A Comparison of Paraconsistent Description Logics

Description logics (DLs) are a family of logic-based knowledge representation formalisms with a number of computer science applications. DLs are especially well-known to be valuable for obtaining logical foundations of web ontology languages (e.g., W3C's ontology language OWL). Paraconsistent (or inconsistency-tolerant) description logics (PDLs) have been studied to cope with inconsistencies which may frequently occur in an open world. In this paper, a comparison and survey of PDLs is presented. It is shown that four existing paraconsistent semantics (i.e., four-valued semantics, quasi-classical semantics, single-interpretation semantics and dual-interpretation semantics) for PDLs are essentially the same semantics. To show this, two generalized and extended new semantics are introduced, and an equivalence between them is proved.



Some recent developments of PDLs may be briefly summarized as follows.An inconsistency-tolerant fourvalued terminological logic was originally introduced by Patel-Schneider [10], three inconsistency-tolerant constructive DLs, which are based on intuitionistic logic, were studied by Odintsov and Wansing [8,9], some paraconsistent four-valued DLs including were studied by Ma et al. [4,5], some quasi-classical DLs were developed and studied by Zhang et al. [12,13], a sequent calculus for reasoning in four-valued DLs was introduced by Straccia [11], and an application of four-valued DL to information retrieval was studied by Meghini et al. [6,7].A PDL called has recently been proposed by Kamide [14,15] based on the idea of Kaneiwa [16] for his multiple-interpretation DL .The logic [4], which is based on four-valued semantics, has a good translation into  [3], and using this translation, the satisfiability problem for is shown to be decidable.But, and its variations have no classical negation (or complement).As mentioned in [17], classical and paraconsistent negations are known to be both useful for some knowledgebased systems.The quasi-classical DLs in [12,13], which are based on quasi-classical semantics, have the classical negation.But, translations of the quasi-classical DLs into the corresponding standard DLs were not proposed.[14], which is based on dual-interpretation semantics, has both the merits of and the quasiclassical DLs, i.e., it has the translation and the classical negation.The semantics of is taken over from the dual-consequence Kripke-style semantics for Nelson's paraconsistent four-valued logic N4 with strong negation [18,19].The constructive PDLs in [8] are based on single-interpretation semantics, which can be seen as a DL-version of the single-consequence Kripke-style semantics for N4 [20].
The following natural question arises: What is the relationship among the single-interpretation semantics of the constructive PDLs, the dual-interpretation semantics the quasi-classical semantics of the quasi-classical DLs?This paper gives an answer to this question: These paraconsistent semantics are essentially the same semantics in the sense that some fragments of these PDLs are logically equivalent.More precisely, we show the following.A new PDL, called  , is introduced based on a generalized quasi-classical semantics.It can be seen that the quasi-classical semantics and the fourvalued semantics are special cases of the  semantics.An equivalence between  and (a slightly modified version of) is proved.A new PDL, called , is introduced based on a modified single-interpretation semantics.An equivalence between and (a slightly modified version of) is proved.These results mean that the existing applications and theoretical results (e.g., decidability, complexity, embeddability and completeness) can be shared in these paraconsistent semantics.
It is remarked that this paper does not give a "comprehensive" comparison, since the existing paraconsistent semantics have some different constructors (or logical connectives), i.e., it is difficult to compare the whole parts of these existing semantics.But, this paper gives an "essential" comparison with respect to the common part with the constructors (paraconsistent negation), (intersection), (union), (universal concept quantification) and (existential concept quantification).To obtain such a comparison with some exact proofs, we need some small modifications of the existing paraconsistent semantics.Since all the logics discussed in this paper are defined as semantics, we will occasionally identify the semantics with the logic determined by it.
The contents of this paper are then summarized as follows.
In Section 2, the essential parts of the existing paraconsistent semantics (i.e., -semantics, four-valued semantics, quasi-classical semantics and single interpretation semantics) are addressed.

 
In Section 3, two new semantics (i.e., the semantics and the  -semantics) are introduced, and the equivalence among the -semantics, the -semantics and the -semantics is proved.It is observed that the essential parts of the four-valued semantics and the quasi-classical semantics are special cases of the -semantics.It is also observed that the -semantics is regarded as a classical version of the -semantics (single-interpretation semantics) for a constructive description logic introduced by

Odintsov and Wansing.
In Section 4, some remarks on constructive PDLs and temporal DLs.
In Section 5, this paper is concluded.

Semantics 
In the following, we present the logic [14], which has dual-interpretation semantics.The concepts are constructed from atomic concepts, roles, ~ (paraconsistent negation),

 
 (classical negation or complement), (intersection), (union), (universal concept quantification) and (existential concept quantification).We use the letters A and i A for atomic concepts, the letter for roles, and the letters and for concepts.
are defined by the following grammar: and to every role R a binary rela The interpretation functions are extended to concepts by the following inductive definitions:   :   :   :   :   :   : An expression . A concept is said to be satisfiable in if there exists a paraconsistent interpretation such that .
The interpretation functions   and    are intended to represent "verification" (or "support of truth") and "falsification" (or "support of falsity"), respectively.It is noted that includes [3] as a subsystem since is constructed based on the following additional axiom schemes for :   It is noted that the interpretations for ~ and  in correspond to the axiom scheme , which means that ~ and are self duals with respect to and ~, respectively.We now give an intuitive example for this axiom.Let does not hold, and hence is paraconsistent with respect to:.It is remarked that is not paraconsistent with respect to   .Next, we explain about some differences and similarities between [16] and . In interpretation functions were used.These interpretation functions include the following characteristic conditions for negations: 1) for any atomic concept A , , 2) for any atomic concept A , 3) for any atomic concept A , It is remarked that the condition 1 above means that  is not paraconsistent with respect to  .The subsystem (or special case) (of ), which adopts two interpretation functions . The conditions for the constructors and R  of are almost the same as those of .The main differences are presented as follows: has the "non-paraconsistent" condition: for any atomic concept adopts the condition: but has no this condition and adopts the condition:

Four-Valued Semantics and Quasi-Classical Semantics
Some four-valued semantics in [4] were based on ,     , DL-Lite, etc., and the quasi-classical semantics in [13] was based on  .The four-valued semantics in [4] has no classical negation, but has some new inclusion constructors such as strong inclusion.In addition, the quasi-classical semantics in [13] has two kinds of definitions called QC weak semantics and QC strong semantics.The following explanation is based on and QC weak semantics.We use the common language based on , and/or R   .We cannot compare the existing paraconsistent semantics (i.e., the four-valued semantics, the quasi-classical semantics, the single-interpretation semantics and the dual-interpretation semantics) themselves since the underlying DLs are different.Moreover, the motivations of introducing the existing semantics are completely different.For example, in the quasi-classical semantics, the main motivation is to satisfy three important inference rules: modus ponens, modus tollens and disjunctive syllogism.These inference rules are strongly dependent on a specific inclusion constructor and a specific QC entailment .Thus, our comparison witho t  is regarded as not so comprehensive or essential in the sense of the original motivation of the quasi-classical semantics.


The following definition is a slight modification of the definition of is defined using a pair , P N of subsets of   and the projection functions proj , : The inter- pretations are then defined as follows: In the four-valued semantics for [4], different kinds of implications were introduced: The interpretations of , and are respectively presented as follows: These implications provide flexible way to model inconsistent ontologies.
The extension of four-valued semantics to the expressive description logic , and the extensions of four-valued semantics to some tractable description logics     , Horn-DLs and DL-Lite family were studied in [5].
Next, we discuss about quasi-classical description logic.The following definition is a slight modification of the definition of quasi-classical description logics [12,13].
of subsets of   without using projection functions.The interpretations are then defined as follows: 21) a role is assigned to a pair The quasi-classical semantics for QC [12] were extended to that of QC [13] to handle inconsistent ontologies.It composes two kinds of semantics, i.e., QC weak semantics and QC strong . QC weak semantics inherits the characteristics of four-valued semantics, and QC strong semantics redefines the in-terpretation for disjunction and conjunction of concepts to make the three important inference rules (i.e., modus ponens, modus tollens and disjunctive syllogism) hold.
Let Q be a QC entailment and be a paraconsistent negation connective, which is represented as ~ in the above definition.Then, the following hold: uery entailment problems (i.e., instance ch section, it is remarked that the pa (modus tollens) 3) e syllogism).
Two basic ecking and subsumption checking) were also defined and discussed in [13].It was also shown that the two basic inference problems can be reduced into the QC consistency problem.
Finally in this sub iring functions used in the four-valued and quasiclassical semantics have been used in some algebraic semantics for Nelson's logics (see e.g., [21] and the references therein).On the other hand, the semantics of  is defined using two interpretation functions    and    instead of the pairing functions.These rpretati functions have been used in some Kripketype semantics for Nelson's logics (see e.g., [22] and the references therein).It will be shown in the next section that the "horizontal" semantics using paring functions and the "vertical" semantics using two kinds of interpretation functions have thus essentially the same meaning.

S
inte on r N4.

ingle-Interpretation
2) for an atomic concept A ,    .


It is remarked that the order relation needs som more conditions.For the details, see [8,9]  :  :  :  :  :  :  : The interpretation function is extended to concepts by:

Next, we show the equivalence between and
For any concept , is in Theorem 3.4 (Equivalence between  and ) For 2, we ha ve: For 2, we have: : For 1, we have: (by induction hypothesis for 1) r 2, we have: (by induction hypothesis for 1) For 2, we ha : ve We define a quasi-classical interpretation  by: 1) :  is a positive ne resp.)p function which assigns to every atomic co t

a interpretation function which assigns to every atomic concept n
A a pair (by induction hypothesis for 2), ent to show the follo It is suffici wing claim which implies the required fact.For any concept , Then, we have the f f act: Since this claim can be show the claim of the direction here.retation idea of this formulation is ed from stent semantics for a con- [8].T ts.
n in the same way as in ) , the proof is omitted □ (

 Semantics
We introduce a new logic  , which has a singleinterp function.The inspir the paraconsi structive PDL proposed in hese single-interpretation semantics can also be adapted to Nelson's paraconsistent logic (see [20]).
Similar notions and terminologies for  are also used for  .The  -concepts are the same as the  -concep Definition 3.5 Let  be the set of atom cepts and ic con tion which assigns to every atomic (or negated atomic) concep  .The interpretation function is extended by the f nductive definitions:   : An expression is defined as .( 11) Next, we show the equivalence between and Theorem 3.6 (Equivalence between and ) For any concept , Proo be the set of atomic concepts, , and  be the set of

5) t following conditions hold:
It is sufficient to show the foll implies the required fact.For any con owing claim which cept C , ( By (simultaneous) induction on C .W show so e me cases.
Case C A  ( A is an atomic co nition. (by induction hypothesis for 1)   : For 1, we have: : For 1, we have: r 2, we have: : For 1, we have: (by induction hypothesis for 1) For 2, we have: (by induction hypothesis for 2),

Remarks
The interpretations of and are rather complex, and hence om .Such retations (2)   :      .: , .Such embeddi theorem is similar to a theorem for embedding  into  .Thus, in an abstract sense,  a  can be viewed as the same kind of embeddable logics.Indeed, the same embedding-based method can be applied to these logics uniform

Conclusions
In this paper, a comparison of paraconsistent description logics was addressed.New paraconsistent description logics  and  were ng nd equivalence among proved.The   , -sem  antics is re  d as a  ralization of both the four-valued semantics [4,5] and the quasi-classical semantics [12,13].The  -semantics is d as a modification of the single-interpretation sema  antics [14], also calle -interpretation semantics, was taken over from the dual-consequence Kripke-style semantics for Nelson's paraconsistent logic N4 [18,1 Finally, some recent developments on paraconsistent logics based on N4 are addressed.In oof theory of N4 and its variations were presented.In [26], completeness and cut-elimination theorems were proved for some trilattice logics which are regarded ations of N4.In [27], a paraconsistent linear-time temporal logic was introduced extending the well-known linear-time temporal logic (LTL).In [28], a paraconsistent computation-tree logic was introduced extending the well-known computation-tree logic (CTL).In [29], a constructive temporal paraconsistent logic was introduced combining N4 and a constructive version of LTL.

)
Since this claim can be shown in the sa e to present the fou quasi-classical semantics and the dual-interp semantics for these constructive PDLs.The notions of constructiveness and paraconsistency are known to be important for logical systems.From the point of view of the truth and falsehood in a logic, the principle of explosion of each other.Paraconsistent logics are logics without the principle of explosion, and paracomplete logics are the logics without the excluded middle.Constructive logics are classified as a paracom he logics with both the paraconsistency and the paracompleteness are called paranormal (or nonalethic) logics.Since the precise definitions of the original semantics for N4  and N4d  are rather complex, we now present only an outline of the (slightly modified versions of the) semantics of N4 interpretation  is a structure .
   where 1)   is a non-empty set, )  and  as in  .Si