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| Paper | |
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 edit
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| description | scientific paper published in CEUR-WS Volume 3197 |
| id | Vol-3197/short5 |
| wikidataid | Q117341737βQ117341737 |
| title | Defeasible Reasoning in RDFS |
| pdfUrl | https://ceur-ws.org/Vol-3197/short5.pdf |
| dblpUrl | https://dblp.org/rec/conf/nmr/CasiniS22 |
| volume | Vol-3197βVol-3197 |
| session | β |
Defeasible Reasoning in RDFS
Defeasible reasoning in RDFS
(Extended Abstract)
Giovanni Casini1,2 , Umberto Straccia1
1
ISTI - CNR, Pisa, Italy
2
CAIR, University of Cape Town, South Africa
Abstract
For non-monotonic logics, the notion of Rational Closure (RC) is acknowledged as one of the main approaches. In this work we
present an integration of RC within the triple language RDFS (Resource Description Framework Schema), which together with
OWL 2 is a major standard semantic web ontology language. To do so, we start from πdf, an RDFS fragment that covers the
essential features of RDFS, and extend it to πππβ₯ , allowing to state that two entities are incompatible/disjoint with each other.
Eventually, we propose defeasible πππβ₯ via a typical RC construction allowing to state default class/property inclusions.
Keywords
RDFS, non-monotonic reasoning, defeasible reasoning, rational closure
1. Introduction 20 years [1, 2, 3, 4, 5, 6]. On the other hand, addressing
non-monotonicity in the context of RDFS, has attracted
RDFS (Resource Description Framework Schema)1 is in comparison little attention so far, and almost all ap-
a main standard semantic web ontology language that proaches we are aware of implement non-monotonicity
consists of triples (π , π, π) (denoting π is related via π by adding a so-called rule-layer on top of RDFS; see
with π). The introduction of non-monotonic formalisms e.g., [7, 8, 9, 2, 10].
in reasoning with ontologies is useful in particular to deal In the following, our aim is to show how to integrate
with situations in which some classes are exceptional Rational Closure (RC), one of the main constructions in
and do not satisfy some typical properties of their super non-monotonic reasoning [11], directly within the triple
classes, as illustrated with the following example. language RDFS. To to do so, we start from πdf [12, 13], a
minimal, but significant RDFS fragment that covers the
Example 1.1 (Running example). Consider the following
essential features of RDFS, and then extend it to πππβ₯ ,
facts (and an intuitive translation into RDFS, where sc is
allowing to state that two entities are incompatible/disjoint
read as βis a subclass ofβ).
with each other. The results in this paper are presented
more in detail in a technical report [14].
- Young people are usually happy; (π¦π, sc, βπ )
- Drug users are usually unhappy; (ππ, sc, π’βπ )
- Drug users are usually young; (ππ, sc, π¦π ) 2. πππβ₯ Graphs
- Controlled drug users are usually happy; (ππ·π, sc, βπ )
- Controlled drug users are drug users; (ππ·π, sc, ππ ) We rely on a fragment of RDFS, called minimal πdf [12,
Def. 15], that covers all main features of RDFS, and it is
We may consider then reasonable to conclude, for example, essentially the formal logic behind RDFS. The vocabulary
that controlled young drug users are usually happy. is composed by two pairwise disjoint alphabets U and L
denoting, respectively, URI references and literals, where
Description Logics provide the logical foundation of
a literal may be a plain literal (e.g., a string) or a typed
formal ontologies of the semantic Web Ontology Lan-
2 literal (e.g., a boolean value) [15]. With UL, the set of
guage (OWL) family and endowing them with non-
terms, we will denote the union of these sets. A πdf-triple
monotonic features has been a main issue in the past
is of the form π = (π , π, π) β UL Γ U Γ UL.3 We
NMR 2022: 20th International Workshop on Non-Monotonic Reason- call π the subject, π the predicate, and π the object. A
ing, August 07β09, 2022, Haifa, Israel graph πΊ is a set of triples. πdf is characterised by the set
$ giovanni.casini@isti.cnr.it (G. Casini); of predicates {sp, sc, type, dom, range} β U, that can
umberto.straccia@isti.cnr.it (U. Straccia)
οΏ½ 0000-0002-4267-4447 (G. Casini); 0000-0001-5998-6757
appear only as second elements in the triples. Informally,
(U. Straccia) (π) (π, sp, π) means that property π is a subproperty of
Β© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
property π; (ππ) (π, sc, π) means that class π is a subclass
CEUR
CEUR Workshop Proceedings (CEUR-WS.org)
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
of class π; (πππ) (π, type, π) means that π is of type π;
1
http://www.w3.org/TR/rdf-schema/
2 3
http://www.w3.org/TR/2009/REC-owl2-profiles-20091027 As in [12], we allow literals for π .
155
οΏ½(ππ£) (π, dom, π) means that the domain of property π is β’ if (π, π) β P[[β₯c β ]] then π, π β ΞC ;
π; and (π£) (π, range, π) means that the range of property
π is π. We also recall that minimal πdf does not consider β’ If (π, π) β P[[β₯c β ]], then (π, π) β P[[β₯c β ]] (sc-
so-called blank nodes [16, 12]. Symmetry);
Concerning the semantics of πdf [12], an interpreta- β’ If (π, π) β P[[β₯c β ]] and (π, π) β P[[scβ ]], then
tion is a tuple β = β¨ΞR , ΞP , ΞC , ΞL , P[[Β·]], C[[Β·]], Β·β β©, (π, π) β P[[β₯c β ]] (sc-Transitivity);
where ΞR , ΞP , ΞC , ΞL are the interpretation domains
of β, which are finite non-empty sets, and P[[Β·]], C[[Β·]], Β·β β’ If (π, π) β P[[β₯c β ]] and π β ΞC then (π, π) β
are the interpretation functions of β. In particular: (π) P[[β₯c β ]] (c-Exhaustive).
ΞR are the resources (the domain or universe of β); (ππ)
ΞP are property names (not necessarily disjoint from These new constraints are such to model relevant prop-
ΞR ); (πππ) ΞC β ΞR are the classes; (ππ£) ΞL β ΞR erties of disjointedness, and allow the definition of an
are the literal values and contains L β© π ; (π£) P[[Β·]] is a entailment relation β¨πππβ₯ . An important feature of πππβ₯
function P[[Β·]] : ΞP β 2ΞR ΓΞR ; (π£π) C[[Β·]] is a function is also that it preserves the πdf property that a graph is
C[[Β·]] : ΞC β 2ΞR ; (π£ππ) Β·β maps each π‘ β UL β© π into always satisfiable, avoiding the possibility of unsatisfia-
a value π‘β β ΞR βͺ ΞP , and such that Β·β is the identity bility and the ex falso quodlibet principle. This is in line
for plain literals and assigns an element in ΞR to each with the πdf semantics [12, 19]. From an inference system
element in L. point of view, new derivation rules are added to the πdf
An interpretation β satisfies a graph πΊ if for each derivation system [14, Sect. 2.3]. The following are just a
(π , π, π) β πΊ, πβ β ΞP and (π β , πβ ) β P[[πβ ]], and few examples:
moreover β satisfies a series of constraints related to the (π΄,β₯c ,π΅)
; (π΄,β₯c ,π΅),(πΆ,sc,π΄)
; (π΄,β₯c ,π΄)
.
(π΅,β₯c ,π΄) (πΆ,β₯c ,π΅) (π΄,β₯c ,π΅)
πdf-predicates. For example, a constraint imposing that
The new derivation relation β’πππβ₯ that we have defined is
P[[scβ ]] is transitive over ΞP indicates that the subclass
correct and complete w.r.t. the entailment relation β¨πππβ₯
relation sc must be transitive. We refer to [12, Def. 15]
[14, Th. 2.1]. Eventually, we say that a graph πΊ has a
for the full definition of the satisfaction relation, and of
conflict if, for some term π‘, either πΊπ β’πππβ₯ (π‘, β₯c , π‘) or
the correspondent entailment relation.
πΊπ β’πππβ₯ (π‘, β₯p , π‘) holds. The intuitive meaning is that
Definition 2.1 (Entailment β¨πππβ₯ ). Given two graphs πΊ πΊ has a conflict if we can derive for some term π‘ that it
and π», we say that πΊ entails π», denoted πΊ β¨πππ π», if is either an empty class, (π‘, β₯c , π‘), or an empty predicate,
and only if every model of πΊ is also a model of π». (π‘, β₯p , π‘).
In [12] the reader can find also a deduction system, con- Example 2.1 (Running example cont.). In Exam-
sistent and complete w.r.t. the πdf entailment relation, that ple 1.1 we could add the triple (π’βπ, β₯c , βπ ) to
is based on rules, such as indicate that βbeing happyβ and βbeing unhappyβ
are incompatible. Notice that from (π’βπ, β₯c , βπ ),
(π΄, sc, π΅), (π΅, sc, πΆ)
(ππ·π, sc, βπ ), (ππ·π, sc, ππ ) and (ππ, sc, π’βπ ) we
(π΄, sc, πΆ)
conclude (ππ·π, β₯c , ππ·π ), that is, that being a con-
encoding the transitivity of sc. trolled drug user is incompatible with being a controlled
Defeasible reasoning can be built only when faced with drug user (that is, ππ·π should be an empty class). Analo-
a conflict between the properties of a class and of a sub- gously, from (π’βπ, β₯c , βπ ), (ππ, sc, π¦π ), (π¦π, sc, βπ )
class. e.g., in Example 1.1,βDrug users are usually un- and (ππ, sc, π’βπ ) we conclude (ππ, β₯c , ππ ).
happyβ appears in conflict with βControlled drug users
are usually happyβ. πdf is not expressive enough to model
such conflicts. So, we need to introduce at least a no- 3. Defeasible πππβ₯
tion of incompatibility, of disjunctiveness [17]. Hence we
Next we show how to model defeasible information. Here
enrich the πdf vocabulary with two new predicates, β₯c
we consider defeasibility w.r.t. the predicates sc and sp
and β₯p , representing incompatible information: (π, β₯c , π)
only, and introduce the notion of defeasible triple:
(resp., (π, β₯p , π)) indicates that the classes π and π (resp.,
the properties π and π) are disjoint. Of course we can πΏ = β¨π , π, πβ© β UL Γ {sc, sp} Γ UL ,
further enrich the language allowing for logically stronger
notions such as negation [18], but it is not necessary for where π , π ΜΈβ πππβ₯ . The intended meaning of
the purpose of the present paper. e.g., β¨π, sc, πβ© is βTypically, an instance of π is also an
We call the new formalism, obtained by adding β₯c and instance of πβ. Analogously, β¨π, sp, πβ© is read as βTypi-
β₯p to πdf, πππβ₯ . Some new constraints are added to the cally, a pair related by π is also related by πβ.
semantics of πdf [14, Sect. 2.2]. Here are a few examples:
156
οΏ½Example 3.1 (Running example cont.). In Exam- Example 3.3 (Running example cont.). We wonder
ple 1.1 the statements containing βusuallyβ can whether β¨ππ·π, sc, π’βπ β© is in the RC of our graph. This
more correctly be modelled using defeasible triples, triple is interesting because it would be derivable in
that is, β¨π¦π, sc, βπ β©, β¨ππ, sc, π’βπ β©, β¨ππ, sc, π¦π β© and the monotonic πππβ₯ -graph we have considered up to
β¨ππ·π, sc, βπ β©. Exmple 2.1, but it is undesirable since we are aware
that β¨ππ·π, sc, βπ β© and that βDrug users are usually
There are various ways of reasoning in a defeasible frame- happyβ, that is a defeasible statement. If we consider
work. Here we take under consideration RC [11], since, our entire graph, we already know (Example 3.2) that
despite having some limits from the inferential point of ππ·π is exceptional, that is, substituting the defeasi-
view [20], it is a main inference relation in conditional ble triples with their πππβ₯ counterparts, we obtain
reasoning on top of which we can define other interesting (ππ·π, β₯c , ππ·π ). The same if we consider the graph
forms of entailment [20, 21, 22]. obtained eliminating all the defeasible triples of rank
We give here only a short overview of the reasoning 0. Only once we eliminate also the triples of rank
procedure, inviting the reader to check [14] for a compre- 1, and we consider only the graph {β¨ππ·π, sc, βπ β©} βͺ
hensive presentation. Given a defeasible graph πΊ and a {(ππ·π, sc, ππ ), (βπ, β₯c , π’βπ )}, we are not able to
query β¨π , π, πβ©, we decide whether β¨π , π, πβ© is in the RC derive (ππ·π, β₯c , ππ·π ) anymore. That is, we do
of πΊ through a two-step procedure: not have a conflict anymore on ππ·π . Our query
1. We rank all the defeasible triples in πΊ, considering β¨ππ·π, sc, π’βπ β© will be decided considering only this
the potential conflicts and the relative logical specificity portion of the original graph: {β¨ππ·π, sc, βπ β©} βͺ
of the first elements of the triples. We give priority (that is, {(ππ·π, sc, ππ ), (βπ, β₯c , π’βπ )}. In order to decide
a higher rank) to more specific triples. To check the pres- whether β¨ππ·π, sc, π’βπ β©, we check whether its πππβ₯ -
ence of potential conflicts in a graph, we translate all the counterpart, (ππ·π, sc, π’βπ ), is derivable from the πππβ₯ -
defeasible triples into the correspondent πππβ₯ triples, that counterpart of the portion of the graph we consider, that
is, we create a new πππβ₯ graph in which every defeasible is, {(ππ·π, sc, βπ ), (ππ·π, sc, ππ ), (βπ, β₯c , π’βπ )}. It
β¨π , π, πβ© is substituted by (π , π, π). is easy to check that there is no way of deriving
(ππ·π, sc, π’βπ ) from this graph.
Example 3.2 (Running example cont.). In Example 2.1 The semantics for defeasible πππβ₯ are defined with a rank-
we have seen that from the πππβ₯ version of our graph ing of πππβ₯ -models: the lowest the rank of the model, the
we obtain (ππ·π, β₯c , ππ·π ) and (ππ, β₯c , ππ ). From more expected the situation it describes is considered. As
this we conclude that all the defeasible triples with for the propositional and DL case [23], given a defeasi-
ππ·π or ππ as first element (e.g., β¨ππ·π, sc, βπ β© and ble graph πΊ its RC is determined by its minimal ranked
β¨ππ, sc, π’βπ β©) have priority (a higher rank) w.r.t. the model, that is, the model of πΊ in which every πππβ₯ -model
other defeasible triples. That is, β¨π¦π, sc, βπ β© has is ranked as low as possible. The technical details can be
rank 0, while the other defeasible triples are excep- found in [14, Sect. 3].
tional. We then reiterate the procedure consider-
ing only the exceptional triples and the πππβ₯ -triples, 4. Conclusions
that is, {β¨ππ, sc, π’βπ β©, β¨ππ, sc, π¦π β©, β¨ππ·π, sc, βπ β©} βͺ The main features of our approach are: (i) the defeasible
{(ππ·π, sc, ππ ), (βπ, β₯c , π’βπ )}. Translating the de- πππβ₯ we propose remains syntactically a triple language
feasible triples into πππβ₯ -triples, the only conflict we by extending it with new predicate symbols with specific
can still derive is (ππ·π, β₯c , ππ·π ), hence we have semantics; (ii) the logic is defined in such a way that any
that β¨ππ, sc, π’βπ β©, β¨ππ, sc, π¦π β© have rank 1, while RDFS reasoner/store may handle the new predicates as
β¨ππ·π, sc, βπ β© is exceptional. From {β¨ππ·π, sc, βπ β©}βͺ ordinary terms if it does not want to take into account of
{(ππ·π, sc, ππ ), (βπ, β₯c , π’βπ )} we cannot derive any- the extra non-monotonic capabilities; (iii) the defeasible
more (ππ·π, β₯c , ππ·π ), hence β¨ππ·π, sc, βπ β© has rank 2 entailment decision procedure is built on top of the πππβ₯
and we have finished the ranking of the graph. entailment decision procedure, which in turn is an exten-
Note that, given a graph πΊ, the ranking procedure needs sion of the one for πdf via some additional inference rules,
to be done once and for all. favouring a potential implementation; (iv) the computa-
tional complexity of deciding entailment in πdf and πππβ₯
2. Given a query β¨π , sc, πβ© (resp., β¨π , sp, πβ©), we check
are the same; and (v) defeasible entailment can be decided
the rank of π , i.e., we check which is the lowest rank in
via a polynomial number of calls to an oracle deciding
which we do not derive (π , β₯c , π ) (resp., (π , β₯p , π )), and
ground triple entailment in πππβ₯ and, in particular, decid-
then we check whether we can derive (π , sc, π) (resp.,
ing defeasible entailment can be done in polynomial time.
(π , sp, π)) considering only the defeasible triples with at
While an extended version of the paper is under review at
least such a rank.
the moment, a technical report is online [14].
157
οΏ½Acknowledgments Semantic Web Conference (ESWC-2009), 2009, pp. 857β
862.
This research was partially supported by TAILOR (Foun- [11] D. Lehmann, M. Magidor, What does a conditional
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