A Categorical Approach to Networks of Aligned Ontologies
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Abstract
Ontology matching and alignment are key mechanism for linking the diverse datasets and ontologies arising in the Semantic Web and other application areas for formalised ontologies. We show that category theory provides the powerful abstractions needed for a uniform treatment of ontology alignment at various levels: semantics, language design, reasoning and tools. The general representation and reasoning framework that we propose includes: (1) an abstract notion of logical system, consisting of a logic syntax and a model theory, based on an extension of institutions with additional features specific to alignments, (2) a declarative language to specify networks of ontologies and alignments, with independent control over specifying local ontologies and complex alignment relations, based on and improving the Distributed Ontology, Model and Specification Language DOL, (3) the possibility to align logically heterogeneous ontologies, and (4) the provision of generic proof support for global reasoning over networks of aligned ontologies, employing different semantics. In particular, we show how the three semantics of Zimmermann and Euzenat can be uniformly and faithfully represented using \(\mathsf {DOL}\) language constructs, by refining them into four different kinds of semantics: simple, integrated (general and inclusive), and contextualised. Finally, we discuss the implementation of the \(\mathsf {DOL}\) alignment features in the Ontohub/Hets tool system.
Keywords
Ontology alignment Networks of ontologies Category theory DOL Semantics Reasoning1 Introduction
Matching, and subsequently aligning and linking the diverse terminologies found in heterogeneous datasets, taxonomies and ontologies, arising in the Semantic Web and related application areas for semantic technologies, are seen as key mechanism towards enabling deeper interoperability between systems and software. Various methods for ontology matching exist, among them linguistic, structural, semantic and statistical methods. The field is quite well developed, with yearly competitions since 2004 comparing the various strengths and weaknesses of existing algorithms [75].
Ontology alignments, produced by ontology matching, express correspondences at the semantic level between the symbols of different ontologies. The correspondences of an alignment can involve various relations, between the entities of the ontologies, like equivalence, subsumption, disjointness or membership. The aligned entities can be named symbols, like classes, object properties, individuals, function symbols and sorts or even complex concepts or terms.
The problem of giving an interpretation to alignments in terms of the semantics of the ontologies is complicated by the fact that the domains of interpretation of the two ontologies are unrelated. Different ways of dealing with this problem exist in the literature. The first solution, called simple semantics in [80], is to assume that the domain of interpretation of the ontologies is uniform [9, 10]. The second solution, called integrated semantics in [80], is to assume the existence of a universal domain together with functions relating the domains of individual ontologies to the universal domain. This approach has been introduced in [78], under the name of integrated distributed description logics (IDDL). Finally, the domains of the individual ontologies can be related among themselves directly instead via a unique universal domain. This approach gives rise to the third semantics, called contextualised semantics in [80]. It was introduced in [80] as an attempt to generalise a number of existing semantic formalisms (distributed firstorder logics (DFOL) [25], distributed description logics (DDL) [5] and contextualised ontologies (COWL) [7]) and later corrected to a relational semantics in [79]. Packagebased description logics (PDL) [3] also fall into this semantic category, as do \(\mathcal {E}\)connections [48]. Moreover, [80] discusses the implications of these possible interpretations of alignments with respect to reasoning and composition of alignments.
A major problem with these approaches is their diversity. There exist some attempts for unification, which however remain unsatisfactory: there is no common syntax, no common semantic framework, and no common tool support. In this work, we show how category theory can provide such a unifying framework at various levels. Our categorical framework provides a precise syntax, semantics and proof mechanisms for alignments and for networks of alignments that follow closely those developed for \(\mathsf {OWL}\) (see [20, 80]) while simultaneously abstracting from the dependency on \(\mathsf {OWL}\) and generalising the approach to an arbitrary logic.
We also improve previous related work at the abstract categorical level [34, 49, 79, 81] which did not spell out all details, and did not make the step from abstract description and case studies to language design and implementation. In particular, the intuitive definition of satisfaction of a correspondence in a pair of aligned ontologies from [20, 80] is not covered by these works, while we cover it and generalise it to an arbitrary logic.^{1}
Our results have greatly influenced the syntax and semantics of alignments, networks of alignments and their combination in the Distributed Ontology, Model and Specification Language \(\mathsf {DOL}\) [59, 62, 68]. While \(\mathsf {DOL}\) has been developed independently, its treatment of alignments is essentially based on the contributions provided in the present paper.
Before giving a detailed outline of the structure of the paper, we describe the general approach followed and the solutions provided in a highlevel fashion.
 1.
It provides a declarative language to specify networks of ontologies and alignments, with independent control over specifying the local ontologies and complex alignment relations.
 2.
It supports the alignment of logically heterogeneous ontologies.
 3.
It allows to independently choose the semantical paradigm to interpret the alignment relations (simple/integrated/contextualised).
 semantic level:

We give a uniform semantics for distributed networks of aligned ontologies while at the same time reflecting properly the semantic variation points indicated above.
 (meta) language level:

Based on the Distributed Ontology, Model and Specification Language \(\mathsf {DOL}\), we provide (and equip \(\mathsf {DOL}\) with) a uniform notation for distributed networks of aligned ontologies, spanning the different possible semantic choices.
 reasoning level:

Using the powerful notion of colimit, we provide reasoning methods for distributed networks of aligned ontologies, again across all semantic choices.^{2}
 tool level:

Hets (https://hets.eu) and Ontohub (https://ontohub.org) provide an implementation of analysis and reasoning for distributed networks of aligned ontologies, again using the powerful abstractions provided by category theory.
 logic level:

Our semantics is defined in a logicindependent way and formulated under rather mild technical assumptions. Moreover, these assumptions are verified by most logical formalisms used in practice. In this paper, we use as running examples the logic \(\mathcal {SROIQ}\) underlying the Web ontology language OWL 2 DL, as well as unsorted firstorder logic.
Our use of category theory is modest, oriented towards the generalisation of results that can, with similar efforts, also be developed for particular logics. Moreover, if one wants to use our general results and instantiate them for a particular logic, there is no free lunch: one still has to provide a number of specific notions for the logic at hand. Still, our approach has benefits over solutions that are handcrafted for each individual logic. Firstly, we gain insights, at a general level, in the nature of systems of aligned ontologies and their properties. Secondly, the individual handcrafted solutions are often similar in their details, for example, the reduction of packagedbased DLs to \(\mathcal {SROIQ}\) in [3], which the authors claim to have adapted from that for DDLs [5]. This calls for abstraction. Indeed, for this kind of reduction, we prove some general theorems that need not to be reproved in each individual case. Thirdly, each approach to alignment and/or to distributed ontologies has its own language and formalism. With \(\mathsf {DOL}\), we provide general language constructs for alignments that do not need to be reinvented for each logic separately. And last but not least, Hets and Ontohub provide logicindependent tool support. While the tool support also does not come for free, it is much less effort to instantiate the general tool framework with additionally needed logicspecific tools than it is to newly create an alignment analysis and reasoning tool from scratch.^{3} A typical user of our framework will mostly benefit from the uniform language design and tool implementation. Both are available and can be used at an intuitive level without a deeper understanding of our framework.
The Distributed Ontology, Model and Specification Language \(\mathsf {DOL}\) ^{4} is a metalanguage in the sense that it enables the reuse of existing ontologies as building blocks for new ontologies using a variety of structuring techniques, as well as the specification of relationships between ontologies. One important feature of \(\mathsf {DOL}\) is the ability to combine ontologies that are written in different languages without changing their semantics. A sketch of the formal specification of the language can be found in [62], with a more extensive description found in [59, 68]. The general theoretical background for \(\mathsf {DOL}\) was given in [51]. This paper gives the first detailed treatment of the syntax and semantics of \(\mathsf {DOL}\) alignments independently of the logics of the aligned ontologies.
The general picture of networks of ontologies in \(\mathsf {DOL}\) is then as follows: existing formalisms for ontologies can be integrated asis into the \(\mathsf {DOL}\) framework. With our new extended \(\mathsf {DOL}\) syntax, we can specify different kinds of alignments. Based on this, we construct networks of ontologies and alignments between them. These networks are called relational. Functional networks of ontologies are possible as well; the links between the ontologies of a functional network are given by ontology morphisms. The latter are similar to alignments that behave like functions (i.e. that relate each symbol in the source ontology with exactly one symbol in the target ontology). Functional networks of ontologies can be combined into an integrated ontology via a colimit. Note that colimits, which can intuitively be considered as quotients of disjoint unions, provide a powerful and flexible way of combining a network of alignments into a single combined ontology. Reasoning in a functional network is then the same as reasoning in the combined ontology. This construction generalises the above mentioned reduction from packagebased DLs and DDLs to \(\mathcal {SROIQ}\).
In order to do the same for relational networks, we first must normalise relational networks into functional ones. This step depends on the particular choice of an alignment framework, specifically on (a) the logics of the ontologies involved, and (b) the assumptions made on the domains, i.e. the type of semantics involved. It may involve transformations on the ontologies, such as relativisation of the domain of interpretation using predicates. During this construction, the correspondences become bridge axioms in a bridge ontology relating the aligned ontologies. Bridge axioms internalise the semantics of correspondences. This transformation is specific to the logics of the aligned ontologies and may require a translation to a more expressive formalism.
Our approach is generic in the following sense: in order to include a new ontology language into the framework, it is sufficient to define the satisfaction of correspondences, the relativisation for local logics, and the transformation of correspondences to sentences for each of the four^{5} kinds of semantics. Also, the construction of colimits must be defined. As a consequence of this construction, the general framework will provide both a treatment of all four semantics for alignments for the newly included ontology language, as well as a way to reason about such alignments via a reduction to reasoning about (unstructured) ontologies.
Outline of the Paper The rest of the paper is structured as follows: in Sect. 2, we present the core technical ingredients for our abstract framework in four parts (i) in Sect. 2.1, we introduce an abstract notion of logical system, based on an extension of institutions with additional infrastructure required to capture aspects specific to alignments, (ii) in Sect. 2.2, we complement this with a notion of logic translation, (iii) in Sect. 2.3, we formalise the notion of relativisation of a logic as a logic translation with certain properties, (iv) and in Sect. 2.4, we introduce a Grothendieck construction over a graph of logics and their translations, meant to provide a foundation for syntactic heterogeneity. In Sect. 3, we formally introduce correspondences, alignments and networks of aligned ontologies over an arbitrary logic. We also present the syntax of alignments and networks in \(\mathsf {DOL}\). In Sect. 4, we recall the different approaches to semantics of networks of aligned ontologies, based on the assumptions made on the domain of interpretation of the involved ontologies. We illustrate the interpretation of correspondence relations, in each case, for \(\mathcal {SROIQ}\) and \(\mathsf {FOL}\). In Sect. 5, we describe the construction of the diagram of an alignment, using a general method that is further instantiated to each of the four semantics of networks of ontologies. The general idea is to build a bridge ontology, internalising the semantics of the correspondences of the alignment, and to relate it with transformations of the aligned ontologies in a Wshaped diagram. In Sect. 6, we address the issue that the logic of the aligned ontologies may not be the same, or that a correspondence cannot be internalised in the logics of the aligned ontologies. Thus, we build a Wdiagram for alignments, similar to the construction of the previous section, for the heterogeneous case. In Sect. 7, we show that the construction of the Wdiagram for alignments captures its intended semantics, for all four cases. Moreover, we discuss when the colimit of the resulting diagram represents exactly the models of the network of aligned ontologies. In Sect. 8, we build on the results of the previous section and study various aspects regarding reasoning with networks of aligned ontologies via reasoning in the derived colimit ontology. We also introduce a notion of satisfiability for NeO s and define modules of networks. Employing modularity is proposed as a pathway to optimise the performance of reasoning in the colimitbased approach. We here also discuss tool support for reasoning with networks of alignments and colimit computation in Hets and Ontohub. In Sect. 9, we outline a roadmap towards a full integration of formalisms like \(\mathcal {E}\)connections, DDL and DFOL into our general framework, sketching the main challenges ahead. Finally, in Sect. 10, we provide a brief summary and outlook to future work.
2 Formal Preliminaries
We will use the Distributed Ontology, Model and Specification Language \(\mathsf {DOL}\) mainly as a means for the specification of ontologies and their relations, networks of ontologies and alignment relations. In this section, we will develop the formal preliminaries for the semantics of \(\mathsf {DOL}\). We will here only introduce the abstract semantics necessary to understand \(\mathsf {DOL}\) ’s treatment of various types of alignments, networks and combinations of networks.
We will use basic concepts of category theory. Readers unfamiliar with category theory may view a category as a graph with a monoidlike composition operation on the arrows. The notion of category is mainly needed to abstract away from the specific details of the notion of signature. Signatures specify the nonlogical vocabulary of an ontology. They form the nodes (called objects in category theory terminology) of the graph. Edges are maps between signatures (called morphisms in category theory terminology). Intuitively, they capture change in notation, e.g. renaming a symbol or extending a signature with new symbols. The monoidlike composition of arrows (morphisms) can be thought of as the usual composition of maps.
We will follow the standard notations of [55], with the exception of composition of morphisms, where we prefer the diagrammatic order and the notation ”;“. That is, if \(f:A\rightarrow B\) and \(g:B\rightarrow C\), while their composition often is written \(g\circ f:A\rightarrow C\), we will write \(f;g:A\rightarrow C\). If C is a category, then C are the objects of C, and \(C^\mathrm{op}\) is its opposite category (i.e. C with the direction of all morphisms reversed).
2.1 Logics
We introduce below a unifying notion of logic, covering the large variety of formalisms in use at an abstract level, following the spirit of Goguen’s and Burstall’s institutions [29]. This allows us to develop results independently of the particularities of a logical system. The definitions presented here are new: while the terminology is inspired by [69], our notions make more assumptions about the features present both in the syntax and the semantics of a logic. That is, we consider institutions with extra infrastructure (like carrier sets and predicates) that is specifically tailored towards alignments, while still staying at a general level. Indeed, our framework of logic (\(=\)institution with extra infrastructure) should cover roughly the same set of logics that are covered by institutions—even though the formalisation of some logics will require some adaptions, see e.g. the formalisation of manysorted logic in Example 9.
We start by introducing the notion of logic syntax. The main idea is that signatures collect the nonlogical symbols of the language. Each signature gets assigned the set of sentences that can be formed with its symbols. Also, for each signature, we provide means for extracting the symbols it consists of, together with their kind. Signature morphisms are mappings between signatures. We do not assume any details except that signature morphisms can be composed and there are identity morphisms; this amounts to a category of signatures.
Definition 1

a category \(\mathbf {Sign}\) of signatures and signature morphisms;

a sentence functor^{6} \(\mathbf {Sen}: \mathbf {Sign}\rightarrow \mathbb {S}et\) assigning to each signature the set of its sentences and to each signature morphism \(\sigma : \Sigma \rightarrow \Sigma '\) a sentence translation function \(\mathbf {Sen}(\sigma ): \mathbf {Sen}(\Sigma ) \rightarrow \mathbf {Sen}(\Sigma ')\);

a set \({\mathbf {Symbols}}\) of symbols and a set \({\mathbf {Kinds}}\) of symbol kinds together with a function \({\mathbf {kind}}: {\mathbf {Symbols}}\rightarrow {\mathbf {Kinds}}\) giving the kind of each symbol;

a faithful functor^{7} \({\mathbf {Sym}}: \mathbf {Sign}\rightarrow \mathbb {S}et\) assigning to each signature \(\Sigma \) a set of symbols \({\mathbf {Sym}}(\Sigma ) \subseteq {\mathbf {Symbols}}\) and to each signature morphism \(\sigma :\Sigma \rightarrow \Sigma '\) a function \({\mathbf {Sym}}(\sigma ):{\mathbf {Sym}}(\Sigma )\rightarrow {\mathbf {Sym}}(\Sigma ')\) such that for each \(s\in {\mathbf {Sym}}(\Sigma )\), \({\mathbf {kind}}(\sigma (s)) = {\mathbf {kind}}(s)\),

a function \({\mathbf {arity}}: {\mathbf {Symbols}}\rightarrow \mathbb {N}\) giving the arity of each symbol.\(\square \)
Before giving examples of logic syntaxes, we introduce the concept of logical theory.
Definition 2
Let L be a logic syntax. A theory of L consists of a signature \(\Sigma \) and a set E of \(\Sigma \)sentences. \(\square \)
For the purposes of this paper, it suffices to regard an ontology as a theory. In \(\mathsf {DOL}\), ontologies can be written using more complex structuring mechanisms.
Example 1
Example 2
The logic \(\mathcal {SROIQ}\) [39], which is the logical core of the Web Ontology Language \(\mathsf {OWL}\) 2 DL,^{9} inherits signatures and symbols from \(\mathcal {ALC} \). Sentences are that of \(\mathcal {ALC} \), extended with the following constructs: (i) complex role inclusions such as \(R \circ S \sqsubseteq S\) as well as simple role hierarchies such as \(R \sqsubseteq S\), assertions for symmetric, transitive, reflexive, asymmetric and disjoint roles (called RBox sentences, denoted by \(s\mathcal {R}\) in the name of the logic), as well as the construct \(\exists R . \mathsf {Self}\) (collecting the set of ‘Rreflexive points’); (ii) nominals, i.e. concepts of the form \(\{a\}\), where \(a\in I\) (denoted by \(\mathcal {O}\)); (iii) inverse roles (denoted by \(\mathcal {I}\)); qualified and unqualified number restrictions (\(\mathcal {Q}\)). For details on the rather complex grammatical restrictions for \(\mathcal {SROIQ}\) (e.g. regular role inclusions, simple roles) compare [39].
Strictly speaking, we have defined \(\mathcal {SROIQ}\) resp. OWL 2 DL without restrictions in the sense of [72]. The reason is that in a logic, sentences can be used for arbitrary formation of theories. By contrast, \(\mathcal {SROIQ}\) imposes specific constraints on theory formation: e.g. a role can only be declared to be either irreflexive or transitive, but not both [39]. Since colimits, similar to unions of theories, can lead to such irreconcilable sets of sentences, the use of OWL 2 DL without restrictions is essential for the existence of colimits, and these are used for combinations of (Sect. 3) and reasoning about (Sect. 7) networks. In particular, this means that in some cases, standard reasoners relying on the restrictions of OWL 2 DL, like Pellet or Fact, cannot be used for reasoning. In such cases, a logic translation to FOL (Sect. 2.2) and a FOL reasoner must be used instead. Note that this limitation is not a specificity of our approach, but applies to all computable sound and complete approaches to reasoning in networks of \(\mathsf {OWL}\) ontologies.^{10} \(\square \)
We now come to firstorder logic (\(\mathsf {FOL}\)), which has been used in the formalisation of many upper ontologies such as DOLCE, BFO, GFO, SUMO and others.
Example 3
In the logic syntax of unsorted firstorder logic \(\mathsf {FOL}\), signatures are pairs of the form (F, P) where \(F = (F^i)_{i\in \mathbb {N}}\) is a set of function symbols such that for each i, \(F^i\) is the set of function symbols of arity i, \(P = (P^i)_{i\in \mathbb {N}}\) is a set of predicate symbols such that for each i, \(P^i\) is the set of function symbols of arity i, and the elements of \(F^i\) and \(P^i\) are from a set of names. Given two signatures \(\Sigma _1 = (F_1,P_1)\) and \(\Sigma _2 = (F_2,P_2)\), a signature morphism \(\varphi : \Sigma _1\rightarrow \Sigma _2\) has two components, \(\varphi ^\mathrm{op}: F_1 \rightarrow F_2\) such that \(\varphi ^\mathrm{op}(f)\in F^i_2\) for each \(f\in F^i_1\) and \(\varphi ^{pred}: P_1 \rightarrow P_2\) such that \(\varphi ^{pred}(p)\in P^i_2\) for each \(p\in P^i_1\). \({\mathbf {Kinds}}\) is the set \(\{ fun , pred \}\). \({\mathbf {Symbols}}\) is the set of all tuples (k, s, i) where k is an element of \({\mathbf {Kinds}}\), s is a name and i is a natural number. For each \((k,s,i)\in {\mathbf {Symbols}}\), \(kind(k,s,i) = k\). For each signature \(\Sigma = (F,P)\), \({\mathbf {Sym}}(\Sigma ) = \{( fun , f, i) \mid f \in F^i\} \cup \{( pred , p, i) \mid p \in P^i\}\). Given a signature \(\Sigma =(F,P)\) and a countable infinite set of variables X, the set \(T_\Sigma (X)\) of terms is the least set that contains X and for each operation symbol \(f\in F_n\), for some \(n\in \mathbb {N}\), and each terms \(t_1,\ldots , t_n \in T_\Sigma (X)\), it also contains \(f(t_1,\ldots ,t_n)\). Formulas are formed starting from atomic sentences, which are predications \(p(t_1,\ldots ,t_n)\) and equations \(t_1 = t_2\), where \(t_1,\ldots , t_n\) are terms and \(p\in P_n\) for some \(n\in \mathbb {N}\), and applying to them the usual Boolean connectives (defined, e.g. in terms of negation and conjunction) and quantification over a subset of X. Sentences are closed formulas. Sentence translation is done by just replacing symbols according to the signature morphism. \(\square \)
Example 4
The logic syntax of manysorted firstorder logic \(\mathsf {MSFOL}\) is similar to that of unsorted \(\mathsf {FOL}\). Signatures are triples (S, F, P) where S is a set of sorts, \(F=(F_{w,s})_{w\in S^*, s\in S}\) is a family of function symbols and \(P=(P_w)_{w\in S^*}\) a family of predicate symbols, both consisting of sets of names. Signature morphisms map sorts, function symbols and predicate symbols in a compatible way. \({\mathbf {Kinds}}\) is the set \(\{ sort , fun , pred \}\). \({\mathbf {Symbols}}\) is the set of all tuples of form (sort, s), \(( fun ,f,w\rightarrow s)\) or \(( pred ,p,w)\), where s, f and p are names and w is a string of names. \(kind(sort,s)=sort\), \(kind(fun,f,w\rightarrow s)=fun\) and \(kind(pred,p,w)=pred\). For each signature \(\Sigma = (S,F,P)\), \({\mathbf {Sym}}(\Sigma ) = \{( sort , s) \mid s\in S\} \cup \{( fun , f, w\rightarrow s) \mid f \in F_{w,s}\} \cup \{( pred , p, w) \mid p \in P_w\}\). Sentences are defined like in \(\mathsf {FOL}\), except that they need to respect typing.\(\square \)
Example 5
The logic syntax of relational database schemes is defined over some fixed set D of datatype sorts with fixed interpretation \(I:D\rightarrow \mathbb {S}et\). For instance, we may have \(D=\{integer, string, text, boolean, blob\}\).
Also, higherorder logic (\(\mathsf {HOL}\)) has been used for ontology design, e.g. in the modelling of universals and roles [4], and is particularly important for foundational ontologies such as UFO [32]. In [54], the authors argue that the focus on \(\mathsf {OWL}\) or \(\mathsf {FOL}\) ontologies does not allow for a sufficiently general analysis and resolution of ontological faults. They advocate \(\mathsf {HOL}\) for the study of ontology evolution in physics.
Example 6
A logic syntax can be complemented with a model theory, which introduces semantics for the language and gives a satisfaction relation between the models and the sentences of a signature. In the context of ontology alignment, we add the assumption that models are interpreted over a universe, and symbols of each kind are interpreted as elements of the same certain set that is formed over the universe.
Definition 3

\(\mathbf {Mod}:\mathbf {Sign}^\mathrm{op}\rightarrow \mathbb {C}at\) ^{13} is a functor giving for each signature \(\Sigma \) the category of \(\Sigma \)models \(\mathbf {Mod}(\Sigma )\) and for each signature morphism \(\varphi :\Sigma _1\rightarrow \Sigma _2\), a functor \(\mathbf {Mod}(\varphi ):\mathbf {Mod}(\Sigma _2) \rightarrow \mathbf {Mod}(\Sigma _1)\) (often called reduct functor), where often \(\mathbf {Mod}(\varphi )(M^2)\) is written \(M^2_{\varphi }\),^{14}
 \({\models } = \{\models _\Sigma \subseteq \mathbf {Mod}(\Sigma ) \times \mathbf {Sen}(\Sigma )\}_{\Sigma \in \mathbf {Sign}}\) is a family of satisfaction relations enjoying the following satisfaction condition:for each \(\varphi :\Sigma \rightarrow \Sigma '\in \mathbf {Sign}\), each \(M'\in \mathbf {Mod}(\Sigma ')\) and each \(e\in \mathbf {Sen}(\Sigma )\),$$\begin{aligned} M'\models _{\Sigma '}\varphi (e) \iff M'_{\varphi } \models _\Sigma e \end{aligned}$$
 \(universe = \{universe_\Sigma :\mathbf {Mod}(\Sigma ) \rightarrow \mathbb {S}et\}_{\Sigma \in \mathbf {Sign}}\) is a family of functors that give the universe of each \(\Sigma \)model, such that for each signature morphism \(\sigma :\Sigma \rightarrow \Sigma '\) and each \(M'\in \mathbf {Mod}(\Sigma ')\),$$\begin{aligned} universe_{\Sigma '}(M') = universe_{\Sigma }(M'_{\sigma }) \end{aligned}$$

\(domain:{\mathbf {Kinds}}\times \mathbb {S}et \rightarrow \mathbb {S}et\) is a function giving the domain of each kind, needed for interpreting symbols of that kind. The domain is given relative to the universe (which of course is a set),
 \(sym:(\Sigma \in \mathbf {Sign})\times (M\in \mathbf {Mod}(\Sigma ))\times (s\in {\mathbf {Sym}}(\Sigma ))\rightarrow domain(kind(s),universe(M))\) gives the interpretation of symbols in the corresponding domain. That is, if \(\Sigma \) is a signature, M is a \(\Sigma \)model and \(s\in {\mathbf {Sym}}(\Sigma )\), the interpretation \(sym(\Sigma ,M,s)\) of s in M is an element of the set domain(kind(s), universe(M)). If \(\Sigma \) is clear from the context, we may shortly denote by \(M_s\) the interpretation of s in M (completely written \(sym(\Sigma ,M,s)\)). Moreover, it is required that for each \(\sigma :\Sigma \rightarrow \Sigma '\), each \(M'\in \mathbf {Mod}(\Sigma ')\) and each \(s\in {\mathbf {Sym}}(\Sigma )\),$$\begin{aligned} sym(\Sigma ', M', {\mathbf {Sym}}(\sigma )(s)) = sym(\Sigma , M'_{\sigma }, s) \end{aligned}$$
The last conditions allows us to define a model by giving its universe and the interpretation of all symbols in the model.
Example 7

to each atomic concept \(A\in \mathcal {A} \) a subset \(A^I\) of \(\Delta \), since \(domain(concept, X) = \mathcal P \left( {X}\right) \),

to each role \(R\in \mathcal {R} \) a binary relation \(R^I:\Delta \times \Delta \), since \(domain(role, X) = \mathcal P \left( {X \times X}\right) \), and

to each individual \(i\in I\) an element \(i^I\) of \(\Delta \), since \(domain(individual, X) = X\).
Example 8
Example 9
Example 10
Example 11
The reduct of a model along a signature morphism \(\varphi \) has the same carrier set as the original model and interprets a table x in the same way as \(\varphi (x)\) is interpreted in the original model. For a signature T and a Tmodel M, the interpretation \(sym(\Sigma , M, s)\) of a symbol \(s = (table, t)\) where t is a name, is defined to be M(t). \(\square \)
A logic combines then a logic syntax with a model theory for it:
Definition 4
A logic consists of a logic syntax L and a model theory for L. \(\square \)
Notice that we follow a modeltheoretic approach to logics, where the central role is played by the satisfaction relation between models and sentences. It is possible to further extend the concept of logic with a provability relation, but this is not needed for our purposes. Moreover, our concept of logic extends the wellknown concept of institution [29], defined below, with additional elements that allow one to speak about the symbols of a signature and with the condition that models are interpreted over a set.
Definition 5

a category \(\mathbf {Sign}^{\mathcal{I}}\) of signatures,

a functor \(\mathbf {Sen}^{\mathcal{I}}:\mathbf {Sign}^{\mathcal{I}}\!\longrightarrow \!\mathbb {S}et\) giving, for each signature \(\Sigma \), the set of sentences \(\mathbf {Sen}^{\mathcal{I}}(\Sigma )\), and for each signature morphism \(\sigma :\Sigma \!\longrightarrow \!\Sigma '\), the sentence translation map \(\mathbf {Sen}^{\mathcal{I}}(\sigma ):\mathbf {Sen}^{\mathcal{I}}(\Sigma )\!\longrightarrow \!\mathbf {Sen}^{\mathcal{I}}(\Sigma ')\), where often \(\mathbf {Sen}^{\mathcal{I}}(\sigma )(e)\) is written as \(\sigma (e)\),

a functor \(\mathbf {Mod}^{\mathcal{I}}:(\mathbf {Sign}^{\mathcal{I}})^\mathrm{op}\!\longrightarrow \!\mathbb {C}at\) giving, for each signature \(\Sigma \), the category of its models \(\mathbf {Mod}^{\mathcal{I}}(\Sigma )\), and for each morphism \(\sigma :\Sigma \!\longrightarrow \!\Sigma '\), the reduct functor \(\mathbf {Mod}^{\mathcal{I}}(\sigma ):\mathbf {Mod}^{\mathcal{I}}(\Sigma ')\!\longrightarrow \!\mathbf {Mod}^{\mathcal{I}}(\Sigma )\), where often \(\mathbf {Mod}^{\mathcal{I}}(\sigma )(M')\) is written as \(M'_{\sigma }\),
 for each \(\Sigma \in \mathbf {Sign}^I\), a satisfaction relation$$\begin{aligned} {\models ^{\mathcal{I}}_{\Sigma }}\subseteq {\mathbf {Mod}^{\mathcal{I}}(\Sigma )\times \mathbf {Sen}^{\mathcal{I}}(\Sigma )} \end{aligned}$$
We can easily obtain an institution underlying a logic.
Definition 6
Given a logic L, we obtain the institution of L, \(Inst(L) = (\mathbf {Sign},\mathbf {Sen},\mathbf {Mod},\models )\), by extracting the corresponding components from the logic syntax and the model theory of L. \(\square \)
This allows us, in the rest of this section, to recall a number of concepts defined over an arbitrary institution and regard them as concepts over an arbitrary logic, in the sense of Definition 4.
In the following, if \(L_i\) is a logic, we will use the subscript \((\_)_i\) for its constituents.
We want to speak of subsignatures, make use of unions of signatures, etc. Note that such settheoretic terms are not available in an arbitrary (signature) category: there, we just have objects (aka signatures) and morphisms. To define subsignatures and signature unions formally, we must extend the notion of institution by requiring the signature category to be inclusive. Informally, this means that we mark some of the morphisms of the signature category to be inclusions. These inclusions then form a partial order, which can be construed as a subcategory having the same objects as the whole signature category, but just fewer morphisms. The effect is that in an inclusive signature category, we can use the usual settheoretic terms like: subsignature or union. This is formalised as follows:
Definition 7
An inclusive category [31] is a category having a broad subcategory^{17} whose arrows are called inclusions, which is a partially ordered class with a least element (denoted \(\emptyset \)), finite infima (categorically: products) and suprema (categorically: coproducts), called intersection (denoted \(\cap \)) and union (denoted \(\cup \)) such that for each pair of objects A, B, \(A \cup B\) is a pushout of \(A \cap B\) in the category.
\(\square \)
For any objects A and B of an inclusive category, we write \(A\subseteq B\) if there is an inclusion from A to B; the unique such inclusion will then be denoted by \(\iota _{A\subseteq B}:A \hookrightarrow B\), or simply \(A \hookrightarrow B\). A functor between two inclusive categories is inclusive if it takes inclusions in the source category to inclusions in the target category.
Definition 8

\(\mathbf {Sign}\) is inclusive and has pushouts which preserve inclusions,

\(\mathbf {Sen}\) is inclusive and preserves intersections,^{18} and

each model category is inclusive, and reduct functors are inclusive.\(\square \)
If \({\mathcal{I}}\) is an inclusive institution and \(\Sigma \subseteq \Sigma '\) is an inclusion of signatures, we write \(M'_{\Sigma }\) for the reduct of a \(\Sigma '\)model \(M'\) along the inclusion \(\iota _{\Sigma \subseteq \Sigma '}\). Also since in such a case \(Sen(\iota _{\Sigma \subseteq \Sigma '})\) is just the settheoretic inclusion, we omit its application.
In the following, let \({\mathcal{I}}\) be an inclusive institution. A theory morphism (also called an ontology morphism) \(\sigma : (\Sigma _1, E_1) \rightarrow (\Sigma _2, E_2)\) is a signature morphism \(\sigma : \Sigma _1 \rightarrow \Sigma _2\) such that \(\sigma (E_1)\) is a logical consequence of \(E_2\). If \(O = (\Sigma ,\Delta )\) is a theory (ontology), we will use the notation \(\mathsf {Sig}(O)\) for \(\Sigma \). A model of a theory \((\Sigma , E)\) is a \(\Sigma \)model that satisfies all sentences in E. The class of all models of an theory will be denoted \(\mathbf {Mod}(O)\).
Definition 9

\(\mathbf {Sign}^{th}\) has as objects all \({\mathcal{I}}\)theories and as arrows \({\mathcal{I}}\)theory morphisms;

for a theory \((\Sigma , E)\), \(\mathbf {Mod}^{th}(\Sigma , E)\) consists of all \(\Sigma \)models that satisfy E;

for each theory \((\Sigma , E)\), each \((\Sigma , E)\)model M and each \(\Sigma \)sentence e, \(\mathbf {Sen}^{th}(\Sigma , E) = \mathbf {Sen}^{\mathcal{I}}(\Sigma )\) and \(M \models ^{th}_{(\Sigma , E)} e \iff M \models ^{\mathcal{I}}_\Sigma e\).\(\square \)
“Given a species of structure, say widgets, then the result of interconnecting a system of widgets to form a superwidget corresponds to taking the colimit of the diagram of widgets in which the morphisms show how they are interconnected.” J. Goguen [28]
Definition 10
A diagram D over \({\mathcal{I}}\) consists of a graph D, ontologies over \({\mathcal{I}}\) \((D_i)_{i\in D}\) for its nodes, and ontology morphisms \((D_m: D_i \rightarrow D_j)_{m: i \rightarrow j \in D}\) between them for its edges. A family of models \((M^i)_{i\in D}\), where \(M^i\) is a model of \(D_i\) for each i, is called compatible with D if for each \(m: i\rightarrow j \in D\) we have that \(M^j_{D_m} = M^i\). \(\square \)
Definition 11
A cocone for a diagram D consists of an ontology O together with a family of ontology morphisms \((\mu _i:D_i\rightarrow O)_{i\in D}\) such that \(\mu _i;D_m = \mu _j\) for each \(m: i \rightarrow j\) in D. A cocone is a colimit, if it can be uniquely naturally embedded into any cocone (and thus can be seen as a minimal cocone). This means that a cocone \((O, (\mu _i)_{i\in D})\) is a colimit iff for any cocone \((O', (\mu '_i)_{i\in D})\) there exists a unique ontology morphism \(\mu :O\rightarrow O'\) such that \(\mu _i;\mu = \mu '_i\) for each \(i\in D\). \(\square \)
The colimit of a diagram is similar to a disjoint union of the object in the diagram, with some identifications of shared parts as specified by the morphisms in the diagram.
An important property of colimits is amalgamation, which can be understood as the fact that families of models compatible with a diagram can be combined uniquely to form a model of the colimit of the diagram.
Definition 12
Let D be a diagram over an institution \({\mathcal{I}}\). A cocone \((O, (\mu _i)_{i\in D})\) for D is called (weakly) amalgamable if for each family of models \((M^i)_{i\in D}\) compatible with D there exists a (not necessarily) unique model M of O such that \(M_{\mu _i} = M^i\) for each \(i\in D\). \(\square \)
An institution \({\mathcal{I}}\) admits (weak) amalgamation if all colimiting cocones are (weakly) amalgamable. For our purposes, amalgamation of all pushouts suffices. A logic admits (weak) amalgamation if its underlying institution does.
Proposition 1
([71], Prop. 4.4.15) An (inclusive^{19}) institution with amalgamation of pushouts has amalgamation of all connected diagrams. \(\square \)
Note that the restriction to connected diagrams is not so harsh, because each diagram can be made connected by adding a node for the empty signature with inclusions to all nodes in the diagram. This addition does not change the colimit signature.
Proposition 2
All institutions introduced in the examples have amalgamation of pushouts. \(\square \)
2.2 Logic Translations
Several types of translations between institutions have been introduced. Institution comorphisms [30] typically express that an institution is included or encoded into another one.
Definition 13
Example 12

each atomic concept \(A\in \mathcal {A} \) is translated to A(x)

\(\top \) is translated to True, and \(\bot \) is translated to False,

\(\lnot C\) is translated to \(\lnot \alpha _x(C)\), \(C_1 \sqcap C_2\) to \(\alpha _x(C_1)\wedge \alpha _x(C_2)\), and \(C_1 \sqcup C_2\) to \(\alpha _x(C_1)\vee \alpha _x(C_2)\),

\(\forall R. C\) is translated to \(\forall y . R(x,y) \implies \alpha _y(C)\)

\(\exists R. C\) is translated to \(\exists y. \alpha _y(C) \wedge R(x,y)\)

\(C_1\subseteq C_2\) becomes \(\forall x . \alpha _x(C_1) \implies \alpha _x(C_2)\),

\(a \in C\) becomes \(\alpha _x(C)[a/x]\),

R(a, b) becomes R(a, b).
Example 13
The comorphism from \(\mathcal {ALC}\) to \(\mathsf {FOL}\) defined in the previous example generalises to a comorphism from \(\mathcal {SROIQ}\) to \(\mathsf {FOL}\). Details can be found in [1]. \(\square \)
Example 14

a predication \(p(t_1,\ldots , t_n)\) to the atomic sentence \(p(t_1,\ldots , t_n) =_{Bool} True\),

an equation \(t_1 = t_2\) to \(t_1 =_{s} t_2\), where s is the sort of the terms \(t_1\) and \(t_2\),

the negation \(\lnot \Phi \) of a formula \(\Phi \) to \(\lnot \alpha _{(S,F)}(\Phi )\),

a conjunction \(\Phi _1 \wedge \Phi _2\) of two formulas \(\Phi _1\) and \(\Phi _2\) to \(\alpha _{(S,F)}(\Phi _1) \wedge \alpha _{(S,F)}(\Phi _2)\),
 a quantification \(\forall \{x_1: s_1,\ldots ,x_n: s_n\}.\Phi \) to$$\begin{aligned} \Pi _*(\lambda x_1:s_1. \Pi _* (\ldots \Pi _*(\lambda x_n:s_n.\alpha _{(S,F)}(\Phi ))\ldots )) \end{aligned}$$
We extend the concept of institution comorphism to translations between logics.
Definition 14

an institution comorphism \((\Phi ,\alpha ,\beta ): Inst(L_1) \rightarrow Inst(L_2)\),

a function \(\kappa : {\mathbf {Kinds}}_1 \rightarrow {\mathbf {Kinds}}_2\)

a function \(\gamma : {\mathbf {Symbols}}_1 \rightarrow {\mathbf {Symbols}}_2\), that for each signature \(\Sigma \in \mathbf {Sign}_1\) restricts to \(\gamma _\Sigma :{\mathbf {Sym}}_1(\Sigma )\rightarrow {\mathbf {Sym}}_2(\Sigma )\)
 for each \(s\in {\mathbf {Symbols}}_1\),$$\begin{aligned} \quad kind_2(\gamma (s)) = \kappa (kind_1(s)) \end{aligned}$$
 for each \(\sigma :\Sigma \rightarrow \Sigma '\) and each \(s\in {\mathbf {Sym}}_1(\Sigma )\),$$\begin{aligned} \quad \gamma ({\mathbf {Sym}}_1(\sigma )(s))= {\mathbf {Sym}}_2(\Phi (\sigma ))(\gamma (s)) \end{aligned}$$
 for each \(k\in {\mathbf {Kinds}}_1\) and each \(S\in \mathbb {S}et\),$$\begin{aligned} \quad domain_2(\kappa (k),S) = domain_1(k, S) \end{aligned}$$
Definition 15
Definition 16
Example 15

\(\kappa (concept) = pred\), \(\kappa (role) = pred\), \(\kappa (individual) = fun \)
 for each name n,

\(\gamma (concept, n) = (pred, n, 1)\),

\(\gamma (role, n) = (pred, n, 2)\), and

\(\gamma (individual,n) = ( fun , n, 0)\).

Example 16

\(\kappa (sort) = sort, \kappa ( fun ) = const\), \(\kappa (pred) = const\)
 for each name n,

\(\gamma (sort, s) = (sort, s)\),

\(\gamma ( fun , f, s_1 \times \cdots \times s_n\rightarrow s) = (const, f, s_1 \rightarrow \cdots \rightarrow s_n \rightarrow s)\), and

\(\gamma (pred, p, s_1 \times \cdots \times s_n) = (const, p, s_1 \rightarrow \cdots \rightarrow s_n \rightarrow Bool)\). \(\square \)

With obvious identities and composition, logics and logic translations form a category named \(\mathbf {Logic}\).
2.3 Relativisation of Ontologies
The general idea of relativisation of ontologies is that we can encode the universe of a model as a predicate that is interpreted as a subset of a larger set, which is the global universe. Relativisations have previously been used in defining Common Logic modules [67], in the reencoding of DDL into OWL [15], or to translate between possibilist and actualist quantification in modal logic [23, 46] or in free logic [52]. They are generally a wellknown tool in mathematical logic used in a variety of contexts [11, 73].^{22}
Definition 17
Let L be a logic. We say that L admits unary and binary predicates if it has two designated kinds unaryPred and binaryPred. The unary predicates of L, denoted \(unary_L\), are all symbols \(s\in {\mathbf {Symbols}}\) of kind unaryPred and arity 1. The binary predicates of L, denoted \(binary_L\), are all symbols \(s\in {\mathbf {Symbols}}\) of kind binaryPred and arity 2. \(\square \)
Example 17
\(\mathcal {ALC}\) admits predicates: unaryPred is concept and binaryPred is role. \(\square \)
Example 18
In \(\mathsf {FOL}\) both unaryPred and binaryPred are of the kind pred. \(\square \)
In the following we will assume that all logics admit predicates.
Definition 18
if \(O'_i\) is the theory obtained by renaming top along \(\sigma _i\) to \(O_i:top\), for \(i=1,2\), then for any two models \(M^1\) of \(O_1\) and \(M^2\) of \(O_2\) and any set U that includes \(universe(M^1)\) and \(universe(M^2)\), there exists a unique model M of \(O'_1\cup O'_2\) with \(universe(M) = U\) and such that, if \(N^i = (M_{\mathsf {Sig}(rel(O_i))})_{\sigma _i}\) for \(i=1,2\), then \(\beta _{\Sigma _1}(N^i) = M^i\).
\(\square \)
The first condition ensures that a correspondence between two ontologies \(O_1\) and \(O_2\) in L is also a correspondence between their relativisations \(rel(O_1)\) and \(rel(O_2)\). The relativisation translation does not preserve universes of models. Note that it is possible to define more than one relativisation for a logic.^{23} We assume that for each logic L, a relativisation translation has been chosen, denoted \(rel_L\).
Example 19

for each \(C\in \mathcal {A} \), \(C \sqsubseteq top\)

for each \(i\in I\), \(i \in top\)

for each \(r\in \mathcal {R} \), \(domain(r) = top\) and \(range(r) = top\)

each occurrence of \(\top \) is replaced by top, and

each concept \(\lnot C\) is replaced by \(top \sqcap \lnot C\)

each concept \(\forall R . C\) is replaced by \(top \sqcap \forall R . C\).

each role has its domain and range, if present, intersected with top.
The model reduct \(\beta _\Sigma : Mod(\Phi (\Sigma )) \rightarrow Mod(\Sigma )\) takes each model \(M' = (U', .^I)\) of \(\Phi (\Sigma )\) to the model \((U, .^I)\) of \(\Sigma \), where \(U = top^I\) (the universe of the reduct is given by the interpretation of top in \(M'\) and each concept of \(\Sigma \) is interpreted as in \(M'\)). The axioms of \(\Phi (\Sigma )\) ensure that the definition is correct.
It is easy to show that the satisfaction condition holds.
Moreover, if \(O_i = (\Sigma _i,E_i)\) and \(M^i\) is a model of \(O_i\), for \(i=1,2\), and U is a set that includes the universes of \(M^1\) and M2, we obtain a model M of the union of the relativisations of \(O_1\) and \(O_2\) by defining its universe to be the set U, \(O_i:\top \) is interpreted as the universe of \(M^i\) for \(i=1,2\) and each concept is interpreted as in \(M^1\) if it comes from \(\Sigma _1\) or as in \(M^2\) if it comes from \(\Sigma _2\).
\(\kappa \) and \(\gamma \) are both identities. \(\square \)
Example 20

for each \(c\in F_0\), top(c),

for each \(f\in F_n\), \( \forall x_1, \ldots , x_n . top(x_1) \wedge \ldots \wedge top(x_n) \implies top(f(x_1,\ldots ,x_n))\)

for each \(p\in P_n\), \( \forall x_1, \ldots , x_n . P(x_1,\ldots ,x_n) \implies top(x_1) \wedge \ldots \wedge top(x_n)\).
\(\kappa \) and \(\gamma \) are both identities. \(\square \)
2.4 Heterogeneity
To deal with the problem of syntactic heterogeneity, i.e. the situation when ontologies expressed in different logical formalisms are aligned, we need a framework that provides principled support for heterogeneous ontologies. Such a framework is given by Grothendieck logics, a concept that adapts Grothendieck institutions [17] to our notion of logic. The idea is to begin with a graph of logics and logic translations, and then to flatten this graph, such that no interaction between logics is made other than those specified via the translations.
Definition 19
Given an index category Ind, an indexed logic is a functor \({\mathcal{L}}:Ind^\mathrm{op}\!\longrightarrow \!\mathbf {Logic}\). \(\square \)
In an indexed logic \({\mathcal{L}}\), we use the notations \({\mathcal{L}}^i\) for the logic \({\mathcal{L}}(i)\) and \(\mu _d\) for the logic translation \({\mathcal{L}}(d)\). Their components are marked with the superscript \(.^i\) or, respectively, \(.^d\). Moreover the constituents of the logic \({\mathcal{L}}^{\#}\) will also marked \((\_)^{\#}\).
Definition 20

signatures in \({\mathcal{L}}^{\#}\) are pairs \((i,\Sigma )\), where \(i\in Ind\) and \(\Sigma \) a signature in \({\mathcal{L}}^i\),

\({\mathbf {Symbols}}^\# = \{(i, s) \mid i \in Ind, s\in {\mathbf {Symbols}}^i\}\) and \({\mathbf {Kinds}}^\# = \{(i, k) \mid i \in Ind, k\in {\mathbf {Kinds}}^i\}\),

\({\mathbf {Sym}}^\#(i, \Sigma ) = \{(i,s) \mid s\in {\mathbf {Sym}}^i(\Sigma )\}\),

\({\mathbf {arity}}^\#(i,s) = {\mathbf {arity}}^i(s)\),

\({\mathbf {kind}}^\#(i, s) = kind^i(s)\),

signature morphisms \((d,\sigma ):(i,\Sigma _1)\!\longrightarrow \!(j,\Sigma _2)\) consist of a morphism \(d:j\!\longrightarrow \!i\in Ind\) and a signature morphism \(\sigma :\Phi ^d(\Sigma _1)\!\longrightarrow \!\Sigma _2\) in \({\mathcal{L}}^j\),
 if \((d_1, \sigma _1): (i, \Sigma ) \rightarrow (j, \Sigma ')\) and \((d_2, \sigma _2): (j, \Sigma ') \rightarrow (k, \Sigma '')\), the composition \((e, \sigma ) := (d_1, \sigma _1);(d_2, \sigma _2): (i, \Sigma ) \rightarrow (k, \Sigma '')\) is obtained by defining e as the composition of \(d_2\) and \(d_1\): and \(\sigma \) as the composition of \(\Phi ^{d_2}(\sigma _1)\) and \(\sigma _2\):

\({\mathbf {Sym}}^\#(d, \sigma )(i,s) = (j, {\mathbf {Sym}}^j(\sigma )(\gamma ^d(s)))\)
 for each signature \((i, \Sigma )\),and$$\begin{aligned} \mathbf {Sen}^\#(i, \Sigma ) = \mathbf {Sen}^i(\Sigma ) \end{aligned}$$$$\begin{aligned} \mathbf {Mod}^\#(i, \Sigma ) = \mathbf {Mod}^i(\Sigma ) \end{aligned}$$

\(universe^\#(M) = universe^i(M)\) for each \((i,\Sigma )\)model M,

\(domain^\#((i, k), S) = domain^i(k,S)\), and similarly for model morphisms,

\(sym^\#((i,\Sigma ), M, s) = sym^i(\Sigma , M, s)\).
That is, sentences, models and satisfaction for a Grothendieck signature \((i,\Sigma )\) are defined component wise, while the sentence and model translations for a Grothendieck signature morphism are obtained by composing the translation given by the interinstitution comorphism with the one given by the intrainstitution signature morphism.
3 Networks of Aligned Ontologies
In this section we recall networks of aligned ontologies and introduce syntax for them in \(\mathsf {DOL}\). Networks of aligned ontologies (here denoted NeO) [20], called distributed systems in [80], consist of a family \((O_i)_{i\in Ind}\) of ontologies over a set of indexes Ind interconnected by a set of alignments \((A_{ij})_{i, j\in Ind}\) between them.
Definition 21
Let L be a logic syntax. A family of correspondence relations, denoted \(\mathfrak {R}\), is a set of relation names, written R. Each relation name has assigned a set of its kinds, which are pairs \((k_1, k_2)\in {\mathbf {Kinds}}\). \(\square \)
Definition 22
Let L be a logic syntax and let \(\mathfrak {R}\) be a family of correspondence relations for L. A correspondence is a triple \((s_1,s_2,R)\) where \(s_1,s_2\in {\mathbf {Symbols}}\) and \(R\in \mathfrak {R}\) such that \((kind(s_1),kind(s_2))\) is in the set of kinds of R. \(\square \)
Definition 23
Example 21
Below are the types of relations that can appear in correspondences between \(\mathcal {ALC}\) symbols, together with their kinds:
\(=\)  \(\{({ concept,\,concept}), ({ role,\,role}), ({ individual, \,individual})\}\) 
\(\bot \)  \(\{({ concept,\,concept}), ({ role,\,role}), ({ individual,\,individual})\}\) 
<  \(\{({ concept,\,concept}), ({ role,\,role})\}\) 
>  \(\{({ concept,\,concept}), ({ role,\,role})\}\) 
\(\in \)  \(\{({ individual,\,concept})\}\) 
\(\ni \)  \(\{({ concept,\,individual})\}\) 
Example 22
\(=\)  \(\{( fun , fun ), ( pred , pred )\}\) 
\(\bot \)  \(\{( fun , fun ), ( pred , pred )\}\) 
<  \(\{( pred , pred )\}\) 
>  \(\{( pred , pred )\}\) 
\(\square \)
Note that in the literature, correspondences are also equipped with a confidence value between 0 and 1 [20]. We omit these values here (and implicitly assume that they are 1), because our semantics maps correspondences to theories in a twovalued logic. Alternatively, one might set a threshold and keep only those correspondences with confidence above this threshold into our framework. More sophisticated reasoning with confidence values other than 1 would need reasoning about uncertainty, which is beyond the scope of the paper. However, to mention just one way to address this problem within our framework, it is possible to extend the notion of bridge rule in the case of contextualised semantics, as has been proposed in [35, 36]. Here, confidence values are internalised into appropriate bridge rule axioms, and the semantics includes ‘fuzzyfied’ connecting relations interpreting these new bridge rule operators. Explicitly studying this version of contextualised semantics for alignments is part of our future work.
3.1 Syntax of \(\mathsf {DOL}\) Alignments

S and T are the ontologies to be aligned,

\(s_1^i\) is the name of an Ssymbol, and \(s_2^i\) is the name of a Tsymbol, for \(i=1,\ldots , n\),

\(s_1^i\ { REL }^i\ s_2^i\) is a correspondence which identifies a relation between the ontology symbols, and

DOMAIN records the type of semantics used for the alignment, using one of SingleDomain, GlobalDomain, GeneralGlobalDomain and Contextualised Domain. If no assumption about the universe is specified, the default is that simple semantics is used. The terminating Open image in new window is also optional.
Example 23
3.2 Syntax of \(\mathsf {DOL}\) Networks
where \(N_i\) are (sub)networks, \(O_i\) are ontologies and \(A_i\) are alignments (with source and target among the \(O_i\) or in the subnetworks \(N_i\)). We will generally assume that all alignments used in a \(\mathsf {DOL}\) network have been specified with the same semantics keyword. (The consideration of \(\mathsf {DOL}\) networks involving heterogeneous alignment semantics is left for future work.)
Thus we can use a \(\mathsf {DOL}\) network for specifying a NeO, by listing all the ontologies and all the alignments of the NeO as elements of a \(\mathsf {DOL}\) network. Conversely, any such (i.e. relational) \(\mathsf {DOL}\) network can be transformed to a NeO by, for each pair of ontologies, uniting multiple alignments or adding the empty alignment if there was none.
where \(N_i\) are (sub)networks, \(O_i\) are ontologies, and \(V_i\) are ontology morphisms (with source and target among the \(O_i\) or in \(N_i\)).
From a functional network, it is straightforward to obtain a diagram in the sense of Definition 10: one has to assemble all ontologies and ontology morphisms of all included subnetworks into one diagram. The semantics of a functional network is then given by the class of compatible families of models for this diagram. The semantics of relational networks is studied in Sect. 4.
\(\mathsf {DOL}\) also provides means for combining a functional network into a new ontology, such that the symbols related in the diagram of the \(\mathsf {DOL}\) network are identified. The syntax of combinations is ontology O = combine D, where D is a functional \(\mathsf {DOL}\) network, named or specified as above.
The semantics of a combination O is the class of models of the colimit ontology of the diagram obtained from the functional network. When the underlying institution has the amalgamation property, this model class is in bijection with the class of models of the diagram.
There is an obvious integration of the concepts of relational and functional network of ontologies into a concept of (general) network of ontologies, which may involve both alignments and ontology morphisms. Actually, \(\mathsf {DOL}\) provides this most general notion of networks. The semantics of these networks follows easily from those of the relational and functional networks. Note however that it is not clear how to give a semantics to combinations of general networks. This question will be addressed in Sect. 5.
In the remainder of this paper, if speaking just of a NeO, we will always mean a relational NeO. Functional or general NeO s will be explicitly qualified as such.
4 Four Semantics of Relational Networks of Ontologies
We will now generalise the three semantics for networks of aligned ontologies introduced in [80] to an arbitrary logic.
A semantics of relational NeOs is given in terms of local interpretation of the ontologies and alignments it consists of. To be able to give such a semantics, one needs to give an interpretation of the relations between symbols that are expressed in the correspondences.
In the following four subsections let \(S = \{ (O_i )_{i\in Ind}, (A_{ij})_{i, j\in Ind} \}\) be a NeO (in any logic) over a set of indexes Ind.
4.1 Simple Semantics
Definition 24
Given a model theory for a logic L, the interpretation of correspondence relations relative to a set is an interpretation function \(.^I\) taking as arguments a relation \(R\in \mathfrak {R}\), a kind \((k_1, k_2)\) of R and a set X and giving as result a relation \(R^I_X\) from \(domain(k_1,X)\) to \(domain(k_2,X)\). \(\square \)
If \(O_1\), \(O_2\) are two ontologies and \(c = (s_1, s_2, R)\) is a correspondence between \(O_1\) and \(O_2\), we say that c is satisfied by the models \(M^1\), \(M^2\) of \(O_1\), \(O_2\), written \(M^1, M^2 \models ^S c\), if and only if \(M^1_{s_1}\ R^I_D \ M^2_{s_2}\). A model of an alignment A between ontologies \(O_1\) and \(O_2\) is then a pair \(M^1\), \(M^2\) of interpretations of \(O_1\), \(O_2\) such that for all \(c \in A\), \(M^1, M^2 \models ^S c\). We denote this by \(M^1, M^2 \models ^S A\). An interpretation of S is a family \((M^i)_{i\in Ind}\) of models \(M^i\) of \(O_i\). A simple interpretation of S is an interpretation \((M^i)_{i\in Ind}\) of S over the same universe D.
Definition 25
[80] A simple model of a NeO S is a simple interpretation \((M^i)_{i\in Ind}\) of S such that for each \(i, j \in I\), \(M^i, M^j \models ^S A_{ij}\). This is written \((M^i)_{i\in Ind} \models ^S S\). We denote by \(\mathbf {Mod}^{sim}(S)\) the class of all simple models of S. \(\square \)
Example 24
\((c_1,c_2, =)\)  \(=\)  \(\mathcal P \left( {D}\right) \times \mathcal P \left( {D}\right) \) 
\((r_1,r_2, =)\)  \(=\)  \(\mathcal P \left( {D\times D}\right) \times \mathcal P \left( {D\times D}\right) \) 
\((i_1,i_2, =)\)  \(=\)  \(D \times D\) 
\((c_1,c_2, \bot )\)  \(M^1_{c_1}\cap M^2_{c_2} = \emptyset \)  \(\mathcal P \left( {D}\right) \times \mathcal P \left( {D}\right) \) 
\((r_1,r_2, \bot )\)  \(M^1_{r_1}\cap M^2_{r_2} = \emptyset \)  \(\mathcal P \left( {D\times D}\right) \times \mathcal P \left( {D\times D}\right) \) 
\((i_1,i_2,\bot )\)  \(\ne \)  \(D \times D\) 
\((c_1,c_2, <)\)  \(\subseteq \)  \(\mathcal P \left( {D}\right) \times \mathcal P \left( {D}\right) \) 
\((r_1,r_2, <)\)  \(\subseteq \)  \(\mathcal P \left( {D\times D}\right) \times \mathcal P \left( {D\times D}\right) \) 
\((c_1,c_2, >)\)  \(\supseteq \)  \(\mathcal P \left( {D}\right) \times \mathcal P \left( {D}\right) \) 
\((r_1,r_2, >)\)  \(\supseteq \)  \(\mathcal P \left( {D\times D}\right) \times \mathcal P \left( {D\times D}\right) \) 
\((c_1, i_2, \ni )\)  \(\ni \)  \(\mathcal P \left( {D}\right) \times D\) 
\((i_1, c_2, \in )\)  \(\in \)  \(D \times \mathcal P \left( {D}\right) \) 
where \(c_k, r_k, i_k\) are class, role and individual symbols from an ontology \(O_k\) and \(M^k \in \mathbf {Mod}(O_k)\) for \(k=1,2\). \(\square \)
Example 25
(Interpretation of correspondence relations in \(\mathsf {FOL}\)) The interpretation of correspondence relations in \(\mathsf {FOL}\) relative to a global universe D is
\((f_1,f_2,=)\)  \(=\)  \(Fun(D) \times Fun(D)\) 
\((f_1,f_2,\bot )\)  \(\ne \)  \(Fun(D) \times Fun(D)\) 
\((p_1,p_2,=)\)  \(=\)  \(Pred(D) \times Pred(D)\) 
\((p_1,p_2,\bot )\)  \(\ne \)  \(Pred(D) \times Pred(D)\) 
\((p_1,p_2,<)\)  \(\subseteq \)  \(Pred(D) \times Pred(D)\) 
\((p_1,p_2,>)\)  \(\supseteq \)  \(Pred(D) \times Pred(D)\) 
where \(f_k, p_k\) are function and predicate symbols from an ontology \(O_k\), with \(k=1,2\). \(\square \)
4.2 Integrated Semantics: General and Inclusive
An integrated interpretation of a NeO S is then a pair \(\{(M^i )_{i\in Ind}, (\gamma _i)_{i\in Ind}\}\) where \((M^i)_{i\in Ind}\) is an interpretation of S and \(\gamma _i:universe(M^i)\rightarrow U\) is a function to a common global universe U for each \(i\in Ind\).
Definition 26
[80] An integrated interpretation of a NeO S, \(\{(M^i )_{i\in Ind}, (\gamma _i)_{i\in Ind}\}\), is an integrated model of S iff for each \(i, j \in Ind\), \( M^i, M^j \models ^I_{\gamma _i, \gamma _j} A_{ij}\). \(\square \)
Example 26
In \(\mathcal {SROIQ}\) the correspondence relations are interpreted like in the case of simple semantics, presented in Example 24, but now relative to the global universe U. \(\square \)
In the sequel, we will distinguish the special case that the \(\gamma _i\) are inclusions as inclusive integrated semantics. Actually, this case will turn out to be technically simpler than general integrated semantics that drops this restriction. General integrated semantics can be useful if different ontologies work with different assumptions about the identity of individuals. For example in \(O_1\), Superman and Clark Kent might be different, while in \(O_2\) they are the same person. The universe U of an integrated model would feature only one element representing both Superman and Clark Kent. Hence, the mapping \(\gamma _1\) from the local \(O_1\)model to U will typically be noninjective.
We denote by \(\mathbf {Mod}^{gint}(S)\) the class of all general integrated models of a NeO S and by \(\mathbf {Mod}^{iint}(S)\) the class of all inclusive integrated models of a NeO S.
4.3 Contextualised Semantics
Contextualised semantics gives up the notion of a global universe, and instead lets each ontology in a network be interpreted with its own local universe. However, in order to give a semantics to alignments, these universes need to be related somehow. The approach of [80] to use mappings between local universes has a number of limitations and has been replaced by a more flexible approach subsequently [20], which uses relations between local universes. This is closely related to the semantics of DDLs [5] and \(\mathcal {E}\)connections [48].
The idea is to relate the universes of the models of the ontologies by a family of relations \(r=(r_{ij})_{i,j\in Ind}\). The relations R in \(\mathfrak {R}\) are interpreted in each universe \(D_i\) of the models \(M^i\) of the ontologies in the NeO.
A contextualised interpretation of S is then a pair \(\{(M^i )_{i\in Ind}, (r_{ij})_{i,j\in Ind}\}\) where \((M^i)_{i\in Ind}\) is an interpretation of S and \((r_{ij})_{i,j\in Ind}\) is a family of relations such that \(r_{ij}\) relates the universe of \(M^i\) to the universe of \(M^j\) and \(r_{ii}\) is the identity (diagonal) relation. Further assumptions about these relations can be added, thus restricting more the class of interpretations of a NeO.
Definition 27
A contextualised model of the NeO S is a contextualised interpretation \(((M^i )_{i\in Ind}, (r_{ij})_{i,j\in Ind})\) of S such that for each \(i, j \in Ind\), \(M^i, M^j \models ^C_r A_{ij}\). We denote by \(\mathbf {Mod}^{con}(S)\) the class of all contextualised models of a NeO S. \(\square \)
Example 27
(Interpretation of correspondence relations in \(\mathcal {SROIQ}\) ) The interpretation of correspondences in \(\mathcal {SROIQ}\) relative to a set D in the contextualised semantics is
\((c_1,c_2, =)\)  \(M^1_{c_1} = r_{21}(M^2_{c_2})\) 
\((r_1,r_2, =)\)  \(M^1_{r_1} = r_{21}(M^2_{r_2})\) 
\((i_1,i_2, =)\)  \(M^1_{i_1} \in r_{21}(M^2_{i_2}) \text { i.e. } (M^2_{i_2},M^1_{i_1})\in r_{21}\) 
\((c_1,c_2, \bot )\)  \(M^1_{c_1} \cap r_{21}(M^2_{c_2}) = \emptyset \) 
\((r_1,r_2, \bot )\)  \(M^1_{r_1} \cap r_{21}(M^2_{r_2}) = \emptyset \) 
\((i_1,i_2,\bot )\)  \((M^2_{i_2}, M^1_{i_1})\not \in r_{21}\) 
\((c_1,c_2, <)\)  \(M^1_{c_1} \subseteq r_{21}(M^2_{c_2})\) 
\((r_1,r_2, <)\)  \(M^1_{r_1} \subseteq r_{21}(M^2_{r_2})\) 
\((c_1,c_2, >)\)  \(M^1_{c_1} \supseteq r_{21}(M^2_{c_2})\) 
\((r_1,r_2, >)\)  \(M^1_{r_1} \supseteq r_{21}(M^2_{r_2})\) 
\((c_1, i_2, \ni )\)  \(r_{21}(M^2_{i_2}) \subseteq M^1_{c_1}\) 
\((i_1, c_2, \in )\)  \(M^1_{i_1} \in r_{21}(M^2_{c_2})\) 
where \(c_1, r_1, i_1\) are class, role and individual symbols from an ontology \(O_1\), \(c_2, r_2, i_2\) are class, role and individual symbols from an ontology \(O_2\), \(M^1\) and \(M^2\) are models of \(O_1\) and \(O_2\) with domains \(D_1\) and \(D_2\) and \(r_{21}\) is the domain relation between \(D_2\) and \(D_1\).^{27} \(\square \)
5 Normalisation of Alignments
In this section we describe how relational (and therefore also general) networks can be normalised into functional ones. Part of this normalisation process generalises to an arbitrary institution, while certain parts (namely relativisation of ontologies and the construction of bridges) are institutionspecific and have to be provided separately for each institution.
A central motivation behind this construction is the following: We will prove representation theorems showing that the semantics of a relational network coincides with that of its normalisation, see Sect. 7. This implies that reasoning in the colimit of the normalised network is complete and (in case of logics admitting amalgamation) also sound for reasoning about the network (Corollary 3).
5.1 Structure of the Normalisation Process
Relational \(\mathsf {DOL}\) networks (i.e. networks involving alignments) can be normalised to purely functional networks. In this section, we lay out the structure of this normalisation process, while in the next four sections, we will provide details for each of the four possible assumptions about the semantics.
Example 28
where we prefix with S : the symbols coming from S and with T : the symbols coming from T.
Note that so far we have not specified which kind of semantics is assumed for Open image in new window . Depending on the choice for the assumed semantics, the normalisation of Open image in new window will be constructed in a different way. \(\square \)
The idea is to introduce for each correspondence a theory that captures its semantics. This is done differently for four possible semantics of the alignment. Using these theories, we then construct a diagram that gives the semantics of the alignment. In all four semantics, the diagram is a Walignment in the sense of [81]:
Definition 28
where the ontologies \(B, {\tilde{S}}, {\tilde{S}}', {\tilde{T}}, {\tilde{T}}'\) and the morphisms \(\iota _1, \iota _2, \sigma _1,\sigma _2\) depend on the choice of semantics for the alignment A, in a way to be made precise for each possible option.
Intuitively, \({\tilde{S}}\) and \({\tilde{T}}\) are either the ontologies S and T being aligned, or a transformation of them, involving their translation along a comorphism. B is a bridge ontology that formalises the intended meaning of the correspondences of A. It will be constructed as a union of smaller theories, each internalising the semantics of a correspondence of A. This means, intuitively, that the models M of a theory that internalises the semantics of \((s_1, s_2, R)\), are precisely those for which the relation \(M_R\) holds for \(M_{s_1}\) and \(M_{s_2}\), in a way that takes into account the possible semantics of the alignment. We will define this formally for each choice of semantics. It is possible that some correspondence cannot be internalised in the logic of the ontologies being aligned. In this case, we will have to look for a more expressive logic, where such a theory internalising the semantics of that correspondence can be constructed. We will discuss the implications of this change of logic in Sect. 6. \({\tilde{S}}'\) and \({\tilde{T}}'\) interface \({\tilde{S}}\) and \({\tilde{T}}\), respectively, with B, meaning that they connect the symbols from the aligned ontologies with their correspondents in the bridge ontology along \(\iota _i\) and \(\sigma _i\).
These diagrams will be used in the construction of the normalisation of a network:
Definition 29
Given a relational NeO, its normalisation is defined as the functional network obtained by uniting all diagrams for the alignments in the NeO. Given a general NeO, its normalisation is defined as the union of its functional part with the normalisation of its relational part. \(\square \)
For each ontology \(O_i\) in a network of aligned ontologies let \(\tilde{O_i}\) be its corresponding ontology in the diagram of the network. Let \(\Sigma _i = \mathsf {Sig}(O_i)\) and \(\Sigma '_i = \mathsf {Sig}(\tilde{O_i})\). In each of the four cases that correspond to the different choices of semantics we can define: (i) a sentence translation functor \(\alpha ^*: \mathbf {Sen}(\Sigma _i) \rightarrow \mathbf {Sen}(\Sigma '_i)\) (ii) a model reduct functor \(\beta ^*: \mathbf {Mod}(\Sigma '_i) \rightarrow \mathbf {Mod}(\Sigma _i)\) such that the condition \(\beta ^*(M')\models e \iff M' \models \alpha ^*(e)\) holds for each \(M\in \mathbf {Mod}(\tilde{O_i})\) and each \(e\in \mathbf {Sen}(\Sigma _i)\). Using these functors allows us to formulate the results about reasoning in a NeO in a uniform way. In all four cases, we can define a signature morphism in a Grothendieck logic from \(\Sigma _i\) to \(\Sigma '_i\) such that \(\alpha ^*\) and \(\beta ^*\) are the sentence translation and model reduction functors corresponding to it. Thus, the expected condition follows from the satisfaction condition of the Grothendieck logic.
We now proceed with discussing how these diagrams are obtained for each of the four possible semantics.
5.2 Simple Semantics
We start with defining what it means for a theory to capture the semantics of a correspondence. In this section, let \(A = \{c_i = (s^i_1,s^i_2,R_i)\mid i\in Ind\}\) be an alignment between two ontologies S and T in a logic L, where Ind is a set of indices. First we define the signature of the theory.
Definition 30

\(\Sigma _1\) is the smallest subsignature of \(\mathsf {Sig}(S)\) such that \({\mathbf {Symbols}}(\Sigma _1)\) includes \(s^i_1\) for each \(i\in Ind\), and \(\mathsf {Sig}_{1}(A)\) is the signature obtained by renaming every symbol \(s\in {\mathbf {Symbols}}(\Sigma _1)\) to S : s and

\(\Sigma _2\) is the smallest subsignature of \(\mathsf {Sig}(T)\) such that \({\mathbf {Symbols}}(\Sigma _2)\) includes \(s^i_2\) for each \(i\in Ind\), and \(\mathsf {Sig}_{2}(A)\) is the signature obtained by renaming every symbol \(s\in {\mathbf {Symbols}}(\Sigma _2)\) to T : s.\(\square \)
We must prefix the symbols occurring in correspondences with the names of the ontology where they come from to avoid unintended identifications when making the union of the involved signatures.
Definition 31
Definition 32

\({\tilde{S}}= S\) and \({\tilde{T}}= T\),

\({\tilde{S}}' = (\mathsf {Sig}_{1}(A), \emptyset )\),

\(\iota _1\) maps each \(S:s\in {\mathbf {Symbols}}(\mathsf {Sig}_{1}(A))\) to s,

\({\tilde{T}}' = (\mathsf {Sig}_{2}(A), \emptyset )\),

\(\iota _2\) maps each \(T:s\in {\mathbf {Symbols}}(\mathsf {Sig}_{2}(A))\) to s,

\(B = (\Sigma _B, \cup _{i\in Ind}\Delta _i)\),

\(\sigma _1\) and \(\sigma _2\) are inclusions. \(\square \)
Example 29
(Simple semantics in \(\mathcal {SROIQ}\))
Example 30
Example 31
Since the original ontologies are not modified in the diagram of the alignments, the signature morphism from \(\mathsf {Sig}(O_i)\) to \(\mathsf {Sig}(\tilde{O_i})\) is the identity, so the functors \(\alpha ^*\) and \(\beta ^*\) are the identities on \(\mathsf {Sig}(O_i)\)sentences, respectively, on \(\mathsf {Sig}(O_i)\)models.
5.3 Inclusive Integrated Semantics
Normalisation of relational networks with integrated semantics is more difficult than for simple semantics, as compatible families of models of connected functional networks always have a unique universe (this follows from model reducts preserving the universe). To remedy this, we use the relativisation of ontologies. Thus the universes of the ontologies appear explicitly as concepts, and they can be interpreted as subsets of the global universe, that subsumes them.
Here internalisation is the same as in the case of simple semantics. For uniformity we use the notation \((\Sigma _B,\Delta ) \approxeq ^{iint} c\), which means that \((\Sigma _B,\Delta ) \approxeq ^{sim} c\).
Definition 33

\({\tilde{S}}= rel_L(S)\) and \({\tilde{T}}= rel_L(T)\), where \(rel_L\) is the relativisation of the logic L of the ontologies being aligned,

\({\tilde{S}}' = (\mathsf {Sig}_{1}(A) \cup \{S:\top \in binary_L\}, \emptyset )\),

\(\iota _1\) maps each \(S:s\in {\mathbf {Symbols}}(\mathsf {Sig}_{1}(A))\) to s and \(S:\top \) to itself,

\({\tilde{T}}' = (\mathsf {Sig}_{2}(A) \cup \{T:\top \in binary_L\}, \emptyset )\),

\(\iota _2\) maps each \(T:s\in {\mathbf {Symbols}}(\mathsf {Sig}_{2}(A))\) to s and \(T:\top \) to itself,

\(B = (\Sigma _B, \cup _{i\in Ind}\Delta _i)\),

\(\sigma _1\) and \(\sigma _2\) are inclusions. \(\square \)
We can instantiate Definition 33 for \(\mathcal {SROIQ}\) using the internalisation of alignments from Example 29 and the relativisation from Example 19. Similarly, we get an instantiation for \(\mathsf {FOL}\) using the internalisation of alignments from Example 30 and the relativisation from Example 20. We expect that in a similar way, also for other logics, internalisation of alignments for simple semantics can be reused for inclusive integrated semantics when a relativisation is added.
Example 32
where \({\tilde{S}}'\) consists of the concepts \(S:\top \), S: Open image in new window and S: Open image in new window and the individual S: Open image in new window and \({\tilde{T}}'\) consists of the concepts \(T:\top \), T: Open image in new window , T: Open image in new window and T: Open image in new window .
The bridge ontology of Open image in new window is the same as for the simple semantics.
If S is a network of aligned ontologies and \(O_i\) an ontology of S for \(i\in I\), we define the heterogeneous signature morphism from \(\mathsf {Sig}(O_i)\) to \(\mathsf {Sig}(rel_L(O_i))\) as \((rel_L, id_{\mathsf {Sig}(rel_L(O_i))})\). Thus \(\alpha ^*\) and \(\beta ^*\) are the components \(\alpha \) and \(\beta \) of the relativisation comorphism \(rel_L\).
5.4 General Integrated Semantics
Since general integrated semantics allow the use of arbitrary equalising functions in the model, the approach of Sect. 5.3 does not work. We have to modify the bridge ontology and the internalisation of alignments: the ontologies of the network can be interpreted using different universes, and these are mapped to a global universe using (functional) binary relations. Moreover, we need to make sure that if an ontology is aligned more than once in a network of aligned ontologies, the global universe and the equalising functions are interpreted in the same way in the entire diagram of the network.
Definition 34
 the signature \(\Sigma _B\) that takes the union of \(\mathsf {Sig}_{1}(A)\) and \(\mathsf {Sig}_{2}(A)\), where

\(\Sigma _1\) is the smallest subsignature of \(\mathsf {Sig}(S)\) such that \({\mathbf {Symbols}}(\Sigma _1)\) includes \(s^i_1\) for each \(i\in Ind\), and \(\mathsf {Sig}_{1}(A)\) takes the signature obtained by renaming every \(s\in {\mathbf {Symbols}}(\Sigma _1)\) to S : s and extends it with \(S:\top , G \in unary_L\) and with \(r_S\) and \(r_T\) in \(binary_L\),

\(\Sigma _2\) is the smallest subsignature of \(\mathsf {Sig}(T)\) such that \({\mathbf {Symbols}}(\Sigma _2)\) includes \(s^i_2\) for each \(i\in Ind\), and \(\mathsf {Sig}_{2}(A)\) takes the signature obtained by renaming every \(s\in {\mathbf {Symbols}}(\Sigma _2)\) to T : s and extends it with a with \(T:\top , G \in unary_L\) and with \(r_S\) and \(r_T\) in \(binary_L\)


logicdependent sentences \(\Delta _B\) that axiomatise \(r_S\) and \(r_T\) to be inversefunctional and right total and having their domains G and their ranges \(S:\top \) and \(T:\top \), respectively.\(\square \)
Besides the symbols involved in the correspondences of A, the bridge ontology contains explicit representation of the involved domains: the universes of the aligned ontologies \(S:\top \) and \(T:\top \) as well as the global domain G. The equalising functions are represented by the relations \(r_S\) and \(r_T\). Note that we work with their inverses to be able to internalise the semantics of correspondences in \(\mathsf {OWL}\).
Definition 35
Let \(c=(s_1,s_2, R)\) be a correspondence of a generalised integrated alignment. Let \((\Sigma _B, \Delta _B)\) be the basic bridge ontology of the alignment and let \(\Delta \) be a set of \(\Sigma _B\)sentences that includes \(\Delta _B\). We say that \((\Sigma _B, \Delta )\) internalises the semantics of c, denoted \((\Sigma _B,\Delta ) \approxeq ^{gint} c\), if for each \(\Sigma _B\)model M, \(M\models _{\Sigma _B} \Delta \) iff we have in \( M \text { that } (x, M_{S:s_1}) \in M_{r_S}\) and \((y, M_{T:s_2}) \in M_{r_T}\) implies \((x, y) \in R^I_{G} \) for \(x,y \in M_{G}\).
\(\square \)
Definition 36

\({\tilde{S}}\) extends \(rel_L(S)\), where \(rel_L\) is the relativisation of the logic L of the ontologies being aligned, with a new symbol G in \(unary_L\) and with a new symbol \(r_S\) in \(binary_L\) and similarly \({\tilde{T}}\) extends \(rel_L(T)\), with a new symbol G in \(unary_L\) and with a new symbol \(r_T\) in \(binary_L\),

\({\tilde{S}}' = (\mathsf {Sig}_{1}(A), \emptyset )\),

\(\iota _1\) maps each \(S:s\in {\mathbf {Symbols}}(\mathsf {Sig}_{1}(A))\) to s, \(S:\top \) to top and \(G, r_S\) identically,

\({\tilde{T}}' = (\mathsf {Sig}_{2}(A), \emptyset )\),

\(\iota _2\) maps each \(T:s\in {\mathbf {Symbols}}(\mathsf {Sig}_{2}(A))\) to s, \(T:\top \) to top and \(G, r_T\)identically,

\(B = (\Sigma _B, \cup _{i\in Ind}\Delta _i)\),

\(\sigma _1\) and \(\sigma _2\) are inclusions. \(\square \)
Example 33
Example 34
For correspondences involving roles, we would need to be able to express equivalences or disjointness axioms involving complex roles, which are beyond the expressivity of \(\mathcal {SROIQ}\). Therefore, the correspondences \((r_1, r_2,=) \), \((r_1, r_2,\bot )\), \((r_1, r_2, <) \) and \((r_1, r_2,>)\) cannot be internalised in \(\mathcal {SROIQ}\). We will give their internalisations in \(\mathsf {FOL}\) in Example 42. \(\square \)
Example 35
If S is a network of aligned ontologies and \(O_i\) an ontology of S for \(i\in Ind\), there is an inclusion morphism from \(\mathsf {Sig}(rel_L(O_i))\) to \(\mathsf {Sig}(\tilde{O_i})\) (recall that this is the ontology corresponding to \(O_i\) in the diagram of S), that we denote \(\iota _i\). We define the heterogeneous signature morphism from \(\mathsf {Sig}(O_i)\) to \(\mathsf {Sig}(\tilde{O_i})\) as \((rel_L, \iota _i)\). Thus \(\alpha ^*(e) = \mathbf {Sen}(\iota _i)(\alpha (e))\) and \(\beta ^*(M') = \beta (M'_{\iota _i})\), where \(rel_L = (\Phi ,\alpha , \beta )\).
5.5 Contextualised Semantics
Recall from Sect. 4.3 that in contextualised semantics, the ontologies of the network can be interpreted using different universes, which however are related using binary relations.
Definition 37
 a signature \(\Sigma _B\) that takes the union of \(\mathsf {Sig}_{1}(A)\) and \(\mathsf {Sig}_{2}(A)\), whereand extends this union with \(r_{TS}\) in \(binary_L\).

\(\Sigma _1\) is the smallest subsignature of \(\mathsf {Sig}(S)\) such that \({\mathbf {Symbols}}(\Sigma _1)\) includes \(s^i_1\) for each \(i\in Ind\), and \(\mathsf {Sig}_{1}(A)\) takes the signature obtained by renaming every \(s\in {\mathbf {Symbols}}(\Sigma _1)\) to S : s and extends it with \(S:\top \in unary_L\),

\(\Sigma _2\) is the smallest subsignature of \(\mathsf {Sig}(T)\) such that \({\mathbf {Symbols}}(\Sigma _2)\) includes \(s^i_2\) for each \(i\in Ind\), and \(\mathsf {Sig}_{2}(A)\) takes the signature obtained by renaming every \(s\in {\mathbf {Symbols}}(\Sigma _2)\) to T : s and extends with \(T:\top \in unary_L\)


a set \(\Delta _B\) of \(\Sigma _B\)sentences that axiomatise in a logicdependent way that the domain of \(r_{TS}\) is \(T:\top \) and the range of \(r_{TS}\) is \(S:\top \).\(\square \)
Definition 38
Definition 39

\({\tilde{S}}= rel_L(S)\) and \({\tilde{T}}= rel_L(T)\), where \(rel_L\) is the relativisation of the logic L of the ontologies being aligned,

\({\tilde{S}}' = (\mathsf {Sig}_{1}(A), \emptyset )\),

\(\iota _1\) maps each \(S:s\in {\mathbf {Symbols}}(\mathsf {Sig}_{1}(A))\) to s and \(S:\top \) to itself,

\({\tilde{T}}' = (\mathsf {Sig}_{2}(A), \emptyset )\),

\(\iota _2\) maps each \(T:s\in {\mathbf {Symbols}}(\mathsf {Sig}_{2}(A))\) to s and \(T:\top \) to itself,

\(B = (\Sigma _B, \cup _{i\in Ind}\Delta _i)\),

\(\sigma _1\) and \(\sigma _2\) are inclusions. \(\square \)
Example 36
However, the following example shows that it is not always possible to express the semantics of a correspondence in the contextualised semantics in the same logic as the one used in the aligned ontologies.
Example 37
(Contextualised semantics in \(\mathcal {SROIQ}\))
For the correspondence \((r_1, r_2, =)\) where \(r_1\) and \(r_2\) are roles, it is not possible to express in \(\mathcal {SROIQ}\) that \(r_1\) and \(r^{1}_{TS};r_2; r_{TS}\) are equivalent roles, where \(r_{TS}\) is the domain relation. A similar problem appears for the correspondence \((r_1, r_2, \bot )\). \(\square \)
To obtain a theory that internalises the semantics of this correspondence, we must use a more expressive logic, like firstorder logic. This will be done in the next section.
Example 38
Note that the correspondences of Open image in new window do not include an equivalence between roles, and thus we can build a bridge ontology in \(\mathcal {SROIQ}\). \(\square \)
The functors \(\alpha ^*\) and \(\beta ^*\) are defined as in the case of inclusive integrated semantics.
6 Heterogeneous Alignments
We now discuss how to generalise the semantics of functional networks to the heterogeneous case, where by heterogeneity we mean the syntactic heterogeneity in the sense of [20], where the ontologies being aligned are not expressed in the same ontology language.^{30} Arrows occurring in functional networks are theory morphisms in the Grothendieck logic, see Definition 20. The semantics using compatible families of models just carries over.
For heterogeneous relational networks, we need to equip the Grothendieck logic (which is induced by an indexed logic) with a family of correspondence relations in the sense of Definition 21. Now typically many or even all of the logics in the indexed logic will come with a family of correspondence relations. Each such relation name R of kind \((k_1,k_2)\) for logic i carries over to a relation name (i, R) of kind \(((i,k_1),(i,k_2))\) for the Grothendieck logic. The interpretation of such relation names in the four different semantics also carries over to the Grothendieck logic.
Naturally, these relation names (and their interpretations) inherited from the individual logics of the indexed logic can be used for homogeneous correspondences and alignments only. In order to be able to write truly heterogeneous alignments, one needs to introduce more relation names to the Grothendieck logic. This leads to a syntax and semantics of heterogeneous relational (and also general) networks of ontologies.
This framework also allows us to align ontologies written in different formalisms, e.g. an \(\mathsf {OWL}\) or RDF ontology with a firstorder ontology. A typical example where this scenario is relevant is the alignment of a domain ontology (expressed in OWL) with a foundational ontology (expressed in FOL) with the general goal of improving the design of the domain ontology (see e.g. [24]).
Example 39
Below are the type of correspondences that can occur in alignments between an \(\mathcal {SROIQ}\) and a \(\mathsf {FOL}\) ontology (in this order):
\(=\)  \(\{({ concept,\,pred}), ({ role,\,pred}),\) \(({ individual}, fun )\}\) 
\(\bot \)  \(\{({ concept,\,pred}), ({ role,\,pred}),\) \(({ individual}, fun )\}\) 
<  \(\{({ concept,\,pred}), ({ role, pred})\}\) 
>  \(\{({ concept,\,pred}), ({ role, pred})\}\) 
\(\in \)  \(\{({ individual,\,pred})\}\) 
\(\ni \)  \(\{({ concept}, fun )\}\) 
Example 40
Below are the type of correspondences that can occur in alignments between a \(\mathsf {MSFOL}\) and a \(\mathsf {HOL}\) ontology (in this order):
\(=\)  \(\{({ sort,\,sort}), ( fun , { const}), ({ pred,\,const})\}\) 
\(\bot \)  \(\{({ sort,\,sort}), ( fun , { const}), ({ pred,\,const})\}\) 
<  \(\{({ sort,\,sort}), ({ pred,\,const})\}\) 
>  \(\{({ sort,\,sort}), ({ pred,\,const})\}\) 
Definition 40

\( (\Phi _1(\mathsf {Sig}_{1}(A)) \cup \Phi _2(\mathsf {Sig}_{2}(A)), \emptyset )\), where \(\mathsf {Sig}_{1}(A)\) and \(\mathsf {Sig}_{2}(A)\) have been defined in Definition 30, for the simple and inclusive integrated semantics,

\(\Sigma _B\) is the extension of \(\Phi _1(\mathsf {Sig}_{1}(A)) \cup \Phi _2(\mathsf {Sig}_{2}(A))\), as defined in Definition 34, with two symbols \(r_1\) and \(r_2\) in \(binary_L\) and one symbol G in \(unary_L\) and \(\Delta _B\) is a set of \(\Sigma _B\)sentences axiomatising in a logicdependent way that the two binary predicates are inversefunctional, right total and have their domain G and the ranges \(S:\top \) and \(T:\top \), respectively, for the general integrated semantics,

\(\Sigma _B\) is the extension of \(\Phi _1(\mathsf {Sig}_{1}(A)) \cup \Phi _2(\mathsf {Sig}_{2}(A))\), as defined in Definition 37, with the symbol \(r_{TS}\) in \(binary_L\) and \(\Delta _B\) is a set of \(\Sigma _B\)sentences axiomatising in a logicdependent way that the domain of \(r_{TS}\) is \(T:\top \) and its range is \(S:\top \) for the contextualised semantics.\(\square \)
The theory that internalises the semantics of a correspondence in an alignment A will then have as signature the translation of the bridge signature of A along a logic translation.
Definition 41
Let S be an ontology in a logic \(L_1\) and T an ontology in a logic \(L_2\), and let \(c = (s_1, s_2, R)\) be a correspondence between the symbols \(s_1\) of S and \(s_2\) of T in an alignment A. Let \((\Sigma _B,\Delta _B)\) be the basic bridge ontology of A along \(\rho _1=(\Phi _1,\alpha _1,\beta _1,\gamma _1,\kappa _1): L_1 \rightarrow L\) and \(\rho _2=(\Phi _2,\alpha _2,\beta _2,\gamma _2,\kappa _2): L_2 \rightarrow L\), introduced in the previous definition, and let \(\rho = (\Phi ',\alpha ',\beta ',\gamma ',\kappa '): L \rightarrow L'\). Let \(\Delta \) be a set of sentences including \(\alpha '(\Delta _B)\). Then the theory \((\Phi '(\Sigma _B), \Delta )\) internalises the semantics of c along \(\rho _1;\rho \) and \(\rho _2;\rho \) if \(M\models ^L_{\Phi '(\Sigma _B)}\Delta \) iff (i) for the simple and inclusive integrated semantics, \((M_{\gamma _1;\gamma '(S:s_1)}, M_{\gamma _2;\gamma '(T:s_2)}) \in R^I_{universe(M)}\), (ii) for the general integrated semantics, for each x, y with \((x, M_{\gamma _1;\gamma '(S:s_1)}) \in M_{r_S}\) and \((y,M_{\gamma _2;\gamma '(T:s_2)}) \in M_{r_T}\), we have that \((x, y) \in R^I_{M_G}\) and (iii) for the contextualised semantics, we have that \((M_{\gamma _1;\gamma '(S:s_1)}, M_{r_{TS}}(M_{\gamma _2;\gamma '(T:s_2)}) ) \in R^I_{M_{S:\top }}\). \(\square \)
Example 41
If \(O_1\) is an \(\mathsf {OWL}\) ontology and \(O_2\) is a \(\mathsf {FOL}\) ontology, and \(c=(s_1, s_2, R)\) is a correspondence in an alignment between \(O_1\) and \(O_2\), a theory internalising the semantics of c along the translation from \(\mathsf {OWL}\) to \(\mathsf {FOL}\) defined in Example 13 and the identity translation on \(\mathsf {FOL}\) is obtained according to the previous definition by internalising the semantics of \((\gamma (s_1), s_2, R)\) in \(\mathsf {FOL}\), where \(\gamma \) is the symbol translation component of the logic translation from \(\mathsf {OWL}\) to \(\mathsf {FOL}\). Similarly, correspondences between a \(\mathsf {MSFOL}\) and a \(\mathsf {HOL}\) ontology can be internalised in \(\mathsf {HOL}\) by translating the \(\mathsf {MSFOL}\) symbols to \(\mathsf {HOL}\).
The construction of the bridge ontology becomes also heterogeneous and increases in complexity. As also described in [21], we look for a logic L that is expressive enough to encode the two logics \(L_1\) and \(L_2\) of the ontologies being aligned. In our previous work we described a graph of logics and ontology languages and translations between them [61]. Given such a graph, L is the least upper bound of \(L_1\) and \(L_2\) in the graph (bound to exist by inclusion of higherorder logic, though typically a less expressive logic suffices). This allows us to compute the bridge ontology B in the logic L.
Definition 42
where \({\tilde{S}}, {\tilde{T}}, {\tilde{S}}', {\tilde{T}}'\) and \(\iota _1, \iota _2\) are constructed as in the previous section, depending on the choice of semantics for A, \(B = (\Phi '(\Sigma _B),\cup _{i\in Ind}\Delta _i)\), and \(\sigma _i\) is an inclusion for \(i=1,2\). \(\square \)
In [13], we discussed how the (approximation of a) colimit of such a heterogeneous diagram can be computed in general.
Example 42
Example 43
(Contextualised semantics in \(\mathcal {SROIQ}\))
Example 44
where \(id_{owl}\) and \(id_{ fol }\) are the identity translations on \(\mathsf {OWL}\) and \(\mathsf {FOL}\). Thus, the OWL concepts on the left are identified with the firstorder counterparts, and in the resulting colimit of the alignment, which will also be a \(\mathsf {FOL}\) theory, they will inherit their full firstorder axiomatisation. \(\square \)
Example 45
Here, we use the correspondences between \(\mathsf {MSFOL}\) and \(\mathsf {HOL}\) introduced in Example 40 and the translation from \(\mathsf {MSFOL}\) to \(\mathsf {HOL}\) introduced in Example 16. \(\square \)
Example 46
Using the obvious generalisation of the institution comorphism from \(\mathsf {MSFOL}\) to \(\mathsf {FOL}\) [18, section 4.1] to a logic translation, we can define an alignment between the unsorted and the multisorted version of DOLCE.
7 Representation Theorems for Networks of Ontologies
In the previous two sections, we have given a normalisation of relational (and general) networks to functional networks. The normalisation constructs a diagram of ontologies linked via ontology morphisms, in a way that depends on the choice of semantics for the alignments in a NeO. Since we assumed that for any two ontologies in a NeO there is an alignment between them, the resulting diagram will always be connected.^{36} In this section, we show that this normalisation process faithfully reflects the semantics, that is, the model class of a normalised network is isomorphic to that of the original network. Moreover, we show that in the case of logics admitting amalgamation, the model class of the normalised network is in turn isomorphic to that of its colimit.
This means that functional networks (and via normalisation also relational and general networks) can be represented by their colimit, and especially that reasoning about the network is equivalent to reasoning about the colimit. Since the colimit is a single ontology, this means that we have established a method for taking standard reasoning tools (designed for single ontologies) and reusing them for reasoning about networks. These reasoning aspects will be discussed in greater detail in the subsequent Sect. 8.
7.1 Representation Theorems for Specific Semantics
In this section let \(S = ((O_i)_{i\in Ind}, (A_{ij})_{i,j\in Ind})\) be a network of aligned ontologies over a logic L. Let \(D^{sim}(S)\) be the union of all diagrams obtained by taking for each alignment \(A_{ij}\) its diagram using simple semantics.
Theorem 1
 1.
\(\mathbf {Mod}^{sim}(S)\)
 2.
the class of all families of models compatible with \(D^{sim}(S)\).
Proof
We prove that an isomorphism between 1. and 2. exists.
Let \(M=(M^i)_{i\in D^{sim}(S)}\) be a family of models compatible with \(D^{sim}(S)\). By construction, \(D^{sim}(S)\) includes all ontologies \(O_i\) and thus we get a family of models \((M^i)_{i\in Ind}\) by restricting M to the indices in Ind. We need to show that for each alignment \(A_{ij} \in S\), \(M^i, M^j\models ^S A_{ij}\), that is, \(M^i, M^j\models ^S c\) for each \(c=(s_1, s_2, R)\) in \(A_{ij}\). By definition this means that \(M^i_{s_1} R^I_U M^j_{s_2}\), where U is the shared universe of the ontologies in S. U is shared because model reducts preserve the universe and \(D^{sim}(S)\) is connected. We know that if \(b \in D^{sim}(S)\) is the node of the bridge ontology for \(A_{ij}\), then \(M^b_{O_i:s_1} R^I_U M^b_{O_j:s_2}\). By compatibility with the diagram we have that \(M^b_{O_i:s_1} = M^{i'}_{O_i:s_1} = M^i_{s_1}\) and \(M^b_{O_j:s_2} = M^{j'}_{O_j:s_2} = M^j_{s_2}\), where we denoted \(i'\) and \(j'\) the nodes of \({\tilde{S}}'\) and \({\tilde{T}}'\) (as in the construction of the diagram of \(A_{ij}\) with the Definition 28) in D. Thus \((M^i)_{i\in I} \in \mathbf {Mod}^{sim}(S)\).
Now let us assume that \(M=(M^i)_{i\in Ind}\) is a model in \(\mathbf {Mod}^{sim}(S)\). For each alignment \(A_{ij}\) we must give interpretations for the ontologies \({\tilde{S}}'\), \({\tilde{T}}'\) and B in its diagram, since models for S and T are already provided by \(M^i\) and \(M^j\). We can interpret \({\tilde{S}}'\) as \(M^i_{\iota _i}\) and \({\tilde{T}}'\) as \(M^j_{\iota _j}\). The symbols from B that come either from \(O'_i\) or from \(O'_j\) are interpreted in \(M^b\) like in \(M^{i'}\) or, respectively, like in \(M^{j'}\). Since M is in \(\mathbf {Mod}^{sim}(S)\), we know that \(M^i_{s^k_1} R^I_U M^j_{s^k_2}\) for each \(c_k=(s^k_1, s^k_2, R)\in A_{ij}\). But by definition \(M^b_{O_i:s^k_1} = M^i_{s^k_1}\) and \(M^b_{O_j:s^k_2} = M^j_{s^k_2}\), which means that \(M^b_{O_i:s^k_1} R^I_U M^b_{O_j:s^k_2}\). Since \((\Sigma _B,\Delta _k)\approxeq ^{sim} c_k\) we get that \(M^b\models \Delta _k\) and since k is arbitrary, we get that \(M^b\in \mathbf {Mod}(B)\).
We need to show that the two constructions are inverse to each other. Let \(M = (M^i)_{i\in D^{sim}(S)}\) be a family of models compatible with \(D^{sim}(S)\). With the first construction, we showed that by restricting to the indices in Ind we get a family of models \((M^i)_{i\in Ind}\) that is in \(\mathbf {Mod}^{sim}(S)\). By applying the second construction to this family of models, for each alignment \(A_{ij}\) we obtain models \(M^{{\tilde{S}}'}\), \(M^{{\tilde{T}}'}\) and \(M^B\) for the ontologies \({\tilde{S}}'\), \({\tilde{T}}'\) and B of the diagram of \(A_{ij}\). Let \(i', j', b\) be the indices of these nodes in \(D^{sim}(S)\). We have that \(M^{{\tilde{S}}'} = M^i_{\iota _i} = M^{i'}\) and \(M^{{\tilde{T}}'} = M^j_{\iota _j} = M^{j'}\), where in both cases the second equality is ensured by the compatibility with the diagram. By compatibility with the diagram and the definition of \(M^B\), \(M^B\) and \(M^b\) have the same universe and interpret each symbol of \(\Sigma _B\) in the same way, and thus must be equal.
In the other direction, let \(M=(M^i)_{i\in Ind}\) be in \(\mathbf {Mod}^{sim}(S)\). With the second construction we extend this family of models to one compatible with \(D^{sim}(S)\). With the first construction we restrict the family obtained to the indices in Ind. Since these models are not modified by the second construction, we get back M. Thus, we have shown that the two constructions are isomorphic. \(\square \)
Theorem 2
 1.
\(\mathbf {Mod}^{iint}(S)\) and
 2.
the class of all families of models compatible with \(D^{iint}(S)\).
Proof
We prove that an isomorphism between 1. and 2. exists.
Let \(M=(M^i)_{i\in D^{iint}(S)}\) be a family of models compatible with \(D^{iint}(S)\). For each index \(i\in Ind\), \(M^i\) is a model of \(rel_L(O_i)\). Let \(N=(\beta _{Sig(O_i)}(M^i))_{i\in Ind}\), where \(\beta \) is the model reduct component of \(rel_L\) and the ith component of N is denoted \(N^i\). We need to show that for each alignment \(A_{ij} \in S\), \(N^i, N^j\models ^I A_{ij}\), that is, \(N^i, N^j\models ^I c\) for each \(c=(s_1, s_2, R)\) in \(A_{ij}\). By definition this means that \(N^i_{s_1} R^I_U N^j_{s_2}\), where U is the global universe (which we know is shared as \(D^{iint}(S)\) is connected and reducts preserve the universe). We know that, if \(b\in D^{iint}(S)\) is the node of the bridge ontology for \(A_{ij}\), \(M^b_{O_i:s_1} R^I_U M^b_{O_j:s_2}\). By compatibility with the diagram we have that \(M^i_{s_1}=M^b_{O_i:s_1}\) and \(M^j_{s_2}=M^b_{O_j:s_2}\). Using the compatibility conditions for the logic translation \( rel _L\) we get that \(N^i_{s_1} = M^i_{s_1}\) and \(N^j_{s_2} = M^j_{s_2}\), and thus \(N\in \mathbf {Mod}^{iint}(S)\).
Now let us assume that \(M=(M^i)_{i\in Ind}\) is a model in \(\mathbf {Mod}^{iint}(S)\) and the global domain is the set G. For each alignment \(A_{ij}\) we must give interpretations for the ontologies \({\tilde{S}}, {\tilde{T}}, {\tilde{S}}', {\tilde{T}}', B\) in its diagram. Using the amalgamation property of \(rel_L\), we know that there exists a unique model N of \(rel_L(O_i)\cup rel_L(O_j)\) with \(universe(N) = G\) such that, if \(N^i = N_{\mathsf {Sig}(rel_L(O_i))}\) and \(N^j = N_{\mathsf {Sig}(rel_L(O_j))}\), we have that \(\beta _{\mathsf {Sig}(O_i)}(N^i) = M^i\) and \(\beta _{\mathsf {Sig}(O_j)}(N^j) = M^j\). We take thus \(N^i\) as the model of \({\tilde{S}}\) and \(N^j\) as the model of \({\tilde{T}}\). We can interpret \({\tilde{S}}'\) as \(N^i_{\iota _i}\) and \({\tilde{T}}'\) as \(N^j_{\iota _j}\). The symbols from B that come either from \({\tilde{S}}'\) or from \({\tilde{T}}'\) are interpreted in \(M^b\) like in \(N^i_{\iota _i}\) or, respectively, like in \(N^j_{\iota _j}\). Since M is in \(\mathbf {Mod}^{iint}(S)\), we know that \(M^i_{s^k_1} R^I_G M^j_{s^k_2}\) for each \(c_k=(s^k_1, s^k_2, R)\in A_{ij}\). By the compatibility conditions of \(rel_L\), we get that \(N^i_{s^k_1} R^I_G N^j_{s^k_2}\) for each \(c_k=(s^k_1, s^k_2, R)\in A_{ij}\). But by definition \(M^b_{O_i:s^k_1} = N^i_{s^k_1}\) and \(M^b_{O_j:s^k_2} = N^j_{s^k_2}\), which means that \(M^b_{O_i:s^k_1} R^I_G M^b_{O_j:s^k_2}\). Since \((\Sigma _B,\Delta _k) \approxeq ^{iint} c_k\) we get that \(M^b\models \Delta _k\) and since k is arbitrary, we get that \(M^b\in \mathbf {Mod}(B)\).
We need to show that the two constructions are inverse to each other. Let \(M = (M^i)_{i\in D^{iint}(S)}\) be a family of models compatible with \(D^{iint}(S)\). With the first construction, we showed that by restricting to the indices in Ind and taking the reduct along \(rel_L\) we get a family of models \((N^i)_{i\in Ind}\) that is in \(\mathbf {Mod}^{iint}(S)\). By applying the second construction to this family of models where the global domain G is set to the universe U of the models in M, for each alignment \(A_{ij}\) we obtain models \(M^{\tilde{S}}\), \(M^{\tilde{T}}\), \(M^{{\tilde{S}}'}\), \(M^{{\tilde{T}}'}\) and \(M^B\) for the ontologies \({\tilde{S}}\), \({\tilde{T}}\), \({\tilde{S}}'\), \({\tilde{T}}'\) and B of the diagram of \(A_{ij}\). Let \(i,j, i', j', b\) be the indices of these nodes in \(D^{iint}(S)\). First, notice that by construction the universes of \(M^{\tilde{S}}\), \(M^{\tilde{T}}\), \(M^{{\tilde{S}}'}\), \(M^{{\tilde{T}}'}\) and \(M^B\) are all equal with U. For each symbol \(s\in {\mathbf {Symbols}}(\mathsf {Sig}(O_i))\), we know that \((M^{\tilde{S}})_s = N^i_s = M^i_s\). We know that \((M^{\tilde{S}})_{top} = universe(N^i) = M^i_{top}\). Thus \(M^{\tilde{S}}= M^i\). Then \(M^{{\tilde{S}}'} = M^{\tilde{S}}_{\iota _1} = M^i_{\iota _1}\) (because \(M^i = M^{\tilde{S}}) = M^{i'}\), where the first and third equality follow from compatibility with \(D^{iint}(S)\). In the same way we obtain that \(M^{\tilde{T}}= M^j\) and \(M^{{\tilde{T}}'} = M^{j'}\). To prove that \(M^B = M^b\), let s be a symbol in \(\Sigma _B\). If \(s = O_i:s_1\), we know that \((M^B)_{O_i:s_1} = (M^{{\tilde{S}}'})_{O_i:s_1} = (M^{i'})_{O_i:s_1} = (M^b)_{O_i:s_1}\). If \(s = O_j:s_2\), we know that \((M^B)_{O_j:s_2} = (M^{{\tilde{T}}'})_{O_j:s_2} = (M^{j'})_{O_j:s_2} = (M^b)_{O_j:s_2}\). If \(s=O_i:\top \), by compatibility with \(D^{iint}(S)\) we have that \((M^B)_{O_i:\top } = (M^{{\tilde{S}}'})_{O_i{:\top }} = (M^{\tilde{S}})_{top} = M^i_{top} = M^b_{O_i:\top }\). Similarly we can show that \((M^B)_{O_j:\top } = M^b_{O_i:\top }\). Thus \(M^B\) and \(M^b\) interpret all symbols of \(\Sigma _B\) in the same way, and since they also have the same universe they are equal.
In the other direction, let \(M=(M^i)_{i\in Ind}\) be a model in \(\mathbf {Mod}^{iint}(S)\) and let G be the global domain (that includes all universes of the models \(M^i\)). With the second construction we obtain a family of models \(N=(N^i)_{i\in D^{iint}(S)}\) compatible with \(D^{iint}(S)\), such that for each \(i\in Ind\), the reduct of \(N^i\) along \(rel_L\) is \(M^i\). By applying the first construction to N, we build a model \(M' = ((M')^i)_{i\in I}\) in \(\mathbf {Mod}^{iint}(S)\) by taking \((M')^i = \beta _{Sig(O_i)}(N^i) = M^i\), so \(M' = M\). \(\square \)
Theorem 3
 1.
\(\mathbf {Mod}^{gint}(S)\) and
 2.
the class of all families of models compatible with \(D^{gint}(S)\).
Proof
Let \(M = (M^i)_{i\in D^{gint}(S)}\) be a family of models compatible with \(D^{gint}(S)\). For each index \(i\in Ind\), \(M^i\) is an expansion of a model \((M_1)^i\) of \(rel_L(O_i)\) with interpretation of G and \(r_{O_i}\). Let \(N^i = \beta _{\mathsf {Sig}(O_i)}((M_1)^i)\) for \(i\in Ind\) and let \(N = (N^i)_{i\in I}\). Using the properties of \(rel_L\) we know that \(M^i_{top} = M^b_{O_i:\top } = universe(N^i)\) and \(M^j_{top} = M^b_{O_j:\top } = universe(N^j)\). This allows us to define \(\gamma _i = (M^i_{r_S})^{1}\) and \(\gamma _j = (M^j_{r_T})^{1}\). The axioms in the basic bridge ontology for the node b ensure that \(\gamma _i\) and \(\gamma _j\) are functions from the universes of the models \(N^i\) and, respectively, \(N^j\) to the global domain \(U = M^b_G\). We need to show that \(N^i, N^j \models _{\gamma _i, \gamma _j} (s_1, s_2, R)\) for each \(c_k = (s_1, s_2, R)\in A_{ij}\). By definition this means that \(( \gamma _i(N^i_{s_1}), \gamma _j(N^j_{s_2}) ) \in R^I_{U} \). Since \(M^b\models \Delta ^b_k\) and \((\Sigma ^b_B,\Delta ^b_k) \approxeq ^{gint} c_k\) we have that \((x, M^b_{O_i:s_1}) \in M^b_{r_S}\) and \((y, M^b_{O_j:s_2}) \in M^b_{r_T}\) implies \((x,y) \in R^I_U\). Using the compatibility of M with the diagram and the properties of \(rel_L\) we deduce that \(N^i_{s_1} = M^b_{O_i:s_1}\) and \(N^j_{s_2} = M^b_{O_j:s_2}\). Since we defined \(\gamma _i\) to be the inverse of \(M^b_{r_S}\) and \(\gamma _j\) the inverse of \(M^b_{r_T}\), we get that \(( \gamma _i(N^i_{s_1}), \gamma _j(N^j_{s_2}) ) \in R^I_{U} \).
Let \(M=((M^i)_{i\in Ind}\) in \(\mathbf {Mod}^{gint}(S)\) with equalising functions \((\gamma _i:universe(M^i)\rightarrow U)_{i\in Ind})\) and X is a set that includes the global domain U and the universes of all models \(M^i\) in M. For each alignment \(A_{ij}\) in S, we must give interpretations for the ontologies \(\tilde{S},\tilde{T},\tilde{S'},\tilde{T'}\) and B in the diagram of \(A_{ij}\). Let \(i, j, i', j', b\) be the indices of these nodes in \(D^{gint}(S)\) and we look for models \(M^i_1, M^j_1, M^{i'}_1, M^{j'}_1\) and \(M^b_1\). Using the amalgamation property of \(rel_L\), we know that there exists a unique model N of \(rel_L(O_i) \cup rel_L(O_j)\) with \(universe(N) = X\) such that, if we denote \(N^i = N_{\mathsf {Sig}(rel_L(O_i))}\) and \(N^j = N_{\mathsf {Sig}(rel_L(O_j))}\), we have that \(\beta _{\mathsf {Sig}(rel_L(O_i))}(N^i) = M^i\) and \(\beta _{\mathsf {Sig}(rel_L(O_j))}(N^j) = M^j\) (since X includes the universes of \(M^i\) and \(M^j\)). We can take thus \(M^i_1\) as the model of \(\tilde{S}\) that interprets all symbols of \(rel_L(O_i)\) as in \(N^i\), the symbol G as U and the symbol \(r_{O_i}\) as the inverse of \(\gamma _i\). Similarly, \(M^j_1\) is the model of \(\tilde{T}\) that interprets all symbols of \(rel_L(O_j)\) as in \(N^j\), the symbol G as U and the symbol \(r_{O_j}\) as the inverse of \(\gamma _j\). The models \(M^{i'}_1\) and \(M^{j'}_1\) are \(M^i_1_{\iota _i}\) and, respectively, \(M^j_2_{\iota _j}\). We have left to define the model \(M^b_1\). Its universe is X. The symbols from B that come from \(\tilde{S}\) or \(\tilde{T}\) are interpreted in \(M^b_1\) like in \(M^i_1\) or, respectively, like in \(M^j_1\). Since \(\gamma _i\) and \(\gamma _j\) are functions, their inverses are inversefunctional and right total. For each correspondence \(c_k = (s,t,R) \in A_{ij}\), we must show that \(M^b_1\models \Delta _k\), where \((\Sigma _B, \Delta _k) \approxeq ^{gint} c_k\). By definition, it suffices to show that if \(x,y \in (M^b_1)_{G}\) such that \((x, (M^b_1)_{O_i:s}) \in (M^b_1)_{r_S}\) and \((y, (M^b_1)_{O_j:t}) \in (M^b_1)_{r_T}\), we have that \((x, y) \in R^I_{(M^b_1)_G}\). But using the definitions of the interpretations of the symbols and of the relations \(r_S\) and \(r_T\) in \(N^b\), this amounts to showing that \((\gamma _i(M^i_s), \gamma _j(M^j_t)) \in R^I_U\) and this holds because M is a model for S.
We need to show that the two constructions are inverse to each other. Let \(M = (M^i)_{i\in D^{gint}(S)}\) be a family of models compatible with \(D^{gint}(S)\). With the first construction we restrict to indices in Ind and take for each such index i the reduct of \(M^i_{\mathsf {Sig}(rel_L(O_i))}\) along \(rel_L\) to obtain a family of models \(N=(N^i)_{i\in Ind}\) that is in \(\mathbf {Mod}^{gint}(S)\) with the equalising functions \(\gamma _i: M^i_{O_i:\top } \rightarrow M^i_G\) defined by \(\gamma _i(x) = y\) iff \((y,x) \in M^i_{r_{O_i}}\). The compatibility of M with \(D^{gint}(S)\) ensures that G is interpreted in the same way in all models and that \(\gamma _i\) are indeed functions. With the first construction for N and the universe of \(M^i\), which we know is the same for each \(i\in D^{gint}(S)\) because \(D^{gint}(S)\) is connected, we obtain a family of models \(M_1\) compatible with \(D^{gint}(S)\). We must show that \(M = M_1\). Let \(i,j, i', j', b\) be the nodes in \(D^{gint}(S)\) of the diagram of the alignment \(A_{ij}\). First notice that by construction the universes of the models in \(M_1\) are equal with the universes of models in M. Moreover, \((M_1)^i_G = M^i_G\) and \((M_1)^i_{r_{O_i}} = M^i_{r_{O_i}}\). For each \(s\in {\mathbf {Symbols}}(\mathsf {Sig}(O_i))\), \((M_1)^i_s = N^i_s = M^i_s\) and moreover \((M_1)^i_{top} = universe(N^i) = M^i_{top}\). Thus \((M_1)^i = M^i\). Similarly \((M_1)^j = M^j\). Thus using the compatibility with \(D^{gint}(S)\) we get \((M_1)^{i'} = (M_1)^i_{\iota _i} = M^i_{\iota _i} = M^{i'}\) and the same for \(j'\). Finally, \(M^b\) and \((M_1)^b\) are equal because by construction they interpret symbols in the same way as in equal models.
In the other direction, let \(M=((M^i)_{i\in Ind})\) be a model in \(\mathbf {Mod}^{gint}(S)\) with equalising functions \((\gamma _i: universe(M^i)\rightarrow U)_{i\in Ind}\). The set X that we use as a parameter for the universe is \(U \cup \cup _{i\in Ind} universe(M^i)\). With the second construction for M and X we obtain a family of models \(N=(N^i)_{i\in D^{gint}(S)}\) compatible with \(D^{gint}(S)\) such that for each \(i\in Ind\) the reduct of \(N^i_{\mathsf {Sig}(rel_L(O_i))}\) along \(rel_L\) is \(M^i\). Thus, by applying the first construction to N, we get back M. \(\square \)
Theorem 4
 1.
\(\mathbf {Mod}^{con}(S)\) and
 2.
the class of all families of models compatible with \(D^{con}(S)\).
Proof
Let \(M=(M^i)_{i\in D^{con}(S)}\) be a family of models compatible with \(D^{con}(S)\). For each index \(i\in Ind\), \(M^i\) is a model of \(rel_L(O_i)\). If \(\beta \) is the model reduct component of \(rel_L\), let \(N=(\beta _{Sig(O_i)}(M^i))_{i\in Ind}\) and let \(N^i\) be the ith component of N. Using the property of \(rel_L\) for each alignment \(A_{ij}\) in S, we have that \(M^i_{top} = M^b_{O_i:\top } = universe(N_i)\) and similarly \(M^j_{top} = M^b_{O_j:\top } = universe(N_j)\), where b is the node of the bridge of \(A_{ij}\). This allows us to define \(r_{ji} = M^b_{r_{TS}}\). We need to show that for each alignment \(A_{ij} \in S\), \(N^i, N^j\models ^C c\) for each \(c=(s_1, s_2, R)\) in \(A_{ij}\). By definition this means \(N^i_{s_1} R^I_U r_{ji}(N^j_{s_2})\), where U is the universe of \(N^i\). But \(N^i_{s_1} =\) (by compatibility properties of \(rel_L\)) \(M^i_{s_1} =\) (by compatibility with the diagram) \(M^b_{O_i:s_1}\). Similarly, \(N^j_{s_2} = M^b_{O_j:s_2}\). Thus, taking into account also the definition of \(r_{ji}\), we need to show that \( M^b_{O_i:s_1} R^I_{M^b_{O_i:\top }} M^b_{r_{TS}}(M^b_{O_j:s_2}) \) This follows from \(M^b\models \Delta \), which in particular means that \(M^b\models \Delta _i\), where \((\Sigma _B,\Delta _i) \approxeq ^{con}c\).
Now, let \(M=\{(M^i)_{i\in Ind}, (r_{ij})_{i,j\in Ind}\}\) be a contextualised model in \(\mathbf {Mod}^{con}(S)\) and let U be a set that includes \(\cup _{i\in Ind} universe(M^i)\). We must define a family of models compatible with \(D^{con}(S)\). For each alignment \(A_{ij}\) we must give models for the ontologies \({\tilde{S}}, {\tilde{T}}, {\tilde{S}}', {\tilde{T}}'\) and B in its diagram. Using the amalgamation property of \(rel_L\), there is a unique model N of \(rel_L(O_i)\cup rel_L(O_j)\) with \(universe(N) = U\) and such that, if we denote \(N^i = N_{\mathsf {Sig}(rel_L(O_i))}\) and \(N^j = N_{\mathsf {Sig}(rel_L(O_j))}\), we have that \(\beta ^{rel_L}_{\mathsf {Sig}(O_i)}(N^i) = M^i\) and \(\beta ^{rel_L}_{\mathsf {Sig}(O_j)}(N^j) = M^j\). This gives us the needed models \(N^i\) for \({\tilde{S}}\) and \(N^j\) for \({\tilde{T}}\). The models for \({\tilde{S}}'\) and \({\tilde{T}}'\) are \(N^i_{\iota _i}\) and, respectively, \(N^j_{\iota _j}\). The model of B is obtained by interpreting each symbol coming from \({\tilde{S}}'\) like in \(N^i_{\iota _i}\), each symbol coming from \({\tilde{T}}'\) like in \(N^j_{\iota _j}\), the relativised top concepts \(O_i:\top \) as \(universe(N_i)\) and \(O_j:\top \) as \(universe(N_j)\), and the relation \(r_{TS}\) as \(r_{ji}\). Thus, we get that the resulting model \(N^b\) is indeed a \(\Delta \)model: we need to show that \(N^b_{O_i:s_1} R^I_{N^b_{O_i:\top }} N^b_{r_{TS}}(N^b_{O_j:s_2})\), which, with a similar argument as for the previous implication, amounts to \(M^i_{s_1} R^I_{universe(M^i)} r_{ji}(M^j_{s_2})\), and this holds because \(\{(M^i)_{i\in Ind}, (A_{ij})_{i,j\in Ind}\} \in \mathbf {Mod}^{con}(S)\).
We need to show that the two constructions are inverse to each other. Let \(M = (M^i)_{i\in D^{con}(S)}\) be a family of models compatible with \(D^{con}(S)\). With the first construction, we showed that by restricting to the indices in I and taking the reduct along \(rel_L\) we get a family of models \((N^i)_{i\in Ind}\) and relations \(r_{ji} = M^b_{r_{TS}}\) for each \(A_{ij}\in S\) where b is the node of the bridge ontology of \(A_{ij}\) such that \(\{(M^i)_{i\in Ind}, (r_{ij})_{i,j\in Ind}\} \in \mathbf {Mod}^{con}(S)\). By applying the second construction to this family of models and to the universe U of \(M^i\), which we know is unique by compatibility with the diagram, for each alignment \(A_{ij}\) we obtain models \(M^{\tilde{S}}\), \(M^{\tilde{T}}\), \(M^{{\tilde{S}}'}\), \(M^{{\tilde{T}}'}\) and \(M^B\) for the ontologies \({\tilde{S}}, {\tilde{T}}, {\tilde{S}}', {\tilde{T}}'\) and B of the diagram of \(A_{ij}\). Let \(i,j, i', j', b\) be the indices of these nodes in \(D^{con}(S)\). First, notice that the universes of \(M^{\tilde{S}}\), \(M^{\tilde{T}}\), \(M^{{\tilde{S}}'}\), \(M^{{\tilde{T}}'}\) and \(M^B\) are all equal with U. For each symbol \(s\in {\mathbf {Symbols}}(\mathsf {Sig}(O_i))\), we know that \((M^{\tilde{S}})_s = N^i_s = M^i_s\). We know that \((M^{\tilde{S}})_{top} = universe(N^i) = M^i_{top}\). Thus \(M^{\tilde{S}}= M^i\). Then \(M^{{\tilde{S}}'} = M^{\tilde{S}}_{\iota _1} = M^i_{\iota _1} (\text { because } M^i = M^{\tilde{S}}) = M^{i'}\), where the first and third equality follow from compatibility with \(D^{con}(S)\). In the same way we obtain that \(M^{\tilde{T}}= M^j\) and \(M^{{\tilde{T}}'} = M^{j'}\). To prove that \(M^B = M^b\), let s be a symbol in \(\Sigma _B\). If \(s = O_i:s_1\), we know that \((M^B)_{O_i:s_1} = (M^{{\tilde{S}}'})_{O_i:s_1} = (M^{i'})_{O_i:s_1} = (M^b)_{O_i:s_1}\). If \(s = O_j:s_2\), we know that \((M^B)_{O_j:s_2} = (M^{{\tilde{T}}'})_{O_j:s_2} = (M^{j'})_{O_j:s_2} = (M^b)_{O_j:s_2}\). If \(s = O_i:\top \), \((M^B)_{O_i:\top } = (M^{\tilde{S}})_{top} = M^i_{top} = M^b_{O_i:\top }\) and similarly for \(s = O_j:\top \). If \(s = r_{TS}\), \((M^B)_{r_{TS}} = r_{ji} = M^b_{r_{TS}}\). Thus \(M^B\) and \(M^b\) interpret all symbols of \(\Sigma _B\) in the same way, and since they also have the same universe they are equal.
In the other direction, let \(M=(M^i)_{i\in Ind}\) be a model in \(\mathbf {Mod}^{con}(S)\). With the second construction where \(U= \cup _{i\in I}universe(M^i)\), we obtain a family of models \(N=(N^i)\) compatible with \(D^{con}(S)\), such that for each \(i\in Ind\), the reduct of \(N^i\) along \(rel_L\) is \(M^i\). By applying the first construction to N, we build a model \(M' = ((M')^i)_{i\in Ind}\) in \(\mathbf {Mod}^{con}(S)\) by taking \((M')^i = \beta _{Sig(O_i)}(N^i) = M^i\), so \(M' = M\). \(\square \)
7.2 General Representation Theorems
We will now summarise the previous theorems to a general representation theorem. In the following we denote by \(\mathbf {Mod}^{sem}(S)\) the class of models of S using the semantics sem and \(D^{sem}(S)\) the corresponding diagram of S, where sem is one of sim, iint, gint, con.
Theorem 5
Proof
By inspecting the proofs of the representation theorems, we can see that the isomorphism is given by \(\beta ^*\).\(\square \)
The colimit ontology \(C^{sem}(S)\) of \(D^{sem}(S)\) exactly represents models of networks in the following sense:
Corollary 1
If we only have weak amalgamation, the construction still works, but it is not necessarily an isomorphism:
Corollary 2
Let S be a \({\textsf {NeO}} \), \(D^{sem}(S)\) its diagram and \(C^{sem}(S)\) the colimit. In all four semantics, for each model C of \(C^{sem}(S)\), \((\beta ^*(C_{\mu _i}))_{i\in Ind}\) is in \(\mathbf {Mod}^{sem}(S)\). Moreover, if the underlying logic is weakly amalgamable (or at least has weak amalgamation of pushouts), for each model \((M^i)_{i\in Ind}\) in \(\mathbf {Mod}^{sem}(S)\), there is a model C of \(C^{sem}(S)\) such that \(\beta ^*(C_{\mu _i}) = M^i\), where \(\mu _i\) is the structural morphism of the colimit of \(D^{sem}(S)\), for each \(i\in Ind\).
In the heterogeneous case, to be able to state representation theorems we must adapt the definitions in Sect. 4 to a heterogeneous setting. With this modification, the proofs of the representation theorem follow using similar arguments as for their homogeneous counterparts, and therefore we do not spell them in detail here. Moreover, since in general we can obtain just a weakly amalgamable cocone for a heterogeneous diagram, Corollary 1 does not hold in all cases, while Corollary 2 still holds in the heterogeneous case, where \(C^{sem}(S)\) is now a weakly amalgamable cocone for \(D^{sem}(S)\).
7.3 Structuring and Nonmonotonic Constructs in DOL
So far, we have assumed that all ontologies used in networks of alignments are flat, i.e. consist of a signature and a flat set of axioms. In reality, large ontologies are formulated in a more structured way. For example, both \(\mathsf {OWL}\) and \(\mathsf {DOL}\) provide an import construct, such that existing ontologies can be reused, avoiding repetition. \(\mathsf {DOL}\) also provides more complex structuring constructs for ontologies such as union, forgetting or module extraction, as well as combination (=colimit) of networks. However, these constructs can be flattened out, which means that any structured ontology can be transformed into an equivalent flat, i.e. unstructured ontology. Hence, one can perform the flattening before applying the above representation theorems (and the reasoning mechanisms of Sect. 8).
\(\mathsf {DOL}\) also provides nonflattenable constructs such as hiding (restriction to an export interface) or minimisation. The latter imposes a nonmonotonic closedworld assumption and is equivalent to McCarthy’s circumscription. Details can be found in [59, 62, 68]. These constructs cannot be flattened. Still, we can make them interact with our diagram and colimit constructions as follows: each nonflattenable ontology in a NeO is replaced by its signature. Moreover, the colimit of the diagram for the NeO is united (using a \(\mathsf {DOL}\) union) with each thus replaced ontology, renamed (using \(\mathsf {DOL}\) renaming) along the injection of its signature into the colimit. Of course, the resulting ontology is a nonflattenable ontology again. This means that the reasoning mechanisms of Sect. 8 are generally not directly applicable. Instead, more specialised reasoning has to be invoked, depending on the nature of the involved \(\mathsf {DOL}\) constructs. Note that our approach still allows the elimination of (networks of) alignments for such reasoning tasks. That is, the remaining problem is that of general reasoning for nonflattenable \(\mathsf {DOL}\) constructs (which is beyond the scope of this paper).
In a similar way, other nonmonotonic logics such as Flogic,^{37} RIF or answer set programming can be combined with our framework. However, the nonmonotonic aspects of these logics can possibly not be expressed in \(\mathsf {DOL}\), but rather require more specialised nonmonotonic constructs.
8 Reasoning with Networks of Ontologies
In the previous section we have established a number of representation results showing that under mild conditions (logics admitting amalgamation), the model class of a normalised network is isomorphic to the model class of its colimit.
As a result, functional networks as well as relational and general networks (via normalisation) can be represented by their colimit. This implies in particular, as we will see below, that determining the logical consequences of a network can be reduced to reasoning in the colimit ontology of the network, a single ontology. This reduction will give an important handle on tool reuse, as we will discuss in greater detail in Sect. 8.6.
8.1 Logical Consequence in Networks of ontologies
We define logical consequence in a NeO as follows:
Definition 43
The four theorems above imply that we can use the colimit to reason about the consequences of a NeO.
Corollary 3
Proof
Assume \(S\models ^{sem}(i,\varphi )\).
Let C be a model of \(C^{sem}(S)\) and let \(M^i=\beta ^*(C_{\mu _i})\). Then by Corollary 2, \((M^i)_{i\in Ind}\) is in \(\mathbf {Mod}^{sem}(S)\). Hence by the assumption \(S\models ^{sem}(i,\varphi )\), we obtain \(\beta ^*(C_{\mu _i})=M^i\models \varphi \). By the satisfaction condition for \(\alpha ^*\) and \(\beta ^*\), we get \(C_{\mu _i}\models \alpha ^*(\varphi )\), and by that for \(\mu _i\), we finally obtain \(C\models \mu _i(\alpha ^*(\varphi ))\). Since C is arbitrarily chosen, we get \(C^{sem}(S)\models \mu _i(\alpha ^*(\varphi )\).
Conversely, assume \(C^{sem}(S)\models \mu _i(\alpha ^*(\varphi ))\). We want to show \(S\models ^{sem}(i,\varphi )\). Let \((M^i)_{i\in Ind}\) be a model of S according to the chosen semantics sem. By weak amalgamation (or by Proposition 1 in case we have only weak amalgamation of pushouts), we can apply Corollary 2 and obtain a model C of the colimit \(C^{sem}(S)\) such that \(\beta ^*(C_{\mu _i}) = M^i\) for each \(i\in D^{sem}(S)\). By the assumption, we obtain \(C\models \mu _i(\alpha ^*(\varphi ))\). By the satisfaction condition for \(\mu _i\) and that for \(\alpha ^*\) and \(\beta ^*\), we get that \(M^i=\beta ^*(C_{\mu _i})\models \varphi \). But this is just \((M^i)_{i\in Ind}\models (i,\varphi )\). \(\square \)
One drawback of reasoning in the colimit ontology is that it includes all symbols and all sentences of the ontologies in the network, and this may lead to inefficient proof search. On the other hand, we can take advantage of the fact that the structure of the NeO has been explicitly specified as a \(\mathsf {DOL}\) network to reason locally, in just one ontology of the network, instead of globally, in the colimit, whenever the goal involves only symbols of the respective ontology.
Proposition 3
Proof
Let \((M^i)_{i\in Ind}\) be a model in \(\mathbf {Mod}^{sem}(S)\). Then \(M^i\) is a model of \(O_i\). By the assumption, \(M^i\models \varphi \). \(\square \)
The converse of Proposition 3 does not hold in general:
Example 47
The converse of Proposition 3 does hold in case of conservativity:
Definition 44
A NeO S is conservative for some ontology \(O_i\) in S, if each model \(M^i\) of \(O_i\) can be expanded to a model \((M^i)_{i\in Ind}\) of S. \(\square \)
Proposition 4
Proof
Assume \(S\models ^{sem}(i,\varphi )\). Let \(M^i\) be a model of \(O_i\). By conservativity, \(M^i\) can be expanded to a model \((M^i)_{i\in Ind}\) in \(\mathbf {Mod}^{sem}(S)\). By assumption, \((M^i)_{i\in I}\models (i,\varphi )\), that is, \(M^i\models \varphi \). \(\square \)
In a sense, this shows that the network S does not propagate any new information into ontology \(O_i\), and hence from the perspective of \(O_i\), the network is not needed. (Note however that still \(O_i\) could propagate information into other ontologies.)
For the next result, we need conservativity of theory morphisms.
Definition 45
A theory morphism \(\sigma :O_1\rightarrow O_2\) is (modeltheoretically) conservative if each \(O_1\) model \(M^1\) has a \(\sigma \)expansion to an \(O_2\)model.
We can reduce conservativity of functional networks to ordinary conservativity of ontology extensions, which are special theory morphisms:
Proposition 5
In institutions with amalgamation (amalgamation of pushouts), conservativity of a functional (connected functional) network S for \(O_i\) is equivalent to conservativity of the inclusion ontology morphism \(\mu _i:O_i\rightarrow C^{sem}(S)\), for each of the four semantics sem.
Proof
By Corollary 1. \(\square \)
Example 47 shows a typical information flow in a network of aligned ontologies. On the network level we obtain new consequences as a result of the alignment of the involved ontologies, which is often the purpose of the alignment and intended, however can also be a surprise to an ontology developer and trigger a revision cycle. In any case, in many application areas new consequences due to an alignment are a desired effect, and conservativity should not be expected to hold in general. Another property, however, that is desirable in general is satisfiability (consistency) of the obtained network, i.e. the existence of a distributed model. We will study this property in the next section.
8.2 Satisfiability
An important property of NeO s is their satisfiability, i.e. existence of a distributed model. Unsatisfiability of a NeO indicates contradictory knowledge in the associated ontologies, or alignments that do not match with the intended meaning of the aligned symbols.
Satisfiability can be shown using the colimit:
Proposition 6
Let S be a \({\textsf {NeO}} \). Satisfiability of the colimit \(C^{sem}(S)\) is sufficient for satisfiability of S. In case of (weak) amalgamation (of at least pushouts), it is also necessary.
Proof
Immediate from Corollary 2. \(\square \)
Let S be a NeO.
Definition 46

simplesatisfiable if there is a simple model of S.

inclusiveintegratedsatisfiable if there is an inclusive integrated model of S.

generalintegratedsatisfiable if there is a general integrated model of S

contextualisedsatisfiable if there is a contextualised model of S.\(\square \)
First, notice that if S is inclusiveintegratedsatisfiable, it is also generalintegratedsatisfiable, as the first notion is more restrictive.
The following proposition allows us to reduce the problem of checking satisfiability of a NeO using any of the four semantics to checking its satisfiability using simple semantics. This has the advantage of generating a less complex colimit ontology, as the relativisations are not involved.
Proposition 7
If S is simplesatisfiable, it is also inclusiveintegratedsatisfiable (and therefore also generalintegratedsatisfiable) as well as contextualisedsatisfiable.
Proof
Let us assume that S is simplesatisfiable. Then there exists a model \((M^i)_{i\in Ind}\) in \(\mathbf {Mod}^{sim}(S)\) and we know that \(universe(M^i) = universe(M^j) = U\) for any \(i,j\in Ind\). Then \(((M^i)_{i\in Ind}, (\mu _i)_{i\in Ind} )\), where \(\mu _i = id_U\) for each \(i \in Ind\), is an inclusive integrated model of S, since for any \(i,j \in Ind\) and any \((s_1,s_2, R) \in A_{ij}\), we have that \(\mu _i(M^i_{s_1}) R^I_U \mu _j(M^j_{s_2})\), because \(M^i_{s_1} R^I_U M^j_{s_2}\). \(((M^i)_{i\in Ind}, (r_{ij})_{i,j\in Ind})\) is a contextualised model for S, where for each \(i, j\in Ind\) \(r_{ij} = \{(x,x) \mid x \in U\}\), since for any \(i,j \in Ind\) and any \((s_1,s_2, R) \in A_{ij}\), we have that \(M^i_{s_1} R^I_U M^j_{s_2}\). \(\square \)
For \(\mathsf {OWL}\) we can also prove that if S is inclusiveintegratedsatisfiable, then it is also simplesatisfiable. The result does not hold in general, e.g. not in \(\mathsf {FOL}\), where quantifiers now range over a larger universe, possibly changing the satisfaction of formulas.
8.3 Reasoning About Correspondences
In Sect. 8.1, we have considered the question whether an individual sentence is a logical consequence of a network of ontologies. Now we extend this to correspondences:
Definition 47
For reasoning about correspondences, we can use the representation theorems of Sect. 7.1 as well. Recall that for a NeO S, in Sect. 7.1 we have constructed the diagram \(D^{sem}(S)\) by collecting all Wdiagrams corresponding to the alignments in S. Recall from Definition 28 that the bridge ontology of such a Wdiagram contains, for each correspondence in the alignment, an internalisation \((\Sigma _B,\Delta )\) according to Definition 31.
Proposition 8
Proof
Assume that \(S\models ^{sem}(i,j,c)\). Let C be a model of the colimit \(C^{sem}(S)\). Let \(M^k=\beta ^*(C_{\mu _k})\) for \(k\in D(S)\). Then by Corollary 2, \((M^k)_{k\in Ind} \in \mathbf {Mod}^{sem}(S)\). Hence by the assumption \(S\models ^{sem}(i,j,c)\), we get that \(M^i,M^j\models ^{sem} c\). By compatibility with the diagram and \((\Sigma ^b,\Delta ) \approxeq ^{sem} c\), this means that \(C_{\mu _b}\models \Delta \). By the satisfaction condition, we finally obtain \(C\models \mu _b(\Delta )\).
Conversely, assume \(C^{sem}(S)\models \mu _b(\Delta )\). We want to show \(S\models ^{sem}(i,j,c)\). Let \((M^k)_{k\in Ind}\) be a model of S according to the chosen semantics sem. By weak amalgamation (or by Proposition 1 in case we have only weak amalgamation of pushouts), we can apply Corollary 2 and obtain a model C of the colimit \(C^{sem}(S)\) such that \(\beta ^*(C_{\mu _k}) = M^k\) for each \(k\in D^{sem}(S)\). By the assumption, we obtain \(C\models \mu _b(\Delta )\). By the satisfaction condition, we get \(C_{\mu _b}\models \Delta \). By \((\Sigma ^b,\Delta ) \approxeq ^{sem} c\) and compatibility with the diagram, we get \(M^i,M^j\models ^{sem} c\), that is, \((M^i)_{i\in Ind}\models ^{sem}(i,j,c)\). \(\square \)
All this easily generalises from single correspondences to sets of these, i.e. alignments. The proof carries over as such for the case of heterogeneous alignments, except that now the morphisms \((\mu _k)_{k\in Ind}\) are heterogeneous signature morphisms and the satisfaction condition is the one in the Grothendieck logic.
8.4 Complexity of Reasoning
In the previous subsections, we have reduced reasoning about a NeO S to reasoning in the colimit ontology \(C^{sem}(S)\), see especially Corollary 3 and Proposition 8. Since reasoning in the colimit ontology can be done using standard tools, this reduction provides us with a reasoning mechanism for NeO s. In the following, we will analyse the computational cost associated with computing the colimit and the resulting complexity of reasoning in the colimit.
In [64], we have examined how colimits in typical signature categories are constructed. Here, “typical” means setlike, that is, signatures consists of several components that are sets of symbols, possibly kinded. This is indeed the case for the vast majority of logical formalisms. A closer inspection of the colimit selection methods in [64] reveals the following:
Proposition 9
Let A be an alphabet of letters, and \(A^*\) the set of words over A. Let the class of setlike categories over \(A^*\) be the smallest class containing \(\mathbb {S}et(A^*)\) (the category of subsets of \(A^*\) and functions between them), and being closed under products and Grothendieck constructions.^{41} Then the number of symbols in a colimit ontology is linear in the number of symbols of the ontologies in the network (diagram). Moreover, colimit computation in such categories requires at most quadratic time.
Proof
The number of symbols is linear because a colimit is a quotient of a disjoint union. Since the equivalence classes in the quotient have at most linear size, the overall computation time is at most quadratic.\(\square \)
Proposition 9 in particular applies to \(\mathsf {OWL}\), unsorted FOL and sorted FOL. Let us add that for practical example ontologies, the size of equivalence classes is usually less than a small constant (also for large ontologies), and hence complexity is even linear. Since Proposition 9 only concerns the colimit computation, we additionally need to take the complexity of computing the Wdiagrams (including the bridge ontology) into account. However, an inspection shows that this is also linear.
The complexity of reasoning even in weak logics can be very high. For example, logical consequence in propositional logic is coNPcomplete. For propositional Horn logic, it is in polynomial time. The standard reasoning problems in the DL \(\mathcal {SROIQ}\) are known to be N2ExpTimecomplete and in the DL \(\mathcal {SHIQ}\) they are NExpTimecomplete [43]. Firstorder logic is known to be undecidable. In all these cases, the computation of the colimit is dominated by the complexity of reasoning. Moreover, the size of the colimit is linear in the size of the input ontologies. This means that reasoning in, e.g. a \(\mathcal {SROIQ}\)based NeO is (like that in \(\mathcal {SROIQ}\) itself) N2ExpTimecomplete. Of course, in the case when a correspondence is not internalisable in the logic of the aligned ontologies, complexity will typically increase when internalising the semantics of that correspondence in a more expressive logic (for \(\mathcal {SROIQ}\), this will not happen for simple semantics). Moreover, when the merging^{42} of different \(\mathcal {SROIQ}\) ontologies is no longer expressible in \(\mathcal {SROIQ}\), complexity can increase as well (even become undecidable). This is not a limitation of our approach, but a problem of reasoning in NeO s in general.
In order to test the running time of our approach in practice, we have tried our reasoning method with the examples from the biomedical domain available at the Bioportal repository.^{43} Bioportal provides about 400, 000 correspondences (called “mappings”). We have organised these into DOL alignments,^{44} organised the alignments into networks, and then tried to show satisfiability of these networks. By Proposition 6, it suffices to compute the colimit ontology and then use a standard \(\mathsf {OWL}\) reasoner to show its satisfiability (w.r.t. simple semantics). However, one major problem was the immense size of many colimits, resulting from the immense size of some input ontologies (some of them 200–500 MB large). Therefore, we decided to use modules of networks, as developed in the next subsection (see especially Corollary 4). This was more successful. Detailed results will be reported in a separate paper.
8.5 Modules of Networks
We have shown that reasoning in a network amounts to reasoning in a colimit ontology. However, the colimit ontology can become very large. A smaller colimit can be obtained by working with modules. A module of an ontology captures all the knowledge that the ontology has about a certain signature \(\Sigma \). Technically, this is ensured by requiring that the whole ontology is a \(\Sigma \)conservative extension of the module. This will be explored now. We restrict ourselves to modules in the modeltheoretic sense, that is, modules using the modeltheoretic notion of conservative extension.
Definition 48
Let O be an ontology and \(\Sigma \subseteq \mathsf {Sig}(O)\). A subontology \(O'\subseteq O\) is called a \(\Sigma \) module for O, if for each \(O'\)model \(M'\), there is an Omodel M with \(M_{\Sigma }=M'_{\Sigma }\). \(\square \)
Definition 49
Let \(S = ((O_i)_{i\in Ind}, (A_{ij})_{i,j\in Ind})\) be a NeO. A NeO \(S' = ((O'_i)_{i\in Ind}, (A_{ij})_{i,j\in Ind})\) is called a module of S if for each \(i\in Ind\), \(O'_i\) is a \(\Sigma _i\)module of \(O_i\),^{45} where \(\Sigma _i\) is the union of all interface signatures related to \(O_i\) stemming from those Wdiagrams occurring in the normalisation of S. \(\square \)
Note that due to the choice of the interface signatures \(\Sigma _i\), each \(A_{ij}\) is still an alignment between \(O'_i\) and \(O'_j\).
The colimit of such a module of a network is generally much smaller than that of the network itself. This smaller colimit can be used for reasoning about inconsistency or about logical consequences, as far as they are formulated in the signature of the smaller colimit:
Proposition 10
Let \(S = ((O_i)_{i\in I}, (A_{ij})_{i,j\in I})\) be a NeO and \(S'= ((O'_i)_{i\in I}, (A'_{ij})_{i,j\in I})\) be a module of S. Then S is a modeltheoretic conservative extension of \(S'\), which means that any \(S'\)model \(((M')^i)_{i\in I}\) can be expanded to an Smodel \((M^i)_{i\in I}\) such that \(M^i_{Sig(O'_i)}=(M')^i\) for all \(i\in I\).
Proof
Given an \(S'\)model \(((M')^i)_{i\in I}\), for each \(i\in I\), \(O'_i\) is a \(\Sigma _i\)module of \(O_i\), where \(\Sigma _i\) is defined as in Definition 49. Hence, \((M')^i\) can be expanded to a model of \(O_i\). Then \((M^i)_{i\in I}\) is a model of S. \(\square \)
Corollary 4
Let \(S'\) be a module of a NeO S. Then satisfiability of S is equivalent to that of \(S'\). \(\square \)
We now show that reasoning over a relational network is equivalent to reasoning over any module.
Proposition 11
Proof
“\(\Rightarrow \)”: Assume \(S\models ^{sem} (i,\iota (e))\) and let \(((M')^i)_{i\in I}\) in \(\mathbf {Mod}^{sem}(S')\). Let \((M^i)_{i\in Ind}\) be an expansion to a model in \(\mathbf {Mod}^{sem}(S)\). By assumption, \(M^i\models \iota (e)\). By the satisfaction condition, \((M')^i\models e\).
“\(\Leftarrow \)”: Assume \(S'\models ^{sem} (i,e)\) and let \((M^i)_{i\in Ind}\) in \(\mathbf {Mod}^{sem}(S)\). Then \((M^i_{Sig(O'_i)})_{i\in Ind}\) is in \(\mathbf {Mod}^{sem}(S')\). By assumption, \(M^i_{Sig(O'_i)}\models e\). By the satisfaction condition, \(M^i\models \iota (e)\). \(\square \)
8.6 Tool Support
DOL is supported by Ontohub (https://ontohub.org, [12, 47]), a Webbased repository engine for managing distributed heterogeneous ontologies. The backend of Ontohub is the Heterogeneous Tool Set Hets [63] which is used for parsing, static analysis and proof management of ontologies. Hets supports alignments and combinations: according to the theory in Sects. 5 and 6, it generates the normalised network of an alignment according to the assumption made on the domain,^{46} and can automatically compute colimits of ontologies in different logics, including \(\mathsf {OWL}\), firstorder logic and multisorted firstorder logic.
Example 48
Figure 1 shows the network of the alignment in Example 28 displayed in Ontohub after analysing the specification of the alignment. The outer nodes form the Wshaped alignment while the central node, labelled with Open image in new window , is the colimit ontology of the NeO. The node Open image in new window is automatically computed by Ontohub via Hets. \(\square \)
Example 49
Figure 2 shows the network of the alignment in Example 32 in Hets. Note that the two ontologies were relativised and they are linked to their relativisation via a double arrow, which in Hets denotes a heterogeneous signature morphism (here, the relativisation translation is applied). The Wshaped diagram of the alignment is visible in the lower part of the figure. The colimit is in the upper part of the figure and was computed automatically by Hets.^{47}
Note that both Hets and Ontohub support reasoning in ontologies via interfacing with standard reasoners, such as Pellet and Hermit (for OWL) and SPASS and Vampire (for firstorder logic). In particular, this means that reasoning in the colimit ontology is possible. By the results of the previous subsections, this means that reasoning for networks is available, too. Implementation of computation of modules is under way; for the initial experiments mentioned above, we have done this computation in an ad hoc way.
9 Extending the Framework
A number of related combination techniques are in principle covered by our approach, in particular those often referred to as modular or distributed ontology languages (see [15, 77] for overviews of some of these languages).^{48} We here discuss only how to deal with the most wellknown cases, namely DDL [5], \(\mathcal {E}\)connections [48], DFOL [25], and multicontext systems (MCS) in general [26]. Note that for all of them, future work is required to incorporate their more expressive alignment or bridging constructs into our abstract framework.

\(i:C {\mathop {\longrightarrow }\limits ^{\sqsubseteq }} j:D\) (‘into’ bridge rule)

\(i:C {\mathop {\longrightarrow }\limits ^{\sqsupseteq }} j:D\) (‘onto’ bridge rule)

\(i:a \mapsto j:b\) (individual correspondence)

\(i:a {\mathop {\longrightarrow }\limits ^{=}} \{j:b_1,\ldots , j:b_n\}\) (complete correspond.)

\(M^i, M^j \models i:C {\mathop {\longrightarrow }\limits ^{\sqsubseteq }} j:D\) iff \(r_{ij}(M^i_C) \subseteq M^j_D\)

\( M^i, M^j \models i:C {\mathop {\longrightarrow }\limits ^{\sqsupseteq }} j:D\) iff \(r_{ij}(M^i_C) \supseteq M^j_D\)

\(M^i, M^j \models i:a \mapsto j:b\) iff \(r_{ij}(M^i_a, M^j_b)\)

\(M^i, M^j \models i:a {\mathop {\longrightarrow }\limits ^{=}} \{j:b_1,\ldots , j:b_n\}\) iff \(r_{ij}(M^i_a) = \{M^j_{b_1},\ldots ,M^j_{b_n}\}\).
Moreover, notice that the contextualised semantics for alignments, introduced in Sect. 4.3, is defined relative to the universe of the source ontology of the alignment, while in the case of DDL, the correspondence relation is defined over the domain of the target ontology. This requires a syntactic modification of DDL alignments in order to capture correctly their semantics within our framework. Namely, we must reverse the source and the target ontology and the direction of the arrows in the bridge rules. For the first three kinds of bridge rules directly supported in our current setting, this transformation then trivially generates valid bridge rules. Alternatively, we could modify the definition in Sect. 4.3 to work from the perspective of the target ontology. We have decided not to do so, in order to stay consistent with the generic definitions in [80].
Similarly, the methodology of \(\mathcal {E}\)connections fits perfectly well into the contextualised semantics paradigm (\(\mathcal {E}\)connections introduce symbols interpreted as relations between the universes of the combined ontologies), however their significantly more expressive alignment constructs and syntactic heterogeneity pose problems.
\(\mathcal {E}\)connections are introduced over abstract description systems, termbased languages that abstract from the specific syntaxes of many typical modal and description logics, and are interpreted over a uniform semantics based on setvalued terms and setvalued functions.^{50} Moreover, \(\mathcal {E}\)connections allow for a variety of bridge logics, essentially variants of typical description logics, including universal and existential restrictions (essentially equivalent to the ‘into’ and ‘onto’ bridge rules introduced for DDL in the case of connecting DLbased ontologies), counting quantifiers, and Boolean operations on links. Grothendieck institutions allow us to manage this kind of heterogeneity. The difficulty is that the expressivity of the language of links (in our terminology, the logic of the bridge ontology in the Wshaped network of the alignment) should be allowed to vary by making a flexible selection of DLlike operators that can be applied to the domain relations. Correspondingly, the computation of heterogeneous colimits will depend on these choices. Moreover, \(\mathcal {E}\)connections generalise the DDLbased paradigm in two other important aspects. First, they allow any number of linking relations between two given domains, which implies that our framework for specifying domain relations needs, for instance, to be extended with means to specify sets of such relations. Secondly, in an network of ontologies based on the \(\mathcal {E}\)connections paradigm, it is also possible to have nary relations between n participating ontologies, with the corresponding obvious generalisations of the alignment operations (see [48] for details). Such an extension requires more significant efforts regarding for instance internalisation of the now nary bridge logic.
Regarding DDL, we have shown that the basic case of standard bridge rules for description logics (into/onto rules and individual correspondences) is directly covered by our approach. As shown in [15, 48], such basic DDLs can be also simulated by basic \(\mathcal {E}\)connections. More precisely, a suitable specialisation of \(\mathcal {E}\)connections to DLbased ontologies, binary relations, and a more basic bridge language, is essentially equivalent to basic DDL. ‘Basic’ is here meant in the technical sense of not involving nonstandard bridge rules, i.e. those beyond the standard ‘into’ and ‘onto’ bridge rules (see [5]). Indeed, the case of such nonstandard bridge rules is the more interesting one. This includes heterogeneous bridge rules relating for instance symbols of different arity (e.g. reifying a relation into a concept by a projection operation) [27], handling default rules [8], or rules for controlling inconsistency propagation [37].
Similarly, distributed firstorder logic (DFOL) [25] can also be incorporated into our general framework with the following qualifications. First, DFOL has a semantics very similar to contextualised semantics for alignments. We can therefore fix the parameters of how the network of the alignment is constructed, namely relativisation of ontologies. Second, we can then obtain the bridge axioms from the correspondences stated in a DFOLalignment. Unlike in the case of \(\mathcal {E}\)connections, in DFOL the bridge language remains in the firstorder logic paradigm.
10 Conclusions
Our theoretical contributions to the foundations of ontology alignment and combination of alignments in networks are intended to provide an abstract, unified view of the most prominent approaches to giving semantics to alignments.
 (1)
the finding/discovery of alignments (based either on human manual effort, or on (semi)automatic structural, linguistic, statistical or logical methods),
 (2)
the construction of the aligned ontology (a global, colimitbased approach, or a localised, distributed approach), and
 (3)
reasoning over the aligned network of ontologies, respectively, debugging, repair and revision of the ontologies or the alignment relations (see e.g. [41, 42, 56]), closing the loop to (1).
Regarding (2), platforms such as Bioportal (with about 400,000 correspondences in 2016) illustrate that mappings between ontologies, ontology modules, and the concepts and definitions living in them, are of great importance to support reuse. They are regarded of particular importance in areas such as clinical and medical terminology, where terminologies are vast, not uniform across user groups, but essential in practice [70]. The importance of heterogeneity in data formats and formal languages used, resulting not in small part from the diversity of user groups and stakeholders involved in the biomedical domains, has been stressed in [50, 76]. In the case of Bioportal, the \(\mathsf {DOL}\) language allows to declaratively manage sets of alignments, and to give them precise semantics. We have started to apply the approach of this paper to Bioportal, see https://ontohub.org/bioportal_mappings for \(\mathsf {DOL}\) formalisations of the Bioportal “mappings” (which are correspondences) as networks of alignments.
The importance of alignments has also been well demonstrated for foundational ontologies in the repository ROMULUS [44]. In the case of ROMULUS, it allows to align ontologies such as Dolce or BFO expressed in firstorder logic with OWL versions of the same ontology. This is a feature that no other alignment framework currently covers and is already supported by the Ontohub platform.
Ontologies developed in the area of applied ontology, in particular foundational and upper ontologies, often need expressiveness beyond \(\mathsf {OWL}\); here, the multilogic nature of Ontohub is essential to allow to host and align these ontologies. A good example for these novel capabilities is given by the recent FOUST initiative (’The FOUndational STance’),^{51} an effort to build a curated, digital library hosted on Ontohub to provide authoritative formalised versions of the leading foundational ontologies (including BFO, DOLCE, GFO, GUM, UFO and YAMATO).^{52} This will include variants of these ontologies given, e.g. in OWL, FOL and HOL, as well as formally establishing their relationships based on theory interpretation and alignment. That is, a standard task is the heterogeneous reasoning problem to establish that the translation of an ontology written in a ‘weaker dialect’ is logically entailed by the version written in the ‘stronger dialect’. Moreover, standard alignment techniques are not sufficient in terms of expressivity to cover the more substantial meaning shifts that occur across terms used in various foundational ontologies. The extensions of our framework discussed in Sect. 9 are therefore essential for the longterm success of a project such as FOUST.
Regarding (3), alignment tools such as LogMap [41], ALCOMO [56] or AgreementMakerLight [22], employ reasoning over aligned ontologies and repair either parts of the input ontologies or revise the mappings (one technique to enable this is to reencode the mappings into a global OWL ontology) to restore global consistency. Using \(\mathsf {DOL}\) and the reasoning capabilities of the Hets/Ontohub ecosystem [47, 63], such tools could be used to directly operate on a NeO, and to update the network structure accordingly.
Note that our approach to reasoning is a global one: the colimit of a NeO integrates all ontologies and all alignments that are part of the NeO. This is in contrast to distributed approaches (like, e.g. DRAGO [74] for DDL or DRAOn [53, 82] for IDDL),^{53} where local reasoning is combined with propagation rules along bridge rules. Distributed approaches can be more efficient than ours in specific cases, although having typically the same worstcase complexity as our global approach (see Sect. 8.4). However, we believe that the global approach in combination with the use of modules of networks (Sect. 8.5) can scale down reasoning times significantly, and thus provide a generic and competitive approach to reasoning in networks of ontologies. Moreover, our approach has the advantage of being very general: it is applicable to the four different semantics of networks of alignments and also to heterogeneous ontologies. The distributed approaches do not provide reasoning methods for all these cases, and they would need ad hoc extensions to cover them. By contrast, we provide a systematic approach that can easily be instantiated to new logics. The approach presented here provides an integration of the major paradigms of ontology alignment into one coherent framework. This currently covers standard alignment relations and basic DDL bridge rules, and, with suitable generalisations as outlined in Sect. 9, also \(\mathcal {E}\)connections, PDL, IDDL, and DFOL.
We provide a generic framework for alignments, orthogonal to the local logics used in the ontologies participating in a network. To instantiate our framework for a particular logic, it suffices to provide constructions of the bridge ontologies and relativisation translations for that logic. Moreover, \(\mathsf {DOL}\) ’s support for heterogeneity allows us not only to handle heterogeneous alignment, but also to move to a more expressive logic when a bridge axiom cannot be expressed in the local logic of the ontologies. Thus, we have seen that contextualised semantics for OWL, without restrictions on correspondences, can be given in our framework.
Future work includes the combination of different alignment paradigms within one network (as principally enabled by our unifying framework) and an integration of techniques for the revision of NeOs [19] into \(\mathsf {DOL}\). In our setting, the propagation of detected repairs into a network could be done by updating the alignment mappings and recomputing the alignment networks. Further work is also needed for the problem of reasoning about the consequences of a NeO; here we expect module extraction to provide an increase in performance of proof search.
We currently pursue to add support for the full \(\mathcal {E}\)connections bridge logic and other major combination paradigms to \(\mathsf {DOL}\) as discussed in the previous section. Proof support for such combinations can either be realised via logic translations to already supported logics, or by adding dedicated reasoning procedures such as [66, 74] to the Hets system. At the tool level, the integration of the four semantics for alignments in Ontohub is currently in progress. Ontohub is already compatible with the OWL API, and its potential for interoperability is increased further by the integration of the Alignment API. Also, scalability to alignments among large ontologies (such as those in Bioportal) is a challenge that should be addressed by future work. While some initial experiments using modules of networks have been promising, we still have to solve more downtoearth problems, e.g. regarding the organisation of prefix maps in the colimit.
Footnotes
 1.
This paper is an extended version of [14]. While in [14] we present the semantics of alignments for the particular case of \(\mathsf {OWL}\), in the present paper we introduce a general construction, independent of the underlying logical formalism. This means that the results of [14] can be obtained as an instantiation of the results of this paper. Note however that we needed to enhance the formalisation of the notion of logical system in order to achieve this goal.
 2.
We do not claim here that the reasoning methods we provide outperform more specialised alignment reasoning methods, say for DDL, or alignment debugging: our main contribution is the provision of a unifying framework that works simultaneously at the various levels.
 3.
Note that this paper does not cover search for alignments.
 4.
\(\mathsf {DOL}\) has been adopted as a standard by the Object Management Group (OMG), see http://www.omg.org/spec/DOL/ and http://dolomg.org.
 5.
Since we split integrated semantics into two versions, we have four semantics, and not just three as in [80].
 6.
Referring to the intuition of categories of graphs, a functor is just a graph homomorphism. However note that also the monoidal composition of arrows must be preserved.
 7.
A functor is faithful if it is injective when restricted to each set of morphisms that have a given source and target.
 8.
According to the \(\mathsf {OWL}\) 2 specification, names of \(\mathsf {OWL}\) symbols should be IRIs; see https://www.w3.org/TR/owl2syntax/.
 9.
See http://www.w3.org/TR/owl2overview/. Note, however, that OWL also includes datatypes, which further enrich the logic. We omit these because they are inessential for the presentation of the logical framework presented in this paper and would unnecessarily complicate exposition.
 10.
An approach overcoming this limitation could easily be used to reason about \(\mathsf {OWL}\) ontologies featuring, e.g. irreflexive transitive roles, which is however undecidable.
 11.
Note that \(\rightarrow \) associates to the right, i.e. we have \(=_t:t\rightarrow (t\rightarrow Bool)\).
 12.
Note that \(=_t\) suffices to define the other constants.
 13.
Strictly speaking, \(\mathbb {C}at\) is not a category but only a socalled quasicategory, which is a category that lives in a higher settheoretic universe [33].
 14.
Model reducts are known from model theory. If \(\varphi \) is an inclusion of a subsignature \(\Sigma _1\) into a larger signature \(\Sigma _2\), then \(M^2_{\varphi }\) is the restriction of model \(M^2\) to \(\Sigma _1\), forgetting those model components interpreting symbols in \(\Sigma _2{\setminus }\Sigma _1\).
 15.
Normally, \(\mathsf {MSFOL}\) models do not feature such a universe. However, for our technical results, having a universe is needed, and it does not change the logic essentially: given a standard manysorted model, it is always possible to let \(M_U\) be the union of all carrier sets.
 16.
Normally, \(\mathsf {HOL}\) models do not feature such a universe, but the same remark as for \(\mathsf {MSFOL}\) applies.
 17.
That is, with the same objects as the original category.
 18.
That is, for each family of signatures \(\mathbb {S} \in \mathbf {Sign}\), we have that \(\mathbf {Sen}(\bigcap \mathbb {S}) = \bigcap _{\Sigma \in \mathbb {S}} \mathbf {Sen}(\Sigma )\).
 19.
Actually, here we only need the existence of an initial (“empty”) signature.
 20.
Recall that a functor is similar to a graph homomorphism.
 21.Given two functors \(F,G:\mathbf {A}\rightarrow \mathbf {B}\), a natural transformation \(\eta :F\rightarrow G\) compares the image of F with that of G by linking them with suitable arrows. More precisely, a natural transformation \(\eta :F\rightarrow G\) consists of a family of arrowssuch that for any \(f:A_1\rightarrow A_2\) in \(\mathbf {A}\), the following diagram commutes (i.e. \(\eta _{A_1};Gf=Ff;\eta _{A_2}\)):$$\begin{aligned} (\eta _A:FA\rightarrow GA)_{A\in \mathbf {A}}, \end{aligned}$$
 22.
Although the intuitions behind relativisations are stable, and for specific logics there usually is a standard form of canonical relativisation, the specifics vary with the syntactic variations of a given logic, and this notion can therefore not be stated fully logicindependently.
 23.
For example, in Example 19, instead of a unary top predicate, one could introduce a binary one, but only use it in the form top(x, x).
 24.
\(\mathsf {DOL}\), in accordance with the Alignment API, has further syntax for cardinality of alignments, which however is not relevant here.
 25.
 26.
 27.
We use the following notations: \(r_{21}(C) = \{ x\in D_1 \mid (y,x)\in r_{21} \text { for some } y\in C \}\) if \(C\subseteq D_2\) and \(r_{21}(R) = \{ (x,y) \mid x,y\in D_1, \exists x',y' \in D_2 \text { with } (x',y')\in R \text { and } (x',x), (y',y)\in r_{21} \}\) if R is a relation on \(D_2\).
 28.
The last four axioms involve general concept inclusions, which can be expressed in OWL, but not in Manchester syntax. We have taken the liberty to keep using Manchester syntax for them.
 29.
Again, this is not valid Manchester syntax, but expressible in \(\mathcal {SROIQ}\) (general concept inclusion).
 30.
Actually, integrated semantics was originally motivated by the need for heterogeneous alignments (J. Euzenat, personal communication).
 31.
We make the simplifying assumption that the logics \(J_i\) of the theories that internalise the semantics of all correspondences are the same. In practice, when this is not the case, there exists a logic J and logic translations \(\gamma _i:J_i\rightarrow J\), for each i, and the theory obtained by translating \((\Sigma _i,\Delta _i)\) along \(\gamma _i\) also internalises the semantics of \(c_i\).
 32.
For example, the first theory is the abbreviation of \(\forall x, y~.~(\exists z_1, z_2~.~r_S(x, z_1) \wedge r_1(z_1, z_2) \wedge r_S(y, z_2)) \iff (\exists w_1, w_2~.~r_T(x, w_1) \wedge r_2(w_1, w_2) \wedge r_T(y, w_2))\).
 33.
 34.
As described in http://www.loa.istc.cnr.it/old/Papers/D18.pdf.
 35.
 36.
Empty bridges, corresponding to empty alignments, can be removed as long as the diagram stays connected.
 37.
See [61] for the formalisation of the monotonic part of Flogic as an institution.
 38.
The \(\mathsf {DOL}\) syntax is: entailment e = i in S entails { \(\varphi \) }, where e is some name for the entailment.
 39.
The proposed \(\mathsf {DOL}\) syntax is: entailment e = i,j in S entails c, where e is some name for the entailment.
 40.
Note that \(\Sigma ^b\) is \(\Sigma _B\), whose definition depends on the choice of sem and was introduced in the corresponding subsection of Sect. 5.
 41.
With these, typed symbols (like in manysorted FOL) can easily be realised.
 42.
That is, a quotient of a disjoint union, as done by the colimit. However note that the problem already arises with simple unions.
 43.
 44.
 45.
The proposed \(\mathsf {DOL}\) syntax for a module of a network S is extract S.
 46.
Full support for all four semantics is currently in progress, the implementation of simple semantics for alignments in Hets is stable.
 47.
The names Open image in new window of the nodes in the diagram of the alignment, as well as the names Open image in new window and Open image in new window for the relativisation of ontologies are generated during the analysis of the alignment Open image in new window .
 48.
These are extensively studied in the Workshops on Modular Ontologies (WoMO), see http://iaoa.org/womo.
 49.
See [60] for an institution independent treatment of terms.
 50.
ADS were first introduced and extensively used in [2] to prove transfer results for fusions of modal and description logics.
 51.
 52.
To give some more detail, DOLCE is available in various DL versions, in FOL, and in an extension using quantified modal logic (see http://www.loa.istc.cnr.it/old/DOLCE.html); GFO is currently in FOL, BFO versions exist in OWL DL, OBO, Isabelle, and CLIF (a syntax for Common Logic [58]) (see https://github.com/BFOontology/BFO), GUM exists in OWL DL and CASL (see http://www.ontospace.unibremen.de/ontology/gum.html), UFO is written in higherorder logic (Coq/Gallina), and YAMATO in FOL and OWL.
 53.
Notes
Acknowledgements
We would like to thank Jérôme Euzenat and Fabian Neuhaus for extensive discussions of ideas found in this paper. We also thank the anonymous reviewers for their substantial feedback and for suggesting a number of improvements, both on a technical level and regarding the presentation of results.
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