One of the main non-monotonic formalism, namely Lehmann and Magidor’s rational closure , is acknowledged as a landmark for non-monotonic reasoning due to its logical properties. Rational closure, that falls under the more general class of the rational entailment relations , has been proposed in the context of Description Logics (DLs) , starting from basic DLs, such as [4, 8, 10, 12, 3, 19, 18], and re-formulated for low-complexity DLs, as [13, 16], as for expressive ones, up to .
Here we show an implementation of a rational entailment relation for an expressive DL such as . The main contribution of this paper is that we re-formulate the decision procedure for rational closure by compiling a non-monotone DL knowledge base into a description logic program (dl-program) . Dl-programs have been proposed to combine DLs with Answer Set Programming , an established approach to implement non-monotonic reasoning for rule-based languages. In this way our approach can be easily be implemented on top of existing reasoners supporting dl-programs, such as DLV 111http://www.dlvsystem.com..
For the sake of completeness and ease the reading, we introduce here a minimum of basic notions.
2.1 Description logic programs
Normal logic programs.
Assume a first-order vocabulary , with a set of constants and a set of predicates , and let be a set of variables . A term is either a variable from or a constant from , and an atom is an expression , where is a -ary predicate in and each is a term. A literal is an atom or its negation (via connective ), while a negation-as-failure literal (NAF-literal) is of the form , where is a literal. A rule is an expression of the form ()
where are literals. Intuitively, a rule has to be read as ‘if we know that are true, but we are not aware that are true, then we can conclude ’. We indicate by (head of ) the literal , by (positive body of ) the set and by (negative body of ) the set . A normal program is a finite set of rules, while a positive program is a finite set of rules in which for every rule .
As usual, atoms, literals, rules and programs are considered ground if they do not contain any variable. The Herbrand Universe of a program () is the set of all the constants that appear in , while the Herbrand Base of () is the set of all the literals that can be constructed from the predicates in and the constants in . A ground instance of a rule is obtained substituting every variable occurring in with a constant symbol in , and, given a program , is the set of all ground instances of rules in .
From the semantics point of view, an interpretation of a program is a consistent subset of , i.e. and there is no atom such that both and are in . The truth value of a literal is true, false, or unknown in iff, respectively, , , or , where is . The satisfiability of a program is reduced to the satisfiability of its rules expressed in ground form: that is, is a model of a program iff it is a model of , i.e. if and , then for every rule in .
In case of a positive program , the answer set of is the least model of with respect to set inclusion: the fact that is positive guarantees the uniqueness of its answer set . If is not positive, the notion of answer set is defined via the so-called Gelfond-Lifschitz transformation (see, e.g. ). Specifically, consider a program and an interpretation . The Gelfond-Lifschitz transformation of relative to gives back a positive program , and it is obtained from with the following procedure:
delete from every rule s.t. ; and
from the remaining rules delete the negative part of the body.
In this way, we end up with a positive program , and is an answer set for iff is the answer set for the positive ground program . We indicate with the set of the answer sets of a program . Eventually, we define a cautious (resp., brave) consequence relation () as follows: () iff the literal is true in any (some) answer set of .
Let a program composed of the following rules:
Consider the interpretation . Then is defined as follows:
It is straightforwardly verified that is the least model of and, thus, is an (indeed, it’s unique) answer set of .
We shall refer here to an expressive DLs, namely (for more details about it, we refer the reader to ). The signature is composed of a set of concept names , a set of role names , and a set of individuals . The set of roles is , where is the inverse of a role ( is ) and is the universal role. We can also compose the roles in into finite chains such as . The set of concepts is defined inductively as:
if , then ;
if , then ;
if , then .
Condition (iii) indicates that the enumerated sets of individuals (nominals) can be used also in the TBox as concepts. An interpretation is a pair , where is a nonempty set, called domain, and the interpretation function assigns to every individual a member of the domain , to every concept name a subset of , and to every role name a subset of . The function is extended to all the concepts and roles in the following way:
where is the cardinality of set . A DL knowledge base is a triple , where is an ABox, containing information about the individuals, is a TBox, containing information about the relations between the concepts, and is a RBox, containing information about the roles. The kind of axioms contained in the ABox, the TBox, and the RBox are described in the table below, with their respective semantics ().
|Concept membership axiom|
|Role membership axiom|
|Concept inclusion axiom|
|Role inclusion axiom|
|Functionality||is a function|
|Disjointness||and are disjoint|
A RBox has further to comply with an additional syntactical restriction: that is, a RBox has to be regular, which essentially prevents a RBox from containing cyclic dependencies among roles that are known to lead to undecidability . For ease of presentation we do not include the definition here and refer the reader to [20, Definition 2] instead. We use as a shorthand of the concept inclusion axiom . With we denote the classical, monotonic, consequence/entailment relation, which is defined as usual.
Note also that every ABox axiom can be reformulated as an equivalent TBox axiom. In particular, can be reformulated as , while is equivalent to . Consequently, in what follows we will not consider ABoxes.
Description logic programs.
A description logic program (dl-program) is composed of a pair , where is a DL knowledge base and is a set of dl-rules , which we are going to specify next. The DL knowledge base is defined over a vocabulary composed of a set of concept names , a set of role names , and a set of individuals, while is defined over a vocabulary , with a set of constants and a set of predicates, and with a set of variables. We assume that the predicative part of the two formalisms are independent, that is is disjoint from , while the same domain of individuals is shared, that is .
Dl-programs use the notions of dl-query and dl-atom to be used in rule bodies to express queries to the DL knowledge base . That is, a dl-query can have various forms, but to what concerns us, it is sufficient to consider the following ones:
a concept membership axiom (so, );
a role membership axiom (so, ).
On the other hand, a dl-atom is an expression of form222The definition given here is again simpler than the original one, as we consider only the form strictly required for our proposal.
with , where each is either a concept or a role (), and each is a predicate symbol from , unary if is a concept, binary otherwise, and is a dl-query. The operator is functional to the updating of the DL knowledge base with factual information obtained from the activation of the rules in the program. That is, each indicates that the extension of is increased by the extension of .
Now, a dl-rule is of the form (1), where any literal may be a dl-atom and a dl-program is a pair , where is a DL knowledge base and is a set of dl-rules.
From a semantics points of view, for an interpretations , we say that is a model of a ground literal or dl-atom under () iff
if , then iff ;
if is a ground dl-atom , where , then iff , where , with, for , ,
As usual, an interpretation is a model of a ground dl-rule iff for all and for all implies . is a model of a dl-program (written ) iff for all . We say that is satisfiable if it has a model.
Let be a dl-program. The strong dl-transform of w.r.t. and (denoted ) is the set of all dl-rules obtained from by deleting
every dl-rule s.t. for some ;
from the remaining dl-rule all literals in .
Note that has only monotonic dl-atoms and no NAF-literals anymore; and a positive dl-program, if satisfiable, has a least model . Now, a strong answer set of is an interpretation s.t. is the least model of . We denote with the set of the strong answer sets of . is a cautious (brave) consequence of , indicated as () iff is true in every (some) strong answer of .
Note that given a dl-program and an answer set of , is a minimal model of .
Consider a dl-program . Let , with
and consider a dl-program composed of the following rules:
It can easily be shown that has an unique answer set
In fact, is the least model of the following set of dl-rules:
2.2 Rational Closure for
For convenience, we recap here some salient notions related to rational closure (RC) for DLs, specifically for the DL (see, e.g. ).
We remind that concepts are inductively defined as (i) ; (ii) ; (iii) if , then ; (iv)] if , then .
Now, a defeasible concept inclusion axiom is of the form , where, without loss of generality, and are assumed to be atomic concepts or their negation. The expression has to be read as ‘if an individual falls under the concept , typically it falls also under the concept ’. A defeasible DL knowledge base is a pair , where (the TBox) is a finite set of concept inclusion axioms of the form , where are concepts, and (the DBox) is a finite set of defeasible concept inclusion of the form .
We next briefly describe the decision procedure for RC for , referring in particular to the one presented in , that in turn has been obtained by refining the one presented in . Consider . The first step of the procedure is to assign a rank to each defeasible axiom in . The rank of the defeasible axioms indicates, in case of conflictual information, which axiom is associated to more specific premises, and has the priority over the axioms associated to more general premises. Central to this step is the exceptionality procedure (see below). The procedure makes use of the notion of materialisation, to reduce concept exceptionality checking to entailment checking, were the materialisation of is defined as .
The ranking of the defeasible axioms is done via the procedure. algocf[ht] In short, the takes as input and gives back a new semantically equivalent knowledge base (with and ), where possibly some defeasible information in has been identified as strict and added to . Also, a sequence of -ordered subsets of , with increasing level of specificity. That is, in case of potential conflicts, the axioms in a set , , have the priority over the axioms in any , . Now, by considering the ranking , we can define a ranking function that associates to every defeasible concept inclusion in a number, representing its level of exceptionality: that is,
Similarly, we may associate a rank to a concept in the following way: consider the result of . Then
Note that . Now, we will say that is entailed by the rational closure of a DL knowledge base (denoted ) iff [23, Theorem 5.17]).
Finally, the procedure determines whether .
We recall that the defined entailment relation is indeed as so-called rational consequence relation , i.e. satisfies the following properties:
|Left Logical Equival.|
|Cut (Cumulative Trans.)|
We refer the reader to  for further explanations and details and limit our presentation to a concluding example.
Assume a DL knowledge base with
By applying the ranking procedure, we end up with
So, for instance, we can conclude that
3 Rational entailment for DLs via dl-programs
In this section we show that, starting from a non-monotone DL () knowledge base , we can compile into a dl-program such that the conditions for rational consequence relations are preserved. So, consider a defeasible knowledge base . Our approach consists of two steps: a ranking step and a compilation step.
Ranking step. To we apply the procedure described in Section 2.2333Of course, and, thus, , can be applied to a DL knowledge base as the classical entailment relation for is decidable. and, thus, we end up with a new defeasible DL knowledge base that correctly separates the strict and the defeasible information contained in the original pair , and a ranking value for every defeasible axiom .
Note that in order to adapt the procedures and to it is sufficient to consider also into the ranking procedure: the inputs of both the procedures is a DL knowledge base instead of , and line 3 in Procedure is modified from to . The set comes out untouched from the ranking procedure, since is the only ranked set, and the only possible new strict information is of the form , with and concepts, hence it can affect only the content of . Hence, starting from a knowledge base we end up with a ranked knowledge base .
Dl-program compilation step.
Given from the ranking step, we now compile the defeasible information in into a a set of dl-rules , which together with defines then the final dl-program .
To alleviate the reading, let ; that is, we assume that has already been ranked via the previous ranking step. Now, define a signature with , while is composed of the predicates () representing at the level of programs the concept names in , i.e. for each concept in we have an unary predicate representing it in the rules. We will use the same name with or without the uppercase initial letter to indicate if it is a concept in DL (e.g., ) or a predicate in (e.g., ), respectively. Let us also recall that for each , and are either atomic concepts or their negation. Given the ranking of the defeasible axioms in , let
be the subset of composed of the axioms with rank value . Now, define the set
as the set of all the antecedents of the defeasible axioms of rank . Moreover, we consider also the set of the consequents of the defeasible axioms
Now, for every axiom of rank , we create a pair of rules of the form444We assume to simplify double negation: that is, for a concept name , is , and similarly, for a logic program predicate , is . See also Example 3.1 later on.
Additionally, for all with , we also consider a rule
In all rules above, . Note that the size of the grounding of the compiled dl-program is polynomially bounded by the size of the defeasible DL knowledge base.
The intuitive meaning of the rule (3) is the following: assume we have an individual that is an instance of concept , which is the antecedent of the defeasible axiom of rank ; if is not an instance of any other -antecedent that is more exceptional than , i.e. holds, and is consistent with our knowledge base, then we can conclude .
On the other hand, the purpose of rule (3) is to update , in case we derive in that the conclusion of a defeasible axiom is negated and, thus, the defeasible axiom cannot be applied. is necessary to update the DL-base with the conclusions drawn at the program level.
Finally, rules of form (4) impose that the individuals we are dealing with are as typical as possible. That is, if we are not aware that an exceptional premise apply to them (any concept in , with ), then we assume that it doesn’t apply (e.g., if we note that is a bird, but we are not aware that it is a penguin, then we presume that it is not a penguin). In the following, we illustrate our technique via an example.
Assume we have a DL vocabulary with , , , and , were the symbols stand for; bird, penguin, flies, insect, fish, has wings, eats insects, eats fishes, while is the relation preys on.
The DL base is composed of
Now, it can be verified that the ranking step returns the following ranking of axioms :
Therefore, , , . The compilation step proceeds now as follows. We define a vocabulary with , while is composed of predicates that represent at the program level the DL atomic concepts and roles: that is, . The program , resulting from the compilation step, is composed of the following rules:
Now, note that the only answer set to the program is the interpretation
In fact, is the least model of the grounded positive program
So, we obtain the intuitive conclusions that, if we are aware about an individual that it is just a bird, we can conclude that, presumably, it flies, eats insects and has wings. On the other hand, if we are informed that it is a penguin, we can conclude that it doesn’t fly and eats fishes.
As well known and already noted in , having nominal concepts may end up in having multiple extensions, i.e., in our context, we may have multiple strong answer sets as shown with following simple example.
Consider a knowledge base , with