Representing and Reasoning About XML with Ontologies

Abstract

The eXtensible Markup Language (XML) has reached a wide acceptance as the relevant standardization for representing and exchanging data on the Web. Unfortunately, XML covers the syntactic level but lacks semantics, and thus cannot be directly used for the Semantic Web. Currently, finding a way to utilize XML data for the Semantic Web is challenging research. As we have known that ontology can formally represent shared domain knowledge and enable semantics interoperability. Therefore, in this paper, we investigate how to represent and reason about XML with ontologies. Firstly, we give formalized representations of XML data sources, including Document Type Definitions (DTDs), XML Schemas, and XML documents. On this basis, we propose formal approaches for transforming the XML data sources into ontologies, and we also discuss the correctness of the transformations and provide several transformation examples. Furthermore, following the proposed approaches, we implement a prototype tool that can automatically transform XML into ontologies. Finally, we apply the transformed ontologies for reasoning about XML, so that some reasoning problems of XML may be checked by the existing ontology reasoners.

Appendix: Proofs of Theorems

Proof of Theorem 1

In the following we give the proof of Theorem 1. As mentioned in Sect. 3.1.3, a XML document d conforms to a DTD D=(r,P) in Definition 1 if d∈d(D), where d(D) is a set of XML document instances over D, which can be inductively defined as follows: If r is a terminal T∈T in Definition 1, then d(r)=T∈d(D); If r is an element type E→(α,A)∈P, then d(r)∈d(D) is a set of sequences <E>d ₁…d _k </E>, where <E> and </E> are the start and end tags of the element E∈E in Definition 1, and d ₁,…,d _k are document instances satisfying the constraints of the content model (α, A).

On this basis, the following first proves the first part of Theorem 1. Let d∈d(D) be a XML document conforming to the DTD D, then a model satisfying the axioms of φ(D)μ(d)=(Δ^μ(d),•^μ(d)) can be inductively defined as follows:

• (a) If d is a terminal T∈T, then Δ^μ(d)=(φ(T))^μ(d);

• (b) If d is a sequence of the form <E>d ₁,…,d _n </E>, where d _i is an instance conforming to the content model E→(α,A) in the DTD D, then the tree-model μ(d) can be constructed as follows:

  $$\begin{aligned} &\Delta ^{\mu (d)} = \{ o,o_{b},o_{1}, \ldots,o_{n},o_{e}\} \cup \bigcup _{1 \le i \le n} \Delta ^{\mu (d_{i})} \\ &\mathit{Tag}^{\mu (d)} = \{ o_{b},o_{e}\} \cup \bigcup _{1 \le i \le n}\mathit{Tag}^{\mu (d_{i})} \\ & \mathit{Start}E^{\mu (d)} = \{ o_{b}\} \cup \bigcup _{1 \le i \le n} \mathit{Start}E^{\mu (d_{i})} \\ &\mathit{End}E^{\mu (d)} = \{ o_{e}\} \cup \bigcup _{1 \le i \le n} \mathit{End}E^{\mu (d_{i})} \\ & f^{\prime \mu ({d})} = \bigl\{ (o,o_{b}),\bigl(o_{1},o_{1}' \bigr),\ldots,\bigl(o_{n},o_{n}'\bigr)\bigr\} \cup \bigcup_{1 \le i \le n} f^{\prime \mu ({d}_{{i}})} \\ &r^{\prime \mu ({d})} = \bigl\{ (o,o_{1}),(o_{1},o_{2}), \ldots,(o_{n - 1},o_{n}),(o_{n},o_{e}) \bigr\} \\ &\hphantom{r^{\prime \mu (d)} =} \cup \bigcup_{1 \le i \le n} r^{\prime \mu (d_{i})} \end{aligned}$$

  where: (i) an class identifier Tag is used to represent all the tags in the XML document d; (ii) for each element type E∈E, two class identifiers StartE and EndE are used to represent the start tag and end tag of E, respectively; (iii) o denotes the root element of d, o _b and o _e denote the start and end tags of the root element, respectively, o _i denotes the ith component of d, and $o_{i}'$ denotes the root element of d _i , i∈{1,…,n}; (iv) in the model μ(d), f′ and r′ are used to denote the property identifiers, where f′ represents the start tag of an element and r′ represents the other components of the element in the tree structure of the XML document d.

The second part of Theorem 1 can be proved similarly for the first part mentioned above, and they are a mutually inverse process. Given a model of $\varphi (D) \mathcal{I} = (\Delta^{ \mathcal{I}}, \bullet^{ \mathcal{I}})$ and an object $o \in \varphi (r)^{ \mathcal{I}} \in \Delta^{ \mathcal{I}}$, a XML document instance λ(o) can be inductively defined as follows:

• (a) If $o \in \varphi (T)^{ \mathcal{I}}$, where φ(T) is an identifier in φ(D), then λ(o)=T∈T;

• (b) If $o \in \varphi (E)^{\mathcal{I}}$, where φ(E) is an identifier in φ(D), and there are some integer n≥0 and objects $o _{{b}}, o _{{i}}, o _{{i}}', o _{{e}}$, such that $o _{{b}} \in \mathit{StartE}^{\mathcal{I}}, o _{{e}} \in \mathit{EndE}^{\mathcal{I}}, (o, o _{{b}}), (o _{1}, o _{1}'), \dots, (o _{{n}}, o _{{n}}') \in f ^{\prime\mathcal{I}}$, and $(o, o _{1}), (o _{1}, o _{2}), \dots, (o_{n-1}, o _{{n}}), (o _{{n}}, o _{{e}} ) \in r ^{\prime\mathcal{I}}$, then λ(o)=<E>$\lambda(o _{1}')\dots \lambda(o _{{n}}')$</E> and E∈E. Moreover, as mentioned in the areas of description logics and ontologies [6, 25], a model of a description logic knowledge base or an ontology can be represented by a tree model. The tree representation of the model $\mathcal{I}$ with the object o is shown in the following Fig. 10.

  Fig. 10

  

  The tree representation of the model $\mathcal{I}$ with the object o mentioned in the second part of Theorem 1

And next we further prove that the XML document instance λ(o) exists and conforms to the DTD D. Let S be a symbol in T∪E, then $o \in \varphi (S)^{\mathcal{I}}$ if and only if λ(o) is defined and λ(o)=d(S)∈d(D). Note that, the symbol d() has been defined at the beginning of the proof of Theorem 1. We proceed by induction on the number of f′-steps on the path from o in $\mathcal{I}$ (see Fig. 10), which may be divided into two cases: (i) Base case, i.e., there are no f′-steps in the path. Then $o \in \varphi (S)^{\mathcal{I}}$ is a terminal node as shown in Fig. 10. The case is easy and direct, according to the proposed transformation approach in Sect. 4.1.1, it shows that a data range identifier φ(S) corresponds to a terminal S∈T in D. Therefore, if S=T∈T, then $o \in \varphi (S)^{\mathcal{I}}$ and also λ(o)=d(S)=T∈d(D) according to the definition of d(D) mentioned above. Otherwise $o \notin \varphi (S)^{\mathcal{I}}$ and also λ(o)∉d(D); (ii) Inductive case. There are f′-steps in the path. Let o ₁,…,o _n be the objects along the r′⋅r ^′∗-path satisfying the constraints in φ(E), and let $o _{{i}}'$ be the f′-successor of o _i , for i∈{1,…,n}, where E is an element type such that λ(o)=<E>$\lambda(o _{1}')\dots \lambda(o _{{n}}')$</E>. If there are symbols S ₁,…,S _n in T∪E such that $o _{{i}}' \in \varphi (S _{{i}})^{\mathcal{I}}$, for i∈{1,…,n}, then by induction hypothesis $\lambda(o _{{i}}') = d(S _{{i}} ) \in d(D)$, and if S∈E and the content model α generates the string S ₁…S _n , then $o \in \varphi (S)^{\mathcal{I}}$ and also λ(o)=d(S)∈d(D). Otherwise $o \notin \varphi (S)^{\mathcal{I}}$ and also λ(o)∉d(D). □

Proof of Theorem 4

Being similar to the DTD mentioned in Theorem 1, for a XML Schema and its transformed OWL ontology as shown in Sect. 4.2.1, there may be mappings between instance documents of the XML Schema and models of the transformed OWL ontology. Formally, assuming that for a XML document d conforming to the XML Schema, there is σ(d) which is a model of the transformed OWL ontology; and for a model $\mathcal{I}$ of the transformed OWL ontology, there is $\tau (\mathcal{I})$ which is a XML document conforming to the XML Schema. Therefore, the proof of this theorem can be shown as follows: If $S _{1}\not\sqsubseteq S _{2}$, according to Definition 7, we have d(S ₁)⊈d(S ₂), i.e., there is at least one XML document instance d with d∈d(S ₁) and d∉d(S ₂). According to the above assumption, σ(d) is a model of O with o∈f(r ₁)^σ(d) and o∉f(r ₂)^σ(d), where o is the individual corresponding to the root node of d. That is, O⊭f(r ₁)⊑f(r ₂), and there is a contradiction, so S ₁⊑S ₂; If O⊭f(r ₁)⊑f(r ₂), then there is $o \in \Delta^{\mathcal{I}}$ such that $o \in (f(r _{1}))^{\mathcal{I}}$ and $o \notin (f(r _{2}))^{\mathcal{I}}$, where $\mathcal{I}$ is a model of O. According to the above assumption again, τ ($\mathcal{I}$) is a XML document instance for S with τ(o)∈d(S ₁) and τ(o)∉d(S ₂). That is, $S _{1} \not\sqsubseteq S _{2}$, and there is a contradiction, so O⊨f(r ₁)⊑f(r ₂). □

Proof of Theorem 5

The proof of this theorem is similar to the proof of Theorem 4. If O⊭f(r ₁)≡f(r ₂), then there is $o \in \Delta^{\mathcal{I}}$ such that $o \in (f(r _{1}))^{\mathcal{I}}$ and $o \notin (f(r _{2}))^{\mathcal{I}}$ or $o \notin (f(r _{1}))^{\mathcal{I}}$ and $o \in (f(r _{2}))^{\mathcal{I}}$, where $\mathcal{I}$ is a model of O. According to the assumption in Theorem 4, τ ($\mathcal{I}$) is a XML document instance for S with τ(o)∈d(S ₁) and τ(o)∉d(S ₂) or τ(o)∉d(S ₁) and τ(o)∈d(S ₂). That is, $S _{1}\not\equiv S _{2}$, and there is a contradiction, so O⊨f(r ₁)≡f(r ₂); If $S _{1} \not\equiv S _{2}$, then there is at least one XML document instance d with d∈d(S ₁) and d∉d(S ₂) or d∉d(S ₁) and d∈d(S ₂). Also, according to the assumption in Theorem 4, σ(d) is a model of O with o∈f(r ₁)^σ(d) and o∉f(r ₂)^σ(d) or o∉f(r ₁)^σ(d) and o∈f(r ₂)^σ(d), where o is the individual corresponding to the root node of d. That is, O⊭f(r ₁)≡f(r ₂), and there is a contradiction, so S ₁≡S ₂. □

Proof of Theorem 6

We will work in a similar way as in the previous proofs. If O⊭f(r ₁)⊓f(r ₂)⊑⊥, then there is at least one $o \in \Delta^{\mathcal{I}}$ such that $o \in (f(r _{1}))^{\mathcal{I}}$ and $o \in (f(r _{2}))^{\mathcal{I}}$, where $\mathcal{I}$ is a model of O. According to the assumption in Theorem 4, τ ($\mathcal{I}$) is a XML document instance for S with τ(o)∈d(S ₁) and τ(o)∈d(S ₂). That is, d(S ₁)∩d(S ₂)≠∅, and there is a contradiction, so O⊨f(r ₁)⊓f(r ₂)⊑⊥; If S ₁ is not disjoint from S ₂, according to Definition 9, we have d(S ₁)∩d(S ₂)≠∅, i.e., there is at least one XML document instance d with d∈d(S ₁) and d∈d(S ₂). Also, according to the assumption in Theorem 4, σ(d) is a model of O with o∈f(r ₁)^σ(d) and o∈f(r ₂)^σ(d), where o is the individual corresponding to the root node of d. That is, O⊭f(r ₁)⊓f(r ₂)⊑⊥, and there is a contradiction, so S ₁⊗S ₂. □