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 df, an RDFS fragment that covers the essential features of RDFS, and extend it to ⊥, allowing to state that two entities are incompatible/disjoint with each other. Eventually, we propose defeasible ⊥ via a typical RC construction allowing to state default class/property inclusions.
RDFS (Resource Description Framework Schema)1 is a main standard semantic web ontology language that consists of triples (, , ) (denoting is related via with ). The introduction of non-monotonic formalisms in reasoning with ontologies is useful in particular to deal with situations in which some classes are exceptional and do not satisfy some typical properties of their super classes, as illustrated with the following example. Example 1.1 (Running example). Consider the following facts (and an intuitive translation into RDFS, where sc is read as “is a subclass of”). - Young people are usually happy; (, sc, ℎ ) - Drug users are usually unhappy; (, sc, ℎ ) - Drug users are usually young; (, sc, ) - Controlled drug users are usually happy; (, sc, ℎ ) - Controlled drug users are drug users; (, sc, ) 20 years [1, 2, 3, 4, 5, 6]. On the other hand, addressing non-monotonicity in the context of RDFS, has attracted in comparison little attention so far, and almost all approaches we are aware of implement non-monotonicity by adding a so-called rule-layer on top of RDFS; see e.g., [7, 8, 9, 2, 10].
In the following, our aim is to show how to integrate Rational Closure (RC), one of the main constructions in non-monotonic reasoning [11], directly within the triple language RDFS. To to do so, we start from df [12, 13], a minimal, but significant RDFS fragment that covers the essential features of RDFS, and then extend it to ⊥, allowing to state that two entities are incompatible/disjoint with each other. The results in this paper are presented more in detail in a technical report [14]. 2. ⊥
() (, dom, ) means that the domain of property is ; and () (, range, ) means that the range of property is . We also recall that minimal df does not consider so-called blank nodes [16, 12].
Concerning the semantics of df [12], an interpretation is a tuple ℐ = ⟨ΔR, ΔP, ΔC, ΔL, P[[· ]], C[[· ]], · ℐ ⟩, where ΔR, ΔP, ΔC, ΔL are the interpretation domains of ℐ, which are finite non-empty sets, and P[[· ]], C[[· ]], · ℐ are the interpretation functions of ℐ. In particular: () ΔR are the resources (the domain or universe of ℐ); () ΔP are property names (not necessarily disjoint from ΔR); () ΔC ⊆ ΔR are the classes; () ΔL ⊆ ΔR are the literal values and contains L ∩ ; () P[[· ]] is a function P[[· ]] : ΔP → 2ΔR× ΔR ; () C[[· ]] is a function C[[· ]] : ΔC → 2ΔR ; () · ℐ maps each ∈ UL ∩ into a value ℐ ∈ ΔR ∪ ΔP, and such that · ℐ is the identity for plain literals and assigns an element in ΔR to each element in L.
An interpretation ℐ satisfies a graph if for each (, , ) ∈ , ℐ ∈ ΔP and (ℐ , ℐ ) ∈ P[[ℐ ]], and moreover ℐ satisfies a series of constraints related to the df-predicates. For example, a constraint imposing that P[[scℐ ]] is transitive over ΔP indicates that the subclass relation sc must be transitive. We refer to [12, Def. 15] for the full definition of the satisfaction relation, and of the correspondent entailment relation.
Definition 2.1 (Entailment ⊨ ⊥ ). Given two graphs and , we say that entails , denoted ⊨ , if and only if every model of is also a model of . • if (, ) ∈ P[[⊥cℐ ]] then , ∈ ΔC; • If (, ) ∈ P[[⊥cℐ ]], then (, ) ∈ P[[⊥cℐ ]] (sc
Symmetry); • If (, ) ∈ P[[⊥cℐ ]] and (, ) ∈ P[[scℐ ]], then
(, ) ∈ P[[⊥cℐ ]] (sc-Transitivity); • If (, ) ∈ P[[⊥cℐ ]] and ∈ ΔC then (, ) ∈
P[[⊥cℐ ]] (c-Exhaustive).
These new constraints are such to model relevant properties of disjointedness, and allow the definition of an entailment relation ⊨ ⊥ . An important feature of ⊥ is also that it preserves the df property that a graph is always satisfiable, avoiding the possibility of unsatisfiability and the ex falso quodlibet principle. This is in line with the df semantics [12, 19]. From an inference system point of view, new derivation rules are added to the df derivation system [14, Sect. 2.3]. The following are just a few examples:
((,,⊥⊥cc,,)) ; (,⊥(c,,⊥),c(,,)sc,) ; ((,,⊥⊥cc,,)) .
The new derivation relation ⊢ ⊥ that we have defined is correct and complete w.r.t. the entailment relation ⊨ ⊥ [14, Th. 2.1]. Eventually, we say that a graph has a conflict if, for some term , either ⊢ ⊥ (, ⊥c, ) or ⊢ ⊥ (, ⊥p, ) holds. The intuitive meaning is that has a conflict if we can derive for some term that it is either an empty class, (, ⊥c, ), or an empty predicate, (, ⊥p, ).
In [12] the reader can find also a deduction system, con- Example 2.1 (Running example cont.). In Examsistent and complete w.r.t. the df entailment relation, that ple 1.1 we could add the triple (ℎ, ⊥c, ℎ ) to is based on rules, such as indicate that ‘being happy’ and ‘being unhappy’ (, sc, ), (, sc, ) are incompatible. Notice that from (ℎ, ⊥c, ℎ ), (, sc, ℎ ), (, sc, ) and (, sc, ℎ ) we (, sc, ) conclude (, ⊥c, ), that is, that being a conencoding 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, should be an empty class). Analoa conflict between the properties of a class and of a sub- gously, from (ℎ, ⊥c, ℎ ), (, sc, ), (, sc, ℎ ) class. e.g., in Example 1.1,“Drug users are usually un- and (, sc, ℎ ) we conclude (, ⊥c, ). happy” appears in conflict with “Controlled drug users are usually happy”. df is not expressive enough to model such conflicts. So, we need to introduce at least a no- 3. Defeasible ⊥ tion of incompatibility, of disjunctiveness [17]. Hence we Next we show how to model defeasible information. Here enrich the df vocabulary with two new predicates, ⊥c we consider defeasibility w.r.t. the predicates sc and sp and ⊥p, representing incompatible information: (, ⊥c, ) only, and introduce the notion of defeasible triple: (resp., (, ⊥p, )) indicates that the classes and (resp., the properties and ) are disjoint. Of course we can = ⟨, , ⟩ ∈ UL × { sc, sp} × UL , further enrich the language allowing for logically stronger notions such as negation [18], but it is not necessary for where , ̸∈ ⊥. The intended meaning of the purpose of the present paper. e.g., ⟨, sc, ⟩ is “Typically, an instance of is also an
We call the new formalism, obtained by adding ⊥c and instance of ”. Analogously, ⟨, sp, ⟩ is read as “Typi⊥p to df, ⊥. Some new constraints are added to the cally, a pair related by is also related by ”. semantics of df [14, Sect. 2.2]. Here are a few examples: 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 ⟨, sc, ℎ ⟩ 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, ⟨, sc, ℎ ⟩, ⟨, sc, ℎ ⟩, ⟨, sc, ⟩ and the monotonic ⊥-graph we have considered up to ⟨, sc, ℎ ⟩. Exmple 2.1, but it is undesirable since we are aware that ⟨, sc, ℎ ⟩ 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 is exceptional, that is, substituting the defeasiview [20], it is a main inference relation in conditional ble triples with their ⊥ counterparts, we obtain reasoning on top of which we can define other interesting (, ⊥c, ). 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 {⟨, sc, ℎ ⟩} ∪ hensive presentation. Given a defeasible graph and a {(, sc, ), (ℎ, ⊥c, ℎ )}, we are not able to query ⟨, , ⟩, we decide whether ⟨, , ⟩ is in the RC derive (, ⊥c, ) anymore. That is, we do of through a two-step procedure: not have a conflict anymore on . Our query 1. We rank all the defeasible triples in , considering ⟨, sc, ℎ ⟩ will be decided considering only this the potential conflicts and the relative logical specificity portion of the original graph: {⟨, sc, ℎ ⟩} ∪ of the first elements of the triples. We give priority (that is, {(, sc, ), (ℎ, ⊥c, ℎ )}. In order to decide a higher rank) to more specific triples. To check the pres- whether ⟨, sc, ℎ ⟩, we check whether its ⊥ence of potential conflicts in a graph, we translate all the counterpart, (, sc, ℎ ), is derivable from the ⊥defeasible triples into the correspondent ⊥ triples, that counterpart of the portion of the graph we consider, that is, we create a new ⊥ graph in which every defeasible is, {(, sc, ℎ ), (, sc, ), (ℎ, ⊥c, ℎ )}. It ⟨, , ⟩ is substituted by (, , ). is easy to check that there is no way of deriving (, sc, ℎ ) from this graph.
Example 3.2 (Running example cont.). In Example 2.1 The semantics for defeasible ⊥ are defined with a rankwe have seen that from the ⊥ version of our graph ing of ⊥-models: the lowest the rank of the model, the we obtain (, ⊥c, ) and (, ⊥c, ). 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 or as first element (e.g., ⟨, sc, ℎ ⟩ and ble graph its RC is determined by its minimal ranked ⟨, sc, ℎ ⟩) have priority (a higher rank) w.r.t. the model, that is, the model of in which every ⊥-model other defeasible triples. That is, ⟨, sc, ℎ ⟩ 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 considering only the exceptional triples and the ⊥-triples, 4. Conclusions that is, {⟨, sc, ℎ ⟩, ⟨, sc, ⟩, ⟨, sc, ℎ ⟩} ∪ The main features of our approach are: (i) the defeasible {(, sc, ), (ℎ, ⊥c, ℎ )}. Translating the de- ⊥ we propose remains syntactically a triple language feasible triples into ⊥-triples, the only conflict we by extending it with new predicate symbols with specific can still derive is (, ⊥c, ), hence we have semantics; (ii) the logic is defined in such a way that any that ⟨, sc, ℎ ⟩, ⟨, sc, ⟩ have rank 1, while RDFS reasoner/store may handle the new predicates as ⟨, sc, ℎ ⟩ is exceptional. From {⟨, sc, ℎ ⟩}∪ ordinary terms if it does not want to take into account of {(, sc, ), (ℎ, ⊥c, ℎ )} we cannot derive any- the extra non-monotonic capabilities; (iii) the defeasible more (, ⊥c, ), hence ⟨, sc, ℎ ⟩ has rank 2 entailment decision procedure is built on top of the ⊥ and we have finished the ranking of the graph. entailment decision procedure, which in turn is an extenNote that, given a graph , the ranking procedure needs sion of the one for df via some additional inference rules, to be done once and for all. favouring a potential implementation; (iv) the computa2. Given a query ⟨, sc, ⟩ (resp., ⟨, sp, ⟩), we check tional complexity of deciding entailment in df and ⊥ the rank of , i.e., we check which is the lowest rank in are the same; and (v) defeasible entailment can be decided which we do not derive (, ⊥c, ) (resp., (, ⊥p, )), and via a polynomial number of calls to an oracle deciding then we check whether we can derive (, sc, ) (resp., ground triple entailment in ⊥ and, in particular, decid(, sp, )) considering only the defeasible triples with at ing defeasible entailment can be done in polynomial time. least such a rank. While an extended version of the paper is under review at the moment, a technical report is online [14].