Overview
Towards sophisticated reasoning capabilities for the Semantic Web, the work [
4
] has
introduced probabilistic description logic programs (or probabilistic dl-programs), which
combine description logics, normal programs under the answer set (resp., well-founded)
semantics, and probabilistic uncertainty. Probabilistic dl-programs are an expressive
formalism, which generalizes Poole’s ICL [
6
], which in turn generalizes (amongst
others) influence diagrams, Bayesian networks, Markov decision processes, normal form
games, and structural causal models. Intuitively, a probabilistic dl-program consists of
(i) a description logic knowledge base L, (ii) a normal program P involving queries
to L [
1
], and (iii) a probability distribution on a set of total choices. It represents a set
of probability distributions on a set of first-order interpretations. Instead of querying L
in P , a variant of dl-programs also allows for using L to constrain the terms in P (which
may e.g. be used to resolve naming inconsistencies between different data sources).
In this paper, we describe how probabilistic dl-programs can be used for modeling
data integration systems with probabilities. A data integration system [
3
], in its most
general form, is a triple hG, S, Mi, where (i) G is the global or mediated schema,
representing the domain of interest of the system, (ii) S is the source schema, representing the
data sources that take part in the system, and (iii) M is a mapping that establishes a
relation between the source schema and the global schema. There exist different approaches
to the specification of the mapping, which is a crucial aspect in a data integration
system. A common issue in data integration is the fact that data may be inconsistent and/or
redundant relative to the global schema G, which in general incorporates constraints
expressed as rules. In other words, the same information may come from different data
sources, with different degrees of certainty, which we model by means of rules in
probabilistic dl-programs (or dl-rules). More formally, we partition the vocabulary Φ into
the pairwise disjoint sets ΦG, ΦS, and Φc: the symbols in ΦG are of arity at least 1,
and represent the (virtual) global predicates; the symbols in ΦS are of arity at least 1,
and represent source predicates; the symbols in Φc are constants. The mapping M
between ΦG and ΦS is then specified by mapping dl-rules, which have only predicates
in ΦS and Φc in the body, and only predicates in ΦG and Φc in the head. A probabilistic
dl-program modeling a data integration system may have: (i) source dl-rules (over ΦS
and Φc): they express properties and constraints of the data sources; (ii) global dl-rules
(over ΦG and Φc): they express properties and constraints on the global schema, which
enhance its expressiveness to better fit the application domain; global dl-rules cannot
comprise ground facts; in fact, such facts specify the contents of data sources, while the
global schema (at least in the “traditional” data integration setting) must remain strictly
virtual; (iii) mapping dl-rules as specified above. To summarize, G consists of ΦG and
the global dl-rules, S consists of ΦS and the source dl-rules, and M consists of the
mapping dl-rules. The fact that the mapping dl-rules are probabilistic allows for a high
flexibility in the treatment of the uncertainty that is present when pieces of data come
from heterogeneous sources whose informative content in general partially overlaps.
2
Example
Consider a typical rule-based data integration, where the global predicate buy (C, X ) is
derived from either the source predicate s1(C, X, Y ) or the source predicates s2(C, D)
and s3(D, X ). Moreover, suppose that C resp. X are restricted to customers resp.
products from a description logic knowledge base L, and that there may be inconsistencies
between the two different ways of deriving buy (C, X ). To consistently integrate them,
we assign to each derivation a total choice from {a, a} along with user-defined
probabilities that depend on the reliability of the derivations (e.g., μ(a) = 0.7 and μ(a) = 0.3).
The following dl-rules then realize the probabilistic data integration:
buy (C, X) ← s1(C, X, Y ), DL[Customer ](C), DL[Product ](X), a ;
buy (C, X) ← s2(C, D), s3(D, X), DL[Customer ](C), DL[Product ](X), a .
So, every fact that holds by the first resp. second dl-rule has the probability 0.7 resp. 0.3,
while every fact that holds by both dl-rules has the probability 1. Note that in addition
to being inconsistent, two data sources may also be independent from each other.