=Paper=
{{Paper
|id=Vol-199/paper-3
|storemode=property
|title=First Steps Towards Revising Ontologies
|pdfUrl=https://ceur-ws.org/Vol-199/wonto-03.pdf
|volume=Vol-199
|dblpUrl=https://dblp.org/rec/conf/wonto/RibeiroW06
}}
==First Steps Towards Revising Ontologies==
https://ceur-ws.org/Vol-199/wonto-03.pdf
First Steps Towards Revising Ontologies
Márcio Moretto Ribeiro and Renata Wassermann
Institute of Mathematics and Statistics
University of São Paulo – Brasil,
{marciomr,renata}@ime.usp.br
Abstract. When modeling an ontology, one very often wants to add
new information and keep the resulting ontology consistent. Belief Re-
vision deals with the problem of consistently adding new formulas to
a knowledge base. In this paper, we present some steps towards apply-
ing belief revision methods to ontologies based on description logics. We
depart from the well known AGM-paradigm and show how it can be
adapted in order to be applied to description logics.
1 Introduction
Recently, with the advent of the Semantic Web [BLHL01], there has been a
growing interest in the use of ontologies for representing domain knowledge.
When ontologies evolve and are re-used in different contexts, inconsistency may
arise.
In [HvHtT05], at least four different reasons for the presence of inconsistency
in ontologies are described: mis-representation of defaults (stating that birds
can fly and then that penguins are birds and cannot fly), polisemy (words with
different meanings), problems of translation between different formalism, and
multiple sources.
There are different approaches concerning how to deal with inconsistencies.
In [HvHH+ 05], four different approaches are described under a unifying frame-
work. The first, consistent evolution, consists in preventing the introduction of
inconsistency in a consistent ontology. The second, repairing, consists in making
an inconsistent ontology consistent. The third reasoning with inconsistency does
not change the inconsistent ontology, but tries to derive meaningful conclusions
from it. Finally, the fourth approach, versioning, keeps track of changes and
compatibility issues between different versions of ontologies.
In this paper, we study the applicability of Belief Revision methods to on-
tologies. Belief Revision [Gär88,Han99] deals with the problem of restoring the
consistence of a knowledge base after the introduction of new, possibly incon-
sistent knowledge. We note that, from the four approaches above, only the last
one (versioning) is not addressed by the Belief Revision literature. Rott has pro-
posed in [Rot01] the classification of methods for Belief Revision into vertical or
horizontal. In the vertical mode, revision is reduced to the simple addition of the
new formula to the knowledge base, while some non-classical notion of inference
is used to answer queries. The inference machinery is responsible for dealing
with the inconsistency, as in the third approach above. On the other hand, the
horizontal mode of belief revision relies on classical notions of inference, but
uses sophisticated operations of revision to keep or restore the consistency in
the knowledge base. This addresses the first two approaches (consistent evolu-
tion and repairing) and will be followed in the rest of the paper. The most well
known paradigm for horizontal belief revision is known as the AGM theory, due
to the initial of the names of the authors of the seminal paper [AGM85].
Although our results are more general, we have in mind the revision of on-
tologies described in OWL [MvH04]. OWL has been a W3C recommendation
since 2004 and is now seen as the standard language for representing ontologies.
It was defined as three different sub-languages, with increasing expressivity (and
complexity for reasoning): OWL-Lite, OWL-DL and OWL-Full. We will concen-
trate on the first two, since there is no complete reasoner for OWL-Full. It was
already shown that OWL-Lite and OWL-DL are equivalent to the description
logics SHIF(D) and SHOIN (D) [HPS04], so we will concentrate our examples
on description logics.
We will first briefly review the AGM-theory. Then, in Section 3 we will present
some results from [FPA04,FPA05] which show that in the description logics
behind OWL there is no operation possible conforming to the AGM-theory.
We then show that the AGM-theory can be slightly adapted so that AGM-like
revision can be applied to SHIF(D) and SHOIN (D).
In the rest of this paper, we call consequence operation any total function
taking sets of formulas to sets of formulas. We use Cn to denote consequence
operators. A Tarskian consequence operator is an operator Cn that satisfies
monotony (A ⊆ B ⇒ Cn(A) ⊆ Cn(B)), inclusion (A ⊆ Cn(A)) and idempo-
tency or iteration (Cn(Cn(A)) = Cn(A)).
2 AGM Belief Change
In this section, we briefly introduce the AGM paradigm for belief change. For a
more complete exposition, the reader is referred to [Gär88],[GR95] or [Han99].
In the AGM model, belief states are represented by theories (possibly to-
gether with some selection mechanism), that is, sets of formulas K such that
Cn(K) = K.These theories are called belief sets. The consequence operator Cn
is assumed to be tarskian, compact, satisfy the deduction theorem and supraclas-
sicality. We will sometimes refer to these properties as the AGM-assumptions.
In AGM theory, there are three operations that can be performed on belief
sets: contraction, expansion and revision. Contraction consists of giving up (at
least) as many beliefs as it is needed so that the new belief set does not imply
(and so, does not contain) a specified sentence. Expansion consists of adding
new information to the belief set. If the old and the new information are not
logically compatible, then the new belief state after expansion will be inconsis-
tent. Revision is consistent incorporation of new information, i.e., if the input
sentence is consistent, then the new belief set will be consistent (even if the old
2
belief set was not). If necessary, consistency is obtained by deleting parts of the
original belief set.
2.1 Postulates
Of the three AGM operations, only expansion is characterised in a unique way.
When a belief set K is expanded with a proposition ϕ, the resulting set K + ϕ
is obtained by simply adding the new belief to the old belief set and taking the
logical consequences of the resulting set:
K + ϕ = Cn(K ∪ {ϕ}).
The name expansion is justified by the fact that K ⊆ K + ϕ.
Contraction and revision operations are not directly defined, but constrained
by a set of rationality postulates.1 For the contraction of a belief set K in relation
to a sentence ϕ (denoted K − ϕ), six basic postulates are given [AGM85] (` is
the consequence relation associated with Cn):
(K-1) K − ϕ is a belief set (closure)
(K-2) K − ϕ ⊆ K (inclusion)
(K-3) If ϕ 6∈ K, then K − ϕ = K (vacuity)
(K-4) If not ` ϕ, then ϕ 6∈ K − ϕ (success)
(K-5) K ⊆ (K − ϕ) + ϕ (recovery)
(K-6) If ` ϕ ↔ ψ, then K − ϕ = K − ψ (extensionality)
These postulates are supposed to capture the intuition behind the operation
of giving up a belief in a rational way. Postulate (K-1) says that the result
of contracting a belief set by a formula should again be a belief set. The next
postulate assures that in an operation of contraction no new formulas are added
to the initial belief set. If the formula to be contracted is not an element of
the initial belief set, then by (K-3) nothing changes. Postulate (K-4) says that
unless the sentence to be contracted is logically valid (and hence, an element
of every theory), it is not an element of the resulting belief set. The recovery
postulate (K-5) is the most controversial one [Mak87]. It says that a contraction
should be recoverable, that is, that the original belief set should be recovered by
expanding by the formula that was contracted. The last postulate assures that
contraction by logically equivalent sentences produces the same output.
1
Originally, expansion was also defined by means of a set of postulates, but it can be
completely determined by the postulates.
3
Contraction and revision can be defined in terms of each other via the Harper
or the Levi identities [Gär88]. Revising with a belief ϕ corresponds to contracting
by the negation of ϕ and then expanding with ϕ:
K ∗ ϕ = (K − ¬ϕ) + ϕ (Levi identity)
The thus defined operator ∗ will be called the revision operator associated with
the contraction operator −. Analogously, the following identity defines a con-
traction operator associated with a revision operator:
K − ϕ = (K ∗ ¬ϕ) ∩ K (Harper identity)
2.2 Construction - Partial Meet
The postulates above do not determine unique contraction or revision operators
for a belief set, but only restrict the set of possible such operators. In [AGM85]
a particular construction is presented that, given a belief set and an input belief,
returns the result of contracting (or revising) the given set by the input.
This construction makes use of the concept of a remainder set, the set of
maximal subsets of a given set not implying a given sentence. Formally:
Definition 1. [AM82] Let X be a set of formulas and α a formula. The re-
mainder set X⊥α of X and α is defined as follows. For any set Y , Y ∈ X⊥α
if and only if:
– Y ⊆X
– Y 6` α
– For all Y 0 such that Y ⊂ Y 0 ⊆ X, Y 0 ` α.
Observation 1 [AGM85] If K is closed under logical deduction, then so are the
elements of K⊥α.
Observation 2 Upper Bound Property [AM81]:2
/ Cn(X) then there is a X 0 such that X ⊆ X 0 inK⊥β.
If X ⊆ K and β ∈
It is assumed that there is some way of picking out the best (in some sense)
elements of a remainder set. This is formalised by means of a selection function:
Definition 2. [AGM85] A selection function for X is a function γ such that:
– If X ⊥ α 6= ∅, then ∅ =
6 γ(X ⊥ α) ⊆ X ⊥ α.
– Otherwise, γ(X ⊥ α) = {X}.
A contraction is obtained by taking the intersection of the best subsets of K
that do not imply α:
2
This property follows from compactness and the axiom of choice.
4
Definition 3. [AGM85] For any sentence α, the operation of partial meet
contraction over a belief set K determined by the selection function γ is given
by:
T
K −γ α = γ(K ⊥ α)
Partial meet revision is obtained from partial meet contraction and expansion
by means of the Levi identity:
Definition 4. Let K be a belief set and γ a selection function. For any sentence
α, the operation of partial meet revision over K determined by γ is given by:
T
K ∗γ α = Cn( γ(K ⊥ α) ∪ {α})
In their paper [AGM85], Alchourrón, Gärdenfors and Makinson show that
partial meet constructions bear a very special relation to the contraction and
revision postulates. They prove the following representation results:
Theorem 1. [AGM85] Let − be a function which, given a formula α, takes a
belief set K into a new belief set K − α. For every theory K, − is a partial
meet contraction operation over K if and only if − satisfies the basic postulates
((K-1)-(K-6)) for contraction.
3 Generalising the Postulates
In this section, we show that not every logic admits a contraction operation sat-
isfying the AGM postulates. We show some results presented in [FPA04,FPA05]
characterising the logics that admit such operations and then propose an alter-
native set of postulates that can be used with a wider family of logics.
Although the AGM theory was formulated having in mind some general
notion of logic, some assumptions were made which limit the kind of logic that
can be really used. In their work [FPA04], Flouris, Plexousakis and Antoniou
have shown that not even all Tarskian logics admit a contraction operation
satisfying the six AGM postulates. As an example, they present a logic containing
only two formulas, a and b and with a consequence operator as follows: C(∅) = ∅,
C({a}) = C({a, b}) = {a, b} and C{b} = {b}. It is easy to see that if we try to
contract b from {a, b}, because of the success postulate, the result must be ∅.
But then, the recovery postulate is not satisfied.
Flouris, Plexousakis and Antoniou define a logic to be AGM-compliant if it
admits a contraction operation satisfying the six AGM postulates. They have
given a condition to test for AGM-compliance:
Definition 5. [FPA04] A logic is called decomposable if and only if for all sets
of formula A, B, such that Cn(∅) ⊂ Cn(B) ⊂ Cn(A), there exists a set of
formulas Cn such that Cn(C) ⊂ Cn(A) and Cn(A) = Cn(B ∪ C).
Theorem 2. [FPA04] A logic is AGM-compliant iff it is decomposable.
5
The logic in the example above is not decomposable (take A = {a} and
B = {b}). But a more interesting negative example for us is the description logic
SHOIN (D), which is shown not to be decomposable. This means that the AGM
theory cannot be directly applied to this logic, which is the basis for OWL. An
example of a logic which is AGM-compliant is the logic ALC 3
We propose here an alternative approach. Instead of accepting the six pos-
tulates as they are, we note that the problem of AGM-compliance is due to the
presence of the recovery postulate. If we consider only the other five postulates,
then every logic admits a contraction operation satisfying them.
The recovery postulate has been criticised in the literature [Mak87,Fer01]
as being the most polemic and less intuitive. The idea behind the postulate is
to guarantee some kind of minimal change, i.e., that as much information as
possible will be preserved. If we want to avoid the recovery postulate, we need
some other condition to preserve information.
In [Han89], Hansson has proposed that minimal change could be captured
by the following intuition: in a contraction operation, when a belief is removed,
it must contribute somehow for the derivation of the contracted belief, that is,
no belief is removed for no reason. This idea is formalised by the postulate:
(relevance) If β ∈ K and β 6∈ K − α, then there is a set K 0 such that
K − α ⊆ K 0 ⊆ K and that α 6∈ Cn(K 0 ), but α ∈ Cn(K 0 ∪ {β}).
For logics which satisfy the AGM-assumptions, Fuhrmann and Hansson have
shown that the relevance postulate is stronger than recovery, but on the presence
of the other postulates, they are equivalent:
Observation 3 [FH94] Let K be a belief set and − a contraction operator for
K. Then:
1. If − satisfies relevance, then it satisfies recovery.
2. If − satisfies closure, inclusion, vacuity and recovery, then it satisfies rele-
vance.
The result above makes the relevance postulate a good candidate to substi-
tute recovery. Although the new set of postulates is equivalent to the original
one in classical logic, this is not the case for any logic, as we show next.
Flouris, Plexousakis and Antoniou have already shown that the only prob-
lematic postulate is indeed recovery:
Theorem 3. [FPA04] Every tarskian logic admits a contraction operator that
satisfies the AGM postulates without recovery.
As we have seen in the example in the beginning of this section, there are
indeed tarskian logics where no contraction with recovery is possible. But it is
not difficult to see that relevance does not present this problem:
3
provided it is equipped with infinite roles and empty ABox, as shown in [Flo06].
6
Theorem 4. Every tarskian logic admits a contraction operator that satisfies
the AGM postulates with relevance instead of recovery.
T
Proof. Take for example the construction K − α = γ(K⊥α), where γ is a
selection function. It is easy to show, based on the proof for classical logic found in
[Gär88], that it satisfies the postulates (K-1)-(K-4) and (K-6) for any tarskian
logic. We only have to show that it also satisfies the relevance postulate. In the
limit case where α is a tautology, K⊥α is empty and K −α = K, so the postulate
is trivially satisfied. If K⊥α is not empty, take any K 0 ∈ γ(K⊥α) and β ∈ K
and β 6∈ K − α. By the definition of K − α and remainder sets, we have that
K − α ⊆ K 0 ⊆ K and that α 6∈ Cn(K 0 ), but α ∈ Cn(K 0 ∪ {β}).
We can now prove the following representation theorem:
Theorem 5. For every belief set K closed under a tarskian logical consequence,
− is a partial meet contraction operation over K if and only if − satisfies the
postulates (K-1)-(K-4), (K-6) and (relevance).
Proof. One side of the implication follows from the proof of theorem 4, namely
that every partial meet contraction satisfies the six postulates.
To see that every operator − satisfying the six postulates can be constructed
as a partial meet contraction, just define the selection function γ as γ(K⊥α) =
{K} when K⊥α = ∅ and γ(K⊥α) = {X ∈ K⊥α : K − α ⊆ X} otherwise.
From extensionality it follows that ` α ↔ β, then K − α = K − β, so the
function γ is well defined. Moreover from success, inclusion and the upper bound
property (Observation 2), it follows that if K⊥α 6= ∅, then γ(K⊥α) 6= ∅, i.e., γ
is a selection function. T
To prove that K − α = γ(K⊥α) we must split the problem T in two. First
suppose that α ∈ Cn(∅) it follows by definition that K − α = γ(K⊥α) = {K}.
Now suppose that α ∈
T / Cn(∅),
T we have by the definition of γ that K − α ⊆
γ(K⊥α).TTo prove that γ(K⊥α) ⊆ K − α we will prove T that if β ∈
/ K −α
then β ∈ / γ(K⊥α). If β ∈ / K then (by definition) β ∈ / γ(K⊥α), so let
us suppose that β ∈ K. By relevance, we have that there is a K 0 such that
K − α ⊆ K 0 ⊆ K, α ∈ / Cn(K 0 ), but α ∈ Cn(K 0 ∪ {β}). By the upper bound
property we have that there is K 00 such that K 0 ⊆ K 00 ∈ K⊥α, but it is easy
to see that β ∈ / K 00 (otherwise we would have α ∈ Cn(K 00 )). ItTfollows from
K − α ⊆ K ⊆ K 00 that K 00 ∈ γ(K⊥α) and we conclude that β ∈
0
/ γ(K⊥α)
4 Applying AGM-Theory to Description Logics
As we have seen in the previous section, there are logics which are not AGM-
compliant, but for which a contraction operator can be constructed which sat-
isfies the AGM postulates with recovery substituted by relevance. We start this
section with an example of a simple logic with these characteristics.
Example 1. Consider a description logic L with a language containing only two
roles R and S, a single concept A, the constructor ∀ and the connective v.
7
The set of tautologies in this logic is given by:
Γ = {R v R, S v S, A v A, ∀R.A ⊆ ∀R.A, ∀S.A v ∀S.A} (1)
With this, we can define the whole language:
L = {R v S, S v R, ∀R.A v ∀S.A, ∀S.A v ∀R.A} ∪ Γ (2)
In this logic, we have that :
Cn({R v S}) = {R v S, ∀R.A v ∀S.A} ∪ Γ (3)
Let K = Cn({R v S, ∀R.A v ∀S.A}). By inclusion, success, and closure, we
have that:
K − (∀R.A v ∀S.A) = Cn(∅) = Γ (4)
Note that recovery is not satisfied, since:
Cn(K − (∀R.A v ∀S.A) ∪ {∀R.A v ∀S.A}) = {∀R.A v ∀S.A} ∪ Γ (5)
On the other hand, to see that the relevance postulate is satisfied, let K 0 = Γ
and consider the two options for β: R v S or ∀R.A v ∀S.A. In both cases,
∀R.A v ∀S.A is in Cn(K 0 ∪ β).
Actually, Flouris, Plexousakis and Antoniou have shown the following result:
Theorem 6. [FPA05] Any description logic which admits:
– At least two role names and one concept name
– At least one of the operators ∀, ∃, (≥n ), (≤n ) for some n
– Any (or none) of the operators ¬, t, u, − , ⊥, >, {...}
– Only the connective v applicable to both concept and roles
is not AGM-compliant.
As the theorem above shows, most expressive description logics are not
AGM-compliant. We are particularly interested in the logics SHIF(D) and
SHOIN (D), which are the underlying logics of OWL-Lite and OWL-DL. It
follows from the theorem above, that these two logics are not AGM-compliant.
What happens when we substitute the recovery postulate for relevance? The
only requirements for the existence of a contraction operator is a tarskian con-
sequence operation and compactness. So it follows that:
Corollary 1. There exists a contraction operation for the logics SHIF(D) and
SHOIN (D) satisfying the AGM postulates with relevance instead of recovery.
In [FPA06], Flouris, Plexousakis and Antoniou have also proposed substitut-
ing the recovery postulate for guaranteeing the existence of contraction opera-
tors. They stated that any candidate substitute for recovery should satisfy two
properties:
8
1. Existence: There should exist a contraction operator satisfying the new set
of postulates in any logic.
2. AGM-Rationality: For logics which are AGM-compliant, the two sets of pos-
tulates should be equivalent.
In their paper, they formulate the following candidate to substitute recovery4 :
(K-5’) If (K − α) + α ⊂ Cn(Y ∪ {α}) for some Y ⊆ K, then Cn(∅) ⊂
Cn({α}) ⊆ Cn(Y ).
The intuition behind this postulate is that instead of requiring that (K −
α) + α is equal to K, the resulting set is only required to be maximal (thus
preserving as much information as possible), in the sense that, if there was some
subset Y of K that when expanded by α would give a “larger” set than (K − α),
the closure of this Y would necessarily contain α and hence not be suitable as a
result of contraction by α.
Flouris, Plexousakis and Antoniou have shown in [FPA06] that although
this postulate satisfies the requirement of AGM-Rationality, it does not satisfy
Existence.
On the other hand, we have shown that relevance satisfies Existence in The-
orem 4. Unfortunately we were not able to prove AGM-Rationality (we plan to
establish whether it holds or not in future work). However, from Observation
3, it follows that in any logic satisfying the AGM-assumptions, the postulates
with relevance in place of recovery are equivalent to the original ones. This is a
weaker result because every logic that satisfies AGM-assumptions also satisfies
AGM-compliance, but the opposite is not true in general.
In [Fuh97], the AGM-postulates were generalised so that the contraction on
belief sets could be applied to remove sets of formulas instead of just a single
formula. For this purpose, Fuhrmann proposed two kinds of contraction: package
contraction (where every formula on the set should be contracted) and choice
contraction (where at least one formula of the set should be contracted). For
each of these contractions he proposed a different generalisation of the postulates
success, extensionality, inclusion and relevance. Furthermore he proved that the
postulates of closure and vacuity follow from the others. Our approach can be
seen as a particular case of his contractions, when we consider only contraction
by singletons.
5 Conclusions and Future Work
In this paper we presented the first steps towards a theory of belief revision
which is applicable to ontologies. We have shown that although the AGM postu-
lates are not compatible with the description logics behind OWL, with a slight
generalisation of the recovery postulate, the theory can be applied.
4
The formulation in [FPA06] is slightly different, as they are dealing with contractions
by multiple sentences.
9
Previous work by Flouris, Plexousakis and Antoniou [FPA06] established
two desiderata for a candidate substitute for recovery: Existence and AGM-
Rationality. They proposed a postulate which satisfies AGM-Rationality, but
does not satisfy Existence. In particular, the postulate does not help with the
logics in which we are interested (SHIF(D) and SHOIN (D)). The postulate
which we use (relevance), satisfies Existence. In particular, it is good for the
revision of ontologies described in OWL-Lite and OWL-DL. On the other hand,
we do not know yet whether it satisfies AGM-Rationality.
Future work includes a similar study of theories for belief revision which
make use of finite bases instead of logically closed sets. This step is essential if
one wants to implement tools to revise inconsistent ontologies.
Acknowledgements: The first author was supported by grants from CNPq
and FAPESP. The second author is partially supported by CNPq grant 304486/2004-
3. This work started during CNPq project 550222/2003-0 and is currently being
developed under FAPESP project 2004/14107-2.
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