The recently introduced Datalog family of tractable knowledge representation formalisms is able to represent and reason over light-weight ontologies. It extends plain Datalog by negative constraints and the possibility of rules with existential quantification and equality in rule heads, and at the same time restricts the rule syntax by the addition of so-called guards in rule bodies to gain decidability and tractability. In this paper, we investigate how a recently proposed probabilistic extension of Datalog can be used for representing ontology mappings in typical information integration settings, such as data exchange, data integration, and peer-to-peer integration. To allow to reconstruct the history of the mappings, to detect cycles, and to enable mapping debugging, we also propose to extend it by provenance annotations.
Information integration aims at querying in a uniform way information that is
distributed over multiple heterogeneous sources. This is usually done via mappings
between logical formalizations of data sources such as database schemas, ontology
schemas, or TBoxes; see also [
Recently, a probabilistic extension of Datalog [
In this paper, we investigate how probabilistic Datalog can be used as a mapping language for information integration and propose to add provenance information to mappings to be able to track the origin of a mapping for trust assessment and debugging. Capturing the provenance of mappings allows to resolve inconsistencies of mappings by considering the history of their creation. Furthermore, it helps to detect whether and how to perform mapping updates in case the information sources have changed or evolved. Finally, it allows to capture mapping cycles, debug mappings and to perform meta-reasoning with mappings and the knowledge bases themselves. 2
We now describe guarded Datalog [
For the elementary ingredients, we assume data constants, nulls, and variables as follows; they serve as arguments in atomic formulas in databases, queries, and dependencies. We assume (i) an infinite universe of data constants (which constitute the “normal” domain of a database), (ii) an infinite set of (labeled) nulls N (used as “fresh” Skolem terms, which are placeholders for unknown values, and can thus be seen as variables), and (iii) an infinite set of variables V (used in queries and dependencies). Different constants represent different values (unique name assumption), while different nulls may represent the same value. We assume a lexicographic order on [ N , with every symbol in N following all symbols in . We denote by X sequences of variables X1; : : : ; Xk with k > 0.
We next define atomic formulas, which occur in databases, queries, and dependencies, and which are constructed from relation names and terms, as usual. We assume a relational schema R, which is a finite set of relation names (or predicate symbols, or simply predicates). A position P [i] identifies the i-th argument of a predicate P . A term t is a data constant, null, or variable. An atomic formula (or atom) a has the form P (t1; :::; tn), where P is an n-ary predicate, and t1; :::; tn are terms. We denote by pred (a) and dom(a) its predicate and the set of all its arguments, respectively. The latter two notations are naturally extended to sets of atoms and conjunctions of atoms. A conjunction of atoms is often identified with the set of all its atoms.
We are now ready to define the notion of a database relative to a relational schema, as well as conjunctive and Boolean conjunctive queries to databases. A database (instance) D for a relational schema R is a (possibly infinite) set of atoms with predicates from R and arguments from . Such D is ground iff it contains only atoms with arguments from . A conjunctive query (CQ) over R has the form Q(X) = 9Y (X; Y), where (X; Y) is a conjunction of atoms with the variables X and Y, and eventually constants, but without nulls. Note that (X; Y) may also contain equalities but no inequalities. A Boolean CQ (BCQ) over R is a CQ of the form Q(). We often write a BCQ as the set of all its atoms, having constants and variables as arguments, and omitting the quantifiers. Answers to CQs and BCQs are defined via homomorphisms, which are mappings : [ N [ V ! [ N [ V such that (i) c 2 implies (c) = c, (ii) c 2 N implies (c) 2 [ N , and (iii) is naturally extended to atoms, sets of atoms, and conjunctions of atoms. The set of all answers to a CQ Q(X) = 9Y (X; Y) over a database D, denoted Q(D), is the set of all tuples t over for which there exists a homomorphism : X [ Y ! [ N such that ( (X; Y)) D and (X) = t. The answer to a BCQ Q() over a database D is Yes, denoted D j= Q, iff Q(D) 6= ;. 2.2
Tuple-generating dependencies (TGDs) describe constraints on databases in the form of generalized Datalog rules with existentially quantified conjunctions of atoms in rule heads; their syntax and semantics are as follows. Given a relational schema R, a tuplegenerating dependency (TGD) is a first-order formula of the form 8X8Y (X; Y) ! 9Z (X; Z), where (X; Y) and (X; Z) are conjunctions of atoms over R called the body and the head of , denoted body ( ) and head ( ), respectively. A TGD is guarded iff it contains an atom in its body that involves all variables appearing in the body. The leftmost such atom is the guard atom (or guard) of . The non-guard atoms in the body of are the side atoms of . We usually omit the universal quantifiers in TGDs. Such is satisfied in a database D for R iff, whenever there exists a homomorphism h that maps the atoms of (X; Y) to atoms of D, there exists an extension h0 of h that maps the atoms of (X; Z) to atoms of D. All sets of TGDs are finite here.
Query answering under TGDs, i.e., the evaluation of CQs and BCQs on databases under a set of TGDs is defined as follows. For a database D for R, and a set of TGDs on R, the set of models of D and , denoted mods(D; ), is the set of all (possibly infinite) databases B such that (i) D B (ii) every 2 is satisfied in B. The set of answers for a CQ Q to D and , denoted ans(Q; D; ), is the set of all tuples a such that a 2 Q(B) for all B 2 mods(D; ). The answer for a BCQ Q to D and is Yes, denoted D [ j= Q, iff ans(Q; D; ) 6= ;. We recall that query answering under TGDs is equivalent to query answering under TGDs with only single atoms in their heads. We thus often assume w.l.o.g. that every TGD has a single atom in its head. 2.3
The chase was introduced to enable checking implication of dependencies [
TGD CHASE RULE. Consider a database D for a relational schema R, and a TGD on R of the form (X; Y) ! 9Z (X; Z). Then, is applicable to D if there exists a homomorphism h that maps the atoms of (X; Y) to atoms of D. Let be applicable to D, and h1 be a homomorphism that extends h as follows: for each Xi 2 X, h1(Xi) = h(Xi); for each Zj 2 Z, h1(Zj ) = zj , where zj is a “fresh” null, i.e., zj 2 N , zj does not occur in D, and zj lexicographically follows all other nulls already introduced. The application of on D adds to D the atom h1( (X; Z)) if not already in D.
The chase algorithm for a database D and a set of TGDs consists of an exhaustive application of the TGD chase rule in a breadth-first (level-saturating) fashion, which leads as result to a (possibly infinite) chase for D and . Formally, the chase of level up to 0 of D relative to , denoted chase0(D; ), is defined as D, assigning to every atom in D the (derivation) level 0. For every k > 1, the chase of level up to k of D relative to , denoted chasek(D; ), is constructed as follows: let I1; : : : ; In be all possible images of bodies of TGDs in relative to some homomorphism such that (i) I1; : : : ; In chasek 1(D; ) and (ii) the highest level of an atom in every Ii is k 1; then, perform every corresponding TGD application on chasek 1(D; ), choosing the applied TGDs and homomorphisms in a (fixed) linear and lexicographic order, respectively, and assigning to every new atom the (derivation) level k. The chase of D relative to , denoted chase(D; ), is then defined as the limit of chasek(D; ) for k ! 1.
The (possibly infinite) chase relative to TGDs is a universal model, i.e., there exists
a homomorphism from chase(D; ) onto every B 2 mods(D; ) [
Note that sets of guarded TGDs (with single-atom heads) are theories in the guarded
fragment of first-order logic [
Another crucial ingredient of Datalog for ontological modeling are negative constraints (NCs, or simply constraints), which are first-order formulas of the form 8X (X) ! ?, where (X) is a conjunction of atoms (not necessarily guarded). We usually omit the universal quantifiers, and we implicitly assume that all sets of constraints are finite here. Adding negative constraints to answering BCQs Q over databases and guarded TGDs is computationally easy, as for each constraint 8X (X) ! ?, we only have to check that the BCQ (X) evaluates to false; if one of these checks fails, then the answer to the original BCQ Q is positive, otherwise the negative constraints can be simply ignored when answering the original BCQ Q. 2.5
A further important ingredient of Datalog for modeling ontologies are equality-generating dependencies (or EGDs) , which are first-order formulas 8X (X) ! Xi = Xj , where (X), called the body of , denoted body ( ), is a (not necessarily guarded) conjunction of atoms, and Xi and Xj are variables from X. We call Xi = Xj the head of , denoted head ( ). Such is satisfied in a database D for R iff, whenever there exists a homomorphism h such that h( (X; Y)) D, it holds that h(Xi) = h(Xj ). We usually omit the universal quantifiers in EGDs, and all sets of EGDs are finite here.
An EGD on R of the form (X) ! Xi = Xj is applicable to a database D for R
iff there exists a homomorphism : (X) ! D such that (Xi) and (Xj ) are different
and not both constants. If (Xi) and (Xj ) are different constants in , then there is
a hard violation of (and, as we will see below, the chase fails). Otherwise, the result
of the application of to D is the database h(D) obtained from D by replacing every
occurrence of a non-constant element e 2 f (Xi); (Xj )g in D by the other element e0
(if e and e0 are both nulls, then e precedes e0 in the lexicographic order). The chase of a
database D, in the presence of two sets T and E of TGDs and EGDs, respectively,
denoted chase(D; T [ E ), is computed by iteratively applying (1) a single TGD
once, according to the standard order and (2) the EGDs, as long as they are applicable
(i.e., until a fixpoint is reached). To assure that adding EGDs to answering BCQs Q
over databases and guarded TGDs along with negative constraints does not increase the
complexity of query answering, all EGDs are assumed to be separable [
We define (guarded) Datalog ontologies as follows. A (guarded) Datalog ontology consists of a database D, a (finite) set of guarded TGDs T , a (finite) set of negative constraints C , and a (finite) set of EGDs E that are separable from T . 3
We consider a probabilistic extension of Datalog based on Markov logic networks
(MLNs) [
We assume an infinite universe of data constants , an infinite set of labeled nulls N , and an infinite set of variables V, as in Datalog . Furthermore, we assume a finite set of random variables X, as in MLNs. Informally, a probabilistic guarded Datalog ontology consists of a finite set of probabilistic atoms, guarded TGDs, negative constraints, and separable EGDs, along with an MLN. We provide the formal details next.
We first define the notion of probabilistic scenario. A (probabilistic) scenario is a (finite) set of pairs (Xi; xi), where Xi 2 X, xi 2 Dom(Xi), and the Xi’s are pairwise distinct. If j j = jXj, then is a full probabilistic scenario. If every random variable Xi has a Boolean domain, then we also abbreviate by the set of all Xi such that (Xi; true) 2 . Intuitively, a probabilistic scenario is used to describe an event in which the random variables in an MLN are compatible with the settings of the random variables described by , i.e., each Xi has the value xi.
If a is an atom, T is a TGD, C is a negative constraint, E is an EGD, and is a probabilistic scenario, then: (i) a : is a probabilistic atom; (ii) T : is a probabilistic TGD (pTGD); (iii) C : is a probabilistic (negative) constraint; and (iv) E : is a probabilistic EGD (pEGD). We also refer to probabilistic atoms, TGDs, (negative) constraints, and EGDs as annotated formulas. Intuitively, annotated formulas hold whenever the events associated with their probabilistic scenarios occur.
A probabilistic (guarded) Datalog ontology is a pair = (O; M ), where O is a finite set of probabilistic atoms, guarded TGDs, constraints, and EGDs, and M is an MLN. In the sequel, we implicitly assume that every such = (O; M ) is separable, which means that E is separable from T , for every 2 Dom(X), where T (resp., E ) is the set of all TGDs (resp., EGDs) such that (i) : 2 O and (ii) is contained in the set of all (Xi; (Xi)) with Xi 2 X. As for queries, we are especially interested in the probabilities of the answers of CQs to probabilistic Datalog ontologies, called probabilistic conjunctive queries (PCQs). 3.2
The semantics of probabilistic Datalog ontologies is given relative to probability distributions over interpretations I = (D; ), where D is a database, and 2 Dom(X). We say I satisfies an annotated formula F : , denoted I j= F : , iff whenever (X) = x, for all (X; x) 2 , then D j= F . A probabilistic interpretation is a probability distribution Pr over the set of all possible interpretations such that only a finite number of interpretations are mapped to a non-zero value. The probability of an annotated formula F : , denoted Pr (F : ), is the sum of all Pr (I) such that I j= F : .
Let Pr be a probabilistic interpretation, and F : be an annotated formula. We say that Pr satisfies (or is a model of) F : iff Pr (F : ) = 1. Furthermore, Pr is a model of a probabilistic Datalog ontology = (O; M ) iff: (i) Pr satisfies all annotated formulas in O, and (ii) 1 Pr (false : ) = Pr M ( ) for all full probabilistic scenarios , where Pr M ( ) is the probability of V(Xi;xi)2 (Xi = xi) in the MLN M (and computed in the same way as P (X = x) in MLNs).
As for the semantics of queries, we begin with defining the semantics of PCQs without free variables. Let be a probabilistic Datalog ontology, and Q be a BCQ. The probability of Q in , denoted Pr (Q), is the infimum of Pr (Q : fg) subject to all probabilistic interpretations Pr such that Pr j= . Note that, as a consequence, the probability of a BCQ Q is the sum of all probabilities of full scenarios where the resulting universal model satisfies Q. We next consider the general case. As usual, given a set of variables V and a set of constants , a substitution of V by is a mapping : V ! ; given a formula F and substitution , we denote by F the formula obtained from F by replacing all variables vi with (vi). We can now define answers to PCQs. Let be a probabilistic Datalog ontology, and Q be a CQ. An answer for Q to is a pair ( ; p), where (i) is a substitution for the free variables of Q, and (ii) p 2 [0; 1] is the probability of Q in . It is positive iff p > 0. 4
As a language integrating the description logics and the logic programming paradigm with TGDs, Datalog allows to nicely tie together the theoretical results on information integration in databases and the work on ontology mediation in the Semantic Web.
When integrating data stored in databases or data warehouses, i.e., data organized by database schemas, usually so-called source-to-target TGDs (s-t TGDs), corresponding to so-called GLAV (global-local-as-view) dependencies, are used as mappings.
According to [
The following two types of dependencies are important special cases of source-totarget TGDs: LAV (local-as-view) and GAV (global as view) as explained below: – A LAV (local as view) dependency is a source-to-target TGD with a single atom in the body, i.e., it has the form 8X AS (X) ! 9Y (X, Y)), where AS is an atom over the source schema, and (X, Y) is a conjunction of atoms over the target schema. – A GAV (global as view) dependency is a source-to-target TGD with a single atom in the head, i.e., of the form 8X (X) ! AT (X0), where (X) is a conjunction of atoms over the source schema, and AT (X0) is an atom over the target schema with X0 X.
The following mappings that are mentioned in [
8x; y S:location(x; y) ! T :address(x; y). – Projection (Column Deletion): Create a target relation or concept or role by deleting one or more columns of a source relation or source concept or source role (of arbitrary arity n 2). Note that this kind of mapping is a LAV and GAV mapping at the same time. Example:
8x; y S:author(x; y) ! T :person(x). – Augmentation (Column Addition): Create a target relation or role (of arbitrary arity n 2) by adding one or more columns to the source relation or role or concept. Note that this is a LAV dependency. Example: 1 Note that all examples are stemming from a consideration of the OAEI benchmark set, more specifically, the ontologies 101 and 301-303. – Decomposition: Decompose a source relation or source role (of arbitrary arity n) into two or more target relations or roles or concepts. Note that this is a LAV dependency. Example:
Only one mapping construct mentioned in [
In ontology mediation, the definition of a mapping or alignment is based on
correspondences between so-called matchable entities of two ontologies. The following
definition is based on [
Semantic Web and ontology mapping languages usually contain a subset of the
above mentioned mapping expressions and in addition constraints, mainly class
disjointness constraints as additional mapping expressions (see also [
With Datalog , such disjointness constraints can be modeled with NCs NC : – Disjointness of ontology entities with the same arity: A source relation (or role or concept) with arity n is disjoint to another relation (or role or concept) with the same arity n. The NC below corresponds to class disjointness that specifies that persons cannot be addresses:
– Disjointness of ontology entities with different arity: A source relation (or role) with arity n 2 is disjoint to another relation (or role or concept) with the arity n > m 1. The example below specifies that persons do not have prices.
EGDs are also part of some mapping languages, especially in the database area, and can be represented by Datalog as long as they are separable from the TGDs. Such kinds of dependencies allow to create mappings like the one of the following form specifying that publishers of the same book or journal in both, the source and target schema (or ontology), have to be the same:
8x; y; z S:publisher(x; y); T :publishes(y; z) ! x = z. 5
A probabilistic (guarded) Datalog mapping has the form M = (S, T, p st, p t; M ), where (i) S and T are the source and the target schemas or ontologies, respectively, (ii) p st is a set of probabilistic (guarded) TGDs, EGDs, and NCs encoding the probabilistic source-to-target dependencies, (iii) p t is a set of probabilistic (guarded) TGDs, EGDs, and NCs encoding the probabilistic target dependencies, and (iv) M is the MLN encoding the probabilistic worlds.
Observe here that the TGDs, EGDs, and NCs are annotated with probabilistic scenarios that correspond to the worlds that they are valid in. The probabilistic dependencies that the annotations are involved in are represented by the MLN. As annotations cannot refer to elements of the ontologies or the mapping except of the MLN itself, there is a modeling advantage of separating the two tasks of ontology modeling and of modeling the uncertainty around the axioms of the ontology.
Note that due to the disconnected representation between the probabilistic
dependencies and the ontology, we can encode part of mapping formulas as predicates
encoding a specific semantics like disjointness, renaming, or decomposition, in a similar way
as done in [
However, another more interesting encoding consists of using a second ontology describing additional features of the generation of the mappings and in this way eventually even do meta reasoning about the mapping generation. A rather general example of such an additional MLN describing the generation of a mapping is shown in Fig. 1. In this example, the MLN describes the generation of a mapping by means of the matcher that it generates and a set of — possibly dependent — applicability conditions as well as additional conditions that influence the probability of the mapping besides the result of the matcher.
With such kind of an MLN describing the dependency of different kinds of conditions (also dependencies between matchers are conceivable in order to combine the results of several different matchers), probabilistic reasoning over data integration settings can be done in more precise settings. To our knowledge, such kinds of probabilistic meta ontologies for the matching process have not yet been proposed. 6
Data provenance information describes the history of data in its life cycle. It adds value to the data by explaining how it was obtained. In information integration, when data from distributed databases or ontologies is integrated, provenance information allows to check the trustworthiness and correctness of the results of queries and debug them as well as trace the errors back to where they have been created. Hence, an information integration framework should be equipped by some form of provenance.
In data provenance, it is mainly distinguished between where-, why- and
howprovenance [
Clearly, such kind of a formalization for our probabilistic Datalog information
integration framework would allow to capture provenance and annotate the mappings
with an id such that the integration paths can be traced back to their origin. In this way,
routes that can be used to debug mappings like in [
In order to use a similar approach as the aforementioned ORCHESTRA system, we need to investigate how to model the application of the chase within probabilistic Datalog with a semiring formalization. It can be expected that in probabilistic data integration with Datalog , the lineage will be restricted by the guards who help to direct the chase towards the answer of a query through the annotated guarded chase forest. 7
By means of probabilistic (guarded) Datalog [
The paper shows how classical and probabilistic (guarded) Datalog can be used to
model information integration settings and sketches a deterministic mapping language
based on Datalog and two different kinds of probabilistic adaptations based on the
rather loosely coupled probabilistic extension of Datalog with worlds represented by
means of an MLN. We also justify why data provenance needs to be captured and
represented within such a probabilistic information integration framework and propose to
use an adaptation of K-relations as proposed by [
As a next step, we will develop the proposed framework for provenance capture and, amongst others, investigate how to model the chase application for reasoning with probabilistic (guarded) Datalog with a semiring-framework.
Acknowledgments. This work was supported by an EU (FP7/2007-2013) Marie-Curie Intra-European Fellowship, the Engineering and Physical Sciences Research Council (EPSRC) grant EP/J008346/1 “PrOQAW: Probabilistic Ontological Query Answering on the Web”, the European Research Council (FP7/2007-2013/ERC) grant 246858 (“DIADEM”), and by a Yahoo! Research Fellowship.