=Paper=
{{Paper
|id=Vol-1517/JOWO-15_OntoChange_paper_3
|storemode=property
|title=Well-founded Paraconsistent Semantics for Hybrid Theories composed of Rules and Ontologies
|pdfUrl=https://ceur-ws.org/Vol-1517/JOWO-15_OntoChange_paper_3.pdf
|volume=Vol-1517
|dblpUrl=https://dblp.org/rec/conf/ijcai/KaminskiKL15a
}}
==Well-founded Paraconsistent Semantics for Hybrid Theories composed of Rules and Ontologies==
https://ceur-ws.org/Vol-1517/JOWO-15_OntoChange_paper_3.pdf
Well-founded Paraconsistent Semantics for Hybrid Theories composed of Rules
and Ontologies∗
Tobias Kaminski and Matthias Knorr and João Leite
NOVA LINCS
Departamento de Informática
Universidade NOVA de Lisboa
2829-516 Caparica, Portugal
Abstract Krötzsch et al., 2011], others follow a hybrid combination
of ontologies with nonmonotonic rules, either providing a
Description Logic (DL) based ontologies and non- modular approach where rules and ontologies use their own
monotonic rules provide complementary features semantics, and allowing limited interaction between them
whose combination is crucial in many applications. [Eiter et al., 2008], or defining a unifying framework for both
In hybrid knowledge bases (KBs), which combine components [Motik and Rosati, 2010; Knorr et al., 2011].
both formalisms, for large real-world applications, Equipped with semantics that are faithful to their constituting
often integrating knowledge originating from dif- parts, these proposals allow for the specification of so-called
ferent sources, inconsistencies can easily occur. hybrid knowledge bases (hybrid KBs) either from scratch,
These commonly trivialize standard reasoning and benefiting from the added expressivity, or by combining ex-
prevent us from drawing any meaningful conclu- isting ontologies and rule bases.
sions. When restoring consistency by changing the
The complex interactions between the ontology component
KB is not possible, paraconsistent reasoning offers
and the rule component of these hybrid KBs – even more so
an alternative by allowing us to obtain meaningful
when they result from combining existing ontologies and rule
conclusions from its consistent part.
bases independently developed – can easily lead to contra-
In this paper, we address the problem of efficiently dictions, which, under classical semantics, trivialize standard
obtaining meaningful conclusions from (possibly reasoning and prevent us from drawing any meaningful con-
inconsistent) hybrid KBs. To this end, we de- clusions, ultimately rendering these hybrid KBs useless.
fine two paraconsistent semantics for hybrid KBs
One way to deal with this problem is to employ some
which, beyond their differentiating properties, are
method based on belief revision (e.g. [Leite, 2003; Oso-
faithful to well-known paraconsistent semantics as
rio and Cuevas, 2007; Slota and Leite, 2012a; 2014; Del-
well as the non-paraconsistent logic they extend,
grande et al., 2013] for LPs, [Flouris et al., 2008; Calvanese
and tractable if reasoning in the DL component is.
et al., 2010; Kharlamov et al., 2013] for DLs, and [Slota et
al., 2011; Slota and Leite, 2012b] for hybrid KBs) to regain
1 Introduction consistency, so that standard reasoning services can be used.
Alternatively, some method based on repairing (e.g. [Are-
In this paper, we address the problem of dealing with incon- nas et al., 1999] for LPs and [Haase et al., 2005] for DLs)
sistent knowledge bases consisting of ontologies and non- can be used, where hypothetical belief revision is employed
monotonic rules, following a paraconsistent reasoning ap- for consistent query answering, without actually changing the
proach with a focus on efficiency. KB. However, this is not always feasible e.g. because (in the
Description Logics (DLs) and Logic Programs (LPs) pro- first case) we may not have permission to change the KB –
vide different strengths when used for Knowledge Represen- as for instance in [Alberti et al., 2011] where the KB en-
tation and Reasoning. While DLs employ the Open World codes laws and norms – or because the usual high complex-
Assumption and are suited for defining ontologies, LPs adopt ity of belief revision and repairing methods simply renders
the Closed World Assumption and are able to express non- their application prohibitive. When these methods are not
monotonic rules with exceptions and preference orders. Com- possible or not feasible, paraconsistent reasoning services,
bining features of both formalisms has been actively pursued typically based on some many-valued logics, offer an alter-
over the last few years, resulting in different proposals with native by being able to draw meaningful conclusions in the
different levels of integration and complexity: while some presence of contradiction. Whereas paraconsistent reasoning
extend DLs with rules [Horrocks and Patel-Schneider, 2004; has been extensively studied in the context of each individ-
∗
Partially supported by Fundação para a Ciência e a Tecnologia ual component of hybrid KBs (see Related Work in Sect. 5),
under project PTDC/EIA-CCO/121823/2010 and strategic project it is still a rather unexplored field in the context of hybrid
PEst/UID/CEC/04516/2013. M. Knorr was also supported by grant KBs. Notable exceptions are [Huang et al., 2011; 2014;
SFRH/BPD/86970/2012. Fink, 2012], yet their computation is not tractable in general
even if reasoning in the DL component is. and ϕ1 ∨ ϕ2 , ∀x : ϕ, >, and ⊥ represent the usual syntactic
In this paper, we investigate efficient paraconsistent seman- short-cuts. Replacing free variables x in ϕ by terms s is repre-
tics for hybrid KBs. We adopt the base framework of [Motik sented by ϕ[s/x]. A closed MKNF formula contains no free
and Rosati, 2010] because of its generality and tight integra- variables. Formulas of the form K ϕ and not ϕ are called,
tion between the ontology and the rules – c.f. [Motik and resp., K-atoms and not-atoms. Intuitively, K ϕ asks if ϕ is
Rosati, 2010] for a thorough discussion – under the semantics known, while notϕ checks if ϕ does not hold, which allows
of [Knorr et al., 2011] because of its computational proper- one to draw conclusions from the absence of information.
ties. We extend such semantics with additional truth values Hybrid KBs combine a set of MKNF rules and a DL on-
to evaluate contradictory pieces of knowledge, following two tology O, which is translatable into a function-free first order
common views on how to deal with contradictory knowledge formula with equality π(O) and for which checking of satis-
bases. According to one view, contradictions are dealt with fiability and instances are decidable [Baader et al., 2003]. An
locally, in a minimally intrusive way, such that a new truth MKNF rule r is of the given form where H, Ai , Bi are atoms:
value is introduced to model inconsistencies, while consistent KH ← KA1 , . . . , KAn , notB1 , . . . , notBm . (1)
pieces of information whose derivation depends on inconsis-
tent information are still considered to be true in the classi- K H is called the rule head, and the sets {K Ai } and
cal sense. This view is adopted in paraconsistent semantics {notBj } are called the positive body and the negative body,
for DLs, e.g. [Maier et al., 2013], LPs, e.g. [Sakama, 1992; respectively. A rule r is positive if m = 0, and r is a fact if
Sakama and Inoue, 1995], and hybrid KBs [Huang et al., n = m = 0. A program P is a finite set of MKNF rules,
2011; Fink, 2012]. The alternative view is to distinguish and positive if all rules in it are positive. A hybrid knowledge
truth which depends on the inconsistent part of a KB, from base (hybrid KB) K is a pair (O, P). Given such K = (O, P),
truth which is derivable without involving any contradictory KG = (O, PG ) is a ground hybrid KB where PG denotes the
knowledge. This view, commonly referred to as Suspicious grounding of P using all constants occurring in K as usual.
Reasoning, is adopted in paraconsistent semantics for LPs, As hybrid KBs can be translated into MKNF formulas using
e.g. [Alferes et al., 1995; Sakama, 1992; Sakama and Inoue, π(O) and the match between → and ⊃ for the MKNF rules,
1995] and hybrid KBs [Huang et al., 2014]. their semantics is derived from that of MKNF formulas, and
We present solutions following both views through the def- we abuse notation and refer to such translation π(K) by K. To
inition of a five-valued and a six-valued paraconsistent se- achieve decidability of reasoning, hybrid KBs are restricted to
mantics for hybrid KBs, the latter implementing Suspicious be DL-safe, basically requiring that variables in rules appear
Reasoning, both of which enjoy the following properties: at least once in the positive body under a predicate which does
• Soundness w.r.t. the three-valued semantics for consis- not occur in O, thus limiting the applicability of constants in
tent hybrid KBs by [Knorr et al., 2011]; rules to those in P. From now on, we only consider DL-safe
• Faithfulness w.r.t. semantics for its base formalisms; hybrid KBs as it can always be ensured [Ivanov et al., 2013].
• Computability by means of a sound and complete fix- Example 1. Consider the following simplified ground hybrid
point algorithm; KB KG for assessing the risk of goods at a port.
• Tractability when a tractable DL is used for the ontology. HasCertif iedSender v ¬IsM onitored (2)
The paper is organized as follows: in Sect. 2, we present KIsM onitored(g) ← Krisk(g). (3)
the formal background; in Sect. 3, we present both semantics,
Krisk(g) ← notisLabelled(g). (4)
starting with common parts, and proceeding with the five-
valued semantics in Sect. 3.1, and the six-valued one in Sect. KisLabelled(g) ← notrisk(g). (5)
3.2; in Sect. 4, we investigate the properties of both semantics KresolvedRisk(g) ← KIsM onitored(g). (6)
and discuss related work and conclude in Sect. 5.1 KHasCertif iedSender(g) ← (7)
(4) and (5) state that good g is either a risk (r) or it is labeled
2 Preliminaries (iL). Any risk is monitored (IM ) (3), thus a resolved risk
In this section, we introduce hybrid knowledge bases, and (rR) (6). As g has a certified sender (HCS) (7), it can be
also recall the syntax of MKNF formulas, originating from proven by means of axiom (2) that it is not monitored. Thus, g
the logic of minimal knowledge and negation as failure can be derived to be monitored and not monitored at the same
(MKNF) [Lifschitz, 1991], into which the former are embed- time if it is considered to be a risk, which can be modeled by
ded. For reasons of space, some details, e.g., on DLs or the means of a paraconsistent evaluation.
semantics for MKNF formulas, are omitted here. These can Let KG = (O, PG ) be a ground hybrid KB. The set of K-
still be found in [Knorr et al., 2011]. atoms of KG , written kA(KG ), is the smallest set that contains
The syntax of MKNF formulas is defined over a function- (i) all ground K-atoms occurring in PG , and (ii) a K-atom
free first-order signature Σ = (Σc , Σp ), where the sets Σc Kξ for each not-atom notξ occurring in PG . For a subset S
and Σp contain, resp., all constants and all predicates includ- of kA(KG ), the objective knowledge of S w.r.t. KG is the set
ing the binary equality predicate ≈. Given a predicate P and of first-order formulas obO,S = {π(O)} ∪ {ξ | Kξ ∈ S}.
terms s over Σc and a set of variables, an atom P (s) is an
MKNF formula. If ϕ, ϕ1 and ϕ2 are MKNF formulas, then 3 Paraconsistent MKNF Semantics
¬ϕ, ∃x : ϕ, ϕ1 ∧ ϕ2 , ϕ1 ⊃ ϕ2 , K ϕ and not ϕ also are,
In this section, we define two paraconsistent semantics for
1
This work has been published in [Kaminski et al., 2015]. hybrid KBs, namely 5- and 6-models, their main difference
being whether Suspicious Reasoning is supported in the rules t ⊃ b t f uf u
of the hybrid KB or not. This requires the integration of dif- b b t f uf u
ferent concepts and assumptions w.r.t. paraconsistency, inde- t b t f uf u
pendently developed for each of the base formalisms. E.g. u b f t t t t t
Suspicious Reasoning has not been considered in DLs, so de- uf uf t t uf t t
veloping a unified semantics that is faithful to the paraconsis- f u t t u t t
tent semantics of the two base formalisms, thus limiting Sus-
picious Reasoning to inconsistencies from the LP, is highly Figure 1: The lattice FIVE and evaluation of the operator ⊃.
non-trivial. Finding a model theory corresponding to some
LP fixed-point based semantics is also very challenging. All
this while maintaining faithfulness to the three-valued seman- It is possible to define a so-called knowledge order k on
tics [Knorr et al., 2011], including properties such as Coher- 5-pairs and 6-pairs, with the intuition that elements which are
ence, i.e. false (first-order) formulas are also default false. We greater allow one to derive more true and false knowledge.
start with common notions of both semantics. Definition 5. For two 5- or 6-pairs (M1 , N1 ) and (M2 , N2 ),
First, we introduce p-interpretations, which extend first- (M1 , N1 ) k (M2 , N2 ) iff M1 ⊆ M2 and N1 ⊇ N2 .
order interpretations2 with the ability to represent that certain
Some 5- and 6-pairs will turn out to be 5- and 6-models as
pieces of information are true and false at the same time.
defined next in Sects. 3.1 and 3.2, and the order k will be
Definition 1. Given two first-order interpretations I and I1 used to compare such 5- and 6-models, and single out those
with I1 ⊆ I, the pair I = hI, I1 i is called a p-interpretation. that are most skeptical.
Intuitively, I indicates what is true (t) and false (f ), while Definition 6. Let ϕ be a closed MKNF formula and (M, N )
the additional interpretation I1 only designates for each (true) a 5-model (6-model) of ϕ such that (M1 , N1 ) k (M, N )
element in I if it is actually inconsistent (b - for both) or not. for all 5-models (6-models) (M1 , N1 ) of ϕ. Then (M, N ) is
No fourth value is assigned, arguably resulting in a simpler a well-founded 5-model (well-founded 6-model) of ϕ.
paraconsistent semantics for first-order formulas, and at the
same time in a stronger consequence relation [Priest, 1979]. In Sects. 3.1 and 3.2, we will now provide a model-based
P-interpretations are the basis for defining structures used account of 5- and 6-models and a procedural characterization
in both semantics, subsequently defined, for interpreting of the (unique) well-founded 5- and 6-model.
MKNF formulas, to which hybrid KBs can be translated. 3.1 Five-Valued Semantics
Definition 2. A 6-structure (I, M, N ) consists of a p- We begin with the five-valued semantics, whose motivation
interpretation I and two pairs M = hM, N i and N = stems from the fact that, for the three-valued MKNF seman-
hM1 , N1 i of sets of p-interpretations. A 5-structure is a 6- tics [Knorr et al., 2011], with truth values true, false and un-
structure where M1 ⊆ M and N1 ⊆ N . defined (t, f and u resp.), two kinds of inconsistencies are
Thus, a 5-structure is a special case of a 6-structure. identified. Namely, either some piece of information is simul-
As common with MKNF semantics, M and N will be taneously considered true and false, or undefined and false.
used, resp., to evaluate K- and not-atoms, while the p- We handle this by introducing for each of the two cases a fur-
interpretation I is used to evaluate first-order expressions. In ther truth value, namely inconsistent (b) and undefined false
particular, the latter applies in both semantics to atoms, ac- (uf ), respectively. The resulting lattice FIVE (Fig. 1) ex-
cording to the intuition given after Def. 1. tends the well-known lattice FOU R, and is not symmetric
Definition 3. Given 5- or 6-structure (I, M, N ), atom P (s): simply because no inconsistencies can occur between t and
u in [Knorr et al., 2011], as t always prevails in this case.
b iff sI ∈ P I , sI1 ∈ P I1 The evaluation of closed MKNF formulas in 5-structures
(hI, I1 i, M, N )(P (s)) = t iff sI ∈ P I , sI1 6∈ P I1 for the truth values in FIVE is shown in Fig. 2 and, in the fol-
f iff sI 6∈ P I , sI1 6∈ P I1 lowing, we will give intuitions and necessary notions. First,
¬ behaves for all values in FOUR as expected. The remain-
The evaluation of complex MKNF formulas differs for 5- and
ing case follows the intuition that uf behaves under negation
6-structures, and is therefore spelled out separately.
like f . The implication ⊃ is defined (Fig. 1) like internal im-
Also based on p-interpretations are the notions that repre-
plication by [Maier et al., 2013] for {b, t, f , u}, apart from
sent potential model candidates for each of the semantics.
the case u ⊃ f , which is no longer mapped to t, as it does
Definition 4. A 6-pair (M, N ) consists of two non-empty correspond to the kind of inconsistency between u and f in
sets M , N of p-interpretations. A 5-pair (M, N ) is a 6-pair [Knorr et al., 2011], for which we introduced uf in the first
where N ⊆ M . If M = N , then (M, N ) is called total. place. Hence, u ⊃ uf is mapped to t, which also corresponds
2 to the idea that uf behaves under ⊃ like a special case of u.
In MKNF semantics, commonly the standard name assumption
For ∧ and ∨, min and max are used like in [Knorr et al.,
(SNA) is imposed on interpretations I, i.e., (1) the universe ∆ of I
contains Σc and an infinite supply of additional constants, (2) con- 2011] instead of the join and meet operations, more common
stants are mapped to themselves, and (3) equality is interpreted as for paraconsistent semantics, and this originates from the fact
a congruence relation. As first-order consequences under SNA and that b and uf should behave like special cases of t and u
the standard first-order semantics are not distinguishable [Motik and respectively, that are only necessary if there is an explicit oc-
Rosati, 2010], we also assume SNA here and in the remainder. currence of an inconsistency. This means that, if a rule body
(I, M, N )(ϕ1 ∧ ϕ2 ) = (I, M, N )(ϕ1 ) ∧ (I, M, N )(ϕ2 )
iff (I, M, N )(ϕ) = b
b
(I, M, N )(ϕ1 ⊃ ϕ2 ) = (I, M, N )(ϕ1 ) ⊃ (I, M, N )(ϕ2 )
t iff (I, M, N )(ϕ) ∈ {f , uf }
(I, M, N )(¬ϕ) =
f iff (I, M, N )(ϕ) = t (I, M, N )(∃x : ϕ) =
_
(I, M, N )(ϕ[α/x])
iff (I, M, N )(ϕ) = u
u
T α∈∆
b iff TM (ϕ) = b T
b iff TM1 (ϕ) = b
t iff TM (ϕ) = t
t T ( N1 (ϕ) = f ) or
iff
(I, M, N )(Kϕ) = f iff TN (ϕ) = f T
T (I, M, N )(notϕ) = ( MT1 (ϕ) = f and N1 (ϕ) = b)
uf iff M (ϕ) = f and N (ϕ) = b
f iff M1 (ϕ) = t
u otherwise
u otherwise
Figure 2: Recursive five-valued evaluation of a closed MKNF formula in a 5-structure (I, M, N ), given that ϕ, ϕ1 , and ϕ2 are
MKNF formulas, and that M = hM, N i and N = hM1 , N1 i. The operator ⊃ is evaluated as shown in Fig. 1. The operators ∧
and ∨ are defined respectively as min and max on the order f < u < uf < t < b.
is b, then its head should be t, unless we can explicitly de- I 0 ∈ M 0 such that (I 0 , hM 0 , N 0 i, hM, N i)(ϕ) 6∈ {b, t}.
rive the negation of the head elsewhere, and similarly for a If ϕ has a 5-model, then ϕ is 5-consistent.
rule body that is uf whose head should be u, unless we can The evaluation of not-atoms is fixed before checking whether
explicitly derive its negation (or alternatively that it is t or M and N are maximal, i.e. whether the evaluation of K-
b which would prevail over the derivation from this rule). atoms is minimal w.r.t. the order f < u < uf < t < b.
Thus, the order f < u < uf < t < b implicitly reduces to
f < u < t in [Knorr et al., 2011]. This means that if a rule Example 2. Consider KG from Ex. 1. For (4) and (5) alone,
body contains (a conjunction of) two elements that are b and there are three 5-models since it is not determined whether g
u, then the head is u from this rule alone, and not f , and, for is labelled or a risk: a), Kr is t and KiL is f , b), vice-versa,
∨, that ∃x : ϕ should be t if there is one replacement of x or c), both are u. Both being b would also 5-satisfy (4) and
which makes it b, one that is u (or uf ), and none that is t. (5), but with the evaluation of not-atoms fixed to b in such
For the evaluation of K and not, we employ intersec- a 5-pair (M, N ), the rule heads can be t to satisfy the rules,
tions over sets of p-interpretations present in 5- and 6- and there is (M 0 , N 0 ) as described in (2) of Def. 8.
structures for which we introduce specific notation. Namely, For 5-satisfying (3)-(5), K IM has to be t or b for a) as
we can intersect p-interpretations component-wise to obtain K r is t, can have any truth value for b) as K r is f , and not
the pieces of information on which they coincide. Given be f for c) as K r is u (cf. Fig. 1). However, e.g. any 5-pair
aT set M ofTp-interpretations Ii = hIi , Ii0 i, we canTdefine mapping K r to f and K IM not to f is not a 5-model since
M = h Ii , Ii0 i, and it can be shown that M is
T it is not minimal by (2) of Def. 8. Accordingly, K IM is
indeed also a p-interpretation. In addition, we abbreviate minimized to t for a), to f for b), and to u for c).
T
( J ∈X J , hM, N i, hM1 , N1 i)(ϕ) for any Taking all of KG , there is a conflict between (2) and (3) if
T ϕ, M , N , M1 , Kr is t (model a)) or u (model c)) since the classical negation
and N1 , and X ∈ {M, N, M1 , N1 }, with X (ϕ).
Regarding the actual evaluation, K ϕ being b can be seen of IM is also derivable. Thus, for the three possible 5-models
as a special case of being t in the sense that no J ∈ M can for (4) and (5) alone, K IM and K rR are resp. minimized
map ϕ to t, and likewise for uf and u w.r.t. M1 , i.e. uf be- to b and t for model a), to f and f for model b), and to uf
haves like (a special case of) u. The evaluation of not ϕ is and u for c). Note that the head of (6) is evaluated as if the
symmetric to that of Kϕ, for all but uf , which, again, behaves body was t and u respectively, w.r.t. models a) and c). Hence,
under negation like f . Using intersections in this evaluation neither kind of inconsistency is propagated, i.e., for a), KIM
slightly deviates, formally, from [Knorr et al., 2011], but the is b, yet KrR is t, and for c) KIM is uf , yet KrR is u.
differences are of no impact for hybrid KBs, and this choice As reasoning – entailment from hybrid KBs – directly with
results in a simpler notation. models that are usually infinite would be unfeasible, we adopt
With the evaluation in 5-structures in place, we can define a common technique for reasoning with hybrid KBs [Motik
5-satisfaction for 5-pairs as follows. and Rosati, 2010; Knorr et al., 2011] and provide a finite rep-
Definition 7. Given a closed MKNF formula ϕ, a 5-pair resentation of 5-models and the well-founded 5-model in par-
(M, N ) 5-satisfies ϕ, written (M, N ) |=5 ϕ, if and only if ticular, which can be directly used for query answering w.r.t.
(I, hM, N i, hM, N i)(ϕ) ∈ {b, t} for each I ∈ M . entailed information. Similarly to [Knorr et al., 2011], this
finite representation is obtained via a fixpoint construction.
For defining 5-models, an additional preference order over To introduce this fixpoint construction for the five-valued
5-pairs is required, minimizing knowledge under operator K. semantics, we first define a variant of the doubling of hybrid
Definition 8. Any 5-pair (M, N ) is a 5-model for a given KBs in [Alferes et al., 2013], for which we introduce a new
closed MKNF formula ϕ iff predicate Ad for each predicate A appearing in KG . Here,
(1) (M, N ) |=5 ϕ and we abuse notation and denote also with Ad the atom resulting
(2) for each 5-pair (M 0 , N 0 ) with M ⊆ M 0 and N ⊆ N 0 , from replacing the original predicate A with Ad . It will be
where at least one of the inclusions is proper, there is clear from the context whether Ad is a predicate or an atom.
Definition 9. Let KG = (O, PG ) be a ground hybrid KB. We t
introduce a new predicate Ad for each predicate A appearing ⊃ b s t f cf u
in KG , and we constructively define u b b b t t f f f
1. Od by substituting each predicate A in O by Ad ; s b t t f f f
2. PGd
by transforming each rule of the form (1) into: t b f t f f f
s f t t t t t t
(a) KH ← KA1 , . . . , KAn , notB1d , . . . , notBmd
, cf
cf t t t t t t
d d d
(b) KH ← KA1 , . . . , KAn , notB1 , . . . , notBm ; u t f t f t t
3. and the ground doubled hybrid KB KG d
= (O, Od , PG d
). f
We now define a monotonic immediate consequence oper- Figure 3: The lattice SIX and evaluation of the operator ⊃.
ator TKG
d ,C . It collects the consequences from the program
component and from the ontology, and subtracts those dou- By Theorem 1, Pdω and Ndω constitute a finite representa-
bled K-atoms of which the classical negation is derivable. tion of the well-founded 5-model and can, for example, be
Definition 10. Let KG = (O, PG ) be a ground positive hy- used for query answering w.r.t. entailed information. Hence,
brid KB. The operators RKG d , DKd , CKd ,C , and TKd ,C are by computing the two fixpoints for KG of Example 1, the
G G G
d
defined on KG and subsets S and C of kA(KG d
) as follows: truth values of all K-atoms appearing in kA(KG ) w.r.t. its
well-founded 5-model can be determined.
RKG
d (S) {KH | PG
= d
contains a rule of the form Example 3. Recall KG from Ex. 1. Then, Pdω contains only
KH ← KA1 , . . . , KAn s.t. KAi ∈ S} K HCS (and K HCS d ), while Ndω = kA(KG d
) \ {K IM d }.
DKG d (S) = {Kξ | Kξ ∈ kA(KG ), obO,S |=5 ξ}∪ Thus, model c) in Ex. 2 is the well-founded 5-model.
{Kξ d | Kξ ∈ kA(KG ), obOd ,S |=5 ξ d } 3.2 Six-Valued Semantics
CKG
d ,C (S) = {Kξ d | Kξ ∈ kA(KG ), obO,C |=5 ¬ξ}
The motivation for the six-valued semantics can be described
TKG
d ,C (S) d (S) ∪ DKd (S)) \ CKd ,C (S)
= (RKG G G as extending the five-valued semantics with the capability for
Suspicious Reasoning. This results in two important changes.
Since TKG d ,C is only defined for positive hybrid KBs, a stan-
First, in order to implement Suspicious Reasoning, we intro-
dard transformation for general hybrid KBs is introduced. duce a further truth value suspicious (s), which will be as-
Definition 11. Let R be a set of ground MKNF rules and S a signed to K-atoms that are not explicitly inconsistent, i.e. not
set of ground K-atoms. The transform R/S contains all rules derivable to be true and false at the same time, but whose
K H ← K A1 , . . . , K An for which there exists a rule of the truth (in the five-valued semantics) is only derivable from a
form (1) in R with KBj 6∈ S for each 1 ≤ j ≤ m. contradiction in the program (i.e. its derivation depends on an
Additionally, let KG = (O, PG ) be a ground hybrid KB. inconsistency in the program). Second, we replace uf with
The transforms KG /S and KG d
/S are defined as KG /S = cf (classically false), which will be assigned to K-atoms that
(O, PG /S) and KG d
/S = (O, Od , PGd
/S) respectively. are neither t nor b, but whose classical negation is derivable
(from the ontology). Differently from uf for the five-valued
We can now define an antitonic operator that computes the semantics, cf can be viewed as a special case of f (rather
least fixpoint of TKG
d ,C w.r.t. the resulting hybrid KB. than u), and no further undefined knowledge can be derived
from it. Thus, classical falsity of K-atoms is propagated in
Definition 12. Let KG be a ground hybrid KB and S ⊆ the semantics. All this results in the lattice SIX (Fig. 3).
d
kA(KG ). We define the operator ΓKG d (S) = TKd /S,S ↑ ω
G The evaluation of closed MKNF formulas in 6-structures
and two sequences Pdi and Ndi as follows. for SIX is shown in Fig. 4 and, subsequently, we will again
provide intuitions and necessary notions. First, ¬ behaves for
Pd0 = ∅ Nd0 = kA(KGd
) values in FIVE (replacing uf by cf ) as in the five-valued
d d d d
Pn+1 = ΓKGd (N ) Nn+1 = ΓKG
d (Pn )
semantics, and the new value s mirrors the behavior of b.
d
S d n d
T d
Pω = Pi Nω = Ni The implication ⊃ is defined (Fig. 3) for all values in
FIVE (again replacing uf by cf ) as in the five-valued se-
The sequence of Pdi is in fact monotonically increasing,
mantics, only that all non-designated truth values have been
maximizing the set of true and inconsistent non-doubled K-
mapped to f for simplicity. The only differing case is cf ⊃ f
atoms, while that of Ndi is monotonically decreasing, mini-
which is now t (while uf ⊃ f is uf in the five-valued se-
mizing the set of non-doubled K-atoms that are not false. It
mantics). This is justified by the fact that cf is understood as
can be shown that both sequences are finite, in fact, even of
a special case of f in general (and not of u). For, s ⊃ x, s
polynomial length, so that both Pdω and Ndω exist for every
behaves like b for any x ∈ SIX , which is motivated by the
ground hybrid KB and are unique.
idea that the propagation of an inconsistency will itself also
Theorem 1 (Soundness and completeness). Let KG be a be propagated. For x ⊃ s, s behaves like t for all x ∈ SIX
5-consistent, ground hybrid KB, and Pdω and Ndω the fix- but t and u. These two cases are mapped to f , based on the
points of the corresponding sequences. Then (IP , IN ) is intuition that a consequent of an implication with true (unde-
the well-founded 5-model of KG , where IP = {I | I |=5 fined) antecedent does not depend on a contradiction since it
obO,Pdω ∩kA(KG ) } and IN = {I | I |=5 obO,Ndω ∩kA(KG ) }. is independently derivable to be true (undefined).
iff (I, M, N )(ϕ) = b
b (I, M, N )(ϕ1 ∧ ϕ2 ) = (I, M, N )(ϕ1 ) ∧ (I, M, N )(ϕ2 )
s iff (I, M, N )(ϕ) = s
(I, M, N )(¬ϕ) = t iff (I, M, N )(ϕ) ∈ {f , cf } (I, M, N )(ϕ1 ⊃ ϕ2 ) = (I, M, N )(ϕ1 ) ⊃ (I, M, N )(ϕ2 )
f iff (I, M, N )(ϕ) = t
_
(I, M, N )(∃x : ϕ) = (I, M, N )(ϕ[α/x])
u iff (I, M, N )(ϕ) = u
α∈∆
T
b iff TM (ϕ) = b
T
s T
iff TM (ϕ) = t and TN (ϕ) = f b
iff TM1 (ϕ) = b T
s iff TM1 (ϕ) = t and N1 (ϕ) = f
iff TM (ϕ) = t and N (ϕ) 6= f
t
t
∃M1 (ϕ) = t]
iff M1 (ϕ) = f s.t. [6 T
f iff T M (ϕ) = f s.t. ∃M (ϕ) = t
(I, M, N )(Kϕ) = (I, M, N )(notϕ) = or [∃M 1 (ϕ) = t and
TN1 (ϕ) = f )]
andT N (ϕ) = f T
f iff TM1 (ϕ) = t and N1 (ϕ) 6= f
iff TM (ϕ) = f s.t. 6 ∃M (ϕ) = t
cf
u iff T M1 (ϕ) = f s.t. ∃M1 (ϕ) = t
u iff T M (ϕ) = f s.t. ∃M (ϕ) = t
and N1 (ϕ) 6= f
and N (ϕ) 6= f
Figure 4: Recursive evaluation of an MKNF formula in a 6-structure (I, M, N ), given that ϕ, ϕ1 , and ϕ2 are MKNF formulas,
and that M = hM, N i and N = hM1 , N1 i. The operator ⊃ is evaluated as shown in Fig. 3. The operators ∧ and ∨ are defined
respectively to be the join and meet operation in the lattice SIX .
For ∧ and ∨, the join and meet operations on SIX are With the contradiction from (2), K IM has to be b for a)
used, which are both standard for paraconsistent semantics. where Kr is t, and KIM has to be cf for b) if Kr is f or c)
For the evaluation of the operators K and not, we may refer where Kr is u, due to the definition of ⊃ and (3) of Def. 14.
to the (non-)existence of p-interpretations in sets of them. For In the first case a), K rR can be minimized to s such that
readability, we abbreviate ∃J ∈ X (resp. 6 ∃J ∈ X) s.t. support on contradiction can be detected. In the other cases
(J , hM, N i, hM1 , N1 i)(ϕ) = y for any ϕ, M , N , M1 , and b) and c), K rR is f . Thus, unlike Ex. 2 where K rR is u if
N1 , X ∈ {M, N, M1 , N1 } and truth value y, with ∃X(ϕ) = KIM is uf , the falsity is propagated to KrR if KIM is cf ,
y (resp. 6 ∃X(ϕ) = y). so no undefined knowledge can be derived from KIM .
Regarding the actual evaluation, the observations for K and As before, we introduce a fixpoint construction which will
not from Sec. 3.1 persist. Then, s can be seen as a second spe- provide a finite representation of the well-founded 6-model.
cial case of t (different from b). The changes w.r.t. the cases We define two operators TKG and TK0 G ,C , where K -atoms
for f , cf , and u occur basically because a) for 6-pairs (M, N )
whose classical negation is derivable are removed in TK0 G ,C .
N ⊆ M does no longer hold, and b) more importantly, cf is
now to be understood as a special case of f and not u. Definition 15. Let KG = (O, PG ) be a ground positive hy-
Since N ⊆ M does not hold for 6-pairs (Def. 4), we have brid KB. The operators RKG , DKG , TKG and TK0 G ,C are de-
to check all p-interpretations in both M ∪N for p-satisfaction. fined on KG and subsets S and C of kA(KG ) as follows.
Definition 13. Given a closed MKNF formula ϕ, a 6-
RKG (S) = {KH | PG contains a rule of the form
pair 6-satisfies ϕ, written (M, N ) |=6 ϕ, if and only if
KH ← KA1 , . . . , KAn s.t. KAi ∈ S}
(I, hM, N i, hM, N i)(ϕ) ∈ {b, t} for each I ∈ M ∪ N .
DKG (S) = {Kξ | Kξ ∈ kA(KG ), obO,S |=p ξ}
The definition of a 6-model only applies to hybrid KBs, so TKG (S) = RKG (S) ∪ DKG (S)
we can refer directly to K-atoms in the program in a new third TK0 G ,C (S) = (RKG (S) ∪ DKG (S))\
condition ensuring that certain 6-models that contain unjusti- {Kξ | Kξ ∈ kA(KG ), obO,C |=p ¬ξ}
fied evaluations to b or cf are removed.
Definition 14. Let (M, N ) be a 6-pair and K = (O, P) a Reusing the transform (Def. 11), we define two operators.
hybrid KB. Any 6-pair (M, N ) is a 6-model of K iff Definition 16. Let KG = (O, PG ) be a ground hybrid KB
(1) (M, N ) |=6 K, and S ⊆ kA(KG ). We define the two operators ΓKG (S) =
(2) for every 6-pair (M 0 , N 0 ) with M ⊆ M 0 and N ⊆ N 0 TKG /S ↑ ω, and Γ0KG (S) = TK0 G /S,S ↑ ω and two sequences
where at least one of the inclusions is proper, there is Pi and Ni as follows.
I 0 ∈ M 0 ∪ N 0 s.t. (I 0 , hM 0 , N 0 i, hM, N i)(K) 6∈ {b, t},
P0 = ∅ N0 = kA(KG )
and 0
Pn+1 = ΓSKG (Nn ) Nn+1 = Γ TKG (Pn )
(3) for every K ξ ∈ kA(KG ) it holds that
Pω = Pi Nω = Ni
(∗, hM, N i, hM, N i)(K ξ) ∈ {b, cf } if and only
if obO,{K ξ0 |(∗,hM,N i,hM,N i)(K ξ0 )∈{b,s,t}} |=6 ¬ξ. In the alternating fixpoint, ΓKG is used in the sequence of Pi ,
If ϕ has a 6-model, then ϕ is 6-consistent. and Γ0KG in that of Ni . This ensures that K -atoms whose
Example 4. Recall KG (Ex. 1) and the three different evalua- classical negation is derivable are not contained in the latter
tions of Kr and KiL in the 5-models a)–c) in Ex. 2. Without sequence that minimizes the set of K-atoms that are t or u.
axiom (2), K IM could also be minimized to cf if K r is u As for the five-valued semantics, the unique well-founded
(model c)), but this would be prevented by (3) of Def. 14 as 6-model can be obtained from the fixpoints Pω and Nω .
¬IM would not be derivable. Condition (3) also removes all Theorem 2 (Soundness and completeness). Let KG be a 6-
6-models where KHCS is b, so it has to be t. consistent, ground hybrid KB, and Pω and Nω the fixpoints
of the corresponding sequences. Then (IP , IN ) is the well- 5 Related Work and Conclusions
founded 6-model of KG , where IP = {I | I |=p obO,Pω }
and IN = {I | I |=p obO,Nω }. We have introduced two novel paraconsistent semantics for
hybrid KBs. They differ in whether inconsistencies and clas-
So, similar to Theorem 1, Theorem 2 provides the desired
sical falsity are propagated in the program component of the
finite representation of the well-founded 6-model for a hy-
KB. We have provided faithfulness results for both semantics
brid KB, and, thus, allows to determine the truth values of the
and have shown that they are efficiently computable if the
atoms in Example 1 w.r.t. its well-founded 6-model.
employed ontology language is tractable.
Example 5. Recall KG from Ex. 1. Then, Pω = {KHCS}, Paraconsistent reasoning has been extensively studied in
while Nω = {K r, K iL}. Thus, the 6-model c) in Ex. 4 is both base formalisms of hybrid KBs. For DLs, most work
the well-founded 6-model (where Kr and KiL are u). [Patel-Schneider, 1989; Straccia, 1997; Ma et al., 2007;
Zhang et al., 2009; Maier et al., 2013] focuses on four-valued
4 Properties semantics, varying which classical rules of inferences they
In this section, we show several important properties of our satisfy. The approach to which both our semantics are faith-
two semantics that demonstrate that both are well-defined. ful, [Ma et al., 2007; Maier et al., 2013], is most general as it
First, both semantics generalize the three-valued semantics covers SROIQ, the DL behind OWL 2, considers tractable
for hybrid KBs [Knorr et al., 2011], in the sense that for each subclasses and truth value removals, and permits re-using
so-called 3-model, a corresponding 5- and 6-model exists. classical reasoners. Also considered are three-valued seman-
Theorem 3 (Faithfulness w.r.t. the three-valued MKNF se- tics for DLs [Zhang et al., 2010] and measuring the degree of
mantics). Let KG = (O, PG ) be a ground hybrid KB, and inconsistency in DL-Lite [Zhou et al., 2012]. For LPs, the sur-
(M, N ) a 3-model of KG . Then, there exists a corresponding vey [Damásio and Pereira, 1998] discusses e.g. a four-valued
5- and 6-model (M 0 , N 0 ) of KG , and both coincide. semantics without default negation [Blair and Subrahmanian,
This 5- and 6-model can in fact be constructed, yet the con- 1989], a four-, six-, and nine-valued semantics [Sakama and
verse does not hold, i.e. there are 5- and 6-models of KG with- Inoue, 1995] for answer sets [Gelfond and Lifschitz, 1991],
out a corresponding 3-model (due to inconsistencies) which and a seven- [Sakama, 1992] and nine-valued [Alferes et al.,
is also why no general correspondence on the unique well- 1995] well-founded semantics [Gelder et al., 1991]. Notably,
founded models for the three semantics exists. the latter, to which our six-valued semantics is faithful, ad-
Both semantics are also faithful w.r.t. ALC4 [Ma et al., mits both the Coherence Principle and Suspicious Reasoning.
2007], provided that, in ALC4, no gaps are admitted, i.e., no More recently, a very general framework for arbitrary bilat-
truth value u, as well as > and ⊥ are represented by short-cuts tices of truth values [Alcântara et al., 2005] and paraconsis-
as shown in [Maier et al., 2013], which is what we assume. tent Datalog [de Amo and Pais, 2007] have been considered.
Theorem 4 (Faithfulness w.r.t. ALC4). Let O be an ALC4 Only two paraconsistent semantics for combinations of
ontology. Models of O in ALC4 without gaps are in a bijec- DLs and LPs directly relate to ours. Both build on an-
tion with p-interpretations that evaluate π(O) to t or b. swer sets, so their computation is not tractable even with
polynomial DLs, and unlike ours, their first-order semantics
Moreover, the six-valued semantics also covers W F SXp
[Alferes et al., 1995], which is defined for LPs with explicit is four-valued, resulting in a weaker consequence relation.
The first, [Huang et al., 2011], is defined for hybrid KBs
negation and implements Suspicious Reasoning, using a so-
in MKNF like ours, but extends [Motik and Rosati, 2010]
called MKNF-translation. Our result applies whenever clas-
and is four-valued and faithful to [Sakama and Inoue, 1995;
sical negation in LPs is limited to unary program atoms.
Ma et al., 2007]. A nine-valued extension to cover Suspi-
Theorem 5 (Faithfulness of the six-valued semantics w.r.t. cious Reasoning is considered in [Huang et al., 2014]. The
W F SXp ). Let Π be an LP with classical negation limited to paraconsistent hybrid semantics [Fink, 2012] is a nine-valued
Π
unary atoms and KG its MKNF-translation. There is a one- extension of semi-equilibrium semantics [Eiter et al., 2010],
to-one correspondence between the well-founded 6-model of and faithful to [Sakama and Inoue, 1995; Maier et al., 2013],
Π
KG and the unique well-founded paraconsistent model as- but Suspicious Reasoning is not considered.
signed by W F SXp . In the future, our fixpoint computations can be used to
No similar result for the five-valued semantics exists since no adapt the Protégé plug-in NoHR [Ivanov et al., 2013] for rea-
corresponding well-founded semantics for LPs exists. soning with our paraconsistent semantics. Future work also
Data complexity of computing the finite representations of includes investigating Suspicious Reasoning in paraconsis-
the well-founded 5- and 6-models depends on the DL used. tent DLs, both standalone and as parts of hybrid KBs.
Theorem 6 (Data Complexity). Let K = (O, P) be a hybrid
KB where entailment of ground atoms in the DL of O is de-
cidable with data complexity C. Then, computing the fixpoints
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