=Paper=
{{Paper
|id=Vol-423/paper-5
|storemode=property
|title=Inference in Probabilistic Ontologies with Attributive Concept Descriptions and Nominals
|pdfUrl=https://ceur-ws.org/Vol-423/paper5.pdf
|volume=Vol-423
|dblpUrl=https://dblp.org/rec/conf/semweb/PolastroC08
}}
==Inference in Probabilistic Ontologies with Attributive Concept Descriptions and Nominals==
https://ceur-ws.org/Vol-423/paper5.pdf
Inference in Probabilistic Ontologies with
Attributive Concept Descriptions and Nominals
Rodrigo Bellizia Polastro and Fabio Gagliardi Cozman
Escola Politécnica, Universidade de São Paulo - São Paulo, SP - Brazil
rodrigopolastro@usp.br, fgcozman@usp.br
Abstract. This paper proposes a probabilistic description logic that
combines (i) constructs of the well-known ALC logic, (ii) probabilistic
assertions, and (iii) limited use of nominals. We start with our recently
proposed logic crALC, where any ontology can be translated into a re-
lational Bayesian network with partially specified probabilities. We then
add nominals to restrictions, while keeping crALC’s interpretation-based
semantics. We discuss the clash between a domain-based semantics for
nominals and an interpretation-based semantics for queries, keeping the
latter semantics throughout. We show how inference can be conducted in
crALC and present examples with real ontologies that display the level
of scalability of our proposals.
Key words: ALC logic, nominals, Bayesian/credal networks.
1 Introduction
Semantic web technologies typically rely on the theory of description logics,
as these logics offer reasonable flexibility and decidability at an computational
cost that seems to be acceptable [1, 2]. Recent literature has examined ways to
enlarge description logics with uncertainty representation and management. In
this paper we focus on two challenges in uncertainty representation: we seek
to define coherent semantics for a probabilistic description logic and to derive
algorithms for inference in this logic. More precisely, we wish to attach sensible
semantics to sentences such as
P (Merlot(a)|∃Color.{red}) = α, (1)
that should refer to the probability that a particular wine is Merlot, given that
its color is red. Note the presence of the nominal red in this expression, a feature
that complicates matters considerably. Also, we wish to compute the smallest
α that makes Expression (1) true with respect to a given ontology. This latter
calculation is an “inference”; that is, an inference is the calculation of a tight
bound on the probability of some assertion.
We have recently proposed a probabilistic description language, referred
to as “credal ALC” or simply crALC [3], that combines the well-known At-
tributive Concept Description (ALC) logic and probabilistic inclusions such as
2 Rodrigo Bellizia Polastro and Fabio Gagliardi Cozman
P (hasWineSugarDry|WineSugar) = β. One of the main features of crALC is that
it adopts an interpretation-based semantics that allows it to handle probabilistic
queries involving Aboxes i.e., sets of assertions. In crALC there is a one-to-one
correspondence between a consistent set of sentences and a relational credal net-
work; that is, a relational Bayesian network with partially specified probabilities.
An inference in crALC is equivalent to an inference in such a network.
In this paper we wish to extend our previous effort [3] by handling realistic
examples, such as the Wine and the Kangaroo ontologies, and by adding to
crALC a limited version of nominals (that is, reference to individuals in concept
descriptions). Nominals often appear in ontologies; besides, the study of nominals
touches on central issues in probabilistic logic, as discussed later.
Section 2 summarizes the main features of crALC and reviews existing
probabilistic description logics, emphasizing the differences between them and
crALC. Section 3 presents the challenges created by nominals, and introduces
our proposal for dealing with (some of) them. Section 4 described our exper-
iments with real ontologies in the literature. While our previous effort [3] was
mainly directed at theoretical analysis of crALC, in this paper we move to eval-
uation of inference methods in real ontologies. Finally, Section 5 evaluates the
results and draws some thoughts on the next steps in the creation of a complete
probabilistic semantic web.
2 Probabilistic Description Logics and crALC
In this section we review a few important concepts, the literature on probabilistic
description logics, and the logic crALC. This section is based on our previous
work [3].
2.1 A few definitions
Assume a vocabulary containing individuals, concepts, and roles [1]. Concepts
and roles are combined to form new concepts using a set of constructors. In ALC
[4], constructors are conjunction (C " D), disjunction (C # D), negation (¬C),
existential restriction (∃r.C) and value restriction (∀r.C). A concept inclusion
is denoted by C % D and concept definition is denoted by C ≡ D, where C
and D are concepts. Usually one is interested in concept subsumption: whether
C % D for concepts C and D. A set of concept inclusions and definitions is
called a terminology. If an inclusion/definition contains a concept C in its left
hand side and a concept D in its right hand side, the concept C directly uses D.
The transitive closure of “directly uses” is indicated by “uses”. A terminology
is acyclic if it is a set of concept inclusions and definitions such that no concept
in the terminology uses itself [1]. Typically terminologies only allow the left
hand side of a concept inclusion/definition to contain a concept name (and no
constructors). Concept C ! #¬C ! is denoted by ' and concept C ! "¬C ! is denoted
by ⊥, where C ! is a dummy concept that does not appear anywhere else; also,
r.' is abbreviated by r (for instance, ∃r).
Inference in Probabilistic Ontologies 3
A set of assertions about individuals may be associated to a terminology. An
assertion C(a) directly uses assertions of concepts (resp. roles) directly used by C
instantiated by a (resp. by (a, b) for b ∈ D), and likewise for the “uses” relation.
As an example, we may have the assertions such as Fruit(appleFromJohn) and
buyFrom(houseBob, John).
The semantics of a description logic is almost always given by a domain D
and an interpretation I. The domain D is a nonempty set; we often assume its
cardinality to be given as input. Note that in description logics the cardinality
of the domain is usually left unspecified, while in probabilistic description logics
this cardinality is usually specified (Section 2.2). The interpretation function I
maps each individual to an element of the domain, each concept name to a subset
of the domain, each role name to a binary relation on D × D. The interpretation
function is extended to other concepts as follows: I(C " D) = I(C) ∩ I(D),
I(C # D) = I(C) ∪ I(D), I(¬C) = D\I(C), I(∃r.C) = {x ∈ D|∃y : (x, y) ∈
I(r) ∧ y ∈ I(C)}, I(∀r.C) = {x ∈ D|∀y : (x, y) ∈ I(r) → y ∈ I(C)}. An
inclusion C % D is entailed iff I(C) ⊆ I(D), and C ≡ D iff I(D) = I(D).
Some logics in the literature offer significantly larger sets of features, such
as numerical restrictions, role hierarchies, inverse and transitive roles (the OWL
language contains several such features [2]). And most description logics have
direct translations into multi-modal logics [5] or fragments of first-order logic [6].
The translation of ALC to first-order logic is: each concept C is interpreted as
a unary predicate C(x); each role r is interpreted as a binary predicate r(x, y);
the other constructs have direct translations into first-order logic, (e.g. ∃r.C is
translated to ∃y : r(x, y) ∧ C(y) and ∀r.C to ∀y : r(x, y) → C(y)).
2.2 Probabilistic description logics
There are several probabilistic description logics in the literature. Heinsohn [7],
Jaeger [8] and Sebastiani [9] consider probabilistic inclusion axioms such as
PD (Plant) = α, meaning that a randomly selected individual is a Plant with
probability α. This interpretation characterizes a domain-based semantics. Se-
bastiani also allows assessments as P (Plant(Tweety)) = α, specifying probabil-
ities over the interpretations themselves, characterizing an interpretation-based
semantics. Most proposals for probabilistic description logics adopt a domain-
based semantics [7–14, 16], while relatively few adopt an interpretation-based
semantics [9, 17].
Direct inference refers to the transfer of statistical information about domains
to specific individuals [18, 19]. Direct inference is a problem for domain-based se-
mantics; for instance, from P (FlyingBird) = 0.3 there is nothing to be concluded
over P (FlyingBird(Tweety)). We discuss direct inference further in Section 3. Due
to the difficulties in solving direct inference, most proposals for probabilistic de-
scription logics with a domain-based semantics simply do not handle assertions.
Dürig and Studer avoid direct inference by only allowing probabilities over as-
sertions [11]. Also note that Lukasiewicz has proposed another strategy, where
expressive logics are combined with probabilities through an entailment relation
with non-monotonic properties, lexicographic entailment [12, 14, 15].
4 Rodrigo Bellizia Polastro and Fabio Gagliardi Cozman
The probabilistic description logics mentioned so far do not encode inde-
pendence relations, neither syntactically nor semantically. A considerable num-
ber of proposals for probabilistic description logics that represent independence
through graphs has appeared in the last decade or so, in parallel with work on sta-
tistical relational models [20, 21]. Logics such as P-CLASSIC [13], Yelland’s Tiny
Description Logic [16], Ding and Peng’s BayesOWL language [10], and Staker’s
logic [22] all employ Bayesian networks and various constructs of description
logics to define probabilities over domains — that is, they have domain-based
semantics. Costa and Laskey’s PR-OWL language [17] uses an interpretation-
based semantics inherited from Multi-entity Bayesian networks (MEBNs) [23].
Related and notable efforts by Nottelmann and Fuhr [24] and Hung et al [25]
should be mentioned (note also the existence of several non-probabilistic variants
of description logics [26]).
The logic crALC, proposed previously by the authors [3], adopts an
interpretation-based semantics, so as to avoid direct inference and to handle
individuals smoothly (this is discussed in more detail later). The closest existing
proposal is Costa and Laskey’s PR-OWL; indeed one can understand crALC as
a trimmed down version of PR-OWL where the focus is on the development of
scalable inference methods. The next section summarizes the main features of
crALC.
2.3 crALC
The logic crALC starts with all constructs of ALC: concepts and roles combined
through conjunction C " D, disjunction C # D, negation ¬C, existential restric-
tion ∃r.C, and value restriction ∀r.C; concept inclusions C % D and concept
definitions C ≡ D; individuals and assertions. An inclusion/definition can only
have a concept name in its left hand side; also, restrictions ∃r.C and ∀r.C can
only use a concept name C (an auxiliary definition may specify a concept C of
arbitrary complexity). A set of assertions is called an Abox. The semantics is
given by a domain D and an interpretation I, just as in ALC.
Probabilistic inclusions are then added to the language. A probability inclu-
sion reads P (C|D) = α, where D is a concept and C is a concept name. If D is
', then we simply write P (C) = α. Probabilistic inclusions are required to only
have concept names in their conditioned concept (that is, an inclusions such as
P (∀r.C|D) is not allowed). Given a probabilistic inclusion P (C|D) = α, say that
C “directly uses” B if B appears in the expression of D; again, “uses” is the
transitive closure of “directly uses”, and a terminology is acyclic if no concept
uses itself. The semantics of a probabilistic inclusion is:
∀x : P (C(x)|D(x)) = α, (2)
where it is understood that probabilities are over the set of all interpretation
mappings I for a domain D. We also allow assessments such as P (r) = β to be
made for roles, with semantics
∀x, y : P (r(x, y)) = β, (3)
Inference in Probabilistic Ontologies 5
where again the probabilities are over the set of all interpretation mappings.
These probabilistic assessments and their semantics allow us to smoothly in-
terpret a query P (A(a)|B(b)) for concepts A and B and individuals a and b. Note
that asserted facts must be conditioned upon; there is no contradiction between
∀x : P (C(x)) = α and observation C(a) holds, as we can have P (C(a)|C(a)) = 1
while P (C(a)) = α. As argued by Bacchus [18], for such a semantics to be useful,
an assumption of rigidity for individuals must be made (that is, an element of
the domain is associated with the same individual in all interpretations).
An inference is the calculation of a query P (A(a)|A), where A is a concept,
a is an individual, and A is an Abox.
Concept inclusions (including probabilistic ones) and definitions are assumed
acyclic: a concept never uses itself. The acyclicity assumption allows one to draw
any terminology T as a directed acyclic graph G(T ) defined as follows. Each
concept (even a restriction) is a node, and if a concept C directly uses concept
D, then D is a parent of C in G(T ). Also, each restriction ∃r.C or ∀r.C also
appears as a node in the graph G(T ), and the graph must contain a node for
each role r, and an edge from r to each restriction directly using it.
The next step in the definition of crALC is a Markov condition. This Markov
condition indicates which independence relations should be read off of a set of
sentences. The Markov condition is similar to Markov conditions adopted in
probabilistic description logics such as P-CLASSIC, BayesOWL and PR-OWL,
but in those logics, a set of sentences is specified with the help of a directed
acyclic graph, while in crALC a set of sentences T specifies a directed acyclic
graph G(T ). The Markov condition for crALC refers to this directed acyclic
graph G(T ). More details on the various possible Markov conditions can be
found elsewhere [3].
The idea in crALC is that the structure of the “directly uses” relation en-
codes stochastic independence through a Markov condition: (i) for every concept
C ∈ T and for every x ∈ D, C(x) is independent of every assertion that does
not use C(x), given assertions that directly use C; (ii) for every (x, y) ∈ D × D,
r(x, y) is independent of all other assertions, except ones that use r(x, y).
A terminology in crALC does not necessarily specify a single probability
measure over interpretations. The following homogeneity condition is assumed.
Consider a concept C with parents D1 , . . . , Dm . For any conjunction of the m
concepts ±Di , where ± indicates that Di may be negated or not, we have that
P (C| ± D1 " ±D2 " · · · " ±Dm ) is a constant. Consequently, any terminology
can be translated into a non-recursive relational Bayesian network [28] where
some probabilities are not fully specified. Indeed, for a fixed finite domain D,
the propositionalization of a terminology T produces a credal network [29].
In this paper we also adopt the unique names assumption (distinct elements of
the domain refer to distinct individuals), and the assumption that the cardinality
of the domain is fixed and known (domain closure). While the rigidity, acyclicity
and Markov conditions are essential to the meaning of crALC, the homogeneity,
unique names, and domain closure assumptions seem less motivated, but are
necessary for computational reasons at this point.
6 Rodrigo Bellizia Polastro and Fabio Gagliardi Cozman
3 crALC and nominals
The logic ALC does not allow nominals; that is, it does not allow individuals to
appear in concept definitions. Nominals are difficult to handle even in standard
description logics. Several optimization techniques employed in description logics
fail with nominals, and indeed few algorithms and packages do support nominals
correctly at this point. For one thing, nominals introduce connections between
a terminology and an Abox, thus complicating inferences. To some extent, nom-
inals cause reasoning to require at least partial grounding of a terminology, a
process that may incur significant cost. Still, nominals appear in many real on-
tologies; an important example is the Wine Ontology that has been alluded to
in the Introduction [30].
In the context of uncertainty handling, nominals are particularly interesting
as they highlight differences between domain-based and interpretation-based se-
mantics. Consider for instance a domain-based semantics, and suppose that a
nominal Tweety is used to define a class {Tweety} such that P ({Tweety}) = 0.3.
Presumably the assessment indicates that Tweety is “selected” with probability
0.3; this is a natural way to interpret nominals. However, now we face the chal-
lenge of direct inference; for instance, what is P (Fly(Tweety))? The difficulty is
that for every interpretation mapping I, Fly(Tweety) either holds or not; that
is, Tweety either flies or not. Once we fix an interpretation mapping, as re-
quired by a domain-based semantics, the probability P (Fly(Tweety)) gets fixed
at 0 or 1. We might then try to consider the set of all interpretation mappings;
this takes us back to an interpretation-based semantics. Worse, with the set of
interpretations mappings we have mappings fixing the behavior of Tweety ei-
ther way (flying or otherwise). Thus we cannot conclude anything about the
probability that Tweety flies, unless we make additional assumptions about the
connection between domains and interpretations. Several proposals exist for con-
necting domains and interpretations, but the matter is still quite controversial
at this point [19].
Our approach is to stay within the interpretation-based semantics of crALC,
allowing some situations to have nominals and interpreting those situations
through an interpretation-based semantics as well. We do not allow general con-
structs such as
WineFlavor ≡ {delicate, moderate, strong}.
Rather, we allow nominals only as domains of roles in restrictions. That is,
the semantics for r.{a} is not based on quantification over the domain, as the
semantics given by Expression (2). Instead, we wish to interpret this construct
directly either as (in existential restrictions):
∃x : r (x, y) ∧ (y = a), (4)
or as (in universal restrictions):
∀x : r (x, y) → (y = a). (5)
Inference in Probabilistic Ontologies 7
In restrictions containing more than one nominal as in r.{a, b, c}, the resulting
restriction considers the disjunction of the various assignments to a, b, c and so
on.
Inference in crALC, as presented previously [3], grounds a terminology into
a credal network. The various conditions previously adopted (acyclicity, domain
closure, homogeneity) guarantee that this is always possible. Inference is then
the calculation of tight lower and upper bounds on some probability P (A(a)|A)
of interest, where A is a concept, a is an individual, and A is an Abox. Inference
can be conducted in the grounded credal network using either exact [31–33] or
approximate [34] algorithms.
In the presence of nominals, this grounding of a terminology in crALC may
generate huge networks. To avoid this problem, the grounded network must be
instantiated only at its relevant nominals; that is, the nominals present in the
roles must have specific domains. So, if the role hasProperty(x, y) indicates that
the element x has one specific property with value y, then x must be one object
being described and y must be a nominal that describes the property indicated
by the role. For instance, ∃hasColor.{red} is interpreted as:
∃x ∈ D : hasColor (x, y) ∧ (y = red), (6)
where D is the domain with the elements being described and y ranges over
all the nominals that “are” colors. This approach is very close to Datatypes,
but its most significant characteristic is the definition of the semantic given by
Expressions 4 and 5.
Nominals are often used to define mutually exclusive individuals. Although
crALC does not have any construct to express this situation, it can be easily
done through the inclusion of a probabilistic node that has the mutually exclusive
nodes as its parents and a conditional probability table that mimics the behavior
of a XOR logic gate. This node must be set as an observed node with value true
so that all of its parents become inter-dependent.
4 Experiments
We now report on two experiments with well-known networks. The first one was
done with the large Wine Ontology, and the second one was done with the not
so famous Kangaroo ontology.
The Wine Ontology was extracted from a OWL file available at the ontol-
ogy repository of the Temporal Knowledge Base Group from Universitat Jaume
I (at http://krono.act.uji.es/Links/ontologies/wine.owl/view). It is a ontology
that relies extensively in nominals for describing the different kind of wines and
their properties. These nominals were represented as indicated in Section 3.
Probability inclusions were added to the terminology; assertions were made on
properties of an unspecified wine and the wine type was then inferred. Figure 1
shows the network generated for a domain of size 1. We have:
8 Rodrigo Bellizia Polastro and Fabio Gagliardi Cozman
Fig. 1. Network generated from the Wine ontology with domain size 1.
Example 1. The probability of a wine to be Merlot given its body is medium, its
color is red, its flavor is moderate, its sugar is dry and it is made from merlot
grape:
P (Merlot(a) | medium(a), red (a), moderate(a), dry(a), merlotGrape(a)) = 1.0.
Example 2. The probability of a wine to be Merlot given its body is medium, its
color is red, its flavor is moderate and its sugar is dry:
P (Merlot(b) | medium(b), red (b), moderate(b), dry(b)) = 0.5.
Example 3. The probability of a wine to be Merlot given it is made from merlot
grape and its sugar is sweet:
P (Merlot(c) | merlotGrape(c), sweet(c)) = 0.0.
The Wine ontology only presents restrictions over roles and properties, not
having any restriction over individuals. That is, there is no connection between
individuals other than the constraints imposed on restrictions by nominals. Con-
sequently, the whole ontology can be translated into a single credal network of
fixed size regardless of the actual size of the domain, as far as inference is con-
cerned. Hence there are no qualms about scalability and computational cost
when the domain grows. In fact, it was possible to run exact inference in this ex-
periment, even with big domains, since we can separate only the necessary nodes
using the Markov condition (we have run exact inferences using the SamIam
package, available at http://reasoning.cs.ucla.edu/samiam/).
The second experiment was done with the Kangaroo ontology, adapted from a
KRSS file available among the toy ontologies for the CEL System1 at
http://lat.inf.tu-dresden.de/meng/ontologies/kangaroo.cl. Although this ontol-
ogy does not contain nominals, it uses restrictions amongst individuals in the
1
A polynomial-time Classifier for the description logic EL+, http://lat.inf.tu-
dresden.de/systems/cel/.
Inference in Probabilistic Ontologies 9
(a) |D| = 1 (b) |D| = 10
Fig. 2. Network generated from kangaroo ontology for various domain sizes.
domain, leading to possible concerns on scalability issues as the domain grows.
For instance, consider some of the definitions in this ontology:
Parent ≡ Human " ∀hasChild.Human.
MaternityKangaroo ≡ Kangaroo " ∀hasChild.Kangaroo.
In this case, the size of the grounded credal network is proportional to |D|2 ; that
is, it is quadratic on domain size.
It was not possible to run exact inference in this ontology with big domains,
but the L2U algorithm [34] produced approximate inferences with reasonable
computational cost. Table 1 shows some results for a growing domain. In Figure
2 we can see the size of the network generated for different domain sizes: in
Fig.2(a) the domain size is 1 while in Fig.2(b) the domain size is 10.
Table 1. Results from the L2U algorithm for the inference P (P arent (0) | Human (1))
for various domain sizes
N 2 5 10 20 30 40 50
L2U 0.2232 0.3536 0.4630 0.5268 0.5377 0.5396 0.5399
5 Conclusion
In this paper we have continued our efforts to develop a probabilistic descrip-
tion logic that can handle both probabilistic inclusions and queries containing
Aboxes. This may seem a modest goal, but it touches on the central question
concerning semantics in probabilistic logics; that is, whether the semantics is a
domain-based or interpretation-based one. We have kept our preference for an
interpretation-based semantics in this paper, as it seems to be the only way to
avoid the challenges of direct inference. Without an interpretation-based seman-
tics, it is hard to imagine how an inference involving Aboxes could be defined.
10 Rodrigo Bellizia Polastro and Fabio Gagliardi Cozman
Most existing probabilistic description logics do adopt domain-based semantics,
but it seems that the cost in avoiding inferences with Aboxes is high.
In this paper we have shown that the algorithms outlined in a previous pub-
lication [3] do scale up to realistic ontologies in the literature. Obviously, there is
a trade-off between expressivity and complexity in any description logic, and it is
difficult to know which features can be added to a description logic before mak-
ing it intractable in practice. In this paper we have examined the challenges in
adding nominals to the crALC logic. Nominals are both useful in practice, and
interesting on theoretical grounds. The discussion of nominals can shed light on
issues of semantics and direct inference, and one of the goals of this paper was to
start a debate in this direction. We have presented relatively simple techniques
that handle nominals in a limited setting; that is, as domains of restrictions.
Much more work must be done before the behavior of nominals in probabilis-
tic description logics becomes well understood. The inclusion of nominals intro
CRALC, however limited, moves us towards the SHOIN logic, and therefore
closer to OWL, the recommended standard for the Semantic Web. We hope to
gradually close the remaining gap and a complete probabilistic version of OWL
in future work
Acknowledgements
This work was partially funded by FAPESP (04/09568-0); the first author is
supported by HP Brazil R&D.; the second author is partially supported by
CNPq. We thank all these organizations.
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