On bisimulations for description logics

doi:10.1016/j.ins.2014.10.022

Information Sciences

Volume 295, 20 February 2015, Pages 465-493

https://doi.org/10.1016/j.ins.2014.10.022 Get rights and content

Abstract

We study bisimulations for useful description logics. The simplest among the considered logics is ${ALC}_{reg}$ (a variant of PDL). The others extend that logic with the features: inverse roles, nominals, qualified number restrictions, the universal role, and/or the concept constructor for expressing the local reflexivity of a role. They also allow role axioms. Our contributions are as follows. We propose to treat named individuals as initial states and give an appropriate bisimulation condition for that. We also give bisimulation conditions for the universal role and the concept constructor $\exists r . Self$ . We prove that all of the bisimulation conditions for the features can be combined together to guarantee invariance of concepts and the Hennessy-Milner property for the whole class of studied description logics. We address and give results on invariance or preservation of ABoxes, RBoxes and knowledge bases in description logics. Independently from Lutz et al. [26], [27] we also give results on invariance of TBoxes. We introduce a new notion called QS-interpretation, which is needed for dealing with minimizing interpretations in description logics with qualified number restrictions and/or the concept constructor $\exists r . Self$ . We formulate and prove results on minimality of quotient interpretations w.r.t. the largest auto-bisimulations. We adapt Hopcroft’s automaton minimization algorithm to give an efficient algorithm for computing the partition corresponding to the largest auto-bisimulation of a finite interpretation in any description logic of the considered family. Using the invariance results we compare the expressiveness of the considered description logics w.r.t. concepts, TBoxes and ABoxes. Our results about separating the expressiveness of description logics are naturally extended to the case when instead of ${ALC}_{reg}$ we have any sublogic of ${ALC}_{reg}$ that extends $ALC$ .

Introduction

Description logics (DLs) are variants of modal logic. They are of particular importance in providing a logical formalism for ontologies and the Semantic Web. DLs represent the domain of interest in terms of concepts, individuals, and roles. A concept is interpreted as a set of individuals, while a role is interpreted as a binary relation among individuals. A DL is characterized by a set of concept constructors, a set of role constructors, and a set of allowed forms of role axioms and individual assertions. A knowledge base in a DL usually has three parts: an RBox consisting of axioms about roles, a TBox consisting of terminology axioms, and an ABox consisting of assertions about individuals. The basic DL $ALC$ allows basic concept constructors listed in Table 1, but does not allow role constructors nor role axioms. The most common additional features for extending $ALC$ are also listed in Table 1.

Given two individuals in an interpretation, sometimes we are interested in the question whether they are “similar” or not, i.e., whether they are indiscernible w.r.t. the considered description language. Indiscernibility is used, for example, in machine learning. In DLs, it is formally characterized by bisimulation. Roughly speaking, two individuals are indiscernible iff they are bisimilar.

Bisimulations arose in modal logic [37], [38], [39] and state transition systems [19], [30]. They were introduced by van Benthem under the name p-relation in [37], [38] and the name zigzag relation in [39]. Bisimulations reflect, in a particularly simple and direct way, the locality of the modal satisfaction definition. The famous Van Benthem Characterization Theorem states that modal logic is the bisimulation invariant fragment of first-order logic. Bisimulations have been used to analyze the expressivity of a wide range of extended modal logics (see, e.g., [4] for details). In state transition systems, bisimulation is viewed as a binary relation associating systems which behave in the same way in the sense that one system simulates the other and vice versa. Kripke models in modal logic are a special case of labeled state transition systems. Hennessy and Milner [19] showed that weak modal languages could be used to classify various notions of process invariance. In general, bisimulations are a very natural notion of equivalence for both mathematical and computational investigations.¹

Bisimilarity between two states is usually defined by three conditions (the states have the same label, each transition from one of the states can be simulated by a similar transition from the other, and vice versa). As shown in [4], the four program constructors of PDL (propositional dynamic logic) are “safe” for these three conditions. That is, we need to specify the mentioned conditions only for atomic programs, and as a consequence, they hold also for complex programs. For bisimulation between two pointed-models, the initial states of the models are also required to be bisimilar. When converse is allowed (the case of CPDL), two additional conditions are required for bisimulation [4]. Bisimulation conditions for dealing with graded modalities were studied in [10], [11], [22]. In the field of hybrid logic, the bisimulation condition for dealing with nominals is well known (see, e.g., [1]).

In this paper we study bisimulations for the family of DLs which extend ${ALC}_{reg}$ (a variant of PDL) with an arbitrary combination of inverse roles, qualified number restrictions, nominals, the universal role, and the concept constructor $\exists r . Self$ for expressing the local reflexivity of a role. Inverse roles are like converse modal operators, qualified number restrictions are like graded modalities, and nominals are as in hybrid logic.

The topic is worth studying due to the following reasons:

1.
Despite that bisimulation conditions are known for PDL and for some features like converse modal operators, graded modal operators and nominals, we are not aware of previous work on bisimulation conditions for the universal role and the concept constructor $\exists r . Self$ . More importantly, without proofs one cannot be sure that all the conditions can be combined together to guarantee standard properties like invariance and the Hennessy-Milner property.
There are many papers on bisimulations, but just a few on bisimulations in DLs:
- •
  In [23] Kurtonina and de Rijke studied expressiveness of concept expressions in some DLs by using bisimulations. They considered a family of DLs that are sublogics of the DL $ALCNR$ , which extend $ALC$ with (unqualified) number restrictions and role conjunction. They did not consider individuals, nominals, qualified number restrictions, the concept constructor $\exists r . Self$ , the universal role, and the role constructors like the program constructors of PDL.
- •
  In [27] Lutz et al. characterized the expressiveness of TBoxes in the DL $ALCQIO$ and its sublogics, including the lightweight DLs such as DL-Lite and $EL$ . They also studied invariance of TBoxes and the problem of TBox rewritability. The logic $ALCQIO$ lacks the role constructors of PDL, the concept constructor $\exists r . Self$ and the universal role.
- •
  In [16] we studied comparisons between interpretations in DLs with respect to “logical consequences” of the form of semi-positive concepts (like semi-positive concept assertions). Such comparisons are called bisimulation-based comparisons and characterized by conditions similar to the ones of bisimulations defined in the current paper. The problems studied in [16] are: preservation of semi-positive concepts with respect to comparisons, the Hennessy-Milner property for comparisons, and minimization of interpretations that preserves semi-positive concepts.
- •
  Bisimulation-based concept learning in DLs was studied in [13], [18], [28], [34], [35], [36]. A survey on these papers is presented in Section 7.3.
The family of DLs studied in this work is large and contains useful DLs. Not only concept constructors and role constructors are allowed, but role axioms are also allowed. In particular, the DL $SROIQ$ , which is the logical basis of the Web Ontology Language OWL 2, belongs to this class.
2.
DLs differ from other logics like modal logics and hybrid logics in the domain of applications and the settings. In DLs, there are special notions like named individual, RBox, TBox, ABox. Also, recall that a knowledge base in a DL usually consists of an RBox, a TBox and an ABox. Invariance of ABoxes and preservation of RBoxes and knowledge bases in DLs were not studied before. On the other hand, invariance of TBoxes was studied in the independent work [26], [27] for the DL $ALCQIO$ and its sublogics. The works [26], [27] use the notion of global bisimulation to characterize invariance of TBoxes, whose condition is the same as the bisimulation conditions introduced in [15] and the current paper for the universal role.
3.
Bisimulation is a very useful notion for DLs. Apart from analyzing expressiveness of DLs, it can be used for minimizing interpretations and concept learning in DLs:
- •
  Roughly speaking, two objects that are bisimilar to each other can be merged. This is the basis for minimizing interpretations. In automated reasoning in DLs, sometimes we want to return a model of a knowledge base (e.g., as a counterexample for a subsumption problem or an instance checking problem). It is expected that the returned model is simple and as small as possible. One can just find some model and minimize it. As another example, given an information system specified by an acyclic knowledge base with a large ABox and a small TBox, one can compute that information system and minimize it to save space and increase efficiency of reasoning tasks.
- •
  Concept learning in DLs is similar to binary classification in traditional machine learning. The difference is that in DLs objects are described not only by attributes but also by relationships between the objects. As bisimulation is the notion for characterizing indiscernibility of objects in DLs, it is useful for concept learning in DLs.

In this paper we present conditions for bisimulation in a uniform way for the whole considered family of DLs. For this, we introduce bisimulation conditions for the universal role and the concept constructor $\exists r . Self$ . A special point of our approach is that named individuals are treated as initial states, which requires an appropriate condition for bisimulation. Our bisimulation condition for qualified number restrictions is simpler than the ones given for graded modalities in [10], [11]. It is weaker than the one given for counting modalities in [22], but is strong enough to guarantee the Hennessy-Milner property for the class of finitely branching (image-finite) interpretations. We prove the standard invariance property (Theorem 3.4) and the Hennessy-Milner property (Theorem 4.1) and address the following problems:

•
When is a TBox invariant for bisimulation? (Corollary 3.5 and Theorem 3.6)
•
When is an ABox invariant for bisimulation? (Theorem 3.7)
•
What can be said about preservation of RBoxes w.r.t. bisimulation? (Theorem 3.8)
•
What can be said about invariance or preservation of knowledge bases w.r.t. bisimulation? (Theorem 3.8)

Furthermore, we give results (Theorem 5.3, Theorem 5.4, Theorem 5.5, Theorem 5.7, Theorem 5.8) on the largest auto-bisimulation of an interpretation in a DL, the quotient interpretation w.r.t. that equivalence relation, and minimality of such a quotient interpretation. To deal with minimizing interpretations for the case when the considered logic allows qualified number restrictions and/or the concept constructor $\exists r . Self$ , we introduce a new notion called QS-interpretation, which is needed for obtaining expected results.

Computing the largest auto-bisimulations in modal logics and state transition systems is standard like Hopcroft’s automaton minimization algorithm [20] and the Paige-Tarjan algorithm [29]. By adapting Hopcroft’s automaton minimization algorithm, we give an efficient algorithm for computing the partition corresponding to the largest auto-bisimulation of a finite interpretation in any DL of the considered family. The adaptation involves the allowed constructors of the considered DLs.

Using the invariance results we compare the expressiveness of the considered DLs w.r.t. concepts, TBoxes and ABoxes. Our results about separating the expressiveness of DLs are naturally extended to the case when instead of ${ALC}_{reg}$ we have any sublogic of ${ALC}_{reg}$ that extends $ALC$ .

In comparison with [16], note that [16] deals with semi-positive concepts, while the current paper deals with (general) concepts, RBoxes, TBoxes, ABoxes and knowledge bases. The overlap between [16] and the current paper is small.

The rest of this paper is structured as follows. In Section 2 we present notation and semantics of the DLs considered in this paper. In Section 3 we define bisimulations in those DLs and give our results on invariance and preservation w.r.t. such bisimulations. In Section 4 we give our results on the Hennessy-Milner property of the considered DLs. Section 5 is devoted to auto-bisimulation and minimization. Section 6 is devoted to computing the partition corresponding to the largest auto-bisimulation of a finite interpretation. Section 7 is devoted to applications of bisimulations. In particular, in Section 7.1 we present our results about separating the expressiveness of DLs w.r.t. concepts, TBoxes and ABoxes, in Section 7.2 we discuss applications of interpretation minimization, and in Section 7.3 we present a survey on bisimulation-based concept learning in DLs. Section 8 concludes this work. All proofs of the results in Sections 2 Notation and semantics of description logics, 3 Bisimulations and invariance results, 4 The Hennessy-Milner property, 5 Auto-bisimulation and minimization, 6 Minimizing interpretations are presented in the appendix.

Section snippets

Notation and semantics of description logics

Our languages use a countable set $Σ_{C}$ of concept names (atomic concepts), a countable set $Σ_{R}$ of role names (atomic roles), and a countable set $Σ_{I}$ of individual names. Let $Σ = Σ_{C} \cup Σ_{R} \cup Σ_{I}$ . We denote concept names by letters like A and B, denote role names by letters like r and s, and denote individual names by letters like a and b.

We consider some (additional) DL-features denoted by I (inverse), O (nominal), Q (qualified number restriction), U (universal role), $Self$ . A set of DL-features is a set

Bisimulations and invariance results

Let $I$ and $I^{'}$ be interpretations. A binary relation $Z \subseteq Δ^{I} \times Δ^{I^{'}}$ is called an $L_{Φ}$ -bisimulation between $I$ and $I^{'}$ if the following conditions hold for every $a \in Σ_{I}, A \in Σ_{C}, r \in Σ_{R}, x, y \in Δ^{I}, x^{'}, y^{'} \in Δ^{I^{'}}$ : $Z (a^{I}, a^{I^{'}})$ $Z (x, x^{'}) \Rightarrow [A^{I} (x) \Leftrightarrow A^{I^{'}} (x^{'})]$ $[Z (x, x^{'}) \land r^{I} (x, y)] \Rightarrow \exists y^{'} \in Δ^{I^{'}} [Z (y, y^{'}) \land r^{I^{'}} (x^{'}, y^{'})]$ $[Z (x, x^{'}) \land r^{I^{'}} (x^{'}, y^{'})] \Rightarrow \exists y \in Δ^{I} [Z (y, y^{'}) \land r^{I} (x, y)],$ if $I \in Φ$ then $[Z (x, x^{'}) \land r^{I} (y, x)] \Rightarrow \exists y^{'} \in Δ^{I^{'}} [Z (y, y^{'}) \land r^{I^{'}} (y^{'}, x^{'})]$ $[Z (x, x^{'}) \land r^{I^{'}} (y^{'}, x^{'})] \Rightarrow \exists y \in Δ^{I} [Z (y, y^{'}) \land r^{I} (y, x)],$ if $O \in Φ$ then $Z (x, x^{'}) \Rightarrow [x = a^{I} \Leftrightarrow x^{'} = a^{I^{'}}],$ if $Q \in Φ$ then $\begin{matrix} if Z (x, x^{'}) holds then, for every role name r, there exists a \end{matrix}$

The Hennessy-Milner property

An interpretation $I$ is finitely branching (or image-finite) w.r.t. $L_{Φ}$ if, for every $x \in Δ^{I}$ and every basic role R in $L_{Φ}$ , the set ${y \in Δ^{I} | R^{I} (x, y)}$ is finite.

Let $I$ and $I^{'}$ be interpretations, and let $x \in Δ^{I}$ and $x^{'} \in Δ^{I^{'}}$ . We say that x is $L_{Φ}$ -equivalent to $x^{'}$ if, for every concept C in $L_{Φ}, x \in C^{I}$ iff $x^{'} \in C^{I^{'}}$ .

Theorem 4.1 The Hennessy-Milner Property

Let $I$ and $I^{'}$ be finitely branching interpretations (w.r.t. $L_{Φ}$ ) such that, for every $a \in Σ_{I}, a^{I}$ is $L_{Φ}$ -equivalent to $a^{I^{'}}$ . Suppose that if $U \in Φ$ then $Σ_{I} \neq \emptyset$ and either both $I, I^{'}$ are finite or both $I, I^{'}$ are

Auto-bisimulation and minimization

An $L_{Φ}$ -bisimulation between $I$ and itself is called an $L_{Φ}$ -auto-bisimulation of $I$ . An $L_{Φ}$ -auto-bisimulation of $I$ is said to be the largest if it is larger than or equal to ( $\supseteq$ ) any other $L_{Φ}$ -auto-bisimulation of $I$ .

Proposition 5.1

For every interpretation $I$ , the largest $L_{Φ}$ -auto-bisimulation of $I$ exists and is an equivalence relation.

This proposition follows from Lemma 3.1.

Given an interpretation $I$ , by $\sim_{Φ, I}$ we denote the largest $L_{Φ}$ -auto-bisimulation of $I$ , and by $\equiv_{Φ, I}$ we denote the binary relation on $Δ^{I}$ with the

Minimizing interpretations

In this section, we adapt Hopcroft’s automaton minimization algorithm [20] to computing the partition corresponding to $\sim_{Φ, I}$ for the case when $I$ is finite. The partition is used to minimize $I$ to obtain $I /_{\sim_{Φ, I}}$ for the case ${Q, Self} \cap Φ = \emptyset$ , or $I /_{\sim_{Φ, I}}^{QS}$ for the other case. We do not require any restrictions on $Φ$ .

The similarity between minimizing automata and minimizing interpretations relies on that equivalence between two states in a finite deterministic automaton is similar to $L_{Φ}$ -equivalence between

Applications

As mentioned in the introduction, bisimulations have applications in analyzing expressiveness of DLs, minimizing interpretations and concept learning in DLs.

Conclusions

We have studied bisimulations in a uniform way for a large class of DLs with useful ones like the DL $SROIQ$ of OWL 2. In comparison with [23], [27], this class allows also the role constructors of PDL, the concept constructor $\exists r . Self$ and the universal role as well as role axioms. Our main contributions are the following:

•
We proposed to treat named individuals as initial states and gave an appropriate condition for bisimulation. We introduced bisimulation conditions for the universal role and the

Acknowledgments

This work was supported by the Polish National Science Center (NCN) under Grant No. 2011/01/B/ST6/02759 and the Vietnamese National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 102.05-2014.08. The first author would like to thank the Institute of Informatics, University of Warsaw for a financial support. We would also like to thank the anonymous reviewers for very helpful comments and suggestions.

References (39)

A. Borgida
On the relative expressiveness of description logics and predicate logics
Artif. Intell.
(1996)
N. Kurtonina et al.
Expressiveness of concept expressions in first-order description logics
Artif. Intell.
(1999)
Z. Pawlak et al.
Rudim. Rough Sets. Inf. Sci.
(2007)
C. Areces et al.
Hybrid logics: characterization, interpolation and complexity
J. Symb. Log.
(2001)
F. Baader
A formal definition for the expressive power of terminological knowledge representation languages
J. Log. Comput.
(1996)
L. Badea et al.
A refinement operator for description logics
P. Blackburn et al.
Modal Logic, Number 53 in Cambridge Tracts in Theoretical Computer Science
(2001)
M. Cadoli, L. Palopoli, M. Lenzerini, Datalog and description logics: expressive power, in: Proceedings of...
S.T. Cao et al.
The web ontology rule language OWL 2 RL + and its extensions
Trans. Comput. Collective Intell.
(2014)
S.T. Cao et al.
WORL: a nonmonotonic rule language for the semantic web
Vietnam J. Comput. Sci.
(2014)

W.W. Cohen, H. Hirsh, Learning the Classic description logic: theoretical and experimental results, in: Proceedings of...

W. Conradie, Definability and changing perspectives: the Beth property for three extensions of modal logic, Master’s...

M. de Rijke

A note on graded modal logic

Studia Logica

(2000)

A.R. Divroodi, Comparing the expressiveness of description logics, in: Informal Proceedings of DL’2014, CEUR Workshop...

A.R. Divroodi et al.

On Clearnability in description logics

A.R. Divroodi, L.A. Nguyen, On bisimulations for description logics, in: Proceedings of CS&P’2011, 2011, pp....

A.R. Divroodi, L.A. Nguyen, A preliminary version of the current paper, 2011....

A.R. Divroodi, L.A. Nguyen, Bisimulation-based comparisons for interpretations in description logics, in: Informal...

N. Fanizzi, C. d’Amato, F. Esposito, T. Lukasiewicz, Representing uncertain concepts in rough description logics via...

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^☆: This is an extended version of the workshop paper [14].

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On bisimulations for description logics☆

Abstract

Introduction

Section snippets

Notation and semantics of description logics

Bisimulations and invariance results

The Hennessy-Milner property

Auto-bisimulation and minimization

Minimizing interpretations

Applications

Conclusions

Acknowledgments

Artif. Intell.

Artif. Intell.

Rudim. Rough Sets. Inf. Sci.

Hybrid logics: characterization, interpolation and complexity

J. Symb. Log.

A formal definition for the expressive power of terminological knowledge representation languages

J. Log. Comput.

A refinement operator for description logics

Modal Logic, Number 53 in Cambridge Tracts in Theoretical Computer Science

The web ontology rule language OWL 2 RL + and its extensions

Trans. Comput. Collective Intell.

WORL: a nonmonotonic rule language for the semantic web

Vietnam J. Comput. Sci.

A note on graded modal logic

Studia Logica

On Clearnability in description logics