printer as a consequence of the fact that he is identified as a professor at PUC.

Considering the base square of Figure 5, the mediator Prof. Silva of the context integration $$\[--><$> UFF\,{\mathop{ \leftarrow }\limits^{{AMS}}} \,Prof.\ Silva\,{\mathop{ \rightarrow }\limits^{{AMS}}} \,PUC <$><!--$$ must capture the fact that Silva is a professor and properly map this information into the ontology of UFF. Diagram of ontologies in Figure 6(a) depicts the ontology for UFF and PUC and shows this alignment. Note that, as the concept Professor at PUC is related to Researcher at UFF, the relation hasAccess(?p,?d) will hold for Prof. Silva and Printer in the resulting context (in Figure 6(b)). Also, note that, in this resulting context, information about Prof. Silva's production is available to be used by AMS_UFF.

Figure 6

Composition concerning the integration of institutions

4 Algorithms and complexities

The contextualization of an entity is the definition of a link between ontologies, where the source is the entity and the target is the context. According to Definition 2, a link preserves hierarchy, relations and logical properties, enabling, thus, a consistent and coherent mapping of meanings of the entity into the context. Contextualization is not an automatic process, as it reflects the intended semantics for the entity. It can, however, be automatically validated, that is, it is possible to define an algorithm to verify if the structure of concepts and relations is preserved and logical constraints are respected (algorithm 1, left). We consider two ontology structures $$\[--><$> O\, = \,{\rm{(}}C,R,{{H}^C}, rel,A{\rm{)}} <$><!--$$ and $$\[--><$> O^{\prime}\, = \,{\rm{(}}C^{\prime},R^{\prime},{{H^{\prime}}^{C^{\prime}}}, rel^{\prime},A^{\prime}{\rm{)}} <$><!--$$ , and a link $$\[--><$> {\rm{(}}f:C\, \rightarrow \,C^{\prime},g:R\, \rightarrow \,R^{\prime},h{\rm{)}} <$><!--$$ .

The complexity of the algorithm for validating a contextualization $$\[--><$> O\,{\mathop{ \rightarrow }\limits^{{{\rm{(}}f,g,h{\rm{)}}}}} \,O^{\prime} <$><!--$$ is dependent of the complexity of the theorem prover for the considered logic. As we are considering $$\[--><$>{\scr A}{\scr L}{\scr C} <$><!--$$ , the complexity is PSPACE Complete, as shown in Baader et al. (Reference Baader, Calvanese, McGuinness, Nardi and Patel-Schneider2003). Considering just the structure of the ontology, the complexity is linear on the size of O .

The contextualization of an entity is the fundamental operation over which all the operations of the algebra are constructed. For example, the alignment of an entity is its contextualization in two ontologies. By duality, the coalignment of two entities is the contextualization of both in the same context. In a similar way, the alignment of a contextualized entity is the consistent combination of alignments in contexts and entity. By duality, the coalignment of two contextualized entities is the consistent combination of coalignments in contexts and entity. By consistent we mean that the ontological structure of the entity (co)alignment is preserved in the context (co)alignment. Referring to Figure 7(a), this means that E_Med is mapped in the same way in C ₁ following E_Med → C_Med → C ₁ or following E_Med → E ₁ → C ₁, and the same for C ₂. Algorithm 1, right, verifies if a square C ₁ ←C_Med ←E_Med → E ₁ → C ₁ is consistent. We consider four links between ontologies forming a square as: $$\[--><$> O^{\prime}\,{\mathop{ \leftarrow }\limits^{{{{{l^{\prime}}}_{\rm{1}}}}}} \,{{O}_{\rm{1}}}\,{\mathop{ \leftarrow }\limits^{{{{l}_{\rm{1}}}}}} \,O\,{\mathop{ \rightarrow }\limits^{{{{l}_{\rm{2}}}}}} \,{{O}_{\rm{2}}}\,{\mathop{ \rightarrow }\limits^{{{{{l^{\prime}}}_{\rm{2}}}}}} O^{\prime} <$><!--$$ , where, $$\[--><$> {{l}_{\rm{1}}}\, = \,{\rm{(}}{{f}_{\rm{1}}},{{g}_{\rm{1}}},{{h}_{\rm{1}}}{\rm{)}},{{l^{\prime}}_{\rm{1}}}\, = \,{\rm{(}}{{f^{\prime}}_{\rm{1}}},{{g^{\prime}}_{\rm{1}}},{{h^{\prime}}_{\rm{1}}}{\rm{)}},{{l}_{\rm{2}}}\, = \,{\rm{(}}{{f}_{\rm{2}}},{{g}_{\rm{2}}},{{h}_{\rm{2}}}),{{l^{\prime}}_{\rm{2}}}\, = \,{\rm{(}}{{f^{\prime}}_{\rm{2}}},{{g^{\prime}}_{\rm{2}}},{{h^{\prime}}_{\rm{2}}}{\rm{)}} <$><!--$$ .

Figure 7

Alignment and coalignment

The algorithm verifies whether concepts and relations are consistently mapped. Axioms are not verified because the transitivity is ensured by the underlying logic (if $$\[--><$> tran{{s}_ {{{{f^{\prime}}}_1}{{{g^{\prime}}}_1}}}(tran{{s}_{{{f}_1}{{g}_1}}}(a)))\, \in \,Th(A^{\prime}) <$><!--$$ then $$\[--><$> tran{{s}_{{{{f^{\prime}}}_2}{{{g^{\prime}}}_2}}}(tran{{s}_{{{f}_2}{{g}_2}}}(a)))\, \in \,Th(A^{\prime}) <$><!--$$ ). As in the previous case, the complexity of the algorithm is linear on the size of O.

Note that (i) the same algorithm can be used to verify the consistency of alignments and coalignments; (ii) Given three contextualized entities and a coalignment of their entities, the consistent coalignment on their contexts is unique and can be trivially computed; thus, for composing contextualized entities it is enough to coalign the entities. In the same way, given three contextualized entities and an alignment of their entities, the consistent alignment on their contexts is unique and can be trivially computed; thus, for decomposing contextualized entities it is enough to align the entities.

The algorithm to compute the collapsed union produces, from an alignment of contextualized entities (Figure 7(a)), a coalignment of contextualized entities (Figure 7(b)). Figure 8 illustrates each step, with the exception of the block in lines 6 to 8, that deals with the particular case when a component of one of the contextualized ontologies is image of more than one component in the alignment mediator. This case must be reflected in the coalignment by collapsing the component introduced with a previously introduced component that is the image of this junction. In the block of lines 13 to 15, we use f_i, g_i to refer to the concept and relation components of each l_i.

Figure 8

Steps of the algorithm to compute the collapsed union

The resulting coalignment embodies the least informative integration of the given alignment. The complexity of the algorithm is linear on the size of the given alignment, as each component of each ontology is visited a unique time.

The context integration of C ₁ ← E → C ₂ can be derived from the collapsed union by contextualizing each entity or context in itself, using the identity as link. Thus, the collapsed union C ₁ → C ₁ and C ₂ ← C ₂ under the mediation of E → E results the context integration of C ₁ ← E → C ₂. The complexity is linear on the size of the ontologies.

The algorithm for Relative Intersection is the dualization of the algorithm for collapsed union.

While the collapsed union algorithm results a coalignment that embodies the least informative integration of the given alignment, this algorithm results an alignment that embodies the more informative ‘intersection’ (common part) of the coalignment. While the collapsed union algorithm requires additional lines (9 to 12) to augment the generated coalignment with information that is particular to the left or right side, the relative intersection algorithm requires the condition of line 1 to restrict the generated alignment to information of the coalignment (avoiding, thus, information that is particular to the left or right side). In the logical part, the disjunction of the axioms ensure the provability in both CE ₁ and CE ₂. The complexity of the algorithm for relative intersection is linear on the size of the given coalignment in the structural part and polynomial on the number of axioms in the logical part.

The entity integration of E ₁ →C ← E ₂ can be derived from the relative intersection by contextualizing each entity or context in itself. Thus, the relative intersection E ₁ → E ₁ and E ₂ → E ₂ under the mediation of C → C results the entity integration of E ₁ → C ← E ₂.

5 Related work

In this section we discuss the most relevant works that are related to ours from different perspectives, namely categorical-based approaches to ontologies integration, logic-based integration, particularly Distributed Description Logic, and other approaches that formalize context modeling.

Zimmermann et al. (Reference Zimmermann, Krötzsch, Euzenat and Hitzler2006) presented a categorically based approach for the alignment of ontologies that use pushouts (a specific co-limit notion) to provide a rigorous semantics to merging, intersection, union and composition of ontologies. Zimmermann et al. define the notion of V-alignment, a span in categorical terminology, raising an abstract notion of merge of two ontologies and an algebra that allows composition of V-alignments, intersection and union also. The approach is then extended with W-alignment, so as to cope also with non-symmetrical relations (like mothership). Both notions, defined in Zimmermann et al. (Reference Zimmermann, Krötzsch, Euzenat and Hitzler2006), have interesting advantages and drawbacks, as pointed out by the authors themselves. Basically, V-alignments are less expressive than W-alignments, while the latter has a quite complex algebra and composition definition. W-alignments are closer to our approach. The categorical notion underlying V- and W-alignments is known as a span diagram, and is built externally to the defined category. Intuitively, this means that A in the V-alignment O ₁ →A → O ₂, is not of the same type as O_i, i = 1,2. This is a major difference to our approach, where the entities and their context are ontologies within the same category. Moreover, our algebra is also simpler, because our definitions of alignment are limits and co-limits in a comma-category over the category of ontologies. This is the main reason why our approach takes advantage of the dualization of notions, which is so important in category theory. Moreover, each morphism in our category represents a set of parallel morphisms in the category of W-alignments. Thus, using contexts we removed the main drawbacks on composition of W-alignments.

In Borgida and Serafini (Reference Borgida and Serafini2003), a purely logical approach to deal with modular ontologies is taken. Following the approach to distributed first-order logic proposed in previous works, the authors define distributed versions of Description Logics (DDLs) able to logically specify the interconnection between concepts/roles between component ontologies. For example, let us consider that one needs to specify that a concept D, described by an ontology i Footnote ¹, is associated to a concept E, described by an ontology j. Their approach says, in this case, that the specifier should set up the (distributed) subsumption $$\[--><$> i\,:\,D\, \sqsubseteq \,j\,:\,E <$><!--$$ interconnecting these two concepts by means of an external ontology.

The DDL approach is an interesting and elegant solution for the general problem of modularizing ontologies. It is somehow related to our approach, but we focus on context-awareness ontological specification. In this sense, we are able to locally identify contextualized entities (concepts/roles descriptions), dealing separately with their respective context as well as context integration. The distributed Description Logic approach, by itself, does not provide an answer to context-awareness. We would adopt the same basic idea proposed in our approach if we used DDLs instead of ordinary DLs. Besides that, our use of an algebra for expressing the structure of contextualized ontologies maintains the information on the structure itself, allowing posterior decomposition and integration at the theories (ontologies) level. Regarding the work discussed in this paragraph, we can say that our approach would add context-awareness and structure representation to DDLs, while DDLs would bring us more elegant and efficient reasoning, regarding memory/storing. Using ordinary DLs (as in OWL-DL), we must either build the whole structured contextualized ontology, as described in Section 2 or implement a backward chain reasoner, aware of this structure. In this later case the reasoner performs the integration at execution time, what implies in a loss of efficiency. One other point is that DL's subsumption is a particular case of the links between ontologies that are used in our approach. Using DDLs, the external ontology should be defined in order to reflect structured contextualized ontologies. This later issue is subject of further research.

There are several other works that deal with formalization of context modeling using different theoretical approaches. In Cafezeiro et al. (Reference Cafezeiro, Viterbo, Rademaker, Haeusler and Endler2008b), we presented a detailed comparison between our approach and Context UNITY (Julien et al., Reference Julien, Payton and Roman2003; Roman et al., Reference Roman, Julien and Payton2004), Context-Aware Action System (Yan & Sere, Reference Yan and Sere2004) and Bigraphical Reactive System (BRS) Models (Birkedal et al., Reference Birkedal, Debois, Elsborg, Hildebrandt and Niss2005). Beside these, we cite CommUnity (Lopes & Fiadeiro, Reference Lopes and Fiadeiro2005) who as well as our approach uses Category Theory as theoretical basis to express integration and contexts. It also emphazises the separation between systems behavior and context modeling. As it evolved from a previous work on distribution and mobility in software architectures, it adopts concepts of communication by channels and location variables. We believe our approach is more general than it, since CommUnity context mappings happen always via channels and location variables while ours is based on any conceptual piece of the ontology.