data integration with uncertainty - vldb endowment inc
TRANSCRIPT
Data Integration with Uncertainty
Xin DongUniversity of Washington
Seattle, WA 98195
Alon Y. HalevyGoogle Inc.
Mountain View, CA 94043
Cong YuUniversity of MichiganAnn Arbor, MI 48109
ABSTRACTThis paper reports our first set of results on managing un-certainty in data integration. We posit that data-integrationsystems need to handle uncertainty at three levels, and doso in a principled fashion. First, the semantic mappings be-tween the data sources and the mediated schema may beapproximate because there may be too many of them to becreated and maintained or because in some domains (e.g.,bioinformatics) it is not clear what the mappings should be.Second, queries to the system may be posed with keywordsrather than in a structured form. Third, the data from thesources may be extracted using information extraction tech-niques and so may yield imprecise data.
As a first step to building such a system, we introduce theconcept of probabilistic schema mappings and analyze theirformal foundations. We show that there are two possiblesemantics for such mappings: by-table semantics assumesthat there exists a correct mapping but we don’t know whatit is; by-tuple semantics assumes that the correct mappingmay depend on the particular tuple in the source data. Wepresent the query complexity and algorithms for answeringqueries in the presence of approximate schema mappings,and we describe an algorithm for efficiently computing thetop-k answers to queries in such a setting.
1. INTRODUCTIONData integration and exchange systems offer a uniform in-
terface to a multitude of data sources and the ability to sharedata across multiple systems. These systems have recentlyenjoyed significant research and commercial success [12, 14,16]. Current data integration systems are essentially a natu-ral extension of traditional database systems in that queriesare specified in a structured form and data is modeled inone of the traditional data models (relational, XML). In ad-dition, the data integration system has exact knowledge ofhow the data in the sources map to the mediated schemaused by the data integration system.
We argue that as the scope of data integration applica-
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tions broadens, such systems need to be able to model uncer-tainty at their core. Uncertainty can arise for multiple rea-sons in data integration. First, the semantic mappings be-tween the data sources and the mediated schema may be ap-proximate. For example, in an application like Google Baseor when mapping millions of sources on the deep web [21], wecannot imagine specifying exact mappings. In some domains(e.g., bioinformatics), we do not necessarily know what theexact mappings are. Second, if the intended users of the ap-plication are not necessarily familiar with schemas, or if thedomain of the system is too broad to offer form-based queryinterfaces (such as web forms), we need to support keywordqueries. Hence, a second source of uncertainty is the trans-formation between keyword queries and a set of candidatestructured queries. Finally, data is often extracted from un-structured sources using information extraction techniques.Since these techniques are approximate, the data obtainedfrom the sources may be uncertain.
Extending data integration systems to handle uncertaintyis also a key step to realizing Dataspace Support Platforms [15].The key idea of dataspaces is that data sources can be addedwith very little (or no) effort and the system still providesuseful search, query and exploration services. Over time,the system evolves in a pay-as-you-go fashion to improvethe quality of semantic mappings where they matter, there-fore improving the quality of query answering. Hence, thedata integration system we describe provides the underlyingmechanisms that enable the dataspace to manage uncer-tainty about the semantic mappings, the queries, and theunderlying data as the system evolves.
This paper takes a first step towards the goal of dataintegration with uncertainty. We first describe how the ar-chitecture of such a system differs from a traditional one(Section 2). At the core, the system models tuples and se-mantic mappings with probabilities associated with them.Query answering ranks answers and typically tries to obtainthe top-k results to a query. These changes lead to a re-quirement for a new kind of adaptivity in query processing.
We then focus on one core component of data integra-tion with uncertainty, namely probabilistic schema map-pings (Section 3). Semantic mappings are the componentof a data integration system that specifies the relationshipbetween the contents of the different sources. The mappingsenable the data integration to reformulate a query posedover the mediated schema into queries over the sources [18,13]. We introduce probabilistic schema mappings, and de-scribe how to answer queries in their presence.
We define probabilistic schema mapping as a set of pos-
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sible (ordinary) mappings between a source schema and atarget schema, where each possible mapping has an asso-ciated probability. We begin by considering a simple classof mappings, where each mapping describes a set of corre-spondences between the attributes of a source table and theattributes of a target table. We argue that there are twopossible interpretations of probabilistic schema mappings.In the first, called by-table semantics, we assume there existsa single correct mapping between the source and the target,but we don’t know which one it is. In the second, calledby-tuple semantics, the correct mapping may depend on theparticular tuple in the source to which it is applied. In bothcases, the semantics of query answers are a generalization ofcertain answers [1] for data integration systems.
We describe algorithms for answering queries in the pres-ence of probabilistic schema mappings and then analyze thecomputational complexity of answering queries (Section 4).We show that the data complexity of answering queries inthe presence of probabilistic mappings is PTIME for by-table semantics and #P-complete for by-tuple semantics.We identify a large subclass of real-world queries for whichwe can still obtain all the by-tuple answers in PTIME. Wethen describe algorithms for finding the top-k answers to aquery (Section 5).
The size of a probabilistic mapping may be quite large,since it essentially enumerates a probability distribution bylisting every combination of events in the probability space.In practice, we can often encode the same probability dis-tribution much more concisely. Our next contribution (Sec-tion 6) is to identify two concise representations of proba-bilistic mappings for which query answering can be done inPTIME in the size of the mapping. We also examine the pos-sibility of representing a probabilistic mapping as a BayesNet, but show that query answering may still be exponentialin the size of a Bayes Net representation of a mapping.
Finally, we consider several more powerful mapping lan-guages, such as complex mappings, where the correspon-dences are between sets of attributes, and conditional map-pings, where the mapping is conditioned on properties of thetuple to which it is applied (Section 7).
2. OVERVIEW OF THE SYSTEMThis section describes the requirements from a data in-
tegration system that supports uncertainty and the overallarchitecture of the system. We frame our specific contribu-tions in context of this architecture.
2.1 Uncertainty in Data IntegrationA data integration system needs to handle uncertainty at
three levels.
Uncertain schema mappings: Data integration systemsrely on schema mappings for specifying the semantic rela-tionships between the data in the sources and the terms usedin the mediated schema. However, schema mappings can beinaccurate. In many applications it is impossible to createand maintain precise mappings between data sources. Thiscan be because the users are not skilled enough to provideprecise mappings, such as in personal information manage-ment [7], because people do not understand the domain welland thus do not even know what correct mappings are, suchas in bioinformatics, or because the scale of the data pre-vents generating and maintaining precise mappings, such asin integrating data of the web scale [21]. Hence, in practice,
Mediated Schema
Q
Q 1 ,...Q m
Keyword Reformulation
Query Reformulation
D 1
D 2
D 3
D 4
D k
Q 11 ,...Q 1n ... Q k1 ,...Q kn
Query Pocessor
Q 11 ,...Q 1n,…, Q k1 ,...Q kn
Figure 1: Architecture of a data-integration system that
handles uncertainty.
schema mappings are often generated by semi-automatictools, and not necessarily verified by domain experts.
Uncertain data: By nature, data integration systems needto handle uncertain data. One reason for uncertainty is thatdata is often extracted from unstructured or semi-structuredsources by automatic methods (e.g., HTML pages, emails,blogs). A second reason is that data may come from sourcesthat are unreliable or not up to date.
Uncertain queries: In some data integration applications,especially on the web, queries will be posed as keywordsrather than as structured queries against a well defined schema.The system needs to translate these queries into some struc-tured form so they can be reformulated with respect to thedata sources. At this step, the system may generate multi-ple candidate structured queries and have some uncertaintyabout which is the real intent of the user.
2.2 System ArchitectureGiven the above requirements, we describe the architec-
ture of a data integration system that manages uncertaintyat its core. We describe the system by contrasting it to atraditional data integration system.
The first and most fundamental characteristic of this sys-tem is that it is based on a probabilistic data model. Thismeans two things. First, as we process data in the systemwe attach probabilities to each tuple. Second, and the focusof this paper, is that schema mappings are also associatedwith probabilities, modeling our uncertainty about whichones are correct. We note that the probabilities associatedwith tuples, mappings, and answers are mostly internal tothe system, and are not meant to be exposed to users. Typ-ically, we will use these probabilities to rank answers.
Second, whereas traditional data integration systems be-gin by reformulating a query onto the schemas of the datasources, a data integration system with uncertainty needsto first reformulate a keyword query into a set of candidatestructured queries. We refer to this step as keyword refor-mulation. Note that keyword reformulation is different fromtechniques for keyword search on structured data (e.g., [17,2]) in that (a) it does not assume access to all the data inthe sources or that the sources support keyword search, and(b) it tries to distinguish different structural elements in thequery in order to pose more precise queries to the sources(e.g., realizing that in the keyword query “chicago weather”,“weather” is an attribute label and “chicago” is an instance
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name). That being said, keyword reformulation should ben-efit from techniques that support answering keyword searchon structured data.
Third, the query answering model is different. Instead ofnecessarily finding all answers to a given query, our goal istypically to find the top-k answers, and rank these answersmost effectively.
The final difference from traditional data integration sys-tems is that our query processing will need to be more adap-tive than usual. Instead of generating a query answeringplan and executing it, the steps we take in query process-ing will depend on results of previous steps. We note thatadaptive query processing has been discussed quite a bit indata integration [19], where the need for adaptivity arisesfrom the fact that data sources did not answer as quickly asexpected or that we did not have accurate statistics abouttheir contents to properly order our operations. In our work,however, the goal for adaptivity is to get the answers withhigh probabilities faster.
The architecture of the system is shown in Figure 1. Thesystem contains a number of data sources and a mediatedschema. When the user poses a query Q, which can be eithera structured query on the mediated schema, or a keywordquery, the system returns a set of answer tuples, each witha probability. If Q is a keyword query, the system firstperforms keyword reformulation to translate it into a setof candidate structured queries on the mediated schema.Otherwise, the candidate query is Q itself.
Consider how the system answers the candidate queries,and assume the queries will not involve joins over multi-ple sources. For each candidate structured query Q0 anda data source S, the system reformulates Q0 according tothe schema mapping (which can be uncertain) between S’sschema and the mediated schema, sends the reformulatedquery (or queries) to S, retrieving the answers. If the userasks for all the answers to the query, then the reformulatedquery is typically a query with grouping and aggregation. IfS does not support grouping and aggregation, then groupingand aggregation can be processed in the integration system.If the user asks for top-k answers, then query processing ismore complex. The system reformulates the query into aset of queries, and uses a middle layer to decide at runtimewhich queries are critical to computing the top-k answersand sends the appropriate queries to S. Note that therecan be multiple iterations of deciding the promising refor-mulated queries and retrieving answers. Furthermore, thesystem can even decide which data sources are more relevantand prioritize the queries to those data sources. Finally, ifthe data in the sources are uncertain, then the sources willreturn answers with probabilities attached to them.
After receiving answers from different data sources, thesystem combines them to get one single set of answer tu-ples. For example, if the data sources are known to beindependent of each other, and we obtain tuple t from ndata sources with probabilities p1, . . . , pn respectively, thenin the final answer set t has probability 1−Πn
i=1(1− pi). Ifwe know that some data sources are duplicates or extensionsof others, a different combination function needs to be used.
2.3 Handling Uncertainty in MappingsAs a first step towards developing such a data integra-
tion system, we introduce in this paper probabilistic schemamappings, and show how to answer queries in their pres-
Possible Mapping Prob{(pname, name), (email-addr, email),
m1 = (current-addr, mailing-addr), 0.5(permanent-addr, home-address)}{(pname, name), (email-addr, email),
m2 = (permanent-addr, mailing-addr), 0.4(current-addr, home-address)}
m3 = {(pname, name), (email-addr, mailing-addr), 0.1(current-addr, home-addr)}
(a)
DS =pname email-addr permanent-addr current-addrAlice alice@ Mountain View SunnyvaleBob bob@ Sunnyvale Sunnyvale
(b)Tuple Prob
(’Sunnyvale’) 0.9(’Mountain View’) 0.5
(’alice@’) 0.1(’bob@’) 0.1
(c)
Figure 2: The running example: (a) a probabilisticschema mapping between S and T ; (b) a source in-stance DS; (c) the answers of Q over DS with respectto the probabilistic mapping.
ence. Before the formal discussion, we illustrate the mainideas with an example.
Example 2.1. Consider a data source S, which describesa person by her email address, current address, and perma-nent address, and the mediated schema T , which describes aperson by her name, email, mailing address, home addressand office address:
S=(pname, email-addr, current-addr, permanent-addr)T=(name, email, mailing-addr, home-addr, office-addr)
A semi-automatic schema-mapping tool may generate threepossible mappings between S and T , assigning each a proba-bility. Whereas the three mappings all map pname to name,they map other attributes in the source and the target differ-ently. Figure 2(a) describes the three mappings using sets ofattribute correspondences. For example, mapping m1 mapspname to name, email-addr to email, current-addr to mailing-addr, and permanent-addr to home-addr. Because of the un-certainty on which mapping is correct, we consider all ofthese mappings in query answering.
Suppose the system receives a query Q composed on themediated schema and asking for people’s mailing addresses:
Q: SELECT mailing-addr FROM T
Using the possible mappings, we can reformulate Q intodifferent queries:
Q1: SELECT current-addr FROM SQ2: SELECT permanent-addr FROM SQ3: SELECT email-addr FROM S
If the user requires all possible answers, the system gen-erates a single aggregation query based on Q1, Q2 and Q3 tocompute the probability of each returned tuple, and sends thequery to the data source. Suppose the data source containsa table DS as shown in Figure 2(b), the system will retrievefour answer tuples, each with a probability, as shown in Fig-ure 2(c).
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If the user requires only the top-1 answer (i.e., the an-swer tuple with the highest probability), the system decidesat runtime which reformulated queries to execute. For ex-ample, after executing Q1 and Q2 at the source, the systemcan already conclude that (‘Sunnyvale’) is the top-1 answerand can skip query Q3. 2
2.4 Source of probabilitiesA critical issue in any system that manages uncertainty is
whether we have a reliable source of probabilities. Whereasobtaining reliable probabilities for such a system is one ofthe most interesting areas for future research, there is quitea bit to build on. For keyword reformulation, it is possibleto train and test reformulators on large numbers of queriessuch that each reformulation result is given a probabilitybased on its performance statistics. In the case of schemamatching, it is standard practice for schema matchers to alsoassociate numbers with the candidates they propose. Theissue here is that the numbers are meant only as a rank-ing mechanism rather than true probabilities. However, asschema matching techniques start looking a larger numberof schemas, one can imagine ascribing probabilities (or ap-proximations thereof) to their measures. Finally, informa-tion extraction techniques are also often based on statisticalmachine learning methods, thereby lending their predictionsa probabilistic interpretation.
3. PROBABILISTIC SCHEMA MAPPINGIn this section we formally define the semantics of proba-
bilistic schema mappings and the query answering problemswe consider. Our discussion is in the context of the rela-tional data model. A schema contains a finite set of rela-tions. Each relation contains a finite set of attributes andis denoted by R = 〈r1, . . . , rn〉. An instance DR of R is afinite set of tuples, where each tuple associates a value witheach attribute in the schema.
We consider select-project-join (SPJ) queries in SQL. Notethat answering such queries is in PTIME in the size of thedata.
3.1 Schema MappingsWe begin by reviewing non-probabilistic schema mappings.
The goal of a schema mapping is to specify the semantic re-lationships between a source schema and a target schema.We refer to the source schema as S̄, and a relation in S̄ asS = 〈s1, . . . , sm〉. Similarly, we refer to the target schemaas T̄ , and a relation in T̄ as T = 〈t1, . . . , tn〉.
The common formalism for semantic mappings, GLAV, isbased on expressions of the form
m : ∀x(φ(x) → ∃yψ(x,y))
In the expression, φ is the body of a conjunctive query overS̄ and ψ is the body of a conjunctive query over T̄ . A pairof instances DS and DT satisfies a GLAV mapping m, if forevery assignment of x in DS that satisfies φ there exists anassignment of y in DT that satisfies ψ.
We consider a limited form of GLAV mappings that in-volve only projection queries on a single table on each sideof the mapping. These mappings have also been referredto as schema matching in the literature [23]. Specifically,we consider GLAV mappings where (1) φ (resp. ψ) is anatomic formula over S (resp. T ), (2) the GLAV mappingdoes not include constants, and (3) each variable occurs at
DS =pname email-addr permanent-addr current-addrAlice alice@ Mountain View SunnyvaleBob bob@ Sunnyvale Sunnyvale
(a)name email mailing-addr home-addr office-addrAlice alice@ Mountain View Sunnyvale officeBob bob@ Sunnyvale Sunnyvale office
(b)name email mailing-addr home-addr office-addrAlice alice@ Sunnyvale Mountain View officeBob email bob@ Sunnyvale office
(c)
Tuple Prob(’Sunnyvale’) 0.9
(’Mountain View’) 0.5(’alice@’) 0.1(’bob@’) 0.1
(d)
Tuple Prob(’Sunnyvale’) 0.94
(’Mountain View’) 0.5(’alice@’) 0.1(’bob@’) 0.1
(e)
Figure 3: Example 3.10: (a) a source instance DS;(b) a target instance that is by-table consistent withDS; (c) a target instance that is by-tuple consistentwith DS; (d) Qtable(DS); (e) Qtuple(DS).
most once on each side of the mapping. We consider thisclass of mappings because they already expose many of thenovel issues involved in probabilistic mappings and becausethey are quite common in practice. We also note that manyof the concepts we define apply to a broader class of map-pings. In Section 7, we briefly discuss a set of extensions toour mappings, such as complex mappings.
Given these restrictions, we can define our mappings interms of attribute correspondences. An attribute correspon-dence is of the form cij = (si, tj), where si is a source at-tribute in the schema S and tj is a target attribute in theschema T . Intuitively, cij specifies that there is a rela-tionship between si and tj . In practice, a correspondencealso involves a function that transforms the value of si tothe value of tj . For example, the correspondence (c-degree,temperature) can be specified as temperature=c-degree∗1.8+32, describing a transformation from Celsius to Fahrenheit.These functions are irrelevant to our discussion, and there-fore we omit them. Formally, we define relation mappingsand schema mappings as follows.
Definition 3.1 (Schema Mapping). Let S̄ and T̄ berelational schemas. A relation mapping M is a triple (S, T,m),where S is a relation in S̄, T is a relation in T̄ , and m is aset of attribute correspondences between S and T .
When each source and target attribute occurs in at mostone correspondence in m, we call M a one-to-one relationmapping.
A schema mapping M is a set of one-to-one relation map-pings between relations in S̄ and in T̄ , where every relationin either S̄ or T̄ appears at most once. 2
Example 3.2. Consider the mappings in Example 2.1.We can view m1 as a GLAV mapping:
∀n, e, c, p(S(n, e, c, p) → ∃o(T (n, e, c, p, o)))
The database in Figure 2 (b) (repeated in Figure 3 (a)) andthe database in Figure 3 (b) satisfy m1. 2
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3.2 Probabilistic Schema MappingsIntuitively, a probabilistic schema mapping describes a
probability distribution of a set of possible schema mappingsbetween a source schema and a target schema.
Definition 3.3 (Probabilistic Mapping). Let S̄ andT̄ be relational schemas. A probabilistic mapping (p-mapping),pM , is a triple (S, T,m), where S ∈ S̄, T ∈ T̄ , and m is aset {(m1,Pr(m1)), . . . , (ml,Pr(ml))}, such that
• for i ∈ [1, l], mi is a one-to-one mapping between Sand T , and for every i, j ∈ [1, l], i 6= j ⇒ mi 6= mj .
• Pr(mi) ∈ [0, 1] andPl
i=1 Pr(mi) = 1.
A schema p-mapping, pM, is a set of p-mappings betweenrelations in S̄ and in T̄ , where every relation in either S̄ orT̄ appears in at most one p-mapping. 2
We refer to a non-probabilistic mapping as an ordinarymapping. Note that a schema p-mapping may contain bothp-mappings and ordinary mappings. Example 2.1 shows a p-mapping (see Figure 2(a)) that contains three possible map-pings.
3.3 Semantics of Probabilistic MappingsIntuitively, a probabilistic schema mapping models the
uncertainty about which of the mappings in pM is the cor-rect one. When a schema matching system produces a setof candidate matches, there are two ways to interpret theuncertainty: (1) a single mapping in pM is the correct oneand it applies to all the data in S, or (2) multiple mappingsare correct and each suitable for a subset of tuples in S,though it is unknown which mapping is the right one fora specific tuple. Example 2.1 illustrates the first interpre-tation. For the same example, the second interpretation isequally valid: the correct mapping may depend on the par-ticular tuple because some people use their current addressas mailing address and some people use their permanentaddress as mailing address.
This paper analyzes query answering under both interpre-tations. We refer to the first interpretation as the by-tablesemantics and to the second one as the by-tuple semanticsof probabilistic mappings. We note that we are not tryingto argue for one interpretation over the other. The needsof the application should dictate the appropriate semantics.Furthermore, our complexity results, which will show ad-vantages to by-table semantics, should not be taken as anargument in the favor of by-table semantics.
We next define the semantics of p-mappings in detail; thedefinitions for schema p-mappings are the obvious exten-sions. The semantics of p-mappings is defined as a naturalextension of that of ordinary ones, which we review now. Amapping defines a relationship between instances of S andinstances of T that are consistent with the mapping.
Definition 3.4 (Consistent Target Instance). LetM = (S, T,m) be a relation mapping and DS be an instanceof S.
An instance DT of T is said to be consistent with DS andM , if DS and DT satisfy m. 2
For a relation mapping M and a source instanceDS , therecan be an infinite number of target instances that are con-sistent with DS and M . We denote by TarM (DS) the set of
all such target instances. The set of answers to a query Q isthe intersection of the answers on all instances in TarM (DS).The following definition is from [1].
Definition 3.5 (Certain Answer). Let M = (S, T,m)be a relation mapping. Let Q be a query over T and let DS
be an instance of S.A tuple t is said to be a certain answer of Q with respect
to DS and M , if for every instance DT ∈ TarM (DS), t ∈Q(DT ). 2
By-table semantics: We now generalize these notions tothe probabilistic setting, beginning with the by-table seman-tics.
Definition 3.6 (By-table Consistent Instance). LetpM = (S, T,m) be a p-mapping and DS be an instance ofS.
An instance DT of T is said to be by-table consistent withDS and pM , if there exists a mapping m ∈ m such that DS
and DT satisfy m. 2
Given a source instance DS and a possible mapping m ∈m, there can be an infinite number of target instances thatare consistent with DS and m. We denote by Tarm(DS) theset of all such instances.
In the probabilistic context, we assign a probability toevery answer. Intuitively, we consider the certain answerswith respect to each possible mapping in isolation. Theprobability of an answer t is the sum of the probabilities ofthe mappings for which t is deemed as a certain answer. Wedefine by-table answers as follows:
Definition 3.7 (By-table Answer). Let pM = (S, T,m) be a p-mapping. Let Q be a query over T and let DS bean instance of S.
Let t be a tuple. Let m̄(t) be the subset of m, such that foreach m ∈ m̄(t) and for each DT ∈ Tarm(DS), t ∈ Q(DT ).
Let p =P
m∈m̄(t) Pr(m). If p > 0, then we say (t, p) is a
by-table answer of Q with respect to DS and pM . 2
By-tuple semantics: The key difference in the definitionof by-tuple semantics is that a consistent target instance isdefined by a mapping sequence that assigns a (possibly dif-ferent) mapping in m to each tuple in DS . (Without losinggenerality, in order to compare between such sequences, weassign some order to the tuples in the instance).
Definition 3.8 (By-tuple Consistent Instance). LetpM = (S, T,m) be a p-mapping and let DS be an instanceof S with d tuples.
An instance DT of T is said to be by-tuple consistent withDS and pM , if there is a sequence 〈m1, . . . ,md〉 such thatfor every 1 ≤ i ≤ d,
• mi ∈ m, and
• for the ith tuple of DS , ti, there exists a target tuplet′i ∈ DT such that ti and t′i satisfy mi. 2
Given a mapping sequence seq = 〈m1, . . . ,md〉, we denoteby Tarseq(DS) the set of all target instances that are consis-tent with DS and seq . Note that if DT is by-table consistentwith DS and m, then DT is also by-tuple consistent with DS
and a mapping sequence in which each mapping is m.
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We can think of every sequence of mappings seq = 〈m1, . . . ,md〉 as a separate event whose probability is Pr(seq) =Πd
i=1Pr(mi). (In Section 7 we relax this independence as-sumption and introduce conditional mappings.) If there arel mappings in pM , then there are ld sequences of length d,and their probabilities add up to 1. We denote by seqd(pM)the set of mapping sequences of length d generated from pM .
Definition 3.9 (By-tuple Answer). Let pM = (S, T,m) be a p-mapping. Let Q be a query over T and DS be aninstance of S with d tuples.
Let t be a tuple. Let seq(t) be the subset of seqd(pM), suchthat for each seq ∈ seq(t) and for each DT ∈ Tar seq(DS),t ∈ Q(DT ).
Let p =P
seq∈seq(t) Pr(seq). If p > 0, we call (t, p) a
by-tuple answer of Q with respect to DS and pM . 2
The set of by-table (resp. by-tuple) answers for Q withrespect to DS is denoted by Qtable(DS) (resp. Qtuple(DS)).
Example 3.10. Consider the p-mapping pM , the sourceinstance DS , and the query Q in the motivating example.
In by-table semantics, Figure 3(b) shows a target instancethat is consistent with DS (repeated in Figure 3(a)) and pos-sible mapping m1. Figure 3(d) shows the by-table answersof Q with respect to DS and pM . As an example, for tuplet =(‘Sunnyvale’), we have m̄(t) = {m1,m2}, so the possibletuple (‘Sunnyvale’, 0.9) is an answer.
In by-tuple semantics, Figure 3(c) shows a target instancethat is by-tuple consistent with DS and the mapping sequence< m2,m3 >. Figure 3(e) shows the by-tuple answers of Qwith respect to DS and pM . 2
4. QUERY ANSWERING COMPLEXITYThis section considers query answering in the presence of
probabilistic mappings. We describe algorithms for queryanswering and study the complexity of query answering interms of the size of the data (data complexity) and the size ofthe mapping (mapping complexity). We also consider casesin which we are not interested in the actual probability ofan answer, just whether or not a tuple is a possible answer.
We show that returning all by-table answers is in PTIMEfor both complexity measures; whereas returning all by-tuple answers in general is #P-complete in data complex-ity. We show that computing the probabilities is the culprithere: even deciding the probability of a single answer tupleunder by-tuple semantics is already #P-complete, whereascomputing all by-tuple answers without returning the prob-abilities is in PTIME. Finally, we identify a large subclass ofcommon queries where returning all by-tuple answers withtheir probabilities is still in PTIME.
4.1 By-table Query AnsweringIn the case of by-table semantics, answering queries is
conceptually simple. Given a p-mapping pM = (S, T,m)and an SPJ query Q, we can compute the certain answersof Q under each of the mappings m ∈ m. We attach theprobability Pr(m) to every certain answer under m. If atuple is an answer to Q under multiple mappings in m, thenwe add up the probabilities of the different mappings.
Algorithm ByTable takes as input an SPJ query Q thatmentions the relations T1, . . . , Tl in the FROM clause. Assumethat we have the p-mapping pMi associated with the tableTi. The algorithm proceeds as follows.
Tuple Prob(’Sunnyvale’) 0.94
(’Mountain View’) 0.5(’alice@’) 0.1(’bob@’) 0.1
(a)
Tuple Prob(’Sunnyvale’) 0.8
(’Mountain View’) 0.8(b)
Figure 4: (a) Qtuple1 (D) and (b) Q
tuple2 (D).
Step 1: Generate the possible reformulations of Q by con-sidering every combination of the form (m1, . . . ,ml), wheremi is one of the possible mappings in pMi. Denote the setof reformulations by Q′
1, . . . , Q′
k. The probability of a refor-mulation Q′ = (m1, . . . ,ml) is Πl
i=1Pr(mi).
Step 2: For each reformulation Q′, retrieve each of theunique answers from the sources. For each answer obtainedby Q′
1 ∪ . . . ∪ Q′
k, its probability is computed by summingthe probabilities of the Q′’s in which it is returned.
Importantly, note that it is possible to express both stepsas a SQL query with grouping and aggregation. Therefore,if the underlying sources support SQL, we can leverage theiroptimizations to compute the answers.
With our restricted form of schema mapping, the algo-rithm takes time polynomial in the size of the data and themappings. We thus have the following complexity result(full proofs are given in [8]).
Theorem 4.1. Let pM be a schema p-mapping and let Qbe an SPJ query.
Answering Q with respect to pM is in PTIME in the sizeof the data and the mapping. 2
GLAV mappings: It is rather straightforward to extendthe above results to arbitrary GLAV mappings. We de-fine general p-mappings to be triples of the form pGM =(S̄, T̄ ,gm), where gm is a set {(gmi, P r(gmi)) | i ∈ [1, n]},such that for each i ∈ [1, n], gmi is a general GLAV mapping.The definition of by-table semantics for such mappings is asimple generalization of Definition 3.7. The following resultholds for general p-mappings.
Theorem 4.2. Let pGM be a general p-mapping betweena source schema S̄ and a target schema T̄ . Let DS be aninstance of S̄. Let Q be an SPJ query with only equalityconditions over T̄ . The problem of computing Qtable(DS)with respect to pGM is in PTIME in the size of the dataand the mapping. 2
4.2 By-tuple Query AnsweringTo extend the by-table query answering strategy to by-
tuple semantics, we would need to compute the certain an-swers for every mapping sequence generated by pM (see Defi-nition 3.8). However, the number of such mapping sequencesis exponential in the size of the input data. The followingexample shows that for certain queries this exponential timecomplexity is not avoidable.
Example 4.3. Suppose that in addition to the tables inExample 2.1, we also have U(city) in the source and V(hightech)in the target. The p-mapping for V contains two possiblemappings: ({(city, hightech)}, .8) and (∅, .2).
Consider the following query Q, which decides if there areany people living in a high-tech city.
Q: SELECT ‘true’FROM T, VWHERE T.mailing-addr = V.hightech
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One may conjecture we can answer the query by first exe-cuting the following two sub-queries Q1 and Q2, then joiningthe answers of Q1 and Q2 and summing up the probabilities.
Q1: SELECT mailing-addr FROM TQ2: SELECT hightech FROM V
Now consider the source instance D, where DS is shownin Figure 2(a), and DU has two tuples (‘Mountain View’)
and (‘Sunnyvale’). Figure 4(a) and (b) show Qtuple1 (D) and
Qtuple2 (D). If we join the results of Q1 and Q2, we obtain for
the true tuple the following probability: 0.94∗0.8+0.5∗0.8 =1.152. However, this is incorrect. By enumerating all con-sistent target tables, we in fact compute 0.864 as the proba-bility. The reason for this error is that generating the tuple(‘Sunnyvale’) as an answer for both Q1 and Q2 and gener-ating the tuple (‘Mountain View’) for both queries are notindependent events; thus, simply adding up their probabili-ties leads to incorrect results. 2
In fact, we show that in general, answering SPJ querieswith by-tuple semantics with respect to schema p-mappingsis hard.
Theorem 4.4. Let Q be an SPJ query and let pM be aschema p-mapping. The problem of finding the probabil-ity for a by-tuple answer to Q with respect to pM is #P-complete with respect to data complexity and is in PTIMEwith respect to mapping complexity. 2
Recall that #P is the complexity class of some hard count-ing problems (e.g., counting the number of variable assign-ments that satisfy a boolean formula). It is believed thatwe cannot solve a #P-complete problem in polynomial time,unless P = NP . The lower bound in Theorem 4.4 is provedby reducing the problem of counting the number of vari-able assignments that satisfy a bipartite monotone 2DNFboolean formula to the problem of finding the answers to Q.
In fact, the reason for the high complexity is exactly thatwe are asking for the probability of the answer. The follow-ing theorem shows that if we only want to know the possibleby-tuple answers, we can do so in polynomial time.
Theorem 4.5. Given an SPJ query and a schema p-mapping, returning all by-tuple answers without probabilities isin PTIME with respect to data complexity. 2
GLAV mappings: Extending by-tuple semantics to ar-bitrary GLAV mappings is a bit trickier than by-table se-mantics. It would involve considering Cartesian productsof mapping sequences, but the length of these products de-pends on the number of tables in the query. We leave thisextension to future work.
4.3 Two Restricted CasesIn this section we identify two restricted but common
classes of queries for which by-tuple query answering takespolynomial time. We conjecture that these are the onlycases where it is possible to answer a query in polynomialtime.
In our discussion we refer to subgoals of a query. Thesubgoals are tables that occur in the FROM clause of a query.Hence, even if the same table occurs twice in the FROM clause,each occurrence is a different subgoal.
4.3.1 Queries with a single p-mapping subgoalThe first class of queries we consider are those that include
only a single subgoal being the target of a p-mapping. Re-lations in the other subgoals are either involved in ordinarymappings or do not require a mapping. Hence, if we onlyhave uncertainty with respect to one part of the domain, ourqueries will typically fall in this class. We call such queriesnon-p-join queries. The query Q in the motivating exampleis an example non-p-join query.
Definition 4.6 (non-p-join queries). Let pM be aschema p-mapping and let Q be an SPJ query.
If at most one subgoal in the body of Q is the target of ap-mapping in pM, then we say Q is a non-p-join query withrespect to pM . 2
For a non-p-join query Q, the by-tuple answers of Q canbe generated from the by-table answers of Q over a set ofdatabases, each containing a single tuple in the source ta-ble. Specifically, let pM = (S, T,m) be the single p-mappingwhose target is a relation in Q, and let DS be an instanceof S with d tuples. Consider the set of tuple databasesT(DS) = {D1, . . . ,Dd}, where for each i ∈ [1, d], Di is aninstance of S and contains only the i-th tuple in DS . Thefollowing lemma shows that Qtuple(DS) can be derived fromQtable(D1), . . . , Q
table(Dd).
Lemma 4.7. Let pM be a schema p-mapping between S̄and T̄ . Let Q be a non-p-join query over T̄ and let DS bean instance of S̄. Let (t, P r(t)) be a by-tuple answer withrespect to DS and pM. Let T̄ (t) be the subset of T(DS) suchthat for each D ∈ T̄ (t), t ∈ Qtable(D). The following twoconditions hold:
1. T̄ (t) 6= ∅;
2. Pr(t) = 1 − ΠD∈T̄(t),(t,p)∈Qtable(D)(1 − p). 2
In practice, answering the query for each tuple databasecan be expensive. We next describe Algorithm NonPJoin,which computes the answers for all tuple databases in onestep. The key idea of the algorithm is to distinguish an-swers generated by different source tuples. To do this, weassume there is an identifier attribute id for the source re-lation whose values are concatenations of values of the keycolumns. We now describe the algorithm in detail.
Algorithm NonPJoin takes as input an SPJ query Q,a schema p-mapping pM , and a source instance DS , andproceeds in three steps to compute all by-tuple answers.
Step 1: Rewrite Q to Q′ such that it returns T .id in addi-tion. Revise the p-mapping such that each possible mappingcontains the correspondence between S.id and T .id.
Step 2: Invoke ByTable with Q′, pM and DS . Note thateach generated result tuple contains the id column in addi-tion to the attributes returned by Q.
Step 3: Project the answers returned in Step 2 on Q’sreturned attributes. Suppose projecting t1, . . . , tn obtainsthe answer tuple t, then the probability of t is 1−Πn
i=1(1−Pr(ti)).
Example 4.8. Consider rewriting Q in the motivatingexample, repeated as follows:
Q: SELECT mailing-addr FROM T
Step 1 rewrites Q into query Q′ by adding the id column:
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Q’: SELECT id, mailing-addr FROM T
In Step 2, ByTable generates the following SQL queryto compute by-table answers for Q′:
Qa: SELECT id, mailing-addr, SUM(pr)FROM (
SELECT DISTINCT id, current-addr AS mailing-addr,0.5 AS pr
FROM SUNION ALLSELECT DISTINCT id, permanant-addr AS mailing-addr,
0.4 AS prFROM SUNION ALLSELECT DISTINCT id, email-addr AS mailing-addr,
0.1 AS prFROM S)
GROUP BY id, mailing-addr
Step 3 then generates the results using the following query.
Qu: SELECT mailing-addr, NOR(pr) AS prFROM QaGROUP BY mailing-addr
where for a set of probabilities pr1, . . . , prn, NOR computes1 − Πn
i=1pri. 2
An analysis of Algorithm NonPJoin leads to the followingcomplexity result for non-p-join queries.
Theorem 4.9. Let pM be a schema p-mapping and let Qbe a non-p-join query with respect to pM .
Answering Q with respect to pM in by-tuple semantics isin PTIME in the size of the data and the mapping. 2
4.3.2 Projected p-join queriesWe now show that query answering can be done in poly-
nomial time for a class of queries, called projected p-joinqueries, that include multiple subgoals involved in p-mappings.In such a query, we say that a join predicate is a p-join pred-icate with respect to a schema p-mapping pM , if at least oneof the involved relations is the target of a p-mapping in pM .We define projected p-join queries as follows.
Definition 4.10 (projected p-join query). Let pMbe a schema p-mapping and Q be an SPJ query over the tar-get of pM . If the following conditions hold, we say Q is aprojected p-join query with respect to pM :
• at least two subgoals in the body of Q are targets ofp-mappings in pM .
• for every p-join predicate, the join attribute (or an at-tribute that is entailed to be equal by the predicates inQ) is returned in the SELECT clause. 2
Example 4.11. Consider the schema p-mapping in Ex-ample 4.3. A slight revision of Q, shown as follows, is aprojected-p-join query.
Q’: SELECT V.hightechFROM T, VWHERE T.mailing-addr = V.hightech
2
Note that in practice, when joining data from multipletables in a data integration scenario, we typically project thejoin attributes, thereby leading to projected p-join queries.
The key to answering a projected-p-join query Q is todivide Q into multiple subqueries, each of which is a non-p-join query, and compute the answer toQ from the answers to
the subqueries. We proceed by considering partitions of thesubgoals in Q. We say that a partitioning J̄ is a refinementof a partitioning J̄ ′, denoted J̄ � J̄ ′, if for each partitionJ ∈ J̄ , there is a partition J ′ ∈ J̄ ′, such that J ⊆ J ′. Weconsider the following partitioning of Q, the generation ofwhich will be described in detail in the algorithm.
Definition 4.12 (Maximal P-Join Partitioning). LetpM be a schema p-mapping. Let Q be an SPJ query and J̄be a partitioning of the subgoals in Q.
We say that J̄ is a p-join partitioning of Q, if (1) eachpartition J ∈ J̄ contains at most one subgoal that is thetarget of a p-mapping in pM, and (2) if neither subgoal ina join predicate is involved in p-mappings in pM , the twosubgoals belong to the same partition.
We say that J̄ is a maximal p-join partitioning of Q, ifthere does not exist a p-join partitioning J̄ ′, such that J̄ �J̄ ′. 2
For each partition J ∈ J̄ , we can define a query QJ asfollows. The FROM clause includes the subgoals in J . TheSELECT clause includes J ’s attributes that occur in (1) Q’sSELECT clause or (2) Q’s join predicates that join subgoalsin J with subgoals in other partitions. The WHERE clause in-cludes Q’s predicates that contain only subgoals in J . WhenJ is a partition in a maximal p-join partitioning of Q, wesay that QJ is a p-join component of Q.
The following is the main lemma underlying our algo-rithm. It shows that we can compute the answers of Qfrom the answers to its p-join components.
Lemma 4.13. Let pM be a schema p-mapping. Let Q bea projected p-join query with respect to pM and let J̄ be amaximal p-join partitioning of Q. Let QJ1, . . . , QJn be thep-join components of Q with respect to J̄.
For any instance DS of the source schema of pM andresult tuple t ∈ Qtuple(DS), the following two conditionshold:
1. For each i ∈ [1, n], there exists a single tuple ti ∈
QtupleJi (DS), such that t1, . . . , tn generate t when joined
together.
2. Let t1, . . . , tn be the above tuples. Then, Pr(t) =Πn
i=1Pr(ti). 2
Lemma 4.13 entails to the query-rewriting algorithm Pro-jectedPJoin, which takes as input a projected-p-join queryQ, a schema p-mapping pM , and a source instance DS , out-puts all by-tuple answers:
Step 1: Generate maximum p-join partitions J1, . . . , Jn asfollows. First, initialize each partition to contain one sub-goal in Q. Then, for each join predicate with subgoals S1
and S2 that are not involved in p-mappings in pM , mergethe partitions that S1 and S2 belong to. Finally, for eachpartition that contains no subgoal involved in pM , merge itwith another partition.
Step 2: For each p-join partition Ji, i ∈ [1, n], generatethe p-join component QJi, and invoke Algorithm NonPJoinwith QJi, pM and DS to compute answers for QJi.
Step 3: Join the results of QJ1, . . . , QJn. If an answer tuplet is obtained by joining t1, . . . , tn, then the probability of tis computed by Πn
i=1Pr(ti).We illustrate the algorithm using the following example.
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Example 4.14. Continue with Example 4.11. Its two p-join components are Q1 and Q2 shown in Example 4.3. Sup-pose we compute Q1 with query Qu (shown in Example 4.8)and compute Q2 with query Q′
u. We can compute by-tupleanswers of Q′ as follows:
SELECT Qu’.hightech, Qu.pr*Qu’.prFROM Qu, Qu’WHERE Qu.mailing-addr = Qu’.hightect
2
Since the number of p-join components is bounded by thenumber of subgoals in a query, and for each of them weinvoke Algorithm NonPJoin, query answering for projectedp-join queries takes polynomial time.
Theorem 4.15. Let pM be a schema p-mapping and letQ be a projected-p-join query with respect to pM.
Answering Q with respect to pM in by-tuple semantics isin PTIME in the size of the data and the mapping. 2
4.3.3 Other SPJ queriesA natural question is whether the two classes of queries we
have identified are the only ones for which query answeringis in PTIME for by-tuple semantics. As Example 4.3 shows,if Q contains multiple subgoals that are involved in a schemap-mapping, but Q is not a projected-p-join query, then Con-dition 1 in Lemma 4.13 does not hold, and query answeringneeds to proceed by enumerating all mapping sequences.
We believe that the complexity of the border case, wherea query joins two relations involved in p-mappings but doesnot return the join attribute, is #P-hard, but currently itremains an open problem.
5. TOP-K QUERY ANSWERINGIn this section, we consider returning the top-k query an-
swers, which are the k answer tuples with the top probabil-ities. The main challenge in designing the algorithm is toonly perform the necessary reformulations at every step andhalt when the top-k answers are found. We first describeour algorithm for by-table semantics. We then show thechallenges for by-tuple semantics and outline our solution,but defer the details to the full version of the paper.
5.1 Returning Top-K By-table AnswersRecall that in by-table query answering, the probability of
an answer is the sum of the probabilities of the reformulatedqueries that generate the answer. Our goal is to reduce thenumber of reformulated queries we execute. Our algorithmproceeds in a greedy fashion: we execute queries in descend-ing order of probabilities. For each tuple t, we maintain theupper bound pmax(t) and lower bound pmin(t) of its proba-bility. This process halts when we find k tuples whose pmin
values are higher than pmax of the rest of the tuples.TopKByTable takes as input an SPJ query Q, a schema
p-mapping pM , an instance DS of the source schema, andan integer k, and outputs the top-k answers in Qtable(DS).The algorithm proceeds in three steps.
Step 1: Rewrite Q according to pM into a set of queriesQ1, . . . , Qn, each with a probability assigned in a similarway as stated in Algorithm ByTable.
Step 2: Execute Q1, . . . , Qn in descending order of theirprobabilities. Maintain the following measures:
• The highest probability, PMax, for the tuples thathave not been generated yet. We initialize PMax to1; after executing query Qi and updating the list of an-swers (see third bullet), we decrease PMax by Pr(Qi);
• The threshold th determining which answers are po-tentially in the top-k. We initialize th to 0; after ex-ecuting Qi and updating the answer list, we set th tothe k-th largest pmin for tuples in the answer list;
• A list L of answers whose pmax is no less than th, andbounds pmin and pmax for each answer in L. Afterexecuting query Qi, we update the list as follows: (1)for each t ∈ L and t ∈ Qi(DS), we increase pmin(t)by Pr(Qi); (2) for each t ∈ L but t 6∈ Qi(DS), wedecrease pmax(t) by Pr(Qi); (3) if PMax ≥ th, foreach t 6∈ L but t ∈ Qi(DS), insert t to L, set pmin toPr(Qi) and pmax(t) to PMax.
• A list T of k tuples with top pmin values.
Step 3: When th > PMax and for each t 6∈ T , th >pmax(t), halt and return T .
Example 5.1. Consider Example 2.1 where we seek thetop-1 answer. We answer the reformulated queries in orderof Q1, Q2, Q3. After answering Q1, for tuple (“Sunnyvale”)we have pmin = .5 and pmax = 1, and for tuple (“MountainView”) we have the same bounds. In addition, PMax = .5and th = .5.
In the second round, we answer Q2. Then, for tuple (“Sun-nyvale”) we have pmin = .9 and pmax = 1, and for tuple(“Mountain View”) we have pmin = .5 and pmax = .6. NowPMax = .1 and th = .9.
Because th > PMax and th is above the pmax for the(“Mountain View”) tuple, we can halt and return (“Sunny-vale”) as the top-1 answer. 2
The next theorem states the correctness of ByTable-TopK.
Theorem 5.2. For any schema mapping pM , SPJ queryQ, instance DS of the source schema of pM , and integerk, Algorithm ByTableTopK correctly computes the top-kanswers in Qtable(DS). 2
Note that in the worst case we need to execute all ofQ1, . . . , Qn in Step 2, so the time complexity of TopKByTableis the same as ByTable. However, in many cases we canreturn top-k answers without executing all of the reformu-lated queries, and thus the greedy strategy can improve theefficiency of query answering.
Our algorithm differs from previous top-k algorithms inthe literature in two aspects. First, we execute the reformu-lated queries only when necessary, so we can return the top-kanswers without executing all reformulated queries therebyleading to significant performance improvements. Fagin etal. [9] have proposed several algorithms for finding instanceswith top-k scores, where each instance has m attributes andthe score of the instance is an aggregation over values ofthese m attributes. However, these algorithms assume thatfor each attribute there exists a sorted list on its values, andthey access the lists in parallel. In our context, this wouldrequire executing all reformulated queries upfront. Li etal. [20] have studied computing top-k answers for aggrega-tion and group-by queries and optimizing query answeringby generating the groups incrementally. Although we can
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also compute by-table answers using an aggregation query,this query is different from those considered in [20] in thatthe WHERE clause contains a set of sub-queries rather thandatabase tables. Therefore, applying [20] here also requiresevaluating all reformulated queries at the beginning.
Second, whereas maintaining upper bounds and lowerbounds for instances has been explored in the literature,such as in Fagin’s NRA (Non-Random Access) algorithmand in [20], our algorithm is different in that it keeps thesebounds only for tuples that have already been generatedby an executed reformulated query and that are potentialtop-k answers (by judging if the upper bound is above thethreshold th). For unseen result tuples, we compute theupper bound of their probability by exploiting the fact thatthe upper bound of each probability score is 1.
5.2 By-tuple Top-K Query AnsweringWe next consider returning top-k answers in by-tuple se-
mantics. In general, we need to consider each mapping se-quence and answer the query on the target instance that isconsistent with the source and the mapping sequence. Al-gorithm TopKByTable can be modified to compute top-kby-tuple answers by deciding at runtime the mapping se-quence to consider next. However, for non-p-join queriesand projected-p-join queries, we can return top-k answersmore efficiently. We outline our method for answering non-p-join queries here, and leave the details of projected-p-joinqueries to the full paper.
For non-p-join queries the probability of an answer tuple tto query Q cannot be expressed as a function of t’s probabil-ities in executing reformulations of Q; rather, it is a functionof t’s probabilities in answering Q on each tuple databaseof the source table. However, retrieving answers on a tuplebase is expensive. Algorithm NonPJoin provides a methodthat computes by-tuple answers on the tuple databases inbatch mode by first rewriting Q into Q′ by returning the idcolumn and then executing Q′’s reformulated queries. Wefind top-k answers in a similar fashion. Here, after execut-ing each reformulated query, we need to maintain two answerlists, one for Q and one for Q′, and compute pmin and pmax
for answers in different lists differently.
6. P-MAPPING REPRESENTATIONSThus far, a p-mapping was represented by listing each
of its possible mappings, and the complexity of query an-swering was polynomial in the size of that representation.Such a representation can be quite lengthy since it essen-tially enumerates a probability distribution by listing everycombination of events in the probability space. Hence, aninteresting question is whether there are more concise rep-resentations of p-mappings and whether our algorithms canleverage them.
We consider three representations that can reduce the sizeof the p-mapping exponentially. In Section 6.1 we considera representation in which the attributes of the source andtarget tables are partitioned into groups and p-mappingsare specified for each group separately. We show that queryanswering can be done in time polynomial in the size of therepresentation. In Section 6.2 we consider probabilistic cor-respondences, where we specify the marginal probability ofeach attribute correspondence. However, we show that sucha representation can only be leveraged in limited cases. Fi-nally, we consider Bayes Nets, the most common method
Mapping Prob{(a,a’), (b,b’), (c,c’)} 0.72
{(a,b’), (c,c’)} 0.18{(a,a’), (b,b’)} 0.08
{(a,b’)} 0.02(a)
Mapping Prob{(a,a’), (b,b’)} 0.8
{(a,b’)} 0.2
(b)
Mapping Prob{(c,c’)} 0.9
∅ 0.1
(c)
Figure 5: The p-mapping in (a) is equivalent to the2-group p-mapping in (b) and (c).
for concisely representing probability distributions, in Sec-tion 6.3, and show that even though some p-mappings canbe represented with them, query answering does not neces-sarily benefit from the representation.
6.1 Group Probabilistic MappingIn practice, the uncertainty we have about a p-mapping
can often be represented as a few localized choices, espe-cially when schema mappings are created by semi-automaticmethods. To represent such p-mappings more concisely, wecan partition the source and target attributes and specifyp-mappings for each partition.
Definition 6.1 (Group P-Mapping). An n-group p-mapping gpM is a triple (S, T, pM), where
• S (resp. T ) is a source (resp. target) relation schemaand {S1, . . . , Sn} (resp. {T1, . . . , Tn}) is a set of dis-joint subsets of attributes in S (resp. T );
• pM is a set of p-mappings {pM1, . . . , pMn}, where foreach 1 ≤ i ≤ n, pMi is a p-mapping between Si andTi. 2
The semantics of an n-group p-mapping gpM = (S, T, pM)is a p-mapping that includes the Cartesian product of themappings in each of the pMi’s. The probability of the map-ping composed of m1 ∈ pM1, . . . ,mn ∈ pMn is Πn
i=1Pr(mi).
Example 6.2. Figure 5(a) shows p-mapping pM betweenthe schemas S(a, b, c) and T (a′, b′, c′). Figure 5(b) and (c)show two independent mappings that together form a 2-groupp-mapping equivalent to pM . 2
Note that a group p-mapping can be considerably morecompact than an equivalent p-mapping. Specifically, if eachpMi includes li mappings, then a group p-mapping can de-scribe Πn
i=1li possible mappings withPn
i=1 li sub-mappings.The important feature of n-group p-mappings is that queryanswering can be done in time polynomial in their size.
Theorem 6.3. Let gpM be a schema group p-mappingand let Q be an SPJ query. The mapping complexity ofanswering Q with respect to gpM in both by-table semanticsand by-tuple semantics is PTIME. 2
Theorem 6.4. Given a p-mapping pM , we can find inpolynomial time in the size of pM the maximal n and an n-group p-mapping gpM , such that gpM is equivalent to pM .2
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Mapping Prob{(a,a’), (b,b’), (c,c’)} 0.8
{(a,b’), (c,c’)} 0.1{(a,b’)} 0.1
(a)
Corr Prob{(a,a’)} 0.8{(a,b’)} 0.2{(b,b’)} 0.8{(c,c’)} 0.9
(b)
Figure 6: The p-mapping in (a) corresponds to thep-correspondence in (b).
6.2 Probabilistic CorrespondencesThe second representation we consider, probabilistic corre-
spondences, represents a p-mapping with the marginal prob-abilities of attribute correspondences. This representation isthe most compact one as its size is proportional to the prod-uct of the schema sizes of S and T .
Definition 6.5 (Probabilistic Correspondences).A probabilistic correspondence mapping (p-correspondence)is a triple pC = (S,T, c), where S = 〈s1, . . . , sm〉 is a sourcerelation schema, T = 〈t1, . . . , tn〉 is a target relation schema,and
• c is a set {(cij ,Pr(cij))|i ∈ [1, m], j ∈ [1, n]}, wherecij = (si, tj) is an attribute correspondence, and Pr(cij)∈ [0, 1];
• for each i ∈ [1, m],Pn
j=1 Pr(cij) ≤ 1;
• for each j ∈ [1, n],Pm
i=1 Pr(cij) ≤ 1. 2
Note that for a source attribute si, we allowPn
j=1 Pr(cij)< 1. This is because in some of the possible mappings, si
may not be mapped to any target attribute. The same istrue for target attributes.
¿From each p-mapping, we can infer a p-correspondenceby calculating the marginal probabilities of each attributecorrespondence. Specifically, for a p-mapping pM = (S, T,m),we denote by pC(pM) the p-correspondence where each marginal probability is computed as follows:
Pr(cij) =X
cij∈m,m∈m
Pr(m)
However, the relationship between p-mappings and p-corres-pondences is many-to-one. For example, The p-correspondencein Figure 6(b) is the one computed both for the p-mappingin Figure 6(a) and for the p-mapping in Figure 5(a).
Given the many-to-one relationship, the question is whenit is possible to compute the correct answer to a query basedonly on the p-correspondence. That is, we are looking fora class of queries Q̄, called p-mapping independent queries,such that for every Q ∈ Q̄ and every database instance DS ,if pC(pM1) = pC(pM2), then the answer of Q with respectto pM1 and DS is the same as the answer of Q with respectto pM2 and DS . Unfortunately, this property holds for avery restricted class of queries, defined as follows:
Definition 6.6 (Single-Attribute Query). Let pC= (S, T, c) be a p-correspondence. An SPJ query Q is saidto be a single-attribute query with respect to pC if T has onesingle attribute occurring in the SELECT and WHERE clausesof Q. 2
Theorem 6.7. Let pC be a schema p-correspondence, andQ be an SPJ query. Then, Q is p-mapping independent withrespect to pC if and only if for each pC ⊆ pC, Q is a single-attribute query with respect to pC. 2
6.3 Bayes NetsBayes Nets are a powerful mechanism for concisely repre-
senting probability distributions and reasoning about proba-bilistic events [22]. The following example shows how BayesNets can be used in our context.
Example 6.8. Consider two schemas S = (s1, . . . , sn,s′1, . . . , s
′
n) and T = (t1, . . . , tn). Consider the p-mappingpM = (S, T,m), which describes the following p-mapping:if s1 maps to t1 then it is more likely that {s2, . . . , sn} mapsto {t2, . . . , tn}, whereas if s′1 maps to t1 then it is more likelythat {s′2, . . . , s
′
n} maps to {t2, . . . , tn}.We can represent the p-mapping using a Bayes Net as
follows. Let c be an integer constant. Then,
1. Pr((s1, t1)) = Pr((s′1, t1)) = 1/2;
2. for each i ∈ [1, n], Pr((si, ti)|(s1, t1)) = 1 − 1c
and
Pr((s′i, ti)|(s1, t1)) = 1c;
3. for each i ∈ [1, n], Pr((si, ti)|(s′
1, t1)) = 1c
and Pr((s′i, ti)
|(s′1, t1)) = 1 − 1c.
Since the p-mapping contains 2n possible mappings, theoriginal representation would take space O(2n); however, theBayes-Net representation takes only space O(n). 2
Although the Bayes-Net representation can reduce the sizeexponentially for some p-mappings, this conciseness may nothelp reduce the complexity of query answering. We formal-ize this result in the following theorem.
Theorem 6.9. There exists a schema p-mapping pM anda query Q, such that answering Q with respect to pM in by-table semantics takes exponential time in the size of pM ’sBayes-Net representation. 2
7. BROADER CLASSES OF MAPPINGSIn this section we briefly show how our results can be
extended to capture two common practical extensions toour mapping language.
Complex mappings: Complex mappings map a set of at-tributes in the source to a set of attributes in the target.For example, we can map the attribute address to the con-catenation of street, city, and state.
Formally, a set correspondence between S and T is a re-lationship between a subset of attributes in S and a subsetof attributes in T . Here, the function associated with therelationship specifies a single value for each of the targetattributes given a value for each of the source attributes.Again, the actual functions are irrelevant to our discussion.A complex mapping is a triple (S, T, cm), where cm is a setof set correspondences, such that each attribute in S or T isinvolved in at most one set correspondence. A probabilisticcomplex mapping is of the form pCM = {(cmi, P r(cmi)) |i ∈ [1, n]}, where
Pn
i=1 Pr(cmi) = 1.
Theorem 7.1. Let pCM be a schema probabilistic com-plex mapping between schemas S̄ and T̄ . Let DS be an in-stance of S̄. Let Q be an SPJ query over T̄ . The datacomplexity and mapping complexity of computing Qtable(DS)w.r.t. pCM are PTIME. The data complexity of computingQtuple(DS) w.r.t. pCM is #P-complete. The mapping com-plexity of computing Qtuple(DS) w.r.t. pCM is PTIME. 2
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Conditional mappings: In practice, our uncertainty isoften conditioned. For example, we may want to state thatdaytime-phone maps to work-phone with probability 60% ifage ≤ 65, and maps to home-phone with probability 90% ifage > 65.
We define a conditional p-mapping as a set cpM = {(pM1,C1), . . . , (pMn, Cn)}, where pM1, . . . , pMn are p-mappings,and C1, . . . , Cn are pairwise disjoint conditions. Intuitively,for each i ∈ [1, n], pMi describes the probability distributionof possible mappings when condition Ci holds. Conditionalmappings make more sense for by-tuple semantics. The fol-lowing theorem shows that our results carry over to suchmappings.
Theorem 7.2. Let cpM be a schema conditional p-mappingbetween S̄ and T̄ . Let DS be an instance of S̄. Let Q be anSPJ query over T̄ . The problem of computing Qtuple(DS)with respect to cpM is in PTIME in the size of the mappingand #P-complete in the size of the data. 2
8. RELATED WORKWe are not aware of any previous work studying the se-
mantics and properties of probabilistic schema mappings.Gal [11] used the top-K schema mappings obtained by asemi-automatic mapper to improve the precision of the topmapping, but did not address any of the issues we consider.Florescu et al. [10] were the first to advocate the use ofprobabilities in data integration. Their work used proba-bilities to model (1) a mediated schema with overlappingclasses (e.g, DatabasePapers and AIPapers), (2) source de-scriptions stating the probability of a tuple being present ina source, and (3) overlap between data sources. Althoughthese are important aspects of many domains and should beincorporated into a data integration system, our focus hereis different. De Rougement and Vieilleribiere [6] consideredapproximate data exchange in that they relaxed the con-straints on the target schema, which is a different approachfrom ours.
There has been a flurry of activity around probabilisticand uncertain databases lately [4, 24, 5, 3]. Our intentionis that a data integration system will be based on a proba-bilistic data model, and we leverage concepts from that workas much as possible. We also believe that uncertainty andlineage are closely related, in the spirit of [4], and that re-lationship will play a key role in data integration. We leaveexploring this topic to future work.
9. CONCLUSIONS AND FUTURE WORKThis paper took the first step towards data integration
systems that handle uncertainty about queries, mappingsand data in a principled fashion. We introduced probabilis-tic semantic mappings, which play a key role in such a sys-tem. We presented query answering algorithms for by-tableand by-tuple semantics and for the case where the user isinterested in the top-k answers. We also considered conciseencoding of probabilistic mappings.
Many other problems need to be solved to realize the dataintegration systems we envision. Aside from the connectionbetween uncertainty and lineage in such systems, we alsowant to incorporate uncertainty about the results of key-word reformulation and about possibly dirty data. In ad-dition, we want to establish a formal connection betweenschema mapping tools and data integration systems that
support uncertainty. In one direction this means extract-ing probabilities about schema mappings from the resultsof schema mapping tools. In the reverse direction, our goalis to reason globally about the uncertainty in a data inte-gration system and its effect on query answers, in order todecide where it is most beneficial to expand more resources(human or otherwise) to improve schema mappings.
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