Interval based Possibilistic Description Logic Programs for the Semantic Web

2018 International Conference on Computational, Modeling, Simulation and Mathematical Statistics (CMSMS 2018) ISBN: 978-1-60595-562-9

Interval-based Possibilistic Description Logic Programs

for the Semantic Web

Ting-ting ZOU

1,*

_{and An-sheng DENG}

1

1_{Dalian Maritime University, Dalian, 116026, China}

*_{Corresponding author}

Keywords: Semantic Web, Description logics, Possibilistic logics, Answer set semantic, Disjunctive logic programs.

Abstract. Integration ontologies and rules has become a central topic in the Semantic Web. In order to deal with uncertainty and inconsistent information, possibilistic description logic program has been investigated in recent years. However, possibilistic description logic program also cannot well model a great deal of real-world problems, because the accurate degrees associated with axioms and atoms are usually difficult to provide for experts. To address this problem, we further extend possibilistic description logic programs so that they can deal with inaccurate degrees associated with axioms and atoms. Therefore, we propose tightly coupled interval-based possibilistic description logic programs under possibilistic answer set semantics, which are a tight integration of disjunctive logic programs, interval-based possibilistic logics and possibilistic description logics. First of all, we define the syntax and semantics. Then, we show some semantic properties. Furthermore, we present three reasoning problems, and present some algorithms to solve these reasoning problems.

Introduction

As an extension of the current World Wide Web, the Semantic Web aims to help computers to understand and process web information automatically [1]. Now, the integration ontologies and rules has become a central topic in the Semantic Web. In fact, standard ontology language is based on Description Logic[2], and the existing proposals for a rule language for use in the Semantic Web originate from Logic Programming[3]. Eiter et al proposed description logic programs under answer set semantics[4]. Subsequently, Lukasiewicz presented a new method to description logic programs under the answer set semantics [5]. Yidong Shen proposed well-justified FLP answer set for description logic programs [6].

Unfortunately, a great deal of real-world problems cannot be modelled by description logic programs, because crisp description logic programs cannot model uncertain and inconsistent information. Nevertheless possibilistic logic can represent and reason on uncertain, incomplete and inconsistent knowledge [7]. Dubois et al. defined the syntax and semantic of possibilistic logic, and also provided some reasoning problems[8]. Dellunde et al. present several fuzzy logics trying to capture different notions of necessity for Gödel logic formulas[9]. Benferhat et al. present an interval-based possibilistic logic[10]. Subsequently, Dubois propose a generalized possibilistic logic[11]. In fact, many researchers have focused their study on possibilistic description logics in recent years [12]. Subsequently, Dubois proposed a new possibilistic description logic [13]. Moreover, Guilin proposed a new possibilistic description logics [14]. Furthermore, he provided a possibilistic description logic reasoner PossDL[15]. Moreover, Lesot et al. improved tableau algorithm and proposed a new algorithm to compute the inconsistency degree of a possibilistic DL knowledge base[16]. We propose an interval-based possibilistic description logic IP-SHOIN(D)[17].

of interval-based possibilistic dl-program. Then we show some semantic properties. Finally, we define three reasoning problems, and present some algorithms to implement these reasoning problems.

Interval-based Possibilistic Description Logic Programs

Syntax

Let be a function-free first-order vocabulary with nonempty finite sets of constant symbols

c

 and predicate symbols p. Suppose c IAID, thus every ground atom made from CA, RA, RD,

and ccan be interpreted in the description logic component. Let be a set of variables. A term is

either a variable from or a constant symbol from c. An atom is an expression of the form

( , , )1

n

h t  t , where h is a predicate symbol of arity n0fromp, andt1,,tn are terms. We use M to

denote a set of atoms. A literal l is an atom a or a negated atom not a, whereaM. Moreover, we

use L=[α, β]⊆[0,1] to denote real number based interval and restrict that α>0, and use ℒ to denote the set of intervals. In the following, we define some operations on intervals.

Definition 2.1. The operations on intervals are defined as follows:

1) Max operation : {L1, L2,…, Ln}=[max{α1, α2,…, αn }, max{β1, β2,…, βn }], where

L1=[α1, β1], L2=[α2, β2],…, and Ln=[αn, βn] are intervals.

2) Min operation : {L1, L2,…, Ln}=[min{α1, α2,…, αn }, min{β1, β2,…, βn }], where

L1=[α1, β1], L2=[α2, β2],…, and Ln=[αn, βn] are intervals.

3) Compare operation ≻: L1≻ L2 if and only if α1>β2, where L1=[α1, β1] and L2=[α2, β2] are

intervals.

4) Subsume operation : L1L2 if and only if α1≥α2 and β1≤β2, where L1=[α1, β1] and

L2=[α2, β2] are intervals.

5) Reverse operation ∼: 1∼L=[1-β, 1-α], where L=[α, β] is an interval.

Definition 2.2. An interval-based possibilistic atom is a pairp=<a L, >, which denotes that the

certainty degree of the atom a is one of the elements of the interval L. Moreover, we define the

classical projection * and necessity degree projection d as follows: *

p =a, d p( )L.

In this paper, we use PM to denote a set of possibilistic atoms in which every atom a occurs at

most one time in PM and always with a strictly positive certainty degree, that is to say,

a M

  ,{a L, PM} 1 .

Definition 2.3. An interval-based possibilistic disjunctive rule (or simply interval-based possibilistic rule) r is of the formr  a1  ak   b1  bl not bl1  not bn, L, where

0

k ,

0

l , k+ >l 0,



a₁, , , , , , a b_{k 1} b b_{l l}₁, , b_n



M , and L∈ℒ. Moreover, we define the classical

projection * and the necessity degree projection d of the possibilistic rule r as follows: d r( )=L

1 k 1 l l1 n

r  a  a    b  b not b   not b .

Then the set



a1, , ak



is called the possibilistic head of r, i.e. PH



a1, , ak



, and the set



b1, , ,b bl l1, , bn



is called the possibilistic body of r, i.e., PBPBPB , PB



b1, ,bl



_{  ,}



l1, , n



PB  b  b . In generally, we can write an interval-based possibilistic rule r as a form of

,

A B B L

   , whereA  a1  ak, B b1 bl  _{   , and}

1

l n

B b   b .

Definition 2.4. An based possibilistic disjunctive logic program (or simply interval-based possibilistic program) IP is a finite set of interval-interval-based possibilistic rules. Let IP{ |r rIP}, then, IP is normal interval-based possibilistic program iff k1for all rules in IP; IP is a positive interval-based possibilistic program iff nlfor all rules in IP.

Definition 2.5. An based possibilistic description logic program (for short, interval-based possibilistic dl-program) IKB=(IL,IP) includes an interval-based possibilistic description

Semantics

A term is ground iff it includes only constant symbols from c. An atom ( , , )

1

n

h t  t is ground iff all

terms t1,,tn are ground.

Definition 2.6. An interval-based possibilistic atom p=<a L, >is a ground atom iff the atom a is

ground. An interval-based possibilistic rule r is ground rule iff all atoms a1, , , , , ,a bk 1 b bl l1, , bn are

ground atoms.

Definition 2.7. A ground instance of an interval-based possibilistic rule r is an interval-based possibilistic ground rule r  a1  ak    b1  bl not bl1  not bn, L , where,

1, , , , , ,k 1 l l1, , n

aa b  b b   bare obtained by substituting constant symbol fromc for every variable

appearing in a1, , , , , , a bk 1b bl l1, , bn respectively. A ground program of an interval-based

possibilistic program IP is a set of all ground instances of interval-based possibilistic rules in IP. In this paper, we use PossG(P) to denote all ground programs of an interval-based possibilistic program IP.

Definition 2.8. The possibilistic Herbrand base relative to , written as PHBF, is a set of all interval-based possibilistic ground atom{ , , }1

u

p  p , for every interval-based possibilistic ground

atom pi h ti( , , ),1i  tni Li, i1, 2,...,u, hi is a predicate symbol of arity n0fromp, andt1i, , tni are constant symbols fromc, and Li.

Definition 2.9. An interval-based possibilistic interpretation { , , }1

v

I p p relative to IKB is a

subset of PHB.

Definition 2.10. An interval-based possibilistic interpretation I is an interval-based possibilistic model of p=<a L, >, denoted I-<a L, >, if and only if a L, I or there exists an interval-based

possibilistic ground atoma L, I such thatLL. An interval-based possibilistic interpretation I

is an interval-based possibilistic model of an interval-based possibilistic ground rule r , denoted byI-r , if and only if, I|  a, { , , , }L L1 Ll  for somea H r( )



 , if I-( , )b Li i , bi B r( )  

 , Li, 1, 2,...,

i l , and I-/( , )b Lj j , bj B r( )  

 , Li , j l 1,l2,...,n . An interval-based possibilistic

interpretation I is an interval-based possibilistic model of IP, denoted byI|=IP , if and only if I-r

for all rPossG(IP).

Definition 2.11. An interval-based possibilistic interpretation I is an interval-based possibilistic model of IL, denoted I|=IL, if and only if there exists an interval-based possibilistic distribution  for knowledge base ILI such that |ILI. I is an interval-based possibilistic model of IKB, denoted I|IKB, if and only if I|=IL and I|=IP.

There may be many interval-based possibilistic models for an interval-based possibilistic dl-program IKB=(IL,IP). Let I1 , I2 be two interval-based possibilistic models of IKB,

thenI1I2 { a,[min{ , }, max{ , }] | 1 2  1 2  a,[ , ] 1 1  I1, a,[ , 2 2]I2}, I1I2if and only if I1 I2

_ _,

or I1 I2

_  _{and for any}

1

aI, if a L, 1I1 and a L, 2I2, then L1L2.

Definition 2.12. An interval-based possibilistic interpretation I is a least interval-based possibilistic model of IKB, if and only if there does not exist an interval-based possibilistic model

I of IKB, such that II.

Definition 2.13. An interval-based possibilistic reduction for IP is defined as follows: IP_PM  { A PMB,L  |r A Bnot B,LP A, PM ,BPM ,BPM} . Moreover, an interval-based possibilistic reduction for IKB is IKB_PM( ,IL IP_PM).

Definition 2.14. Let I be an interval-based possibilistic interpretation relative to IKB such that

*

Semantic Properties

Theorem 3.1. I is an interval-based possibilistic answer set of KB=(IL,IP) if and only if I is an

interval-based possibilistic answer set of IP, where IL .

Proof. It is known that I is an interval-based possibilistic answer set of IKB if and only if I is a least interval-based possibilistic model of IKB_I( ,IL IP_I) such thatIKB_I|I. So, I is an

interval-based possibilistic model of IKB_I , iff I|=IL, and I-r for rPossG(IP_I). Because IL , then I is

an interval-based possibilistic model of IKB_I, iff I-r for rPossG(IP_I) iff I is an interval-based

possibilistic model of IP_I. Thus, I is a least interval-based possibilistic model of IKB_I iff I is a least

interval-based possibilistic model of IP_I. Moreover, IKB_I|I iff IP_I|I.

Theorem 3.2. Let ,L be an interval-based possibilistic ground atom of PHB_. Then for all interval-based possibilistic answer set I of IKB, I| ,L if and only if for all interval-based possibilistic distribution  such that |ILPossG( )IP , |,L.

Proof. Because IKB is a positive interval-based possibilistic dl-program, then the set of all

possibilistic answer set of IKB is equivalent to the set of all least interval-based possibilistic model of KB. Moreover, for ,LPHB , for all least interval-based possibilistic model of IKB,

| ,

I   L iff for all interval-based possibilistic model of IKB, J| ,L. So, for all interval-based possibilistic answer set I| ,L iff for all interval-based possibilistic model J of KB,

| ,

J   L. Now, we need to prove that for all interval-based possibilistic modelJ| ,L iff for all interval-based possibilistic distribution  such that |ILPossG(IP), |,L.

( ) Suppose that for all interval-based possibilistic model J of IKB, J| ,L. Let be any interval-based possibilistic distribution such that |ILPossG(IP). Now, we define an

interval-based possibilistic interpretation I PHB such that I ,Liff |,L. Let ILILI, then

I is an interval-based possibilistic model of IL. Because | PossG( IP), then |r for rPossG(IP).

Thus, I r for rPossG(IP). So, I is also an interval-based possibilistic model of IP. Therefore, I

is an interval-based possibilistic model of IKB. According to I  | ,L. So, |,L. Therefore, for all interval-based possibilistic distribution  such that |ILPossG(IP), |,L.

( ) Suppose that for all interval-based possibilistic distribution  such that |ILPossG(IP),

| ,L

  . Let IPHB be any interval-based possibilistic model of IKB. So, I=IL and I-r for

all rPossG(IP). Thus, there exists an interval-based possibilistic such that  | ILI. So,  | IL,

| I

  . Moreover,  | r for all rPossG(IP) , and thus rPossG(IP) . So,  | ILPossG( )IP .

According to known condition,  | ,L. Thus, ,LI , or there exists an interval-based

possibilistic ground atom ,LI, such thatLL. So, I| ,L. Therefore, for all interval-based possibilistic answer set I| ,L.

Theorem 3.3. Let ,L be an interval-based possibilistic ground atom of PHB_, andIP .

Then for all interval-based possibilistic answer set I of IKB, I| ,L iff for all interval-based possibilistic distribution  such that |IL, |,L.

Proof. It is obvious.

Reasoning Problems for Interval-based Possibilistic Description Logic Programs

Let IKB=(IL,IP) be an interval-based possibilistic dl-program and L be an interval. Then the

classical DL knowledge base associated with IL is IL{ |  ,LIL}, and the L-cut set of IL

isILL{ |i i,Li IL and L, i L}. The classical logic program associated with IP is IP { |r r IP}

_  _{ ,}

and the L-cut set of IP isIPL {r P d r| ( )L}. The classical dl-program associated with IKB is

( , )

Definition 4.1. Let IKB=(IL,IP) be an interval-based possibilistic dl-program. Then a

possibilistic description logic program KB is compatible with ∑ if and only if there exists a bijection function f IKB: KB, such that for any,LIKB, f(,L )  , KB,  L, and

for any , KB, f( ,  ) ,LIKB,  L.

For any interval-based possibilistic dl-program IKB, we can obtain a compatible possibilistic

description logic programKB {  ,  | ,LIKBandL}. Thus, we use CPB(IKB) to denote the

infinite set of all possibility description logic program knowledge bases that are compatible with

IKB.

Definition 4.2. Let IKB=(IL,IP) be an interval-based possibilistic dl-program. Then a lower

bound compatible possibilistic description logic program knowledge base KBlband an upper bound

compatible possibilistic description logic program knowledge base KBubare defined as follows:





{ , | , , }

lb

KB       IKB , KBub {  ,  |   , ,





IKB}.

Definition 4.3. Let IKB=(IL,IP) be an based possibilistic dl-program. Then the

interval-based consistency degree of IKB , denoted by Icon IKB( ), is defined as follows:

( ) { ( ) | CPB( )}

Icon IKB  Icon KB KB IKB .

Theorem 4.4. Let IKB=(IL,IP) be an interval-based possibilistic dl-program. Then

( ) ( )

( ) [ min ( ), max ( )]

KB CPB IKB KB CPB IKB

Icon IKB Icon KB Icon KB

 

 .

Proof. According to Definition 4.1, for any KBCPB(IKB) , Icon KB( )Icon IKB( ) . Thus,

CPB( )

KB IKB

  ,

( ) ( )

min ( ) ( ) max ( )]

KB CPB IKB Icon KB Icon KB KB CPB IKB Icon KB .

Therefore,

( ) ( )

( ) [ min ( ), max ( )]

KB CPB IKB KB CPB IKB

Icon IKB Icon KB Icon KB

 

 .

Theorem 4.5. Let IKB=(IL,IP) be an interval-based possibilistic dl-program, and KBlb be a lower

bound compatible possibilistic description logic program knowledge base of IKB, and KBub be an

upper bound compatible possibilistic description logic program knowledge base of IKB. Then

( ) [ ( lb), ( ub)]

Icon IKB = Icon KB Icon KB .

Proof. KBCPB(IKB),KB {  i, i |   ,



i, i



IKB andi



 i, i



}. So, the consistency degree of

KB is Icon KB( )=min{ |di KB>d_iis consistent} . Moreover,





{ , | , , }

lb i i i i i

KB        IKB , KBub {  i, i |   i,



i, i



IKB} .Thus,

( lb) min{ |i lb _i is consistent}

Icon KB = a KB >a , Icon KB( ub)=min{ |bi KBub>biis consistent}.Because  i ii, So

CPB( )

KB IKB

  , Icon KB( lb)Icon KB( )Icon KB( ub) . Therefore, _{KB CPB IKB}min_{( )}Icon KB( )Icon KB( lb)] ,

( )

max ( ) ( ub)

KB CPB IKB Icon KB Icon KB .According to Theorem 4.4, we have Icon IKB( ) [Icon KB( lb),Icon KB( ub)]

= .

Theorem 4.6. Let IKB=(IL,IP) be an interval-based possibilistic dl-program, Icon IKB( )be an

interval-based consistency degree of IKB. Then,

1) KBCPB(IKB), IKBIcon IKB( )KBIcon KB( ).

2) IKBIcon IKB( )is consistent.

Proof. 1) IKBIcon IKB( ){ | i  i,Li IKB and LiIcon IKB( )} .For any KBCPB(IKB) , we have

{ i, i | , i i i}

KB     L IKB and L . So, Icon KB( )=min{ |di KB>d_iis consistent} ,

( ) { | , ( )}

Icon KB i i i i

KB     IKB and Icon KB . According to Definition 4.3, for any KBCPB(IKB),

( ) ( )

Icon KB Icon IKB . Thus, for any ,LIKB and LIcon IKB( ), there exists a possibilistic atom

, KB

 

  such that L andIcon IKB( ) . So, for any IKBIcon IKB_{( )}, we have KBIcon KB( ) .

2) According to the definition of consistency degree of interval-based possibilistic dl-program knowledge base, we can obtain, KBCPB(IKB), KBIcon KB( )is consistent. Therefore, IKBIcon IKB( ) is

[image:6.612.112.507.110.405.2]

consistent.

Figure 1. Algorithm ICONIP.

According to the above theorems, we propose an algorithm ICONIP is to compute an interval-based consistency degree of an interval-interval-based possibilistic dl-program knowledge base IKB as shown in Figure 1. The main idea of this algorithm is to compute the consistency degree of two compatible possibilistic dl-program knowledge bases. Firstly, we obtain a lower bound compatible possibilistic dl-program knowledge base KBlb of IKB, and an upper bound compatible possibilistic

dl-program knowledge base KBub of IKB. Then we compute the consistency degree of KBlb, i.e.

( lb)

Icon KB , and the consistency degree of KBub, i.e. Icon KB( ub). Finally, we get an interval-based

consistency degree of IKB,. Icon IKB( )=[Icon KB( lb),Icon KB( ub)].

Definition 4.7. An interval-based possibilistic axiom or atom ,L is called credutions

possibilistic consequence of IKB, denoted by IKB| c ,L, if and only if, there exists an interval-based possibilistic answer set I of IKB such thatI| ,L.

Definition 4.8. An interval-based possibilistic axiom or atom ,L is called skeptical

possibilistic consequence of IKB, denoted by IKB| s ,L, if and only if, for all interval-based possibilistic answer set I I1, ,...,2 Im of IKB such that I1I2 ... Im| ,L.

Conclusion

possibilistic answer set semantics. We prove that the interval-based possibilistic answer set semantics of an interval-based possibilistic dl-program faithfully extends the possibilistic semantics of an interval-based possibilistic program and interval-based possibilistic description logic. Finally, we present three reasoning problems: computing consistency degree, credutions possibilistic consequence and skeptical possibilistic consequence, and propose some algorithms to solve these reasoning problems. In a word, possibilistic dl-program can well represent and reason a great deal of real-word problems. An interesting topic of future research is to implement of the presented approach. Another interesting issue is to extend interval-based possibilistic dl-programs by a new semantics.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (No. 61272171), and the Fundamental Research Funds for the Central Universities of China (No. 3132018199).

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