1 Introduction
Belief revision is at the core of artificial intelligence and philosophy questionings. It is defined as the process of changing an agent belief with a new acquired knowledge. Three change operations are usually considered:
expansion, contraction and revision. We focus here on the revision, i.e. the process of adding consistently the new belief sets. Belief revision has been intensively studied in classical logics (e.g. propositional logic) mostly under the prism of AGM theory [Alchourrón et al.1985]. With the growing interest in nonclassical logics, such as Horn Logics and Description Logics [Baader2003], several attempts to generalize AGM theory, making it compliant to the metalogical flavor of these logics, have been introduced recently [Flouris et al.2005, Delgrande and Peppas2015, Ribeiro et al.2013, Ribeiro and Wassermann2014].In this paper, we are interested in defining concrete revision operators in Description Logics (DLs). DLs are now pervasive in many knowledgebased representation systems, such as ontological reasoning, semantic web, scene understanding, cognitive robotics, to mention a few. In all these domains, the expert knowledge is rather a flux evolving through time, requiring hence the definition of rational revision operators. Revision is then a cornerstone in ontology engineering lifecycle where the expert knowledge is prone to change and inconsistency. This paper contributes to the effort of defining such rational revision operators, in line with the recent art of the domain
[Qi et al.2006b, Qi et al.2006a, Flouris et al.2005, Flouris et al.2006]. In Section 2 we discuss the adaptation of AGM theory to nonclassical logics, including DLs, and introduce, as a first contribution, a modeltheoretic rewriting of AGM postulates. In Section 3, we introduce our general framework of relaxationbased revision operators. As a second contribution, we demonstrate that they satisfy the AGM postulates and lead to a faithful assignment. Our third contribution is detailed in Section 4, by providing concrete theory relaxation operators in different DLs (namely and its fragments and ). Section 5 positions our contributions with regards to the literature and finally Section 6 draws some conclusions and perspectives.2 Preliminaries
2.1 Description Logics
Description logics are a family of knowledge representation formalisms (see e.g. [Baader2003] for more details). We consider, in this paper, the logic and its fragments and . In the following we provide the syntax and semantics of , from which those of and are easily deducible. Signatures in DLs are triplets where , and are nonempty pairwise disjoint sets such that elements in , and are concept names, role names and individuals, respectively. Given a signature , contains all the sentences of the form , and where , and is an concept inductively defined from and binary and unary operators in and in , respectively. The set of concept descriptions provided by is denoted by . The semantics of concept descriptions is defined using interpretations. An interpretation is a pair consisting of an interpretation domain and an interpretation function which maps concepts to subsets of the domain and role names to binary relations on the domain. A concept description is said to subsume a concept description (denoted by ) if holds for every interpretation . Two concepts and are equivalent (denoted by ) if both and hold. An interpretation is a model of a sentence (TBox or ABox axiom) if it satisfies this sentence (e.g. ).
A DL knowledge base is a set of sentences (i.e. ). An interpretation is a model of a DL knowledge base if it satisfies every sentence in . In the following, we use (or ) to denote the set of all the models of a sentence (or DL knowledge base ). A knowledge base is said to be a theory if and only if , where is the consequence operator defined as: and satisfying monotonicity, inclusion and idempotence. Hence DLs can be considered as Tarskian logics, i.e. pairs .
Classically, consistency of a theory in DLs is defined as . Such a definition raised several issues in adapting revision postulates to Description Logics (see [Ribeiro et al.2013, Ribeiro and Wassermann2014]). We consider in this paper a more general definition of consistency the meaning of which is that there is at least a sentence which is not a semantic consequence: is consistent if .
2.2 AGM theory and Description Logics
AGM theory [Alchourrón et al.1985]
is probably the most influential paradigm in belief revision theory
[Gärdenfors2003]. It provides, at an abstract logical level, a set of postulates that a revision operator should satisfy so that the old belief is changed minimally and rationally to become consistent with the new one. These postulates require the logic to be closed under negation and usual propositional connectives which prevents its use in many nonclassical logics, including DLs. Indeed, many DLs do not allow for negation of concepts (e.g. ) and a fortiori disjunction between TBox and ABox sentences is not defined in all DLs. Recently, many papers have addressed the adaptation of AGM theory to nonclassical logics, e.g. [Flouris et al.2005, Ribeiro et al.2013, Delgrande and Peppas2015, Ribeiro and Wassermann2014]. The first efforts concentrated on the adaptation of contraction postulates, but more recently, [Ribeiro and Wassermann2014] discussed the adaptation of revision postulates and introduced new minimality criteria, not necessarily related to the contraction operator, throwing out the need for negation. However, one can find in [Qi et al.2006b] an attempt to adapt the AGM revision postulates to DL in a modeltheoretic way, following the seminal work of [Katsuno and Mendelzon1991] that translated the AGM postulates in propositional logic semantics. The translation in [Qi et al.2006b] is provided with the classical notion of consistency (a theory is consistent if ) which is not adequate to revision purposes (see [Flouris et al.2006] for a discussion). In this paper we consider a modeltheoretic translation of AGM postulates, similar to the ones in [Qi et al.2006b], with the notable difference that consistency is defined through the consequence operator as introduced in the previous section. This translation is in accordance with the postulates as introduced in [Ribeiro and Wassermann2014].Given two knowledge bases , denotes the revision of the old belief by the new one . The modeltheoretic translation of AGM postulates writes:
 (G1)

.
 (G2)

If is consistent, then .
 (G3)

If is consistent, then so is .
 (G4)

If and , then .
 (G5)

.
 (G6)

If is consistent, then .
Besides these postulates, we consider a minimality criterion introduced in [Ribeiro and Wassermann2014]:
(Relevance) If , then there exists , , such that and .
A classical construction in belief theory is to characterize the revision operator in terms of faithful assignments [Katsuno and Mendelzon1991, Grove1988]. We provide here a similar representation theorem for the postulates defined above. The proof can be found in [Aiguier et al.2015].
Definition 1 (Faithful assignment)
Let be a knowledge base. Let be a total preorder. is a faithful assignment (FA) if the following three conditions are satisfied:

If , .

For every and every , .

For every , if , then .
Theorem 1
A revision operator satisfies the postulates (G1)(G6) if and only if for any DL knowledge base , there exists a wellfounded (i.e. the min is well defined) FA such that , with .
3 Relaxation of theories and associated revision operator
The notion of relaxation has been introduced in [Distel et al.2014b, Distel et al.2014c] to define dissimilarity measures between DL concept descriptions. In this paper we generalize this notion to formula relaxation and subsequently to theory relaxation in order to define revision operators.
Definition 2 (Concept Relaxation [Distel et al.2014c])
A (concept) relaxation is an operator that satisfies the following two properties^{1}^{1}1The nondecreasingness property in the original definition is removed here, since it is not needed in our construction. for all .

is extensive, i.e. ,

is exhaustive, i.e. ,
where denotes applied times, and is the identity mapping.
Our idea to define revision operators is to relax the set of models of the old belief until it becomes consistent with the new pieces of knowledge. This is illustrated in Figure 1 where theories are represented as sets of their models. Intermediate steps to define the revision operators are then the definition of formula and theory relaxations. The whole scheme of our framework is provided in Figure 2.
Definition 3 (Formula Relaxation)
Given a signature , a formula relaxation is a mapping satisfying the following properties:

Extensivity: .

Exhaustivity: , where denotes applied times, and is the identity mapping.
Definition 4 (theory relaxation)
Let be a theory, , a formula relaxation and a set . Then a theory relaxation is a mapping defined as:
Proposition 1
is extensive (), and exhaustive ().
Definition 5 (Relaxationbased revision)
Let be a theory relaxation. We define the revision operator as follows:
for a set such that is consistent, and , and (disjoint union) such that:

,

,

.
Partitioning into and is not unique and the only constraint is that is of maximal size. The set may not be unique either.
Theorem 2
From any theory relaxation and every knowledge base , the binary relation defined by if:
is a wellfounded faithful assignment such that for every , .
Proof: By construction, is obviously a total preorder. Wellfoundness follows from exhaustivity. The two first conditions follow from the fact that . The third one is obvious.
It remains to show that . To simplify the proof, we suppose that (i.e. if , then ), the more general case where being easily obtained from this more simple case.
(i) Let . By definition of , there exists a set such that , and then . Let . If , then and . Otherwise and such that we have . Then , which implies . We can hence conclude that .
(ii) Conversely, let . By definition of , there exists a set of minimal sum such that is consistent and . As , this means that, for every , , and then .
Proposition 2
The revision in Definition 5 satisfies the Relevance minimality criterion.
The proof is direct by setting .
4 Concrete theory relaxations in different fragments
In this section, we introduce concrete relaxation operators suited to the syntax of the logic , as defined in Section 2.1, and its fragments and . concept description constructors are existential restriction (), conjunction (), and , while concept constructors are those of enriched with disjunction ().
Formulas in DL are of the GCI form: , where and are any two complex concepts, or Abox assertions: (, with a role. We propose to define a formula relaxation in two ways (other definitions may also exist). For GCIs, a first approach consists in relaxing the set of models of while another one amounts to “retract” the set of models of .
Definition 6
Let and be any two complex concepts defined over the signature . The concept relaxation based formula relaxation, denoted is defined as follows:
where and is a concept relaxation as in Definition 2.
Proposition 3
is a formula relaxation, that is extensive and exhaustive.
The proof directly follows from the extensivity and exhaustivity of .
Definition 7 (Concept Retraction)
A (concept) retraction is an operator that satisfies the following three properties for all .

is antiextensive, i.e. , and

is exhaustive, i.e. ,
where denotes applied times, and is the identity mapping.
Definition 8
Let and be any two complex concepts defined over the signature . The concept retraction based formula relaxation, denoted is defined as follows:
where is a concept retraction.
The definition for Abox assertions is similar as in Definition 6.
Proposition 4
is a formula relaxation..
For coming up with revision operators, it remains to define concrete relaxation and retraction operators at the concept level (cf. Figure 2). Some examples of retraction and relaxation operators are given below.
4.1 Relaxation and retraction in
Concept Retractions.
A trivial concept retraction is the operator that maps every concept to . This operator is particularly interesting for debugging ontologies expressed in [Schlobach et al.2007]. Let us illustrate this operator through the following example adapted from [Qi et al.2006b] to restrict the language to .
Example 1
Let and . Clearly is inconsistent. The retractionbased formula relaxation amounts to apply to the concept Tweety resulting in the following new knowledge base which is now consistent with . An alternative solution is to retract the concept bird in which results in the following knowledge base which is also consistent with . The sets of minimal sum and in Definition 5 are , (i.e. , where ) and . A good final solution could be based on an additional preference relation among the solutions defined from the minimality of the “size” of the modified concepts.
Concept Relaxations.
A trivial relaxation is the operator that maps every concept to . Other nontrivial concept description relaxations have been introduced in [Distel et al.2014b]. We summarize here some of these operators.
concept descriptions can appropriately be represented as labeled trees, often called description trees [Baader et al.1999]. An description tree is a tree whose nodes are labeled with sets of concept names and whose edges are labeled with role names. An concept description
(1) 
with , can be translated into a description tree by labeling the root node with , creating an successor, and then proceeding inductively by expanding for the successor node for all .
An concept description relaxation then amounts to apply simple tree operations. Two relaxations can hence be defined [Distel et al.2014b]: (i) that reduces the role depth of each concept by , simply by pruning the description tree, and (ii) that removes all leaves from a description tree.
4.2 Relaxations in
The relaxation defined above exploits the strong property that an concept description is isomorphic to a description tree. This is arguably not true for more expressive DLs. Let us try to go a one step further in expressivity and consider the logic . A relaxation operator as introduced in [Distel et al.2014b] requires a concept description to be in a special normal form, called normal form with grouping of existentials, defined recursively as follows.
Definition 9
We say that an concept is written in normal form with grouping of existential restrictions if it is of the form
(2) 
where is a set of concept names and the concepts are of the form
(3) 
where no subsumption relation holds between two distinct conjuncts and is a set of complex concepts that are themselves in normal form with grouping of existential restrictions.
The purpose of terms is simply to group existential restrictions that share the same role name. For an concept we say that is in normal form if it is of the form () and each of the is an concept in normal form with grouping of existential restrictions.
Definition 10
[Distel et al.2014b] Given an concept description we define an operator recursively as follows. For and for we define . For , where is a group of existential restrictions as in (3), we need to distinguish two cases:

if we define , and

if then we define .
Notice that in the latter case since is in normal form. For as in (2) we define , where . Finally for we set .
The proof of being a relaxation, i.e. satisfying exhaustivity and extensivity is detailed in [Distel et al.2014a].
Let us illustrate this operator on an example.
Example 2
Suppose an agent believes that a person Bob is married to a female judge: . Suppose now that due to some obscurantist law, it happens that females are not allowed to be judges. This new belief is captured as . By applying one can resolve the conflict between the two belief sets. To ease the reading, let us rewrite the concepts as follows: . Hence , with and
Then
The new agent’s belief, up to a rewriting, becomes
Another possibility for defining a relaxation in is obtained by exploiting the disjunction constructor by augmenting a concept description with a set of exceptions.
Definition 11
Given an exception set , we define a relaxation of degree of an concept description as follows: for a finite set with
Extensivity of this operator follows directly from the definition. However, exhaustivity is not necessarily satisfied unless the exception set includes the concept or the disjunction of some or all of its elements entails the concept.
If we consider again Example 1, a relaxation of the formula using the operator over the concept flies with the exception set results in the formula . The new revised knowledge base is then which is consistent.
Another example involving this relaxation will be discussed in the case (cf. Example 3).
4.3 Relaxation and retraction in
We consider here operators suited to language. Of course, all the operators defined for and remain valid.
Concept Retractions.
A first possibility for defining retraction is to remove iteratively from an concept description one or a set of its subconcepts. A similar construction has been introduced in [Qi et al.2006b] by transforming Abox assertions to nominals and conjuncting their negations from the concept they belong to. Interestingly enough, almost all the operators defined in [Qi et al.2006b, Gorogiannis and Hunter2008] are relaxations.
Definition 12
Let be any concept description, we define .
Consider again Example 1. . The resulting revised knowledge base is then which is consistent.
Another possibility, suggested in [Gorogiannis and Hunter2008] and related to operators defined in propositional logic as introduced in [Bloch and Lang2002], consists in applying the retraction at the atomic level. This captures somehow the Dalal’s idea of revision operators in propositional logic [Dalal1988].
Definition 13
Let be an concept description of the form , where is a quantifier and is quantifierfree and in CNF form, i.e. with being disjunctions of possibly negated atomic concepts. Define, as in the propositional case [Bloch and Lang2002], . Then .
This idea can be generalized to consider any retraction defined in .
Definition 14
Let be an concept description of the form , where is a quantifier and is a quantifierfree. Then .
Another possible concept description retraction is obtained by substituting the existential restriction by an universal one. This idea has been sketched in [Gorogiannis and Hunter2008] for defining dilation operators (then by transforming into ), i.e. special relaxation operators enjoying additional properties [Distel et al.2014b, Distel et al.2014b]. We adapt it here to define retraction in DL syntax.
Definition 15
Let be an concept description of the form , where is a quantifier and is quantifierfree, then
Concept Relaxations.
Let us now introduce some relaxation operators suited to language.
Definition 16
Let be an concept description of the form , where is a quantifier and is quantifierfree and in DNF form, i.e. with being conjunction of possibly negated atomic concepts. Define, as in the propositional case [Bloch and Lang2002], , then .
As for retraction, this idea can be generalized to consider any relaxation defined in .
Definition 17
Let be an concept description of the form , where is a quantifier and is quantifierfree, then
Let us consider another example adapted from the literature to illustrate these operators [Qi et al.2006b].
Example 3
Let us consider the following knowledge bases: and . Relaxing the formula by applying to the concept on the right hand side results in the following formula which resolves the conflict between the two knowledge bases.
A last possibility, dual to the retraction operator given in Definition 15, consists in transforming universal quantifiers to existential ones.
Definition 18
Let be an concept description of the form , where is a quantifier and is quantifierfree, then
If we consider again Example 3, relaxing the formula by applying to the concept on the right hand side results in the following formula , which resolves the conflict between the two knowledge bases.
The following proposition summarizes the properties of the introduced operators.
Proposition 5
The operators are extensive and exhaustive. The operators are extensive but not exhaustive. The operators are antiextensive and exhaustive. The operators is antiextensive but not exhaustive.
These properties are directly derived from the definitions and from properties of and detailed in [Bloch and Lang2002]. Note that for exhaustivity can be obtained by further removing recursively the remaining universal quantifiers and apply at the final step any retraction defined above on the concept .
5 Related works
In the last decade, several works have studied revision operators in Description Logics. While most of them concentrated on the adaptation of AGM theory, few works have concerned the definition of concrete operators [Meyer et al.2005, Qi et al.2006a, Qi et al.2006b]. A closely related field is inconsistency handling in ontologies (e.g. [Schlobach and Cornet2003, Schlobach et al.2007]), with the main difference that the rationality of inconsistency repairing operators is not investigated, as suggested by AGM theory.
Some of our relaxation operators are closely related to the ones introduced in [Qi et al.2006b] for knowledge bases revision and in [Gorogiannis and Hunter2008] for merging firstorder theories. Our relaxationbased revision framework, being abstract enough (i.e. defined through easily satisfied properties), encompasses these operators. Moreover, the revision operator defined in [Qi et al.2006b] considers only inconsistencies due to Abox assertions. Our operators are general in the sense that Abox assertions are handled as any formula of the language.
The relaxation idea originates from the work on MorphoLogics, initially introduced in [Bloch and Lang2002, Bloch et al.2004]. In this seminal work, revision operators (and explanatory relations) were defined through dilation and erosion operators. These operators share some similarities with relaxation and retraction as defined in this paper. Dilation is a suppreserving operator and erosion is infpreserving, hence both are increasing. Some particular dilations and erosions are exhaustive and extensive while relaxation and retraction operators are defined to be exhaustive and extensive but not necessarily sup and infpreserving.
Another contribution in this paper concerns the generalization of AGM postulates and their translation in a modeltheoretic writing with a definition of inconsistency, allowing using them in a wide class of nonclassical logics. This follows recent works on the adaptation of AGM theory (e.g. [Ribeiro et al.2013, Ribeiro and Wassermann2014, Delgrande and Peppas2015, Flouris et al.2005]). Our generalization is closely related to the one recently introduced in [Ribeiro and Wassermann2014] and could be seen as its counterpart in a modeltheoretic setting. It also extends the one introduced in [Qi et al.2006b].
6 Conclusion
The contribution of this paper is threefold. First, we provided a generalization of AGM postulates so as they become applicable to a wide class of nonclassical logics. Secondly we proposed a general framework for defining revision operators based on the notion of relaxation. We demonstrated that such a relaxationbased framework for belief revision satisfies the AGM postulates and leads to a faithful assignment. Thirdly, we introduced a bunch of concrete relaxations, discussed their properties and illustrated them through simple examples. Future work will concern the study of the complexity of these operators, the comparison of their induced ordering, and their generalization to other nonclassical logics such as Horn logic.
Acknowledgments.
This work was partially funded by the French ANR project LOGIMA.
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