Vol-3197/short5
Wolfgang Fahl
Paper
| Paper | |
|---|---|
 edit
| |
| description | scientific paper published in CEUR-WS Volume 3197 |
| id | Vol-3197/short5 |
| wikidataid | Q117341737→Q117341737 |
| title | Defeasible Reasoning in RDFS |
| pdfUrl | https://ceur-ws.org/Vol-3197/short5.pdf |
| dblpUrl | https://dblp.org/rec/conf/nmr/CasiniS22 |
| volume | Vol-3197→Vol-3197 |
| session | → |
Paper[edit]
| Paper | |
|---|---|
| edit
| |
| description | scientific paper published in CEUR-WS Volume 3197 |
| id | Vol-3197/short5 |
| wikidataid | Q117341737→Q117341737 |
| title | Defeasible Reasoning in RDFS |
| pdfUrl | https://ceur-ws.org/Vol-3197/short5.pdf |
| dblpUrl | https://dblp.org/rec/conf/nmr/CasiniS22 |
| volume | Vol-3197→Vol-3197 |
| session | → |
Defeasible Reasoning in RDFS[edit]
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 𝜌df, an RDFS fragment that covers the
essential features of RDFS, and extend it to 𝜌𝑑𝑓⊥ , allowing to state that two entities are incompatible/disjoint with each other.
Eventually, we propose defeasible 𝜌𝑑𝑓⊥ 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 (𝑠, 𝑝, 𝑜) (denoting 𝑠 is related via 𝑝 by adding a so-called rule-layer on top of RDFS; see
with 𝑜). 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 𝜌df [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 𝜌𝑑𝑓⊥ ,
facts (and an intuitive translation into RDFS, where sc is
allowing to state that two entities are incompatible/disjoint
read as “is a subclass of”).
with each other. The results in this paper are presented
more in detail in a technical report [14].
- Young people are usually happy; (𝑦𝑃, sc, ℎ𝑃 )
- Drug users are usually unhappy; (𝑑𝑈, sc, 𝑢ℎ𝑃 )
- Drug users are usually young; (𝑑𝑈, sc, 𝑦𝑃 ) 2. 𝜌𝑑𝑓⊥ Graphs
- Controlled drug users are usually happy; (𝑐𝐷𝑈, sc, ℎ𝑃 )
- Controlled drug users are drug users; (𝑐𝐷𝑈, sc, 𝑑𝑈 ) We rely on a fragment of RDFS, called minimal 𝜌df [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 𝜌df-triple
monotonic features has been a main issue in the past
is of the form 𝜏 = (𝑠, 𝑝, 𝑜) ∈ UL × U × UL.3 We
NMR 2022: 20th International Workshop on Non-Monotonic Reason- call 𝑠 the subject, 𝑝 the predicate, and 𝑜 the object. A
ing, August 07–09, 2022, Haifa, Israel graph 𝐺 is a set of triples. 𝜌df is characterised by the set
$ giovanni.casini@isti.cnr.it (G. Casini); of predicates {sp, sc, type, dom, range} ⊆ U, that can
umberto.straccia@isti.cnr.it (U. Straccia)
� 0000-0002-4267-4447 (G. Casini); 0000-0001-5998-6757
appear only as second elements in the triples. Informally,
(U. Straccia) (𝑖) (𝑝, sp, 𝑞) means that property 𝑝 is a subproperty of
© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
property 𝑞; (𝑖𝑖) (𝑐, sc, 𝑑) means that class 𝑐 is a subclass
CEUR
CEUR Workshop Proceedings (CEUR-WS.org)
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
of class 𝑑; (𝑖𝑖𝑖) (𝑎, type, 𝑏) means that 𝑎 is of type 𝑏;
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 𝑠.
155
�(𝑖𝑣) (𝑝, dom, 𝑐) means that the domain of property 𝑝 is • if (𝑐, 𝑑) ∈ P[[⊥c ℐ ]] then 𝑐, 𝑑 ∈ ΔC ;
𝑐; and (𝑣) (𝑝, range, 𝑐) means that the range of property
𝑝 is 𝑐. We also recall that minimal 𝜌df does not consider • If (𝑐, 𝑑) ∈ P[[⊥c ℐ ]], then (𝑑, 𝑐) ∈ P[[⊥c ℐ ]] (sc-
so-called blank nodes [16, 12]. Symmetry);
Concerning the semantics of 𝜌df [12], an interpreta- • If (𝑐, 𝑑) ∈ P[[⊥c ℐ ]] and (𝑒, 𝑐) ∈ P[[scℐ ]], then
tion is a tuple ℐ = ⟨ΔR , ΔP , ΔC , ΔL , P[[·]], C[[·]], ·ℐ ⟩, (𝑒, 𝑑) ∈ P[[⊥c ℐ ]] (sc-Transitivity);
where ΔR , ΔP , ΔC , ΔL are the interpretation domains
of ℐ, which are finite non-empty sets, and P[[·]], C[[·]], ·ℐ • If (𝑐, 𝑐) ∈ P[[⊥c ℐ ]] and 𝑑 ∈ ΔC then (𝑐, 𝑑) ∈
are the interpretation functions of ℐ. In particular: (𝑖) P[[⊥c ℐ ]] (c-Exhaustive).
ΔR are the resources (the domain or universe of ℐ); (𝑖𝑖)
ΔP are property names (not necessarily disjoint from These new constraints are such to model relevant prop-
ΔR ); (𝑖𝑖𝑖) ΔC ⊆ ΔR are the classes; (𝑖𝑣) ΔL ⊆ ΔR erties of disjointedness, and allow the definition of an
are the literal values and contains L ∩ 𝑉 ; (𝑣) P[[·]] is a entailment relation ⊨𝜌𝑑𝑓⊥ . An important feature of 𝜌𝑑𝑓⊥
function P[[·]] : ΔP → 2ΔR ×ΔR ; (𝑣𝑖) C[[·]] is a function is also that it preserves the 𝜌df property that a graph is
C[[·]] : ΔC → 2ΔR ; (𝑣𝑖𝑖) ·ℐ maps each 𝑡 ∈ UL ∩ 𝑉 into always satisfiable, avoiding the possibility of unsatisfia-
a value 𝑡ℐ ∈ ΔR ∪ ΔP , and such that ·ℐ is the identity bility and the ex falso quodlibet principle. This is in line
for plain literals and assigns an element in ΔR to each with the 𝜌df semantics [12, 19]. From an inference system
element in L. point of view, new derivation rules are added to the 𝜌df
An interpretation ℐ satisfies a graph 𝐺 if for each derivation system [14, Sect. 2.3]. The following are just a
(𝑠, 𝑝, 𝑜) ∈ 𝐺, 𝑝ℐ ∈ ΔP and (𝑠ℐ , 𝑜ℐ ) ∈ P[[𝑝ℐ ]], and few examples:
moreover ℐ satisfies a series of constraints related to the (𝐴,⊥c ,𝐵)
; (𝐴,⊥c ,𝐵),(𝐶,sc,𝐴)
; (𝐴,⊥c ,𝐴)
.
(𝐵,⊥c ,𝐴) (𝐶,⊥c ,𝐵) (𝐴,⊥c ,𝐵)
𝜌df-predicates. For example, a constraint imposing that
The new derivation relation ⊢𝜌𝑑𝑓⊥ that we have defined is
P[[scℐ ]] is transitive over ΔP indicates that the subclass
correct and complete w.r.t. the entailment relation ⊨𝜌𝑑𝑓⊥
relation sc must be transitive. We refer to [12, Def. 15]
[14, Th. 2.1]. Eventually, we say that a graph 𝐺 has a
for the full definition of the satisfaction relation, and of
conflict if, for some term 𝑡, either 𝐺𝑠 ⊢𝜌𝑑𝑓⊥ (𝑡, ⊥c , 𝑡) or
the correspondent entailment relation.
𝐺𝑠 ⊢𝜌𝑑𝑓⊥ (𝑡, ⊥p , 𝑡) holds. The intuitive meaning is that
Definition 2.1 (Entailment ⊨𝜌𝑑𝑓⊥ ). Given two graphs 𝐺 𝐺 has a conflict if we can derive for some term 𝑡 that it
and 𝐻, we say that 𝐺 entails 𝐻, denoted 𝐺 ⊨𝜌𝑑𝑓 𝐻, if is either an empty class, (𝑡, ⊥c , 𝑡), or an empty predicate,
and only if every model of 𝐺 is also a model of 𝐻. (𝑡, ⊥p , 𝑡).
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 𝜌df entailment relation, that ple 1.1 we could add the triple (𝑢ℎ𝑃, ⊥c , ℎ𝑃 ) to
is based on rules, such as indicate that ‘being happy’ and ‘being unhappy’
are incompatible. Notice that from (𝑢ℎ𝑃, ⊥c , ℎ𝑃 ),
(𝐴, sc, 𝐵), (𝐵, sc, 𝐶)
(𝑐𝐷𝑈, sc, ℎ𝑃 ), (𝑐𝐷𝑈, sc, 𝑑𝑈 ) and (𝑑𝑈, sc, 𝑢ℎ𝑃 ) we
(𝐴, sc, 𝐶)
conclude (𝑐𝐷𝑈, ⊥c , 𝑐𝐷𝑈 ), 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, 𝑐𝐷𝑈 should be an empty class). Analo-
a conflict between the properties of a class and of a sub- gously, from (𝑢ℎ𝑃, ⊥c , ℎ𝑃 ), (𝑑𝑈, sc, 𝑦𝑃 ), (𝑦𝑃, sc, ℎ𝑃 )
class. e.g., in Example 1.1,“Drug users are usually un- and (𝑑𝑈, sc, 𝑢ℎ𝑃 ) we conclude (𝑑𝑈, ⊥c , 𝑑𝑈 ).
happy” appears in conflict with “Controlled drug users
are usually happy”. 𝜌df is not expressive enough to model
such conflicts. So, we need to introduce at least a no- 3. Defeasible 𝜌𝑑𝑓⊥
tion of incompatibility, of disjunctiveness [17]. Hence we
Next we show how to model defeasible information. Here
enrich the 𝜌df vocabulary with two new predicates, ⊥c
we consider defeasibility w.r.t. the predicates sc and sp
and ⊥p , representing incompatible information: (𝑐, ⊥c , 𝑑)
only, and introduce the notion of defeasible triple:
(resp., (𝑝, ⊥p , 𝑞)) indicates that the classes 𝑐 and 𝑑 (resp.,
the properties 𝑝 and 𝑞) are disjoint. Of course we can 𝛿 = ⟨𝑠, 𝑝, 𝑜⟩ ∈ UL × {sc, sp} × UL ,
further enrich the language allowing for logically stronger
notions such as negation [18], but it is not necessary for where 𝑠, 𝑜 ̸∈ 𝜌𝑑𝑓⊥ . The intended meaning of
the purpose of the present paper. e.g., ⟨𝑐, sc, 𝑑⟩ is “Typically, an instance of 𝑐 is also an
We call the new formalism, obtained by adding ⊥c and instance of 𝑏”. Analogously, ⟨𝑝, sp, 𝑞⟩ is read as “Typi-
⊥p to 𝜌df, 𝜌𝑑𝑓⊥ . Some new constraints are added to the cally, a pair related by 𝑝 is also related by 𝑞”.
semantics of 𝜌df [14, Sect. 2.2]. Here are a few examples:
156
�Example 3.1 (Running example cont.). In Exam- Example 3.3 (Running example cont.). We wonder
ple 1.1 the statements containing ‘usually’ can whether ⟨𝑐𝐷𝑈, sc, 𝑢ℎ𝑃 ⟩ 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, ⟨𝑦𝑃, sc, ℎ𝑃 ⟩, ⟨𝑑𝑈, sc, 𝑢ℎ𝑃 ⟩, ⟨𝑑𝑈, sc, 𝑦𝑃 ⟩ and the monotonic 𝜌𝑑𝑓⊥ -graph we have considered up to
⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩. Exmple 2.1, but it is undesirable since we are aware
that ⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩ and that ‘Drug users are usually
There are various ways of reasoning in a defeasible frame- happy’, 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 𝑐𝐷𝑈 is exceptional, that is, substituting the defeasi-
view [20], it is a main inference relation in conditional ble triples with their 𝜌𝑑𝑓⊥ counterparts, we obtain
reasoning on top of which we can define other interesting (𝑐𝐷𝑈, ⊥c , 𝑐𝐷𝑈 ). 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 {⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩} ∪
hensive presentation. Given a defeasible graph 𝐺 and a {(𝑐𝐷𝑈, sc, 𝑑𝑈 ), (ℎ𝑃, ⊥c , 𝑢ℎ𝑃 )}, we are not able to
query ⟨𝑠, 𝑝, 𝑜⟩, we decide whether ⟨𝑠, 𝑝, 𝑜⟩ is in the RC derive (𝑐𝐷𝑈, ⊥c , 𝑐𝐷𝑈 ) anymore. That is, we do
of 𝐺 through a two-step procedure: not have a conflict anymore on 𝑐𝐷𝑈 . Our query
1. We rank all the defeasible triples in 𝐺, considering ⟨𝑐𝐷𝑈, sc, 𝑢ℎ𝑃 ⟩ will be decided considering only this
the potential conflicts and the relative logical specificity portion of the original graph: {⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩} ∪
of the first elements of the triples. We give priority (that is, {(𝑐𝐷𝑈, sc, 𝑑𝑈 ), (ℎ𝑃, ⊥c , 𝑢ℎ𝑃 )}. In order to decide
a higher rank) to more specific triples. To check the pres- whether ⟨𝑐𝐷𝑈, sc, 𝑢ℎ𝑃 ⟩, we check whether its 𝜌𝑑𝑓⊥ -
ence of potential conflicts in a graph, we translate all the counterpart, (𝑐𝐷𝑈, sc, 𝑢ℎ𝑃 ), is derivable from the 𝜌𝑑𝑓⊥ -
defeasible triples into the correspondent 𝜌𝑑𝑓⊥ triples, that counterpart of the portion of the graph we consider, that
is, we create a new 𝜌𝑑𝑓⊥ graph in which every defeasible is, {(𝑐𝐷𝑈, sc, ℎ𝑃 ), (𝑐𝐷𝑈, sc, 𝑑𝑈 ), (ℎ𝑃, ⊥c , 𝑢ℎ𝑃 )}. It
⟨𝑠, 𝑝, 𝑜⟩ is substituted by (𝑠, 𝑝, 𝑜). is easy to check that there is no way of deriving
(𝑐𝐷𝑈, sc, 𝑢ℎ𝑃 ) from this graph.
Example 3.2 (Running example cont.). In Example 2.1 The semantics for defeasible 𝜌𝑑𝑓⊥ are defined with a rank-
we have seen that from the 𝜌𝑑𝑓⊥ version of our graph ing of 𝜌𝑑𝑓⊥ -models: the lowest the rank of the model, the
we obtain (𝑐𝐷𝑈, ⊥c , 𝑐𝐷𝑈 ) and (𝑑𝑈, ⊥c , 𝑑𝑈 ). 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-
𝑐𝐷𝑈 or 𝑑𝑈 as first element (e.g., ⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩ and ble graph 𝐺 its RC is determined by its minimal ranked
⟨𝑑𝑈, sc, 𝑢ℎ𝑃 ⟩) have priority (a higher rank) w.r.t. the model, that is, the model of 𝐺 in which every 𝜌𝑑𝑓⊥ -model
other defeasible triples. That is, ⟨𝑦𝑃, sc, ℎ𝑃 ⟩ 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 𝜌𝑑𝑓⊥ -triples, 4. Conclusions
that is, {⟨𝑑𝑈, sc, 𝑢ℎ𝑃 ⟩, ⟨𝑑𝑈, sc, 𝑦𝑃 ⟩, ⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩} ∪ The main features of our approach are: (i) the defeasible
{(𝑐𝐷𝑈, sc, 𝑑𝑈 ), (ℎ𝑃, ⊥c , 𝑢ℎ𝑃 )}. Translating the de- 𝜌𝑑𝑓⊥ we propose remains syntactically a triple language
feasible triples into 𝜌𝑑𝑓⊥ -triples, the only conflict we by extending it with new predicate symbols with specific
can still derive is (𝑐𝐷𝑈, ⊥c , 𝑐𝐷𝑈 ), hence we have semantics; (ii) the logic is defined in such a way that any
that ⟨𝑑𝑈, sc, 𝑢ℎ𝑃 ⟩, ⟨𝑑𝑈, sc, 𝑦𝑃 ⟩ have rank 1, while RDFS reasoner/store may handle the new predicates as
⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩ is exceptional. From {⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩}∪ ordinary terms if it does not want to take into account of
{(𝑐𝐷𝑈, sc, 𝑑𝑈 ), (ℎ𝑃, ⊥c , 𝑢ℎ𝑃 )} we cannot derive any- the extra non-monotonic capabilities; (iii) the defeasible
more (𝑐𝐷𝑈, ⊥c , 𝑐𝐷𝑈 ), hence ⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩ has rank 2 entailment decision procedure is built on top of the 𝜌𝑑𝑓⊥
and we have finished the ranking of the graph. entailment decision procedure, which in turn is an exten-
Note that, given a graph 𝐺, the ranking procedure needs sion of the one for 𝜌df 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 𝜌df and 𝜌𝑑𝑓⊥
2. Given a query ⟨𝑠, sc, 𝑜⟩ (resp., ⟨𝑠, sp, 𝑜⟩), we check
are the same; and (v) defeasible entailment can be decided
the rank of 𝑠, 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 (𝑠, ⊥c , 𝑠) (resp., (𝑠, ⊥p , 𝑠)), and
ground triple entailment in 𝜌𝑑𝑓⊥ and, in particular, decid-
then we check whether we can derive (𝑠, sc, 𝑜) (resp.,
ing defeasible entailment can be done in polynomial time.
(𝑠, sp, 𝑜)) 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
�Acknowledgments Semantic Web Conference (ESWC-2009), 2009, pp. 857–
862.
This research was partially supported by TAILOR (Foun- [11] D. Lehmann, M. Magidor, What does a conditional
dations of Trustworthy AI – Integrating Reasoning, Learn- knowledge base entail?, Artif. Intell. 55 (1992) 1–60.
ing and Optimization), a project funded by EU Horizon [12] S. Muñoz, J. Pérez, C. Gutierrez, Simple and Efficient
2020 research and innovation programme under GA No Minimal RDFS, Web Semantics: Science, Services and
Agents on the World Wide Web 7 (2009) 220–234. URL:
952215.
http://dx.doi.org/10.1016/j.websem.2009.07.003. doi:10.
1016/j.websem.2009.07.003.
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158
�
Defeasible Reasoning in RDFS[edit]
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 𝜌df, an RDFS fragment that covers the
essential features of RDFS, and extend it to 𝜌𝑑𝑓⊥ , allowing to state that two entities are incompatible/disjoint with each other.
Eventually, we propose defeasible 𝜌𝑑𝑓⊥ 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 (𝑠, 𝑝, 𝑜) (denoting 𝑠 is related via 𝑝 by adding a so-called rule-layer on top of RDFS; see
with 𝑜). 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 𝜌df [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 𝜌𝑑𝑓⊥ ,
facts (and an intuitive translation into RDFS, where sc is
allowing to state that two entities are incompatible/disjoint
read as “is a subclass of”).
with each other. The results in this paper are presented
more in detail in a technical report [14].
- Young people are usually happy; (𝑦𝑃, sc, ℎ𝑃 )
- Drug users are usually unhappy; (𝑑𝑈, sc, 𝑢ℎ𝑃 )
- Drug users are usually young; (𝑑𝑈, sc, 𝑦𝑃 ) 2. 𝜌𝑑𝑓⊥ Graphs
- Controlled drug users are usually happy; (𝑐𝐷𝑈, sc, ℎ𝑃 )
- Controlled drug users are drug users; (𝑐𝐷𝑈, sc, 𝑑𝑈 ) We rely on a fragment of RDFS, called minimal 𝜌df [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 𝜌df-triple
monotonic features has been a main issue in the past
is of the form 𝜏 = (𝑠, 𝑝, 𝑜) ∈ UL × U × UL.3 We
NMR 2022: 20th International Workshop on Non-Monotonic Reason- call 𝑠 the subject, 𝑝 the predicate, and 𝑜 the object. A
ing, August 07–09, 2022, Haifa, Israel graph 𝐺 is a set of triples. 𝜌df is characterised by the set
$ giovanni.casini@isti.cnr.it (G. Casini); of predicates {sp, sc, type, dom, range} ⊆ U, that can
umberto.straccia@isti.cnr.it (U. Straccia)
� 0000-0002-4267-4447 (G. Casini); 0000-0001-5998-6757
appear only as second elements in the triples. Informally,
(U. Straccia) (𝑖) (𝑝, sp, 𝑞) means that property 𝑝 is a subproperty of
© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
property 𝑞; (𝑖𝑖) (𝑐, sc, 𝑑) means that class 𝑐 is a subclass
CEUR
CEUR Workshop Proceedings (CEUR-WS.org)
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
of class 𝑑; (𝑖𝑖𝑖) (𝑎, type, 𝑏) means that 𝑎 is of type 𝑏;
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 𝑠.
155
�(𝑖𝑣) (𝑝, dom, 𝑐) means that the domain of property 𝑝 is • if (𝑐, 𝑑) ∈ P[[⊥c ℐ ]] then 𝑐, 𝑑 ∈ ΔC ;
𝑐; and (𝑣) (𝑝, range, 𝑐) means that the range of property
𝑝 is 𝑐. We also recall that minimal 𝜌df does not consider • If (𝑐, 𝑑) ∈ P[[⊥c ℐ ]], then (𝑑, 𝑐) ∈ P[[⊥c ℐ ]] (sc-
so-called blank nodes [16, 12]. Symmetry);
Concerning the semantics of 𝜌df [12], an interpreta- • If (𝑐, 𝑑) ∈ P[[⊥c ℐ ]] and (𝑒, 𝑐) ∈ P[[scℐ ]], then
tion is a tuple ℐ = ⟨ΔR , ΔP , ΔC , ΔL , P[[·]], C[[·]], ·ℐ ⟩, (𝑒, 𝑑) ∈ P[[⊥c ℐ ]] (sc-Transitivity);
where ΔR , ΔP , ΔC , ΔL are the interpretation domains
of ℐ, which are finite non-empty sets, and P[[·]], C[[·]], ·ℐ • If (𝑐, 𝑐) ∈ P[[⊥c ℐ ]] and 𝑑 ∈ ΔC then (𝑐, 𝑑) ∈
are the interpretation functions of ℐ. In particular: (𝑖) P[[⊥c ℐ ]] (c-Exhaustive).
ΔR are the resources (the domain or universe of ℐ); (𝑖𝑖)
ΔP are property names (not necessarily disjoint from These new constraints are such to model relevant prop-
ΔR ); (𝑖𝑖𝑖) ΔC ⊆ ΔR are the classes; (𝑖𝑣) ΔL ⊆ ΔR erties of disjointedness, and allow the definition of an
are the literal values and contains L ∩ 𝑉 ; (𝑣) P[[·]] is a entailment relation ⊨𝜌𝑑𝑓⊥ . An important feature of 𝜌𝑑𝑓⊥
function P[[·]] : ΔP → 2ΔR ×ΔR ; (𝑣𝑖) C[[·]] is a function is also that it preserves the 𝜌df property that a graph is
C[[·]] : ΔC → 2ΔR ; (𝑣𝑖𝑖) ·ℐ maps each 𝑡 ∈ UL ∩ 𝑉 into always satisfiable, avoiding the possibility of unsatisfia-
a value 𝑡ℐ ∈ ΔR ∪ ΔP , and such that ·ℐ is the identity bility and the ex falso quodlibet principle. This is in line
for plain literals and assigns an element in ΔR to each with the 𝜌df semantics [12, 19]. From an inference system
element in L. point of view, new derivation rules are added to the 𝜌df
An interpretation ℐ satisfies a graph 𝐺 if for each derivation system [14, Sect. 2.3]. The following are just a
(𝑠, 𝑝, 𝑜) ∈ 𝐺, 𝑝ℐ ∈ ΔP and (𝑠ℐ , 𝑜ℐ ) ∈ P[[𝑝ℐ ]], and few examples:
moreover ℐ satisfies a series of constraints related to the (𝐴,⊥c ,𝐵)
; (𝐴,⊥c ,𝐵),(𝐶,sc,𝐴)
; (𝐴,⊥c ,𝐴)
.
(𝐵,⊥c ,𝐴) (𝐶,⊥c ,𝐵) (𝐴,⊥c ,𝐵)
𝜌df-predicates. For example, a constraint imposing that
The new derivation relation ⊢𝜌𝑑𝑓⊥ that we have defined is
P[[scℐ ]] is transitive over ΔP indicates that the subclass
correct and complete w.r.t. the entailment relation ⊨𝜌𝑑𝑓⊥
relation sc must be transitive. We refer to [12, Def. 15]
[14, Th. 2.1]. Eventually, we say that a graph 𝐺 has a
for the full definition of the satisfaction relation, and of
conflict if, for some term 𝑡, either 𝐺𝑠 ⊢𝜌𝑑𝑓⊥ (𝑡, ⊥c , 𝑡) or
the correspondent entailment relation.
𝐺𝑠 ⊢𝜌𝑑𝑓⊥ (𝑡, ⊥p , 𝑡) holds. The intuitive meaning is that
Definition 2.1 (Entailment ⊨𝜌𝑑𝑓⊥ ). Given two graphs 𝐺 𝐺 has a conflict if we can derive for some term 𝑡 that it
and 𝐻, we say that 𝐺 entails 𝐻, denoted 𝐺 ⊨𝜌𝑑𝑓 𝐻, if is either an empty class, (𝑡, ⊥c , 𝑡), or an empty predicate,
and only if every model of 𝐺 is also a model of 𝐻. (𝑡, ⊥p , 𝑡).
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 𝜌df entailment relation, that ple 1.1 we could add the triple (𝑢ℎ𝑃, ⊥c , ℎ𝑃 ) to
is based on rules, such as indicate that ‘being happy’ and ‘being unhappy’
are incompatible. Notice that from (𝑢ℎ𝑃, ⊥c , ℎ𝑃 ),
(𝐴, sc, 𝐵), (𝐵, sc, 𝐶)
(𝑐𝐷𝑈, sc, ℎ𝑃 ), (𝑐𝐷𝑈, sc, 𝑑𝑈 ) and (𝑑𝑈, sc, 𝑢ℎ𝑃 ) we
(𝐴, sc, 𝐶)
conclude (𝑐𝐷𝑈, ⊥c , 𝑐𝐷𝑈 ), 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, 𝑐𝐷𝑈 should be an empty class). Analo-
a conflict between the properties of a class and of a sub- gously, from (𝑢ℎ𝑃, ⊥c , ℎ𝑃 ), (𝑑𝑈, sc, 𝑦𝑃 ), (𝑦𝑃, sc, ℎ𝑃 )
class. e.g., in Example 1.1,“Drug users are usually un- and (𝑑𝑈, sc, 𝑢ℎ𝑃 ) we conclude (𝑑𝑈, ⊥c , 𝑑𝑈 ).
happy” appears in conflict with “Controlled drug users
are usually happy”. 𝜌df is not expressive enough to model
such conflicts. So, we need to introduce at least a no- 3. Defeasible 𝜌𝑑𝑓⊥
tion of incompatibility, of disjunctiveness [17]. Hence we
Next we show how to model defeasible information. Here
enrich the 𝜌df vocabulary with two new predicates, ⊥c
we consider defeasibility w.r.t. the predicates sc and sp
and ⊥p , representing incompatible information: (𝑐, ⊥c , 𝑑)
only, and introduce the notion of defeasible triple:
(resp., (𝑝, ⊥p , 𝑞)) indicates that the classes 𝑐 and 𝑑 (resp.,
the properties 𝑝 and 𝑞) are disjoint. Of course we can 𝛿 = ⟨𝑠, 𝑝, 𝑜⟩ ∈ UL × {sc, sp} × UL ,
further enrich the language allowing for logically stronger
notions such as negation [18], but it is not necessary for where 𝑠, 𝑜 ̸∈ 𝜌𝑑𝑓⊥ . The intended meaning of
the purpose of the present paper. e.g., ⟨𝑐, sc, 𝑑⟩ is “Typically, an instance of 𝑐 is also an
We call the new formalism, obtained by adding ⊥c and instance of 𝑏”. Analogously, ⟨𝑝, sp, 𝑞⟩ is read as “Typi-
⊥p to 𝜌df, 𝜌𝑑𝑓⊥ . Some new constraints are added to the cally, a pair related by 𝑝 is also related by 𝑞”.
semantics of 𝜌df [14, Sect. 2.2]. Here are a few examples:
156
�Example 3.1 (Running example cont.). In Exam- Example 3.3 (Running example cont.). We wonder
ple 1.1 the statements containing ‘usually’ can whether ⟨𝑐𝐷𝑈, sc, 𝑢ℎ𝑃 ⟩ 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, ⟨𝑦𝑃, sc, ℎ𝑃 ⟩, ⟨𝑑𝑈, sc, 𝑢ℎ𝑃 ⟩, ⟨𝑑𝑈, sc, 𝑦𝑃 ⟩ and the monotonic 𝜌𝑑𝑓⊥ -graph we have considered up to
⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩. Exmple 2.1, but it is undesirable since we are aware
that ⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩ and that ‘Drug users are usually
There are various ways of reasoning in a defeasible frame- happy’, 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 𝑐𝐷𝑈 is exceptional, that is, substituting the defeasi-
view [20], it is a main inference relation in conditional ble triples with their 𝜌𝑑𝑓⊥ counterparts, we obtain
reasoning on top of which we can define other interesting (𝑐𝐷𝑈, ⊥c , 𝑐𝐷𝑈 ). 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 {⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩} ∪
hensive presentation. Given a defeasible graph 𝐺 and a {(𝑐𝐷𝑈, sc, 𝑑𝑈 ), (ℎ𝑃, ⊥c , 𝑢ℎ𝑃 )}, we are not able to
query ⟨𝑠, 𝑝, 𝑜⟩, we decide whether ⟨𝑠, 𝑝, 𝑜⟩ is in the RC derive (𝑐𝐷𝑈, ⊥c , 𝑐𝐷𝑈 ) anymore. That is, we do
of 𝐺 through a two-step procedure: not have a conflict anymore on 𝑐𝐷𝑈 . Our query
1. We rank all the defeasible triples in 𝐺, considering ⟨𝑐𝐷𝑈, sc, 𝑢ℎ𝑃 ⟩ will be decided considering only this
the potential conflicts and the relative logical specificity portion of the original graph: {⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩} ∪
of the first elements of the triples. We give priority (that is, {(𝑐𝐷𝑈, sc, 𝑑𝑈 ), (ℎ𝑃, ⊥c , 𝑢ℎ𝑃 )}. In order to decide
a higher rank) to more specific triples. To check the pres- whether ⟨𝑐𝐷𝑈, sc, 𝑢ℎ𝑃 ⟩, we check whether its 𝜌𝑑𝑓⊥ -
ence of potential conflicts in a graph, we translate all the counterpart, (𝑐𝐷𝑈, sc, 𝑢ℎ𝑃 ), is derivable from the 𝜌𝑑𝑓⊥ -
defeasible triples into the correspondent 𝜌𝑑𝑓⊥ triples, that counterpart of the portion of the graph we consider, that
is, we create a new 𝜌𝑑𝑓⊥ graph in which every defeasible is, {(𝑐𝐷𝑈, sc, ℎ𝑃 ), (𝑐𝐷𝑈, sc, 𝑑𝑈 ), (ℎ𝑃, ⊥c , 𝑢ℎ𝑃 )}. It
⟨𝑠, 𝑝, 𝑜⟩ is substituted by (𝑠, 𝑝, 𝑜). is easy to check that there is no way of deriving
(𝑐𝐷𝑈, sc, 𝑢ℎ𝑃 ) from this graph.
Example 3.2 (Running example cont.). In Example 2.1 The semantics for defeasible 𝜌𝑑𝑓⊥ are defined with a rank-
we have seen that from the 𝜌𝑑𝑓⊥ version of our graph ing of 𝜌𝑑𝑓⊥ -models: the lowest the rank of the model, the
we obtain (𝑐𝐷𝑈, ⊥c , 𝑐𝐷𝑈 ) and (𝑑𝑈, ⊥c , 𝑑𝑈 ). 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-
𝑐𝐷𝑈 or 𝑑𝑈 as first element (e.g., ⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩ and ble graph 𝐺 its RC is determined by its minimal ranked
⟨𝑑𝑈, sc, 𝑢ℎ𝑃 ⟩) have priority (a higher rank) w.r.t. the model, that is, the model of 𝐺 in which every 𝜌𝑑𝑓⊥ -model
other defeasible triples. That is, ⟨𝑦𝑃, sc, ℎ𝑃 ⟩ 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 𝜌𝑑𝑓⊥ -triples, 4. Conclusions
that is, {⟨𝑑𝑈, sc, 𝑢ℎ𝑃 ⟩, ⟨𝑑𝑈, sc, 𝑦𝑃 ⟩, ⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩} ∪ The main features of our approach are: (i) the defeasible
{(𝑐𝐷𝑈, sc, 𝑑𝑈 ), (ℎ𝑃, ⊥c , 𝑢ℎ𝑃 )}. Translating the de- 𝜌𝑑𝑓⊥ we propose remains syntactically a triple language
feasible triples into 𝜌𝑑𝑓⊥ -triples, the only conflict we by extending it with new predicate symbols with specific
can still derive is (𝑐𝐷𝑈, ⊥c , 𝑐𝐷𝑈 ), hence we have semantics; (ii) the logic is defined in such a way that any
that ⟨𝑑𝑈, sc, 𝑢ℎ𝑃 ⟩, ⟨𝑑𝑈, sc, 𝑦𝑃 ⟩ have rank 1, while RDFS reasoner/store may handle the new predicates as
⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩ is exceptional. From {⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩}∪ ordinary terms if it does not want to take into account of
{(𝑐𝐷𝑈, sc, 𝑑𝑈 ), (ℎ𝑃, ⊥c , 𝑢ℎ𝑃 )} we cannot derive any- the extra non-monotonic capabilities; (iii) the defeasible
more (𝑐𝐷𝑈, ⊥c , 𝑐𝐷𝑈 ), hence ⟨𝑐𝐷𝑈, sc, ℎ𝑃 ⟩ has rank 2 entailment decision procedure is built on top of the 𝜌𝑑𝑓⊥
and we have finished the ranking of the graph. entailment decision procedure, which in turn is an exten-
Note that, given a graph 𝐺, the ranking procedure needs sion of the one for 𝜌df 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 𝜌df and 𝜌𝑑𝑓⊥
2. Given a query ⟨𝑠, sc, 𝑜⟩ (resp., ⟨𝑠, sp, 𝑜⟩), we check
are the same; and (v) defeasible entailment can be decided
the rank of 𝑠, 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 (𝑠, ⊥c , 𝑠) (resp., (𝑠, ⊥p , 𝑠)), and
ground triple entailment in 𝜌𝑑𝑓⊥ and, in particular, decid-
then we check whether we can derive (𝑠, sc, 𝑜) (resp.,
ing defeasible entailment can be done in polynomial time.
(𝑠, sp, 𝑜)) 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
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