Storing fuzzy description logic ontology knowledge bases in fuzzy relational databases

Abstract

In the context of the Semantic Web, fuzzy extensions to OWL (the W3C standard ontology language) and Description Logics (DLs, the logical foundation of OWL) have been extensively investigated, and there are many real fuzzy DL ontology knowledge bases. Therefore, how to store fuzzy DL ontology knowledge bases has become an important issue. In this paper, we propose an approach and implement a tool for storing fuzzy DL ontology knowledge bases in fuzzy relational databases. Our chosen formalism is a fuzzy extension of the very expressive DL SHOIN(D), which is the main logical foundation of the standard ontology language OWL, so that our storage approach can store not only fuzzy DL-knowledge bases but also fuzzy ontology knowledge bases. Firstly, we give a formal definition of fuzzy DL-knowledge bases. In the definition, we consider the constructors of both fuzzy SHOIN(D) DL and fuzzy OWL ontology and add some common fuzzy datatypes (e.g., trapezoidal values, interval values, approximate values, and labels) into the knowledge bases. On this basis, we propose an approach which can store fuzzy DL-knowledge bases in fuzzy relational databases, and provide an example to well explain the approach. The correctness and quality of the storage approach are proved and analyzed. Furthermore, following the proposed approach, we implemented a prototype tool, which can automatically store fuzzy DL-knowledge bases. Finally, we make a discussion about the query problem and make a comparison with the existing works.

Appendices

Appendix A: The fuzzy DL syntax, fuzzy OWL syntax, and semantics of a fuzzy DL-KB

Fuzzy DL Syntax	Fuzzy OWL Syntax	Semantics
Fuzzy concepts	Fuzzy class descriptions
A	Class (A)	A FI :ΔFI → [0, 1]
⊤	owl:Thing	$\top ^{FI}\left (d \right )=1$
⊥	owl:Nothing	$\bot ^{FI}\left (d \right )=0$
C 1 ⊓ ... ⊓ C n	intersectionOf (C 1…C n )	(C 1 ⊓ ... ⊓ C n )FI(d) = min{$C_{\mathrm {1}}^{\text {FI}}(d)$,…, $C_{\mathrm {n}}^{\text {FI}}(d)$}
C 1 ⊔ ... ⊔ C n	unionOf (C 1... C n)	(C 1 ⊔ ... ⊔ C n )FI(d) = max{$C_{1}^{\text {FI}}(d)$,..., $C_{\mathrm {n}}^{\text {FI}}(d)$}
¬C	complementOf (C)	(¬C)FI(d) = 1 − C FI(d)
{o 1}⊔ ... ⊔{o n }	oneOf(o 1... o n)	({o 1}⊔ ... ⊔{o n })FI(d) = 1 if $d\in \{o_{1}^{\text {FI}}$,..., $o_{\mathrm {n}}^{\text {FI}}$}, 0 otherwise
∃P.E	restriction (P someValuesFrom(E))	$(\exists P.E)^{\text {FI}}(d)=\sup _{a\in {\Delta }^{\text {FI}}} \{\min \{P^{\text {FI}}(d,a),E^{\text {FI}}(a)\}\}$
∀P.E	restriction (P allValuesFrom(E))	${(\forall P.E)}^{\text {FI}}(d)=\inf _{a\in {\Delta }^{\text {FI}}} \{\max \{1-P^{\text {FI}}(d,a),E^{\text {FI}}(a)\}\}$
∃P. {a}	restriction (P hasValue(a))	(∃P.{a})FI(d) = P FI(d,a)
≥nP	restriction (P minCardinality (n))	$(\ge nP)^{\text {FI}}(d)=\sup _{a_{1} ,...,a_{\mathrm {n}} \in {\Delta }^{\text {FI}}} \wedge _{\mathrm {i}=1}^{\mathrm {n}} P^{\text {FI}}(d,a_{\mathrm {i}} )$
≤nP	restriction (P maxCardinality(n))	$(\le nP)^{\text {FI}}(d)=\inf _{a_{1} ,...,a_{\mathrm {n}+1} \in {\Delta }^{\text {FI}}} \vee _{\mathrm {i}=1}^{\mathrm {n}+1} (1-P^{\text {FI}}(d,a_{\mathrm {i}} ))$
= nP	restriction (P cardinality(n) )	(≥ n P⊓≤ n P)FI(d)
Fuzzy axioms	Fuzzy class axioms
$A\sqsubseteq C_{1} \sqcap ...\sqcap C_{n}$	Class (A partial C 1… C n)	A FI(d) ≤ min{$C_{\mathrm {1}}^{\text {FI}}(d)$,…, $C_{\mathrm {n}}^{\text {FI}}(d)$ }
A ≡ C 1 ⊓ ... ⊓ C n	Class (A complete C 1… C n)	A FI(d) = min{$C_{\mathrm {1}}^{\text {FI}}(d)$,…, $C_{\mathrm {n}}^{\text {FI}}(d)$ }
$<C_{1} \sqsubseteq C_{2} ,n>$	SubClassOf (C 1 C 2 n)	inf$_{\mathrm {}}a\in {\Delta }^{\text {FI}}${ max{ 1-$C_{\mathrm {1}}^{\text {FI}}(a)$, $C_{\mathrm {2}}^{\text {FI}}(a)$}} ≥ n
C 1 ≡ ... ≡ C n	EquivalentClasses (C 1… C n)	$C_{\mathrm {1}}^{\text {FI}}(d) \quad =$…$= \quad C_{\mathrm {n}}^{\text {FI}}(d)$
$C_{1} \sqcap C_{2} \sqsubseteq \thinspace \bot$	DisjointClasses (C 1… C n)	min{$C_{\mathrm {i}}^{\text {FI}}(d)$, $C_{\mathrm {j}}^{\text {FI}}(d)$} = 0 1 ≤ i < j ≤ n
A ≡{o 1...o n }	EnumeratedClass (A o 1… o n)	A FI(d) = 1 if d ∈{$o_{\mathrm {1}}^{\mathrm {FI\thinspace }}$,..., $o_{\mathrm {n}}^{\text {FI}}$} , A FI(d) = 0 otherwise
Fuzzy axioms	Fuzzy property axioms
	DatatypeProperty (T
$\exists \top .T\sqsubseteq C_{i}$	domain(C 1)...domain(C m)	T FI(d, $v) \le C_{\mathrm {i}}^{\text {FI}}(d) i =$ 1,..., m
$\top \sqsubseteq \forall T$. D i	range(D 1)...range(D k)	T FI(d, $v) \le D_{\mathrm {i}}^{\text {FI}}(v) i =$ 1,..., k
$\top \sqsubseteq \le$1T	[Functional] )	∀d ∈ΔFI #... v ∈ΔD: T FI(d, v) ≥ 0... ≤ 1
	ObjectProperty (R
∃R.$\top \sqsubseteq C_{\mathrm {i}}$	domain(C 1)...domain(C m)	R FI(d 1, $d_{2}) \le C_{\mathrm {i}}^{\text {FI}}(d_{\mathrm {1}}) i =$ 1,..., m
$\top \sqsubseteq \forall R$. C i	range(C 1)...range(C k)	R FI(d 1, $d_{2}) \le C_{\mathrm {i}}^{\text {FI}}(d_{2}) i =$ 1,..., k
$\top \sqsubseteq \le$1 R	[Functional])	$\forall d_{1}\in {\Delta }^{\text {FI}}$ #$\{d_{2}\in {\Delta }^{\text {FI}}$: R FI(d 1, d 2) ≥ 0}≤ 1
R = (R 0)−	[InverseOf(R 0)]	R FI(d 1, $d_{2}) \le R_{\mathrm {0}}^{\text {FI}}(d_{\mathrm {2}}$, d 1)
$\top \sqsubseteq \le$1 R −	[InverseFunctional]	$\forall d_{1}\in {\Delta }^{\text {FI}}$ #$\{d_{2}\in {\Delta }^{\text {FI}}$: (R −)FI(d 1, d 2) ≥ 0}≤ 1
Trans(R)	[Transitive])	sup d ∈ΔFI{ min{ R FI(d 1, d), R FI(d, d 2)}} ≤ R FI(d 1, d 2)
$E_{\mathrm {1}}\sqsubseteq E_{\mathrm {2}}$	SubPropertyOf(E 1, E 2)	$E_{\mathrm {1}}^{\text {FI}}(d$,$a) \quad \le \quad E_{\mathrm {2}}^{\text {FI}}(d$, a)
E 1 ≡ ... ≡ E n	EquivalentProperties(E 1, ..., E n)	$E_{\mathrm {1}}^{\text {FI}}(d$, a) = ... $= \quad E_{\mathrm {n}}^{\text {FI}}(d$, a)
Fuzzy assertions	Fuzzy individual axioms
o: C i⋈m i	Individual (o type(C i) [ ⋈m i]...	$C_{\mathrm {i}}^{\text {FI}}(o) \bowtie m_{\mathrm {i}} \quad m_{\mathrm {i}} \in$ [0,1] 1 ≤ i ≤ n
(o, o i): R i⋈k i	value(R i, o i) [ ⋈k i]…	$R_{\mathrm {i}}^{\text {FI}}(o$, o i)⋈k i k i ∈ [0,1] 1 ≤ i ≤ n
(o, v i): T i⋈l i	value(T i, v i) [ ⋈l i]…)	$T_{\mathrm {i}}^{\text {FI}}(o$, v i) ⋈ l i l i ∈ [0,1] 1 ≤ i ≤ n
o 1 = … = o n	SameIndividual (o 1… o n)	$o_{\mathrm {1}}^{\mathrm {FI\thinspace }}=$…$= \quad o_{\mathrm {n}}^{\text {FI}}$
o i≠o j	DifferentIndividuals (o 1… o n)	$o_{\mathrm {i}}^{\mathrm {FI\thinspace }}\ne o_{\mathrm {j}}^{\mathrm {FI\thinspace \thinspace \thinspace \thinspace \thinspace }}$ 1 ≤ i < j ≤ n

1. Comments: where P ∈{R, T}is an object property R or a datatype property T; E ∈ {C, D}is a class C or a concrete datatype D; d and o are abstract individuals, v is a concrete individual, a ∈{d, v}, and ⋈ ∈ { ≥, >, ≤, <}

Appendix B: The detailed explanations of the storage architecture in Table 3

Tables	Structure and Description
Resource_Table	Storing all resources in a fuzzy DL KB, i.e., fuzzy classes, fuzzy properties, and individuals. FKBName describes the stored fuzzy DL KB name; ID identifies uniquely a resource in the fuzzy DL KB, and ID attribute is the primary key of the table; Namespace and localname describe a URIref of any resource; type describes the type of a resource, which may be a fuzzy class, a fuzzy property, or an individual. In the following, ProID ∈ID denotes property, ClassID ∈ID denotes class, and IndID ∈ID denotes individual.
Property_Field_Table	Storing the domain and range of a property. ProID is the primary key of the table, and it is also a foreign key that references the primary key ID in Resource_Table. The fields domain and range store the domain and range of a property.
Property_Character_Table	Storing the characters of a property. ProID is similar to the ProID above; type, which describes the type of a property, may be a datatype property or an object property; characters, which describes the characters of a property, may be Symmetric, Functional, and Transitive.
Property_Restriction_Table	Storing the restrictions of a property. ClassID and ProID reference the primary key IDs in Resource_Table. (ClassID, ProID) is the primary key of the table; type, which describes restrictions, may be allValuesFrom, someValuesFrom, hasValue, minCardinality, maxCardinality; value denotes the value of restriction of a property, where: value = ID (i.e., ID in Resource_Table) if type = allValuesFrom |someValuesFrom |hasValue; otherwise, value is greater than or equal to 0 (which can be represented by adding a Check constraint to the ”value” field).
Class_Relation_Table	Storing the relationships among fuzzy classes (for simplicity Class_Relation_Table gives only two fuzzy classes). Class 1 ID and Class 2 ID reference the primary key IDs in Resource_Table. (Class 1 ID, Class 2 ID) is the primary key of table; relationship may be SubClassOf, EquivalentClasses, or DisjointClasses; u denotes that a fuzzy class is a subclass of another fuzzy class with membership degree of [0, 1].
Property_Relation_Table	Storing the relationships among fuzzy properties (similarly for the Class_Relation_Table above). Pro 1 ID and Pro 2 ID are similar to the Class 1 ID and Class 2 ID above; relationship may be Sub Property Of, Equivalent Properties, or inverse Of; u denotes that a fuzzy property is a subproperty of another fuzzy property with membership degree of [0, 1].
Class_Operation_Table	Storing the operations among fuzzy classes. ClassID and Class i ID reference the primary key IDs in Resource_Table. (ClassID, Class i ID, ...) is the primary key; type is the operation of fuzzy classes such as intersection Of and union Of.
Class_oneOf_Table	Storing the operation oneOf between a fuzzy class and individuals. ClassID and Ind i ID reference the primary key IDs in Resource_Table. (ClassID, Ind i ID, ...) is the primary key of the table; type is the operation one Of.
Individual_Class_Relation_Table	Storing the relationships between fuzzy classes and individuals. IndID and ClassID reference the primary key IDs in Resource_Table. (IndID and ClassID) is the primary key of the table; u denotes that an individual is an instance of a fuzzy class with membership degree of [0, 1].
Individuals_Relation_Table	Storing the relationships among individuals (for simplicity only two individuals are listed). Ind 1 ID andInd 2 ID reference the primary key IDs in Resource_Table. (Ind 1 ID andInd 2 ID) is the primary key of the table; type denotes the relationships between individuals such as Same Individual or Different Individuals.
Individual_Crisp_Property_Value_Table	Storing the values of crisp properties(for simplicity only one property is listed). IndID is the primary key, and IndID and ProID reference the primary key IDs in Resource_Table; value is the value of the property, and the crisp value can be easily stored in the current database systems.
Individual_Fuzzy_Property_Value_Table	Storing the values of fuzzy properties in fuzzy relational databases (for simplicity only one property is listed). IndID, which denotes an individual with fuzzy properties, references the primary key ID in Resource_Table. ProID is a fuzzy property, i.e., the domain of the property may be any fuzzy datatype in Table 2. Fuzzy Property Value is the value of the fuzzy property ProID.

Appendix C: The Resource_Table created in Section 5.2.1

Resource_Table.
FKBName	ID	namespace	localname	type
FKB_1	c_1	http://www.neu.edu.cn/ailab/	Department	class
FKB_1	c_2	http://www.neu.edu.cn/ailab/	Staff	class
FKB_1	c_3	http://www.neu.edu.cn/ailab/	AdminStaff	class
FKB_1	c_4	http://www.neu.edu.cn/ailab/	AcademicStaff	class
FKB_1	c_5	http://www.neu.edu.cn/ailab/	Student	class
FKB_1	c_6	http://www.neu.edu.cn/ailab/	Course	class
FKB_1	p_1	http://www.neu.edu.cn/ailab/	study_in	property
FKB_1	p_2	http://www.neu.edu.cn/ailab/	work_in	property
FKB_1	p_3	http://www.neu.edu.cn/ailab/	teach	property
FKB_1	p_4	http://www.neu.edu.cn/ailab/	choosecourse	property
FKB_1	p_5	http://www.neu.edu.cn/ailab/	staffname	property
FKB_1	p_6	http://www.neu.edu.cn/ailab/	title	property
FKB_1	p_7	http://www.neu.edu.cn/ailab/	email	property
FKB_1	p_8	http://www.neu.edu.cn/ailab/	age	property
FKB_1	p_9	http://www.neu.edu.cn/ailab/	height	property
FKB_1	p_10	http://www.neu.edu.cn/ailab/	ability	property
FKB_1	i_1	http://www.neu.edu.cn/ailab/	staffid_11001	individual
FKB_1	i_2	http://www.neu.edu.cn/ailab/	depid_0206	individual
FKB_1	i_3	http://www.neu.edu.cn/ailab/	courid_309	individual