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A Distance-based Paraconsistent Semantics for DL-Lite

by   Xiaowang Zhang, et al.

DL-Lite is an important family of description logics. Recently, there is an increasing interest in handling inconsistency in DL-Lite as the constraint imposed by a TBox can be easily violated by assertions in ABox in DL-Lite. In this paper, we present a distance-based paraconsistent semantics based on the notion of feature in DL-Lite, which provides a novel way to rationally draw meaningful conclusions even from an inconsistent knowledge base. Finally, we investigate several important logical properties of this entailment relation based on the new semantics and show its promising advantages in non-monotonic reasoning for DL-Lite.


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1 Introduction

The DL-Lite [1] is a family of lightweight description logics (DLs), the logical foundation of OWL 2.0 QL, one of the three profiles of OWL 2.0 for Web ontology language recommended by W3C. In description logics, an ontology is expressed as a knowledge base (KB). Inconsistency is not rare in ontology applications and may be caused by several reasons, such as errors in modeling, migration from other formalisms, ontology merging, and ontology evolution. Therefore, handling inconsistency is always considered an important problem in DL and ontology management communities. However, DL-Lite reasoning mechanism based on classical DL semantics faces problem when inconsistency occurs, which is referred to as the triviality problem. That is, any conclusions, that are possibly irrelevant or even contradicting, will be entailed from an inconsistent DL-Lite ontology under the classical semantics.

In many practical ontology applications, there is a strong need for inferring (only) useful information from inconsistent ontologies. For instance, consider a simple DL-Lite KB where and . Under the classical semantics for DLs, anything can be inferred from . Intuitively, one might wish to still infer and , while they are useless to derive both and from .

There exist several proposals for reasoning with inconsistent DL-Lite KBs in the literature. These approaches usually fall into one of two fundamentally different streams. The first one is based on the assumption that inconsistencies are caused by erroneous data and thus, they should be removed in order to obtain a consistent KB ([2, 3, 4, 5]). In most approaches in this stream, the task of repairing inconsistent ontologies is actually reduced to finding a maximum consistent subset of the original KB. A shortcoming of these approaches is the so-called multi-extension problem. That is, in many cases, an inconsistent KB may have several different sub-KBs that are maximum consistent. The other stream, based on the idea of living with inconsistency, is to introduce a form of paraconsistent reasoning or inconsistency-tolerant reasoning by employing non-standard reasoning methods (e.g., non-standard inference and non-classical semantics). [7, 8] introduce some strategies to select consistent subsets from an inconsistent KB as substitutes of the original KB in reasoning. [11] present the Belnap’s four-valued semantics of DLs where two additional logical values besides “true” and “false” are introduced to indicate contradictory conclusions. [12] present the Hunter’s quasi-classical semantics of DLs whose strong semantics strengthens the inference power of four-valued semantics. However, the reasoning capability of such paraconsistent methods is not strong enough for many practical applications. For instance, a conclusion , that can inferred from a consistent KB under the classical semantics, may become not derivable under their paraconsistent semantics. We argue that approaches in these two streams are mostly coarse-grained in the sense that they fail to fully utilize semantic information in the given inconsistent KB. For instance, when two interpretations make a concept unsatisfiable, one interpretation may be more reasonable than the other. But existing approaches to paraconsistent semantics in DLs do not take this into account usually.

Recently a distance-based semantics presented by [14] has been proposed to deal with inconsistent KBs in propositional logic. However, it is not straightforward to generalize this approach to DLs because a DL KB can have infinite number of models and a model can also be infinite. Additionally, it is also a challenge in adopting distance-based semantics for DL complex constructors.

To overcome these difficulties, in this paper we first use the notion of features [15] and introduce a distance-based semantics for paraconsistent reasoning with DL-Lite. Feature in DL-Lite are Herbrand interpretations extended with limited structure, which provide a novel semantic characterization for DLs. In addition, features also generalize the notion of types for TBoxes [16] to general KBs. Each KB in DL-Lite has a finite number of features and each feature is finite. This makes it possible to cast Arieli’s distance-based semantics to DL-Lite.

The main innovations and contributions of this paper can be summarized as follows. We introduce distances on types of KBs, which avoids the problem of domain infiniteness and model infiniteness in defining the distance in terms of models of KBs. Based on the new distance on types, we develop a way of measuring types that are closest to a TBox and the notion of minimal model types is introduced. This notion is also extended to minimal model features for KBs. We propose a distance-based semantics for so that useful information can still be inferred when a KB is inconsistent. This is accomplished by introducing a novel entailment relation (i.e. distance-based entailment) between a KB and an axiom in terms of minimal model features. Our results show that the distance-based entailment is paraconsistent, non-monotonic, cautious as the paraconsistent based on multi-valued semantics. We also show that the distance-based entailment is not over-skeptical in the sense that for a classically consistent KB, the distance-based entailment coincides with the classical entailment, which is missing in most existing paraconsistent semantics for DLs.

2 The DL-Lite Family and Features

A signature is a finite set where is the set of atomic concepts, the set of atomic roles, the set of individual names (or, objects) and the set of natural numbers in . We use capital letters ( with subscripts ) to denote concept names, (with subscripts ) to denote role names, lowercase letters to denote individual names and assume 1 is always in . and will not be considered as concept names or role names.

Formally, given a signature , the language is inductively constructed by syntax rules: r1: , r1: , r3: . We say a basic concept and a general concept. Other standard concept constructs such as , , and can be introduced as abbreviations: for , for , for and for . For any , .

A TBox is a finite set of (concept) inclusions of the form where and are general concepts. An ABox is a finite set of concept assertions and role assertions . Concept inclusions, concept assertions and role assertions are axioms. A KB is composed of a TBox and an ABox, written by . denotes the signature of .

An interpretation is a pair , where is a non-empty set called the domain and is an interpretation function such that , and . General concepts are interpreted as follows: , , and . The definition of interpretation is based on the unique name assumption (UNA), i.e., for two different individual names and .

An interpretation is a model of a concept inclusion (a concept assertion , or a role assertion ) if (, or ); and is called a model of a TBox (an ABox ) if is a model of each inclusion of (each assertion of ). is called a model of a KB if is a model of both and . We use to denote the set of models of . A KB entails an axiom , if . Two KBs and are equivalent if , denoted by . A KB is consistent if it has at least one mode, inconsistent otherwise.

As we all known, models of a KB are often infinite and the number of models of a KB is possibly infinite. To characterize infinite models in a finite expression, two important notions, namely, type and feature, are respectively defined by [16] and [15] for TBoxes and general KBs in DL-Lite.

Let be a signature. A -type (or simply a type) is a set of basic concepts over , s.t., , and for any with , , implies . As for any type , we omit it in examples for simplicity. denotes the set of all -types. Note that if (or ) occurs in a general concept then (or ) should be also considered as a new concept independent of (or ) in computing types of respectively. We say a type set as a set of types , denoted as and a type group as a set of type sets , denoted as . Then we denote and . A type satisfies a basic concept if , satisfies if does not satisfy , and satisfies if satisfies both and . denotes a collection of all -types of . In this way, each general concept over corresponds to a set of all -types satisfying . satisfies a concept inclusion if . And is a model type a TBox iff it satisfies each inclusion in . Model type sets and model type groups are analogously defined. If is a model type set of a TBox then iff . This property is called role coherence which can be used to check whether a type set is the model type set of some TBox. denotes the model type group of where is a TBox over . It appears that is the collection of model -types of .

A -Herbrand set (or simply Herbrand set) is a finite set of member assertions satisfying: (1) for each , if , where are all the concept assertions about in , then the set is a -type; (2) for each , if are all the role assertions about in , then for any with , is in ; (3) for each , if are all the role assertions in , then for any with , is in . We simply write where . Moreover, given a set of types , denotes without confusion. In this case, we say is in if . A Herbrand set satisfies a concept assertion (a role assertion or ) if is in and ( or ). A Herbrand set satisfies an ABox if satisfies all assertions in .

A -feature (or simply a feature) is a pair , where is a non-empty set of -types and a -Herbrand set, if satisfies: (1) for each , iff (i.e., holds role coherence); and (2) for each and in , s.t., is a -type, . A feature satisfies an inclusion over , if ; satisfies a concept assertion over , if and ; and satisfies a role assertion (resp., ) over , if . A feature is a model feature of KB if satisfies each inclusion and each assertion in . denotes the set of all model features of . It easily concludes that is consistent iff . Given two KBs and , let , F-entails if , written by ; and is F-equivalent if , written by . [15] conclude that: (1) iff ; (2) iff . Intuitively, F-entailment relation is equal to classical entailment relation. In the remainder of this paper, we directly use to express F-entailment relation ().

3 Distance-based Semantics for TBoxes

In this section, we introduce distances between types of TBoxes.

Definition 1 (Type Distance)

Let be a signature, a total function is a pseudo-distance (for short, distance) on if it satisfies: (1) iff ; and (2) .

Given a type and a type set , the distance between and is defined as .

If , then we set where is a default value of distance greater than any value be to considered.

There are two representative distance functions on types, namely, Hamming distance where and drastic distance where if and otherwise.

An aggregation function is a total function that accepts a multi-set of real numbers and returns a real number, satisfying: (1) is non-decreasing in the values of its argument; (2) iff ; and (3) , . There exist some popular aggregation functions (see [18]):

  • The summation function: ;

  • The maximun function: ;

  • The -voting function (): if ; if and otherwise, where is the number of zeros in . is called the voting index of .

Using aggregation functions, the distance between two types can be extended to a type and a type group.

Definition 2 (Minimal Type)

Let be a signature, a type and a type group. Given and , the distance between and is defined as . Furthermore, is called -minimal (for short, minimal) w.r.t. if for any , . Given a type set . denotes a set of all -minimal types w.r.t. in .

First, minimal types have the following simple properties.

Proposition 1

Let be a signature and a type group. For any and , we have

  • If for some (), then ;

  • If , then .

The first statement guarantees that if a type group contains a non-empty type set then a minimal type of it always exists and the second shows that each type belong to all members of a type group is exactly a minimal type.

Let be a signature and a TBox over . Each axiom is of the form where () are concepts. We simply write as instead of if . Thus we conclude that Proposition 1 holds in TBoxes.

Corollary 1

Let be a signature and a TBox over . For any and , we have

  • ;

  • if is consistent then .

The above second result is not true if is inconsistent.

Example 1

Let is a TBox and a signature. Assume that is the Hamming distance and is the summation function. So and is inconsistent. has eight possible types: , , , , , , and . Thus, while .

Unfortunately, does not always satisfy the role coherence as the following example shows.

Example 2

Let and . If is the Hamming distance and is the summation function, then . We note that .

The reason that the role coherence might be absent in is that and are taken as two independent concepts so that the relation of satisfiability between and can not be captured when minimal types are computed. In other words, given an arbitrary type set, there is not always a TBox such that it is a model type set of the TBox (see [16]).

Given an arbitrary type set , if it is not a model type set of any TBox, there are two possible options to recovery the role coherence. For instance, if such that and for some role , then we can either remove from or add a new type such that in to . In Example 2, if we remove the type , then will be empty, which is not desirable. So we will extend the type violating the role coherence. Consider Example 2 again, there are three possible types and such that () where , and . So we can pick as the desired minimal type. Furthermore, this extension is an iterative process since newly added types possibly contains new role names and role incoherence is not yet satisfied at every step. To construct a model type set of from a random type set , we introduce an iterative operator and its fixpoint written by .

Formally, let be a signature and a type group, given a type set , Let , where and for some , and , and for any , implies . We use to denote the fixpoint of , i.e., . For any , and , always exists since is inflationary (i.e., ) and monotonic (i.e., if ).

Given a signature and a TBox over , we say is the minimal model type set of . Intuitively speaking, a minimal model type set is a set of minimal types with maintaining role coherence. In Example 2, = = .

We show that minimal model type sets meet our motivation.

Proposition 2

Let be a signature and a TBox over . For any and , we have

  • ;

  • , if is consistent;

  • iff for any .

The first result states that there always exist minimal model types for any non-empty TBox; the second shows that when a TBox is consistent, each minimal model type is exactly model type; and the third ensures that minimal model type sets always hold role coherence.

Definition 3 (TBox Distance-based Entailment)

Let be a TBox and an inclusion. Assume be a signature. Given and , distance-based entails (d-entails) , denoted by , if .

In Example 1, and in Example 2, . Additionally, contradiction can not be d-entailed by any TBox .

4 Distance-based Semantics for Knowledge Bases

This section define minimal model features of KBs. Compared with inconsistency of TBoxes, inconsistency occurring in KBs is complicated since it contains two extra cases: inconsistency of ABoxes and inconsistency between TBoxes and ABoxes. In more detail, these inconsistencies occur in among concept assertions, between concept assertions and role assertions, between assertions and inclusions, even between a single inclusion and a single assertion. For instance,

Example 3

Let be a KB (see [15]) and . It concludes that is inconsistent and thus has no model feature, i.e., .

To deal with those various inconsistencies in a unified way, we adopt the technique of computing minimal model types to construct minimal model features instead of directly introducing distance over features. We argue that this adoption can not only uniformly deal with all inconsistencies but also overcome difficulty of defining distance over pairs of features (see [15]).

We first introduce concept profiles to use type distance can describe how far apart features are. Let be a signature and an ABox over . Assume that a set of all named individuals in . or . A concept profile of in , denoted by , defined as follows: .

Intuitively speaking, a set of concept profiles is a partition of concepts occurring in an ABox w.r.t. individuals. For instance, let and . Thus , and .

Let be a KB. We extend the signature of as where . Indeed, is obtained from by adding all possible natural numbers occurring all concept profiles but not occurring . In other words, and are no different except . In the above example, while .

Without loss of generality, we assume that in the remain of this paper, unless otherwise stated.

Next, we will define the notion of minimal model features.

Definition 4 (Minimal Model Feature)

Let be a KB. Denote . Given and , a -minimal model feature of is a feature satisfying the following three conditions:

  • ;

  • for each , iff ;

  • for each and ;

  • for any , either and , or and .

is the set of -minimal model features of .

In the above definition, a minimal model feature is a feature which contains two parts, namely, type set and Herbrand set . The first condition requires that all types of be minimal; the second says that should be a model type set, i.e., it satisfies the property of role coherence; the third guarantees that each type of satisfying each concept assertion in has the minimal distance to its corresponding concept profile, that is, if a concept assertion satisfied by then types satisfying are minimal w.r.t. type group of concept profile ; and the last ensures that is consistent by those role assertions conflicting with concept assertions.

Example 4

In Penguin KB, we abbreviate to , to , to , to , to and to . Thus , and . Assume that is the Hamming distance and is the aggregation function. We have , shown in the following table.

1 0 1 1 1
1 1 2 1 1
1 1 2 1 1
1 1 2 0 0
0 1 1 1 1
0 1 1 0 0

Here = , = , = , and = . From the above table, it concludes that = and = . Thus, , , and .

We find that minimal model features can reach our aim.

Proposition 3

Let be a KB. For any and , we have

  • ;

  • , if is consistent.

The intuition behind Proposition 3 is that the minimal model features of a KB are features closet to it’s classical semantics. Remember that, while an inconsistent KB does not have any model feature, each KB has at least one minimal model feature. An expected result is that the second statement of Proposition 3 does not necessarily hold if is inconsistent. For instance, in Example 3, where and while .

Now, based on minimal model features, we are ready to define the distance-based entailment for KBs, written , under which meaningful information can be entailed from an inconsistent KB.

Definition 5 (KB Distance-based Entailment)

Let be a KB and an axiom. Assume be a signature. Given and , distance-based entails (d-entails) , still denoted by , if .

Distance-based entailment brings a new semantics (called distance-based semantics) for inconsistent KBs by weakening classical entailment. It is not hard to see that no contradiction can be entailed in this semantics. For instance, in Penguin KB, can not entailed but can under this semantics.

In the rest of this section, we exemplify that distance-based semantics is suitable for reasoning with inconsistent KBs.

Consequences are intuitive and reasonable under the distance-based semantics. In Penguin KB, while and . We further analyze those conclusions under distance-based semantics. The inconsistency of is caused by statement about . On the one hand, is a penguin which can not fly, i.e., . On the other hand, penguin is a bird which can fly, i.e., . Moreover, there exists no more argument for either or . In this sense, neither nor can be entailed under distance-based semantics. However, the statement about in contains no conflict. Thus can be entailed under distance-based semantics.

Furthermore, distance-based semantics also embodies the idea of “majority vote consideration” (see [14]), that is, conclusions must hold more argument for them, for resolving contradictions so that the results are more stable. For instance,

Example 5

Let be a KB , . Thus and . Then , , . and . Therefore, can d-entails , and except for . Intuitively, compared with , there is an extra argument for .

5 Properties of Distance-based Semantics

In general, our distance-based semantics can be taken as a framework in which many logical consequences are defined by selecting various distances and aggregation functions. In this section, we enumerate several good properties of distance-based semantics and several interesting relationships among them.

If is inconsistent and there exists an axiom such that where is an entailment relation, then we say is paraconsistent. It is well known that classical entailment is not paraconsistent.

Proposition 4 (Paraconsistency)

For any and , is paraconsistent.

To show the paraconsistency of the distance-based entailment, consider Example 3 and we have while .

Most existing semantics for paraconsistent reasoning in DLs are much weaker than the classical semantics in this sense that there exists a consistent KB and an axiom such that (also called consistency preservation) but is not entailed by under the paraconsistent semantics. The following result shows that the distance-based semantics does not have such shortcoming.

Proposition 5 (Consistency Preservation)

Let be a KB and an axiom. For any and , if is consistent, then iff .

In classical semantics, a property that iff for any inclusion is called TBox-preservation where the problem of subsumption checking is irrelevant to ABoxes. Our distance-based semantics satisfies such a property.

Proposition 6 (TBox Preservation)

Let be a KB and an inclusion. For any and , iff .

The closure w.r.t. of an arbitrary KB is always consistent.

Proposition 7