=Paper=
{{Paper
|id=Vol-477/paper-28
|storemode=property
|title=Relativizing Concept Descriptions to Comparison Classes
|pdfUrl=https://ceur-ws.org/Vol-477/paper_12.pdf
|volume=Vol-477
|dblpUrl=https://dblp.org/rec/conf/dlog/KlarmanS09
}}
==Relativizing Concept Descriptions to Comparison Classes==
https://ceur-ws.org/Vol-477/paper_12.pdf
Relativizing Concept Descriptions to
Comparison Classes in ALC
Szymon Klarman and Stefan Schlobach
Department of Artificial Intelligence, Vrije Universiteit Amsterdam,
De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands
sklarman@few.vu.nl, schlobac@few.vu.nl
Abstract. Context-sensitivity has been for long a subject of study in
linguistics, logic and computer science. Recently the problem of repre-
senting and reasoning with contextual knowledge has been brought up in
the research on the Semantic Web. In this paper we introduce a conserva-
tive extension to the Description Logic ALC which supports representa-
tion of ontologies containing relative terms, such as ‘big’ or ‘tall’, whose
meaning depends on the reference to a particular comparison class (con-
text). We specify the syntax and semantics of the language and present
a sound and complete tableau decision procedure.
1 Introduction
It is a commonplace observation that the same expressions might have different
meanings when used in different contexts. A trivial example might be that of the
concept The Biggest. Figure 1 presents three snapshots of the same knowledge
base that focus on different parts of the domain. The extension of the concept
visibly varies across the three takes. Intuitively, there seem to be no contradic-
tion in the fact that individual Moscow is an instance of The Biggest, when
considered in the context of European cities, an instance of ¬The Biggest,
¬
¬ ¬
Fig. 1. Example of a relative concept The Biggest.
when contrasted with all cities, and finally, not belonging to any of these when
the focus is only on Asian cities. Natural language users resolve such superficial
incoherencies simply by recognizing that certain terms, call them relative 1 , such
as The Biggest, acquire definite meanings only when put in the context of
other denoting expressions — in this case, expressions denoting specific collec-
tions of objects, so-called comparison classes, with respect to which the relative
terms are used [2,3,4].
The problem of context-sensitivity has been for a long time a subject of
studies in linguistics, logic and even computer science. Recently, it has been
also encountered in the research on the Semantic Web [5,6], where the need
for representing and reasoning with imperfect information becomes ever more
pressing. Relativity of meaning appears as one of common types of such imper-
fection. Alas, Description Logics (DLs), which form the foundation of the Web
Ontology Language [7], the basic knowledge representation formalism on the
Semantic Web, were originally developed for modeling crisp, static and unam-
biguous knowledge, and as such, are incapable of handling the task seamlessly.
Consequently, it has become highly desirable to look for more expressive, ideally
backward compatible languages to meet the new application requirements [8,9].
In this paper we define a simple, conservative extension to the DL ALC,
which is intended for representation of context-sensitive terminologies, where by
contexts we understand specifically comparison classes with respect to which
the relative terms acquire precise meanings. In the following section we formally
define the language, next we present a tableau calculus for deciding satisfiability,
and finally, in the last two sections, we shortly position our work in a broader
perspective and conclude the presentation.
2 Representation Language
The language CALC , introduced in this section, extends the basic ALC logic
with two modal-like operators, enabling construction of contextualized concept
descriptions, which internalize the use of comparison classes in the language.
The classes are denoted by means of arbitrary concept descriptions. Formally,
the novel feature of CALC is founded on an extra modal dimension incorporated
into DL interpretations, whose possible states are represented by subsets of the
object domain.
In the following subsections we first shortly recap the basic notions concerning
DLs and then give a detailed account of the syntax and semantics of CALC .
2.1 Description Logic ALC
A DL language is specified by a signature Σ = (NI , NC , NR ), where NI is a set
of individual names, NC a set of concept names, and NR a set of role names,
and a set of logical operators enabling construction of complex formulas [10].
1
Philosophy of language qualifies such terms generically as syncategorematic. More
precisely, a term is syncategorematic if it does not form a denoting expression by
itself. See e.g. [1].
The DL ALC sanctions concept descriptions defined by means of concept names
(atomic concepts), special symbols >, ⊥ and the following concept constructors:
C, D, r → ¬C | C u D | C t D | ∃r.C | ∀r.C
A knowledge base K = (T , A) in ALC, consists of the terminological and the
assertional component. A (general) TBox T contains concept inclusion axioms
C v D (abbreviated to C ≡ D whenever C v D and D v C) for arbitrary
concept descriptions C and D. An ABox A contains axioms of possibly two forms:
concept assertions C(a) and role assertions r(a, b), where a, b are individual
names, C a concept, and r a role.
The semantics is defined in terms of an interpretation I = (∆I , ·I ), where
∆ is a non-empty domain of individuals, and ·I is an interpretation function,
I
which specifies the meaning of the vocabulary by mapping every a ∈ NI to an
element of ∆I , every C ∈ NC to a subset of ∆I and every r ∈ NR to a subset
of ∆I × ∆I . The function is inductively extended over complex terms in a usual
way, according to the fixed semantics of the logical operators. An interpretation
I satisfies an axiom in either of the following cases:
– I |= C v D iff C I ⊆ DI
– I |= C(a) iff aI ∈ C I
– I |= r(a, b) iff haI , bI i ∈ rI
Finally, I is said to be a model of a DL knowledge base, i.e. it makes the
knowledge base true, if and only if it satisfies all its axioms.
2.2 Description Logic CALC
The syntax of ALC is extended in CALC with two additional concept constructors,
based on modal-like operators h·i and [·]:
C, D → hDiC | [D]C
A contextualized concept description consists of a relative concept C and
a specified comparison class D, which co-determines the meaning of C. Intu-
itively, hDiC denotes all objects which are C as considered in the context of all
objects which are D, whereas [D]C denotes all objects which are either not D
or otherwise (like in the former case) are C as considered in the context of all
objects which are D. For instance, hCityiThe Biggest describes the individ-
uals that are the biggest as considered in the context of (all and only) cities,
while [European City]¬The Biggest describes all those individuals that are
either not European cities, or if they are, they are not the biggest ones in that
context. Other than that CALC does not differ from ALC on the syntactic level.
More interesting changes appear in the semantics of the language, which is
essentially augmented with an additional modal dimension, with possible states
— comparison classes — corresponding to subsets of the (global) domain of
interpretation, and the accessibility relation corresponding to the ⊇-ordering of
those subsets. In each state the vocabulary (of the relevant part) of the language
is interpreted independently from the others. Definition 1 introduces the notion
of context structure which characterizes an interpretation of a CALC language.
Definition 1. A context structure for a set of languages {Lw }w∈W is a tuple
C = hW, C, ∆, {Iw }w∈W i, where:
– W ⊆ ℘(∆) is a set of comparison classes, with ∆ ∈ W and ∅ 6∈ W ,
– C ⊆ W × W is an accessibility relation, s.t. w C v iff v ⊆ w
– ∆ is a global domain of interpretation,
– Iw = (∆Iw , ·Iw ) is an interpretation of Lw with respect to w:
• ∆Iw = w is a non-empty domain of individuals,
• ·Iw is an interpretation function defined as usual.
Instead of speaking of one language L, in many cases it is more convenient
to refer to particular sublanguages {Lw }w∈W , which are based on the parts
of the vocabulary of L deemed meaningful in particular contexts.2 We assume
that individual names are interpreted rigidly, i.e. for every a ∈ NI and every
w, v ∈ W , such that a belongs to the vocabulary of Lw and Lv , the context
structure has to satisfy aIw = aIv . It is easy to observe that C imposes a partial
order (reflexive, asymmetric and transitive) on the set of contexts, with the root
ŵ = ∆ ∈ W as its least element. Thus context structures correspond to rooted,
partially ordered Kripke frames.
Given a context structure C = hW, C, ∆, {Iw }w∈W i we can now properly
define the semantics of contextualized concept descriptions.
(hDiC)Iw = {x ∈ ∆Iw | ∃w C v, ∆Iv = DIw : x ∈ C Iv }
([D]C)Iw = {x ∈ ∆Iw | ∀w C v, ∆Iv = DIw : x ∈ ∆Iv implies x ∈ C Iv }
As usual the operators can be defined in terms of their dual counterparts,
i.e. [D]C = ¬hDi¬C and hDiC = ¬[D]¬C. Furthermore, another less common
relationship between the operators can also be derived, namely: [D]C = ¬D t
hDiC and hDiC = D u [D]C.
As expected, the notion of satisfaction in CALC is relativized to context struc-
ture and a particular comparison class in that structure. A context structure C
is a model of a knowledge base if and only if all the axioms in that knowledge
base are satisfied at the root of C.
– C, ŵ |= C v D iff C Iŵ ⊆ DIŵ
– C, ŵ |= C(a) iff aIŵ ∈ C Iŵ
– C, ŵ |= r(a, b) iff haIŵ , bIŵ i ∈ rIŵ
Note, that we interpret all axioms only at the roots of models. It follows that
both syntactically and semantically CALC is a conservative extension of ALC,
i.e. every satisfiable ALC knowledge base is a satisfiable CALC knowledge base.
2
For instance, the name Moscow is not meaningful when the focus is only on the Asian
cities. Similarly, some concepts can be applicable only with respect to particular
types of objects, e.g. natural numbers can be neither Hot nor ¬Hot.
3 Tableau Calculus
The tableau calculus for CALC , presented in this section, is an extension of the
well-known procedures for ALC [11,12]. The proof of satisfiability of a formula
ϑ is a process of finding a complete and clash-free constraint system for ϑ (a set
of logical constraints) by means of tableau rules. If such a system exists then ϑ
is satisfiable. The constraint systems are constructed by iterative application of
inference rules to the constraints in the system.
Apart from terms representing domain objects ΛI = NI ∪ {x, y, . . .}, the cal-
culus involves a set of context labels ΛC = {γ, δ, . . .} used for marking comparison
classes, where each label is a finite sequence of concept descriptions separated
with a vertical line: γ = C1 | C2 | . . . | Cn . The empty label ∈ ΛC refers to the
root of the context structure. The labels can be rendered into CALC by means of
the function p : ΛC 7→ L, such that for every γ = C1 | . . . | Cn−1 | Cn ∈ ΛC :
p(γ) = hC1 i . . . hCn−1 iCn ; p() = >
⇒u if γ : (C u D)(x) ∈ S
then set S 0 := S ∪ {γ : C(x), γ : D(x)}
⇒t if γ : (C t D)(x) ∈ S
then set S 0 := S ∪ {γ : C(x)} or S 0 := S ∪ {γ : D(x)}
⇒∃ if γ : (∃r.C)(x) ∈ S and x is not blocked in γ
then set S 0 := S ∪ {γ : r(x, y), γ : C(y)}, for a new -minimal y
⇒∀ if γ : (∀r.C)(x) ∈ S and γ : r(x, y) ∈ S
then set S 0 := S ∪ {γ : C(y)}
⇒≡ if : (> ≡ C) ∈ S and : φ(x) ∈ S
then set S 0 := S ∪ { : C(x)}
⇒6≡ if : (C 6≡ D) ∈ S
then set S 0 := S ∪ { : C(x), ¬D(x)}
or S 0 := S ∪ { : ¬C(x), D(x)}, for a a new -minimal x
⇒B if {C | γ : C(y) ∈ S} ⊆ {C | γ : C(x) ∈ S}
for a fixed γ and y x then mark y as blocked in γ by x
Table 1. ALC tableau rules.
V
A tableau proof for ϑ = i ϑi , where every ϑi is a DL axiom, is initiated
by setting a constraint system containing : ϑi for all i. At every stage of the
procedure, the system contains only elements of the form: (1) γ : C(x); (2)
γ : r(x, y); (3) : > ≡ C; (4) : C 6≡ D; for some γ ∈ ΛC , x, y ∈ ΛI and
concept/role descriptions C, D, r. Note that TBox
d axioms can be rewritten in a
usual manner into a single constraint : > ≡ CvD∈T ¬C t D. We moreover
assume that all concepts descriptions are given in the Negation Normal Form,
and that all occurrences of [D]C are replaced with the equisatisfiable ¬DthDiC.
The standard ALC inference rules are restated in Table 1. As usual we assume
a well-ordering of the individual variables used in a proof for a proper ap-
plication of the blocking mechanism. We always require that applications of the
⇒∃ rule are deferred until no other rules apply. We say that a system contains
a clash if for some γ, x and A it contains both γ : A(x) and γ : ¬A(x). Besides
⇒h·i if γ : hCiD(x) ∈ S
then set S 0 := S ∪ {γ | C : D(x)}
⇒⊃ if γ | C : φ(x) ∈ S
then set S 0 := S ∪ {γ : C(x)}
⇒6= if γ : C(x) ∈ S and δ : ¬C(x) ∈ S
then set S 0 := S ∪ { : p(γ) 6≡ p(δ)}
Table 2. CALC tableau rules.
the ALC component, the inference engine comprises three additional rules, pre-
sented in Table 2. The meaning of the ⇒h·i rule is straightforward: it introduces
a relative concept assertion within the scope of a newly generated context label,
this way marking a transition of the proof into a different comparison class. The
two remaining rules require some more comment. The ⇒⊃ rule accounts for the
Q I
nnnnn QQQ∆QQQ
n QQQ
nnn
n QQQ
Q(
wnn I
hAi hBi hBt⊥i
nnn QQQ
Ae M ∆ A =AI B fC ∆IB =B I B8 t ⊥
nn
q{9 C
S Y k {{ CC
? e {{ C = ∆IBt⊥ =(Bt⊥)I
hAi hBiC
CC
!
{{ CC
{{
B | A ∆ B|A =AIB B | B ∆ B|B =B IB
} {
I I
Fig. 2. A tree of context labels and the underlying context structure.
fact that the accessibility relation between comparison classes has to follow their
⊇-ordering. Hence all individual terms introduced in a proof within the scope
of a particular context label have to be simultaneously represented within the
scope of its predecessor, and moreover they have to occur there as instances of
the concept which denotes the successor comparison class. The ⇒⊃ rule ensures
that exactly these conditions are satisfied. The ⇒6= rule resolves synonymity
between certain context labels. Observe, that the labeling generated by ⇒h·i is
tree-shaped (see Figure 2), while the C ordering of the comparison classes does
not have to be such in principle. In fact, different context labels (or rather their
p-translations) might denote exactly the same comparison classes (e.g. consider
operators hBi and hB t ⊥i in the picture). This is not a problem as long as
there is no potential clash between the assertions occurring within the scope of
different labels. In case there is such a possibility, the rule ensures that the labels
indeed represent non-equivalent comparison classes.
It can be shown that the basic results of soundness, termination and com-
pleteness hold for the calculus. Below we prove termination and sketch the proofs
of the remaining theorems. For their full versions we refer to the accompanying
technical report [13].
Lemma 1 (Soundness). Let S be a complete clash-free constraint system for
ϑ. Then ϑ is satisfiable.
Sketch of proof. As in the typical proofs of soundness for tableau procedures for
DLs, we use S as ‘a guide’ to show that there exists a model for ϑ. To this end
we first define a structure C = hW, C, ∆, {Iw }w∈W i as follows:
– wγ ∈ W for every γ occurring in S, where
wγ = ∆Iwγ comprises all terms x such that γ : φ(x) ∈ S
– γ C δ for every γ and δ = γ | C occurring in S
– x ∈ AIwγ for every γ, x and A such that γ : A(x) ∈ S
– aIwγ = a for every γ and a ∈ obj (ϑ) such that γ : φ(a) ∈ S
– rIwγ (x, y) for every γ, x, y and r such that γ : r(x, y) ∈ S (or γ : r(z, y) ∈ S
in case x is blocked in γ by z)
Next, we show by induction on the structure of ϑ that if ŵ is the root of C then
C, ŵ |= ϑ. Finally, we acknowledge that C, as defined above, might not necessarily
be a context structure in the sense of Definition 1. Nevertheless, we show that
there always exists a way of transforming C into a proper context structure C 0 ,
such that if ϑ is satisfied in C then it is also satisfied in C 0 . This proves the
existence of a model for ϑ and thus concludes the proof of soundness.
Lemma 2 (Termination). There is no infinite sequence of inference steps via
the tableau rules.
Proof. Consider a formula ϑ in NNF with all occurrences of [D]C transformed
into ¬D t hDiC. Clearly, there is only a finite number of h·i operators used in ϑ,
and hence, there can be only a finite number of unique context labels occurring
in the tableau proof for ϑ due to application of the ⇒h·i rule. Given that, there
can be also only finite number of inference steps via the ⇒6= rule, as well as via
the ⇒⊃ rule for any individual variable. Note, that other than occurrences of h·i
ϑ does not contain any symbols from beyond the ALC, hence the only problem
for termination is posed by application of the ⇒∃ rule (clearly, upon suspending
it there can be always only a finite number of possible inference steps). But given
a finite number of context labels it has to be the case that at some point the ⇒B
rule applies, and all -minimal individual variables occurring in S are blocked.
Hence the tableau procedure for ϑ terminates in a finite number of steps.
Lemma 3 (Completeness). If ϑ is satisfiable then there exists a complete
clash-free constraint system for ϑ.
Sketch of proof. We assume ϑ is satisfiable, with C = hW, C, ∆, {Iw }w∈W i being
a context structure such that C, ŵ |= ϑ. In a usual manner, we use C as an oracle
in determining the construction of a complete clash-free constraint system for
ϑ. Essentially, we argue by induction on the possible applications of the tableau
rules, that given ϑ is satisfiable, it is impossible to derive a clash in the process
of construction of a constraint system: if a constraint system for ϑ is clash-free
before the application of a rule, it has to remain such also after that. Since,
by Lemma 2 the number of inference steps applicable to S has to be finite,
therefore at some point one has to obtain a complete constraint system, which
is clash-free.
4 Relative Terminologies — Example
For a small example of a CALC knowledge base we will formalize part of Figure 1
depicted in the introduction. Consider knowledge base K = (T , A), where TBox
and ABox are defined as follows:
T = { (1) City ≡ European City t Asian City,
(2) European City u Asian City v ⊥,
(3) hCityiThe Biggest ≡ hAsian CityiThe Biggest }
A = { (4) hCityiThe Biggest(Tokyo),
(5) hEuropean CityiThe Biggest(Moscow) }
The TBox states that every city is either a European or an Asian city, and
that, in fact, the two classes are disjoint. The third axiom ensures that the
concept The Biggest has the same instances in the context of all cities, and
in the context of Asian cities. The axioms in the ABox assert that Tokyo is
the biggest among all cities, whereas Moscow is the biggest in the context of
European cities. Given this setup it can be shown, for instance, that the following
entailments hold:
K |= hCity u ¬European CityiThe Biggest(Tokyo)
K |= hCityi¬The Biggest(Moscow)
Due to limited space we will not present full tableau proofs and resort only
to informal arguments. The validity of the first entailment rests on the fact that
according to the TBox Asian cities are exactly those that are non-European cities
(from 1). The comparison class denoted by Asian City is therefore equivalent
to that described by City u ¬European City, and so the two descriptions
represent in fact the same state in the context structure. Consequently, since
Tokyo is an instance of The Biggest in the former context (from 4 and 3), it
has to be such also in the latter.
To see that the second entailment follows as well, assume on the contrary that
¬hCityi¬The Biggest(Moscow), i.e. ¬City t hCityiThe Biggest(Moscow).
Clearly Moscow is a city (from 5 and 1), hence it would have to be an instance
of The Biggest in the context of all cities, and consequently, in the context of
Asian cities (from 3). But this would mean that Moscow is an Asian city, which
is not true (from 2 and 5).
Note that, as intended, there is no contradiction between the fact that Moscow
is an instance of The Biggest in the context of European cities and an instance
of ¬The Biggest in the context of all cities. Finally we can define a context
structure C = hW, C, ∆, {Iw }w∈W i which models K. We pose:
W = {ŵ = {Moscow, Tokyo}, w1 = {Moscow}, w2 = {Tokyo}}
C = {hŵ, w1 i, hŵ, w2 i}
∆ = ∆Iŵ = {Moscow, Tokyo}
CityIŵ = {Moscow, Tokyo}
European CityIŵ = {Moscow}
Asian CityIŵ = {Tokyo}
The BiggestIŵ = {Tokyo}
MoscowIŵ = Moscow
TokyoIŵ = Tokyo
∆Iw1 = {Moscow}
The BiggestIw1 = {Moscow}
MoscowIw1 = Moscow
∆Iw2 = {Tokyo}
The BiggestIw2 = {Tokyo}
TokyoIw2 = Tokyo
5 Related Work
The logic discussed in the previous sections can be seen as a special case of multi-
dimensional DLs [14,15], and more generally, as an instance of multi-dimensional
modal logics [16,12], in which next to the standard object dimension we introduce
a second one, referring to the subsets of the object domain as the possible states
in the model. The scope of multi-dimensionality involved here, however, is very
limited, thus discharging a number of computational problems, which otherwise
are inherent to richer multi-dimensional formalisms. Notably, we were able to
define a terminating decision procedure without resorting to certain advanced
techniques such as based on quasimodels [17,12].
In general, the problem of representing and reasoning with contextual knowl-
edge, in particular in relation to DL-like knowledge bases, has been quite broadly
studied in the literature, for instance in [6,18,19,20,21]. However, the vast ma-
jority of authors considers the notion of context on a very abstract level, merely
as a specific (limited) view on the application domain, without explicating the
formal character of that specificity. In particular there has been no attempt of
formalizing contexts as comparison classes in DL. On the other hand, the general
semantic intuition of introducing an additional modal dimension, in an explicit
(e.g. by context indexing [19]) or an implicit (e.g. by reference to subsets of
possible models of knowledge base [20]) manner, remains the same as in our
approach.
Finally, from a different perspective, CALC shares also some significant sim-
ilarities with dynamic epistemic logics, and in particular, with the public an-
nouncement logic (PAL) [22], which studies the dynamics of information flow
in epistemic models. Interestingly, the two special operators used in CALC can
be to some extent interpreted as public announcements (in the sense used in
PAL), whose role is to reduce the description (epistemic) model to only those
individuals (epistemic states) that satisfy given description (formula). Unlike
in PAL, however, we allow for much deeper revisions of the models, involving
also the interpretation function, e.g. it is possible that after contextualizing the
representation by hCi there are no individuals that are C, simply because C
gets essentially reinterpreted in the accessed context. For that reason it is also
not possible to reduce reasoning in CALC to the PAL case, for which there exist
tableau proof procedures, e.g. [23].
6 Conclusions and Future Work
Providing a sound formal account of context-sensitivity and related phenomena,
which abound in the real-life knowledge representation and reasoning scenarios,
is a longstanding challenge in Artificial Intelligence, and quite recently, also in
the research on the Semantic Web. In this paper we have addressed a very
specific form of that problem, namely, representation of relative terms, whose
meaning depends on the selection of comparison classes, to which the terms are
applied. For such tasks we have proposed language CALC , a simple extension
of the DL ALC, which by utilizing an additional modal dimension introduced
into standard ALC models allows for a roughly independent interpretation of
the vocabulary of the language in the context of different comparison classes.
This way the resulting reasoning regime complies much closer to the intuitions
associated with the use of relative terms in the natural language, e.g. it does not
allow for inferring contradiction based on existence of complementary statements
about an object, as long as the statements apply to different comparison classes.
Admittedly, the scope of the proposal is very limited and in no way can
it pretend to have solved the problem of context-sensitivity in the DL-based
representations of knowledge. Nevertheless, we have showed that by a careful
use of supplementary modal dimensions one can obtain extra expressive power,
which on the one hand is sufficient to handle certain interesting representation
problems, while on the other does not require deep revisions on the syntactic, se-
mantic nor, most importantly, the proof-theoretic side of the basic DL paradigm.
Our belief is that in a similar manner, aspects of multi-dimensionality can offer
convenient formal means for addressing other phenomena related to imperfect
knowledge, such as uncertainty or vagueness, which currently are approached on
the grounds of formalisms involving a thorough reconstruction of the semantics
and the proof theory of DLs, e.g. probabilistic, possibilistic or fuzzy DLs [8].
In the future work on CALC we want to focus on defining suitable mechanisms
of support for both relative and absolute (rigid) terms, preferably by distin-
guishing between local (contextualized) and global (applicable to all contexts)
terminological constraints, so that some parts of vocabulary can be rendered
context-independent. In a broader perspective, we intend to investigate ways of
extending the approach to other types of context-sensitivity as well as to other
aspects of imperfect knowledge representations.
Acknowledgements The first author is funded by a Hewlett-Packard Labs Inno-
vation Research Program.
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