UNCERTAINTY IN DATA INTEGRATION

Chapter 1 UNCERTAINTY IN DATA INTEGRATION Anish Das Sarma Stanford University California, USA [email protected]

Xin Dong AT&T Labs-Research New Jersey, USA [email protected]

Alon Halevy Google Inc. California, USA [email protected]

Abstract

Data integration has been an important area of research for several years. In this chapter, we argue that supporting modern data integration applications requires systems to handle uncertainty at every step of integration. We provide a formal framework for data integration systems with uncertainty. We define probabilistic schema mappings and probabilistic mediated schemas, show how they can be constructed automatically for a set of data sources, and provide techniques for query answering. The foundations laid out in this chapter enable bootstrapping a pay-as-you-go integration system completely automatically.

Keywords:

data integration, uncertainty, pay-as-you-go, mediated schema, schema mapping

1.

Introduction

Data integration and exchange systems offer a uniform interface to a multitude of data sources and the ability to share data across multiple systems. These systems have recently enjoyed significant research and commercial success [18, 19]. Current data integration systems are essentially a natural extension of traditional database systems in that queries are specified in a structured form and data are modeled in one of the traditional data models (relational, XML). In addition, the data integration system has exact knowledge of how the data in the sources map to the schema used by the data integration system. In this chapter we argue that as the scope of data integration applications broadens, such systems need to be able to model uncertainty at their core. Uncertainty can arise for multiple reasons in data integration. First, the semantic mappings between the data sources and the mediated schema may be approximate. For example, in an application like Google Base [17] that enables anyone to upload structured data, or when mapping millions of sources on the deep web [28], we cannot imagine specifying exact mappings. In some domains

2 (e.g., bioinformatics), we do not necessarily know what the exact mapping is. Second, data are often extracted from unstructured sources using information extraction techniques. Since these techniques are approximate, the data obtained from the sources may be uncertain. Third, if the intended users of the application are not necessarily familiar with schemata, or if the domain of the system is too broad to offer form-based query interfaces (such as web forms), we need to support keyword queries. Hence, another source of uncertainty is the transformation between keyword queries and a set of candidate structured queries. Finally, if the scope of the domain is very broad, there can even be uncertainty about the concepts in the mediated schema. Another reason for data integration systems to model uncertainty is to support pay-as-you-go integration. Dataspace Support Platforms [20] envision data integration systems where sources are added with no effort and the system is constantly evolving in a pay-as-you-go fashion to improve the quality of semantic mappings and query answering. This means that as the system evolves, there will be uncertainty about the semanantic mappings to its sources, its mediated schema and even the semantics of the queries posed to it. This chapter describes some of the formal foundations for data integration with uncertainty. We define probabilistic schema mappings and probabilistic mediated schemas, and show how to answer queries in their presence. With these foundations, we show that it is possible to completely automatically bootstrap a pay-as-you-go integration system. This chapter is largely based on previous papers [10, 6]. The proofs of the theorems we state and the experimental results validating some of our claims can be found in there. We also place several other works on uncertainty in data integration in the context of the system we envision. In the next section, we begin by describing an architecture for data integration system that incorporates uncertainty.

2.

Overview of the System

This section describes the requirements from a data integration system that supports uncertainty and the overall architecture of the system.

2.1

Uncertainty in data integration

A data integration system needs to handle uncertainty at three levels. Uncertain mediated schema: The mediated schema is the set of schema terms in which queries are posed. They do not necessarily cover all the attributes appearing in any of the sources, but rather the aspects of the domain that the application builder wishes to expose to the users. Uncertainty in schema mappings can arise for several reasons. First, as we describe in Section 4, if the mediated schema is automatically inferred from the data sources in a pay-as-

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you-go integration system, there will be some uncertainty about the results. Second, when domains get broad, there will be some uncertainty about how to model the domain. For example, if we model all the topics in Computer Science there will be some uncertainty about the degree of overlap between different topics. Uncertain schema mappings: Data integration systems rely on schema mappings for specifying the semantic relationships between the data in the sources and the terms used in the mediated schema. However, schema mappings can be inaccurate. In many applications it is impossible to create and maintain precise mappings between data sources. This can be because the users are not skilled enough to provide precise mappings, such as in personal information management [11], because people do not understand the domain well and thus do not even know what correct mappings are, such as in bioinformatics, or because the scale of the data prevents generating and maintaining precise mappings, such as in integrating data of the web scale [27]. Hence, in practice, schema mappings are often generated by semi-automatic tools and not necessarily verified by domain experts. Uncertain data: By nature, data integration systems need to handle uncertain data. One reason for uncertainty is that data are often extracted from unstructured or semi-structured sources by automatic methods (e.g., HTML pages, emails, blogs). A second reason is that data may come from sources that are unreliable or not up to date. For example, in enterprise settings, it is common for informational data such as gender, racial, and income level to be dirty or missing, even when the transactional data is precise. Uncertain queries: In some data integration applications, especially on the web, queries will be posed as keywords rather than as structured queries against a well defined schema. The system needs to translate these queries into some structured form so they can be reformulated with respect to the data sources. At this step, the system may generate multiple candidate structured queries and have some uncertainty about which is the real intent of the user.

2.2

System architecture

Given the previously discussed requirements, we describe the architecture of a data integration system we envision that manages uncertainty at its core. We describe the system by contrasting it to a traditional data integration system. The first and most fundamental characteristic of this system is that it is based on a probabilistic data model. This means that we attach probabilities to: tuples that we process in the system, schema mappings,

4 Q Keyword Reformulation

Mediated Schema

Q 1,...Q m Query Reformulation

Q 11,...Q 1n,…,Qk1,...Q kn Query Pocessor

Q 11,...Q 1n

...

Q k1,...Q kn

Dk

D1 D4

D2 D3

Figure 1.1.

Architecture of a data-integration system that handles uncertainty.

mediated schemas, and possible interpretations of keyword queries posed to the system. In contrast, a traditional data integration system includes a single mediated schema and we assume we have as single (and correct) schema mapping between the mediated schema and each source. The data in the sources is also assumed to be correct. Traditional data integration systems assume that the query is posed in a structured fashion (i.e., can be translated to some subset of SQL). Here, we assume that queries can be posed as keywords (to accommodate a much broader class of users and applications). Hence, whereas traditional data integration systems begin by reformulating a query onto the schemas of the data sources, a data integration system with uncertainty needs to first reformulate a keyword query into a set of candidate structured queries. We refer to this step as keyword reformulation. Note that keyword reformulation is different from techniques for keyword search on structured data (e.g., [22, 1]) in that (a) it does not assume access to all the data in the sources or that the sources support keyword search, and (b) it tries to distinguish different structural elements in the query 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 name). That being said, keyword reformulation should benefit from techniques that support answering keyword search on structured data. The query answering model is different. Instead of necessarily finding all answers to a given query, our goal is typically to find the top-k answers, and rank these answers most effectively. The final difference from traditional data integration systems is that our query processing will need to be more adaptive than usual. Instead of generating a query answering plan and executing it, the steps we take in query

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processing will depend on results of previous steps. We note that adaptive query processing has been discussed quite a bit in data integration [12], where the need for adaptivity arises from the fact that data sources did not answer as quickly as expected or that we did not have accurate statistics about their contents to properly order our operations. In our work, however, the goal for adaptivity is to get the answers with high probabilities faster. The architecture of the system is shown in Figure 1.1. The system contains a number of data sources and a mediated schema (we omit probabilistic mediated schemas from this figure). When the user poses a query Q, which can be either a structured query on the mediated schema or a keyword query, the system returns a set of answer tuples, each with a probability. If Q is a keyword query, the system first performs keyword reformulation to translate it into a set of candidate structured queries on the mediated schema. Otherwise, the candidate query is Q itself.

2.3

Source of probabilities

A critical issue in any system that manages uncertainty is whether we have a reliable source of probabilities. Whereas obtaining reliable probabilities for such a system is one of the most interesting areas for future research, there is quite a bit to build on. For keyword reformulation, it is possible to train and test reformulators on large numbers of queries such that each reformulation result is given a probability based on its performance statistics. For information extraction, current techniques are often based on statistical machine learning methods and can be extended to compute probabilities of each extraction result. Finally, in the case of schema matching, it is standard practice for schema matchers to also associate numbers with the candidates they propose (e.g., [3, 7–9, 21, 26, 34, 35]). The issue here is that the numbers are meant only as a ranking mechanism rather than true probabilities. However, as schema matching techniques start looking at a larger number of schemas, one can imagine ascribing probabilities (or estimations thereof) to their measures.

2.4

Outline of the chapter

We begin by discussing probabilistic schema mappings in Section 3. We also discuss how to answer queries in their presence and how to answer top-k queries. In Section 4 we discuss probabilistic mediated schemas. We begin by motivating them and showing that in some cases they add expressive power to the resulting system. Then we describe an algorithm for generating probabilistic mediated schemas from a collection of data sources.

6

m1 = m2 = m3 =

Possible Mapping {(pname, name), (email-addr, email), (current-addr, mailing-addr), (permanent-addr, home-addr)} {(pname, name), (email-addr, email), (permanent-addr, mailing-addr), (current-addr, home-addr)} {(pname, name), (email-addr, mailing-addr), (current-addr, home-addr)}

Prob 0.5 0.4 0.1

(a) pname Alice Bob

email-addr alice@ bob@

current-addr Mountain View Sunnyvale

permanent-addr Sunnyvale Sunnyvale

(b) Tuple (mailing-addr) (’Sunnyvale’) (’Mountain View’) (’alice@’) (’bob@’)

Prob 0.9 0.5 0.1 0.1

(c) Figure 1.2. The running example: (a) a probabilistic schema mapping between S and T ; (b) a source instance DS ; (c) the answers of Q over DS with respect to the probabilistic mapping.

3.

Uncertainty in Mappings

The key to resolving heterogeneity at the schema level is to specify schema mappings between data sources. These mappings describe the relationship between the contents of the different sources and are used to reformulate a query posed over one source (or a mediated schema) into queries over the sources that are deemed relevant. However, in many applications we are not able to provide all the schema mappings upfront. In this section we introduce probabilistic schema mappings (p-mappings) to capture uncertainty on mappings between schemas. We start by presenting a running example for this section that also motivates p-mappings (Section 3.1). Then we present a formal definition of probabilistic schema mapping and its semantics (Section 3.2). Then, Section 3.3 describes algorithms for query answering with respect to probabilistic mappings and discusses the complexity. Next, Section 3.4 shows how to leverage previous work on schema matching to automatically create probabilistic mappings. In the end, Section 3.5 briefly describes various extensions to the basic definition and Section 3.6 describes other types of approximate schema mappings that have been proposed in the literature.

3.1

Motivating probabilistic mappings

Example 1.1 Consider a data source S, which describes a person by her email address, current address, and permanent address, and the mediated

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schema T , which describes a person by her name, email, mailing address, home address and 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 three possible mappings between S and T , assigning each a probability. Whereas the three mappings all map pname to name, they map other attributes in the source and the target differently. Figure 1.2(a) describes the three mappings using sets of attribute correspondences. For example, mapping m1 maps pname to name, email-addr to email, current-addr to mailing-addr, and permanent-addr to home-addr. Because of the uncertainty about which mapping is correct, we consider all of these mappings in query answering. Suppose the system receives a query Q composed on the mediated schema and asking for people’s mailing addresses: Q: SELECT mailing-addr FROM T Using the possible mappings, we can reformulate Q into different queries: Q1: SELECT current-addr FROM S Q2: SELECT permanent-addr FROM S Q3: SELECT email-addr FROM S If the user requires all possible answers, the system generates a single aggregation query based on Q1 , Q2 and Q3 to compute the probability of each returned tuple, and sends the query to the data source. Suppose the data source contains a table DS as shown in Figure 1.2(b), the system will retrieve four answer tuples, each with a probability, as shown in Figure 1.2(c). If the user requires only the top-1 answer (i.e., the answer tuple with the highest probability), the system decides at runtime which reformulated queries to execute. For example, after executing Q1 and Q2 at the source, the system can already conclude that (‘Sunnyvale’) is the top-1 answer and can skip query Q3 . 

3.2

Definition and Semantics

3.2.1 Schema mappings. We begin by reviewing non-probabilistic schema mappings. The goal of a schema mapping is to specify the semantic relationships between a source schema and a target schema. We refer to the ¯ and a relation in S¯ as S = hs1 , . . . , sm i. Similarly, we source schema as S, refer to the target schema as T¯, and a relation in T¯ as T = ht1 , . . . , tn i. We consider a limited form of schema mappings that are also referred to as schema matching in the literature. Specifically, a schema matching contains a set of attribute correspondences. An attribute correspondence is of the form

8 cij = (si , tj ), where si is a source attribute in the schema S and tj is a target attribute in the schema T . Intuitively, cij specifies that there is a relationship between si and tj . In practice, a correspondence also involves a function that transforms the value of si to the 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 therefore we omit them. This class of mappings are quite common in practice and already expose many of the novel issues involved in probabilistic mappings and In Section 3.5 we will briefly discuss extensions to a broader class of mappings. Formally, relation mappings and schema mappings are defined as follows.

Definition 1.2 (Schema Mapping) Let S¯ and T¯ be relational schemas. ¯ T is a A relation mapping M is a triple (S, T, m), where S is a relation in S, ¯ relation in T , and m is a set of attribute correspondences between S and T . When each source and target attribute occurs in at most one correspondence in m, we call M a one-to-one relation mapping. A schema mapping M is a set of one-to-one relation mappings between relations in S¯ and in T¯, where every relation in either S¯ or T¯ appears at most once.  A pair of instances DS and DT satisfies a relation mapping m if for every source tuple ts ∈ DS , there exists a target tuple tt ∈ Dt , such that for every attribute correspondence (s, t) ∈ m, the value of attribute s in ts is the same as the value of attribute t in tt .

Example 1.3 Consider the mappings in Example 1.1. The source database in Figure 1.2(b) (repeated in Figure 1.3(a)) and the target database in Figure 1.3(b) satisfy m1 .  3.2.2 Probabilistic schema mappings. Intuitively, a probabilistic schema mapping describes a probability distribution of a set of possible schema mappings between a source schema and a target schema. Definition 1.4 (Probabilistic Mapping) Let S¯ and T¯ be relational schemas. A probabilistic mapping (p-mapping), pM , is a triple (S, T, m), ¯ T ∈ T¯, and m is a set {(m1 , Pr (m1 )), . . . , (ml , Pr (ml ))}, where S ∈ S, such that for i ∈ [1, l], mi is a one-to-one mapping between S and T , and for every i, j ∈ [1, l], i 6= j ⇒ mi 6= mj . P Pr (mi ) ∈ [0, 1] and li=1 Pr (mi ) = 1.

9

Uncertainty in Data Integration pname Alice Bob

email-addr alice@ bob@

current-addr Mountain View Sunnyvale

permanent-addr Sunnyvale Sunnyvale

(a) name Alice Bob

email alice@ bob@

mailing-addr Mountain View Sunnyvale

home-addr Sunnyvale Sunnyvale

office-addr office office

(b) name Alice Bob

email alice@ email

mailing-addr Sunnyvale bob@

home-addr Mountain View Sunnyvale

office-addr office office

(c) Tuple (mailing-addr) (’Sunnyvale’) (’Mountain View’) (’alice@’) (’bob@’)

(d)

Prob 0.9 0.5 0.1 0.1

Tuple (mailing-addr) (’Sunnyvale’) (’Mountain View’) (’alice@’) (’bob@’)

Prob 0.94 0.5 0.1 0.1

(e)

Figure 1.3. Example 1.11: (a) a source instance DS ; (b) a target instance that is by-table consistent with DS and m1 ; (c) a target instance that is by-tuple consistent with DS and < m2 , m3 >; (d) Qtable (DS ); (e) Qtuple (DS ).

A schema p-mapping, pM , is a set of p-mappings between relations in S¯ and in T¯, where every relation in either S¯ or T¯ appears in at most one p-mapping.  We refer to a non-probabilistic mapping as an ordinary mapping. A schema p-mapping may contain both p-mappings and ordinary mappings. Example 1.1 shows a p-mapping (see Figure 1.2(a)) that contains three possible mappings.

3.2.3 Semantics of probabilistic mappings. Intuitively, a probabilistic schema mapping models the uncertainty about which of the mappings in pM is the correct one. When a schema matching system produces a set of candidate matches, there are two ways to interpret the uncertainty: (1) a single mapping in pM is the correct one and it applies to all the data in S, or (2) several mappings are partially correct and each is suitable for a subset of tuples in S, though it is not known which mapping is the right one for a specific tuple. Figure 1.3(b) illustrates the first interpretation and applies mapping m1 . For the same example, the second interpretation is equally valid: some people may choose to use their current address as mailing address while others use their permanent address as mailing address; thus, for different tuples we may apply different mappings, so the correct mapping depends on the particular tuple.

10 We define query answering under both interpretations. The first interpretation is referred to as the by-table semantics and the second one is referred to as the by-tuple semantics of probabilistic mappings. Note that one cannot argue for one interpretation over the other; the needs of the application should dictate the appropriate semantics. Furthermore, the complexity results for query answering, which will show advantages to by-table semantics, should not be taken as an argument in the favor of by-table semantics. We next define query answering with respect to p-mappings in detail and the definitions for schema p-mappings are the obvious extensions. Recall that given a query and an ordinary mapping, we can compute certain answers to the query with respect to the mapping. Query answering with respect to pmappings is defined as a natural extension of certain answers, which we next review. A mapping defines a relationship between instances of S and instances of T that are consistent with the mapping.

Definition 1.5 (Consistent Target Instance) Let M = (S, T, m) be a relation mapping and DS be an instance of S. An instance DT of T is said to be consistent with DS and M , if for each tuple ts ∈ DS , there exists a tuple tt ∈ DT , such that for every attribute correspondence (as , at ) ∈ m, the value of as in ts is the same as the value of at in tt .  For a relation mapping M and a source instance DS , there can be an infinite number of target instances that are consistent with DS and M . We denote by Tar M (DS ) the set of all such target instances. The set of answers to a query Q is the intersection of the answers on all instances in Tar M (DS ).

Definition 1.6 (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 ∈ Tar M (DS ), t ∈ Q(DT ).  By-table semantics: We now generalize these notions to the probabilistic setting, beginning with the by-table semantics. Intuitively, a p-mapping pM describes a set of possible worlds, each with a possible mapping m ∈ pM . In by-table semantics, a source table can fall in one of the possible worlds; that is, the possible mapping associated with that possible world applies to the whole source table. Following this intuition, we define target instances that are consistent with the source instance.

Definition 1.7 (By-table Consistent Instance) Let pM = (S, T, m) be a p-mapping and DS be an instance of S.

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An instance DT of T is said to be by-table consistent with DS and pM , if there exists a mapping m ∈ m such that DS and DT satisfy m.  Given a source instance DS and a possible mapping m ∈ m, there can be an infinite number of target instances that are consistent with DS and m. We denote by Tar m (DS ) the set of all such instances. In the probabilistic context, we assign a probability to every answer. Intuitively, we consider the certain answers with respect to each possible mapping in isolation. The probability of an answer t is the sum of the probabilities of the mappings for which t is deemed to be a certain answer. We define by-table answers as follows:

Definition 1.8 (By-table Answer) Let pM = (S, T, m) be a p-mapping. Let Q be a query over T and let DS be an instance of S. Let t be a tuple. Let m(t) ¯ be the subset of m, such that for each m ∈ m(t) ¯ and for each D ∈ Tar (D ), t ∈ Q(D ). T m S T P Let p = m∈m(t) Pr(m). If p > 0, then we say (t, p) is a by-table answer ¯ of Q with respect to DS and pM .  By-tuple semantics: If we follow the possible-world notions, in by-tuple semantics, different tuples in a source table can fall in different possible worlds; that is, different possible mappings associated with those possible worlds can apply to the different source tuples. Formally, the key difference in the definition of by-tuple semantics from that of by-table semantics is that a consistent target instance is defined by a mapping sequence that assigns a (possibly different) mapping in m to each source tuple in DS . (Without losing generality, in order to compare between such sequences, we assign some order to the tuples in the instance).

Definition 1.9 (By-tuple Consistent Instance) Let pM = (S, T, m) be a p-mapping and let DS be an instance of S with d tuples. An instance DT of T is said to be by-tuple consistent with DS and pM , if there is a sequence hm1 , . . . , md i such that d is the number of tuples in DS and for every 1 ≤ i ≤ d, mi ∈ m, and for the ith tuple of DS , ti , there exists a target tuple t′i ∈ DT such that for each attribute correspondence (as , at ) ∈ mi , the value of as in ti is the same as the value of at in t′i .  Given a mapping sequence seq = hm1 , . . . , md i, we denote by Tar seq (DS ) the set of all target instances that are consistent with DS and seq. Note that if DT is by-table consistent with DS and m, then DT is also by-tuple consistent with DS and a mapping sequence in which each mapping is m.

12 We can think of every sequence of mappings seq = hm1 , . . . , md i as a separate event whose probability is Pr (seq ) = Πdi=1 Pr (mi ). (Section 3.5 relaxes this independence assumption and introduces conditional mappings.) If there are l 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 1.10 (By-tuple Answer) Let pM = (S, T, m) be a p-mapping. Let Q be a query over T and DS be an instance of S with d tuples. Let t be a tuple. Let seq(t) be the subset of seqd (pM ), such that for each seq ∈ seq(t)Pand for each DT ∈ Tar seq (DS ), t ∈ Q(DT ). Let 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 .  The set of by-table answers for Q with respect to DS is denoted by Qtable (DS ) and the set of by-tuple answers for Q with respect to DS is denoted by Qtuple (DS ).

Example 1.11 Consider the p-mapping pM , the source instance DS , and the query Q in the motivating example. In by-table semantics, Figure 1.3(b) shows a target instance that is consistent with DS (repeated in Figure 1.3(a)) and possible mapping m1 . Figure 1.3(d) shows the by-table answers of Q with respect to DS and pM . As an example, for tuple t =(‘Sunnyvale’), we have m(t) ¯ = {m1 , m2 }, so the possible tuple (‘Sunnyvale’, 0.9) is an answer. In by-tuple semantics, Figure 1.3(c) shows a target instance that is by-tuple consistent with DS and the mapping sequence < m2 , m3 >. Figure 1.3(e) shows the by-tuple answers of Q with respect to DS and pM . Note that the probability of tuple t=(’Sunnyvale’) in the by-table answers is different from that in the by-tuple answers. We describe how to compute the probabilities in detail in the next section. 

3.3

Query Answering

This section studies query answering in the presence of probabilistic mappings. We start with describing algorithms for returning all answer tuples with probabilities, and discussing the complexity of query answering in terms of the size of the data (data complexity) and the size of the p-mapping (mapping complexity). We then consider returning the top-k query answers, which are the k answer tuples with the top probabilities.

3.3.1 By-table query answering. In 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 answers of Q under each of the mappings m ∈ m. We attach the probability P r(m) to every

13

Uncertainty in Data Integration Tuple (mailing-addr) (’Sunnyvale’) (’Mountain View’) (’alice@’) (’bob@’)

Pr 0.94 0.5 0.1 0.1

Tuple (mailing-addr) (’Sunnyvale’) (’Mountain View’)

Pr 0.8 0.8

(b)

(a) Figure 1.4.

Example 1.13: (a) Qtuple (D) and (b) Qtuple (D). 1 2

certain answer under m. If a tuple is an answer to Q under multiple mappings in m, then we add up the probabilities of the different mappings. Algorithm B Y TABLE takes as input an SPJ query Q that mentions the relations T1 , . . . , Tl in the FROM clause. Assume that we have the p-mapping pMi associated with the table Ti . The algorithm proceeds as follows. Step 1: We generate the possible reformulations of Q (a reformulation query computes all certain answers when executed on the source data) by considering every combination of the form (m1 , . . . , ml ), where mi is one of the possible mappings in pMi . Denote the set of reformulations by Q′1 , . . . , Q′k . The probability of a reformulation Q′ = (m1 , . . . , ml ) is Πli=1 P r(mi ). Step 2: For each reformulation Q′ , retrieve each of the unique answers from the sources. For each answer obtained by Q′1 ∪ . . . ∪ Q′k , its probability is computed by summing the probabilities of the Q′ ’s in which it is returned. Importantly, note that it is possible to express both steps as an SQL query with grouping and aggregation. Therefore, if the underlying sources support SQL, we can leverage their optimizations to compute the answers. With our restricted form of schema mapping, the algorithm takes time polynomial in the size of the data and the mappings. We thus have the following complexity result.

Theorem 1.12 Let pM be a schema p-mapping and let Q be an SPJ query. Answering Q with respect to pM in by-table semantics is in PTIME in the size of the data and the mapping.  3.3.2 By-tuple query answering. To extend the by-table query-answering strategy to by-tuple semantics, we would need to compute the certain answers for every mapping sequence generated by pM . However, the number of such mapping sequences is exponential in the size of the input data. The following example shows that for certain queries this exponential time complexity is inevitable. Example 1.13 Suppose that in addition to the tables in Example 1.1, we also have U(city) in the source and V(hightech) in the target. The p-mapping for V contains two possible mappings: ({(city, hightech)}, .8) and (∅, .2).

14 Consider the following query Q, which decides if there are any people living in a high-tech city. Q: SELECT ‘true’ FROM T, V WHERE T.mailing-addr = V.hightech An incorrect way of answering the query is to first execute the following two sub-queries Q1 and Q2 , then join the answers of Q1 and Q2 and summing up the probabilities. Q1: SELECT mailing-addr FROM T Q2: SELECT hightech FROM V Now consider the source instance D, where DS is shown in Figure 1.2(a), and DU has two tuples (‘Mountain View’) and (‘Sunnyvale’). Figure 1.4(a) and (b) show Qtuple (D) and Qtuple (D). If we join the results of Q1 and Q2 , 1 2 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 consistent target tables, we in fact compute 0.864 as the probability. The reason for this error is that on some target instance that is by-tuple consistent with the source instance, the answers to both Q1 and Q2 contain tuple (‘Sunnyvale’) and tuple (‘Mountain View’). Thus, generating the tuple (‘Sunnyvale’) as an answer for both Q1 and Q2 and generating the tuple (‘Mountain View’) for both queries are not independent events, and so simply adding up their probabilities leads to incorrect results. Indeed, it is not clear if there exists a better algorithm to answer Q than by enumerating all by-tuple consistent target instances and then answering Q on each of them.  In fact, it is proved that in general, answering SPJ queries in by-tuple semantics with respect to schema p-mappings is hard.

Theorem 1.14 Let Q be an SPJ query and let pM be a schema p-mapping. The problem of finding the probability for a by-tuple answer to Q with respect to pM is #P-complete with respect to data complexity and is in PTIME with respect to mapping complexity.  Recall that #P is the complexity class of some hard counting problems (e.g. , counting the number of variable assignments that satisfy a Boolean formula). It is believed that a #P-complete problem cannot be solved in polynomial time, unless P = N P . Although by-tuple query answering in general is hard, there are two restricted but common classes of queries for which by-tuple query answering takes polynomial time. The first class of queries are those that include only a

Uncertainty in Data Integration

15

single subgoal being the target of a p-mapping; here, we refer to an occurrence of a table in the FROM clause of a query as a subgoal of the query. Relations in the other subgoals are either involved in ordinary mappings or do not require a mapping. Hence, if we only have uncertainty with respect to one part of the domain, our queries will typically fall in this class. The second class of queries can include multiple subgoals involved in p-mappings, but return the join attributes for such subgoals. We next illustrate these two classes of queries and query answering for them using two examples.

Example 1.15 Consider rewriting Q in the motivating example, repeated as follows: Q: SELECT mailing-addr FROM T To answer the query, we first rewrite Q into query Q′ by adding the id column: Q’: SELECT id, mailing-addr FROM T We then invoke B Y TABLE and generate the following SQL query to 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 S UNION ALL SELECT DISTINCT id, permanent-addr AS mailing-addr, 0.4 AS pr FROM S UNION ALL SELECT DISTINCT id, email-addr AS mailing-addr, 0.1 AS pr FROM S) GROUP BY id, mailing-addr Finally, we generate the results using the following query. Qu:

SELECT mailing-addr, NOR(pr) AS pr FROM Qa GROUP BY mailing-addr

where for a set of probabilities pr1 , . . . , prn , N OR computes 1−Πni=1 (1−pri ). 

16 Example 1.16 Consider the schema p-mapping in Example 1.13. If we revise Q slightly be returning the join attribute, shown as follows, we can answer the query in polynomial time. Q’: SELECT V.hightech FROM T, V WHERE T.mailing-addr = V.hightech We answer the query by dividing it into two sub-queries, Q1 and Q2 , as shown in Example 1.13. We can compute Q1 with query Qu (shown in Example 1.15) and compute Q2 similarly with a query Q′u . We compute by-tuple answers of Q′ as follows: SELECT Qu’.hightech, Qu.pr*Qu’.pr FROM Qu, Qu’ WHERE Qu.mailing-addr = Qu’.hightect 

3.3.3 Top-K Query Answering. The main challenge in designing the algorithm for returning top-k query answers is to only perform the necessary reformulations at every step and halt when the top-k answers are found. We focus on top-k query answering for by-table semantics and the algorithm can be modified for by-tuple semantics. Recall that in by-table query answering, the probability of an answer is the sum of the probabilities of the reformulated queries that generate the answer. Our goal is to reduce the number of reformulated queries we execute. The algorithm we describe next proceeds in a greedy fashion: it executes queries in descending order of probabilities. For each tuple t, it maintains the upper bound pmax (t) and lower bound pmin (t) of its probability. This process halts when it finds k tuples whose pmin values are higher than pmax of the rest of the tuples. T OP KB Y TABLE takes as input an SPJ query Q, a schema p-mapping pM , an instance DS of the source schema, and an 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 queries Q1 , . . . , Qn , each with a probability assigned in a similar way as stated in Algorithm B Y TABLE. Step 2: Execute Q1 , . . . , Qn in descending order of their probabilities. Maintain the following measures: The highest probability, P M ax, for the tuples that have not been generated yet. We initialize P M ax to 1; after executing query Qi and updating the list of answers (see third bullet), we decrease P M ax by P r(Qi );

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The threshold th determining which answers are potentially in the top-k. We initialize th to 0; after executing Qi and updating the answer list, we set th to the k-th largest pmin for tuples in the answer list; A list L of answers whose pmax is no less than th, and bounds pmin and pmax for each answer in L. After executing query Qi , we update the list as follows: (1) for each t ∈ L and t ∈ Qi (DS ), we increase pmin (t) by P r(Qi ); (2) for each t ∈ L but t 6∈ Qi (DS ), we decrease pmax (t) by P r(Qi ); (3) if P M ax ≥ th, for each t 6∈ L but t ∈ Qi (DS ), insert t to L, set pmin to P r(Qi ) and pmax (t) to P M ax. A list T of k tuples with top pmin values. Step 3: When th > P M ax and for each t 6∈ T , th > pmax (t), halt and return T.

Example 1.17 Consider Example 1.1 where we seek for top-1 answer. We answer the reformulated queries in order of Q1 , Q2 , Q3 . After answering Q1 , for tuple (“Sunnyvale”) we have pmin = .5 and pmax = 1, and for tuple (“Mountain View”) we have the same bounds. In addition, P M ax = .5 and th = .5. In the second round, we answer Q2 . Then, for tuple (“Sunnyvale”) we have pmin = .9 and pmax = 1, and for tuple (“Mountain View”) we have pmin = .5 and pmax = .6. Now P M ax = .1 and th = .9. Because th > P M ax and th is above the pmax for the (“Mountain View”) tuple, we can halt and return (“Sunnyvale”) as the top-1 answer. 

3.4

Creating P-mappings

We now address the problem of generating a p-mapping between a source schema and a target schema. We begin by assuming we have a set of weighted correspondences between the source attributes and the target attributes. These weighted correspondences are created by a set of schema matching modules. However, as we explain shortly, there can be multiple p-mappings that are consistent with a given set of weighted correspondences, and the question is which of them to choose. We describe an approach to creating p-mappings that is based on choosing the mapping that maximizes the entropy of the probability assignment.

3.4.1 Computing weighted correspondences. A weighted correspondence between a pair of attributes specifies the degree of semantic similarity between them. Let S(s1 , . . . , sm ) be a source schema and T (t1 , . . . , tn ) be a target schema. We denote by Ci,j , i ∈ [1, m], j ∈ [1, n], the weighted correspondence between si and tj and by wi,j the weight of Ci,j . The first step is

18 to compute a weighted correspondence between every pair of attributes, which can be done by applying existing schema matching techniques. Although weighted correspondences tell us the degree of similarity between pairs of attributes, they do not tell us which target attribute a source attribute should map to. For example, a target attribute mailing-address can be both similar to the source attribute current-addr and to permanent-addr, so it makes sense to map either of them to mailing-address in a schema mapping. In fact, given a set of weighted correspondences, there could be a set of pmappings that are consistent with it. We can define the one-to-many relationship between sets of weighted correspondences and p-mappings by specifying when a p-mapping is consistent with a set of weighted correspondences.

Definition 1.18 (Consistent p-mapping) A p-mapping pM is consistent with a weighted correspondence Ci,j between a pair of source and target attributes if the sum of the probabilities of all mappings m ∈ pM containing correspondence (i, j) equals wi,j ; that is, X wi,j = Pr(m). m∈pM,(i,j)∈m

A p-mapping is consistent with a set of weighted correspondences C if it is consistent with each weighted correspondence C ∈ C.  However, not every set of weighted correspondences admits a consistent pmapping. The following theorem shows under which conditions a consistent p-mapping exists, and establishes a normalization factor for weighted correspondences that will guarantee the existence of a consistent p-mapping.

Theorem 1.19 Let C be a set of weighted correspondences between a source schema S(s1 , . . . , sm ) and a target schema T (t1 , . . . , tn ). There exists a consistent P p-mapping with respect to C if and only if (1) for every i ∈ [1, m], nj=1 wi,j ≤ 1 and (2) for every j ∈ [1, n], Pm i=1 wi,j ≤ 1. Let n m X X M ′ = max{maxi { wi,j }, maxj { wi,j }}. j=1

Then, for each i ∈ [1, m], Pm wi,j i=1 M ′ ≤ 1.

Pn

wi,j j=1 M ′

i=1

≤ 1 and for each j ∈ [1, n], 

Based on Theorem 1.19, we normalize the weighted correspondences we generated as described previously by dividing them by M ′ ; that is, wi,j ′ wi,j = . M′

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3.4.2 Generating p-mappings. To motivate our approach to generating p-mappings, consider the following example. Consider a source schema (A, B) and a target schema (A′ , B ′ ). Assume we have computed the following weighted correspondences between source and target attributes: wA,A′ = 0.6 and wB,B ′ = 0.5 (the rest are 0). As we explained above, there are an infinite number of p-mappings that are consistent with this set of weighted correspondences and below we list two: pM1 : m1: (A,A’), (B,B’): 0.3 m2: (A,A’): 0.3 m3: (B,B’): 0.2 m4: empty: 0.2 pM2 : m1: (A,A’), (B,B’): 0.5 m2: (A,A’): 0.1 m3: empty: 0.4 In a sense, pM1 seems better than pM2 because it assumes that the similarity between A and A′ is independent of the similarity between B and B ′ . In the general case, among the many p-mappings that are consistent with a set of weighted correspondences C, we choose the one with the maximum entropy; that is, thePp-mappings whose probability distribution obtains the maximum value of li=1 −pi ∗ logpi . In the above example, pM1 obtains the maximum entropy. The intuition behind maximum entropy is that when we need to select among multiple possible distributions on a set of exclusive events, we choose the one that does not favor any of the events over the others. Hence, we choose the distribution that does not introduce new information that we didn’t have apriori. The principle of maximum entropy is widely used in other areas such as natural language processing. To create the p-mapping, we proceed in two steps. First, we enumerate all possible one-to-one schema mappings between S and M that contain a subset of correspondences in C. Second, we assign probabilities on each of the mappings in a way that maximizes the entropy of our result p-mapping. Enumerating all possible schema mappings given C is trivial: for each subset of correspondences, if it corresponds to a one-to-one mapping, we consider the mapping as a possible mapping. Given the possible mappings m1 , . . . , ml , we assign probabilities p1 , . . . , pl to m1 , . . . , ml by solving the following constraint optimization problem (OPT): P maximize lk=1 −pk ∗ log pk subject to: 1 ∀k ∈ [1, l], 0 ≤ pk ≤ 1, Pl 2 k=1 pk = 1, and

20 3 ∀i, j :

P

k∈[1,l],(i,j)∈mk

pk = wi,j .

We can apply existing technology in solving the OPT optimization problem. Although finding maximum-entropy solutions in general is costly, the experiments described in [6] show that the execution time is reasonable for a one-time process.

3.5

Broader Classes of Mappings

In this section we describe several practical extensions to the basic mapping language. The query answering techniques and complexity results we have described carry over to these techniques. GLAV mappings: The common formalism for schema mappings, GLAV (a.k.a. tuple-generating dependencies), is based on expressions of the form m : ∀x(ϕ(x) → ∃yψ(x, y)). In the expression, ϕ is the body of a conjunctive query over S¯ and ψ is the body of a conjunctive query over T¯. A pair of instances DS and DT satisfies a GLAV mapping m if for every assignment of x in DS that satisfies ϕ there exists an assignment of y in DT that satisfies ψ. ¯ T¯, gm), We define general p-mappings to be triples of the form pGM = (S, 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 a simple generalization of Definition 1.8 and query answering can be conducted in PTIME. Extending by-tuple semantics to arbitrary GLAV mappings is much trickier than by-table semantics and would involve considering mapping sequences whose length is the product of the number of tuples in each source table, and the results are much less intuitive.

Theorem 1.20 Let pGM be a general p-mapping between a source schema ¯ Let Q be an SPJ query S¯ and a target schema T¯. Let DS be an instance of S. ¯ with only equality conditions over T . The problem of computing Qtable (DS ) with respect to pGM is in PTIME in the size of the data and the mapping.  Complex mappings: Complex mappings map a set of attributes in the source to a set of attributes in the target. For example, we can map the attribute address to the concatenation of street, city, and state. Formally, a set correspondence between S and T is a relationship between a subset of attributes in S and a subset of attributes in T . Here, the function associated with the relationship specifies a single value for each of the target attributes 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),

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where cm is a set of set correspondences, such that each attribute in S or T is involved in at most one set correspondence. A complex P p-mapping is of the form pCM = {(cmi , P r(cmi )) | i ∈ [1, n]}, where ni=1 Pr (cmi ) = 1.

Theorem 1.21 Let pCM be a complex schema p-mapping between schemas ¯ Let Q be an SPJ query over T¯. The data S¯ and T¯. Let DS be an instance of S. complexity and mapping complexity of computing Qtable (DS ) with respect to pCM are PTIME. The data complexity of computing Qtuple (DS ) with respect to pCM is #P-complete. The mapping complexity of computing Qtuple (DS ) with respect to pCM is in PTIME.  Union mapping: Union mappings specify relationships such as both attribute home-address and attribute office-address can be mapped to address. Formally, a union mapping is a triple (S, T, m), ¯ where m ¯ is a set of mappings between S and T . Given a source relation DS and a target relation DT , we say DS and DT are consistent with respect to the union mapping if for each source tuple t and m ∈ m, ¯ there exists a target tuple t′ , such that t and t′ satisfy m. A union p-mapping is of the form pU M = {(m ¯ i , P r(m ¯ i )) | i ∈ [1, n]}, where P n Pr ( m ¯ ) = 1. i i=1 Both by-table and by-tuple semantics apply to probabilistic union mappings.

Theorem 1.22 Let pUM be a union schema p-mapping between a source ¯ Let Q be a schema S¯ and a target schema T¯. Let DS be an instance of S. table ¯ conjunctive query over T . The problem of computing Q (DS ) with respect to pUM is in PTIME in the size of the data and the mapping; the problem of computing Qtuple (DS ) with respect to pUM is in PTIME in the size of the mapping and #P-complete in the size of the data.  Conditional mappings: In practice, our uncertainty is often conditioned. For example, we may want to state that daytime-phone maps to work-phone with probability 60% if age ≤ 65, and maps to home-phone with probability 90% if age > 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 distribution of possible mappings when condition Ci holds. Conditional mappings make more sense for by-tuple semantics. The following theorem shows that the complexity results carry over to such mappings.

Theorem 1.23 Let cpM be a conditional schema p-mapping between S¯ and ¯ Let Q be an SPJ query over T¯. The problem T¯ . Let DS be an instance of S. tuple of computing Q (DS ) with respect to cpM is in PTIME in the size of the mapping and #P-complete in the size of the data. 

22

3.6

Other Types of Approximate Schema Mappings

There have been various models proposed to capture uncertainty on mappings between attributes. [15] proposes keeping the top-K mappings between two schemas, each with a probability (between 0 and 1) of being true. [16] proposes assigning a probability for matching of every pair of source and target attributes. This notion corresponds to weighted correspondences described in Section 3.4. Magnani and Montesi [29] have empirically shown that top-k schema mappings can be used to increase the recall of a data integration process and Gal [14] described how to generate top-k schema matchings by combining the matching results generated by various matchers. The probabilistic schema mappings we described above are different as they contain all possible schema mappings that conform to the schema matching results and assigns probabilities to these mappings to reflect the likelihood that each mapping is correct. Nottelmann and Straccia [32] proposed generating probabilistic schema matchings that capture the uncertainty on each matching step. The probabilistic schema mappings we create not only capture our uncertainty on results of the matching step, but also take into consideration various combinations of attribute correspondences and describe a distribution of possible schema mappings where the probabilities of all mappings sum up to 1. There have also been work studying how to use probabilistic models to capture uncertainty on mappings of schema object classes, such as DatabasePapers and AIPapers. Query answering can take such uncertainty into consideration in computing the coverage percentage of the returned answers and in ordering information sources to maximize the likelihood of obtaining answers early. In the relational model, an object class is often represented using a relational table; thus, these probabilistic models focus on mapping between tables rather than attributes in the tables. Specifically, consider two object classes A and B. The goal of the probabilistic models is to capture the uncertainty on whether A maps to B. One method [13] uses probability P (B|A), which is the probability that an instance of A is also an instance of B. Another method [29] uses a tuple < A, B, R, P >, where R is a set of mutually exclusive relationships between A and B, and P is a probability distribution over R. The possible relationships considered in this model include equivalent =, subset-subsumption ⊂, superset-subsumption ⊃, overlapping ∩, disjointness 6 ∩, and incompatibility 6∼.

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23

Uncertainty in Mediated Schema

The mediated schema is the set of schema terms (e.g., relations, attribute names) in which queries are posed. They do not necessarily cover all the attributes appearing in any of the sources, but rather the aspects of the domain that are important for the integration application. When domains are broad, and there are multiple perspectives on them (e.g., a domain in science that is constantly under evolution), then there will be uncertainty about which is the correct mediated schema and about the meaning of its terms. When the mediated schema is created automatically by inspecting the sources in a payas-you-go system, there will also be uncertainty about the mediated schema. In this section we first motivate the need for probabilistic mediated schemas (p-med-schemas) with an example (Section 4.1). In Section 4.2 we formally define p-med-schemas and relate them with p-mappings in terms of expressive power and semantics of query answering. Then in Section 4.3 we describe an algorithm for creating a p-med-schema from a set of data sources. Finally, Section 4.4 gives an algorithm for consolidating a p-med-schema into a single schema that is visible to the user in a pay-as-you-go system.

4.1

P-Med-Schema Motivating Example

Let us begin with an example motivating p-med-schemas. Consider a setting in which we are trying to automatically infer a mediated schema from a set of data sources, where each of the sources is a single relational table. In this context, the mediated schema can be thought of as a “clustering” source attributes, with similar attributes being grouped into the same cluster. The quality of query answers critically depends on the quality of this clustering. Because of the heterogeneity of the data sources being integrated, one is typically unsure of the semantics of the source attributes and in turn of the clustering.

Example 1.24 Consider two source schemas both describing people: S1(name, hPhone, hAddr, oPhone, oAddr) S2(name, phone, address) In S2, the attribute phone can either be a home phone number or be an office phone number. Similarly, address can either be a home address or be an office address. Suppose we cluster the attributes of S1 and S2. There are multiple ways to cluster the attributes and they correspond to different mediated schemas. Below we list a few (in the mediated schemas we abbreviate hPhone as hP, oPhone as oP, hAddr as hA, and oAddr as oA): M1({name}, {phone, hP, oP}, {address, hA, oA}) M2({name}, {phone, hP}, {oP}, {address, oA}, {hA})

24 Possible Mapping {(name, name), (hP, hPP), (oP, oP), (hA, hAA), (oA, oA)} {(name, name), (hP, hPP), (oP, oP), (oA, hAA), (hA, oA)} {(name, name), (oP, hPP), (hP, oP), (hA, hAA), (oA, oA)} {(name, name), (oP, hPP), (hP, oP), (oA, hAA), (hA, oA)}

Probability 0.64 0.16 0.16 0.04

(a) Possible Mapping {(name, name), (oP, oPP), (hP, hP), (oA, oAA), (hA, hA)} {(name, name), (oP, oPP), (hP, hP), (hA, oAA), (oA, hA)} {(name, name), (hP, oPP), (oP, hP), (oA, oAA), (hA, hA)} {(name, name), (hP, oPP), (oP, hP), (hA, oAA), (oA, hA)}

Probability 0.64 0.16 0.16 0.04

(b) Answer (’Alice’, ’123-4567’, ’123, A Ave.’) (’Alice’, ’765-4321’, ’456, B Ave.’) (’Alice’, ’765-4321’, ’123, A Ave.’) (’Alice’, ’123-4567’, ’456, B Ave.’)

Probability 0.34 0.34 0.16 0.16

(c) Figure 1.5. The motivating example: (a) p-mapping for S1 and M3 , (b) p-mapping for S1 and M4 , and (c) query answers w.r.t. M and pM. Here we denote {phone, hP} by hPP, {phone, oP} by oPP, {address, hA} by hAA, and {address, oA} by oAA.

M3({name}, {phone, hP}, {oP}, {address, hA}, {oA}) M4({name}, {phone, oP}, {hP}, {address, oA}, {hA}) M5({name}, {phone}, {hP}, {oP}, {address}, {hA}, {oA}) None of the listed mediated schemas is perfect. Schema M1 groups multiple attributes from S1. M2 seems inconsistent because phone is grouped with hPhone while address is grouped with oAddress. Schemas M3 , M4 and M5 are partially correct but none of them captures the fact that phone and address can be either home phone and home address, or office phone and office address. Even if we introduce probabilistic schema mappings, none of the listed mediated schemas will return ideal answers. For example, using M1 prohibits returning correct answers for queries that contain both hPhone and oPhone

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because they are taken to be the same attribute. As another example, consider a query that contains phone and address. Using M3 or M4 as the mediated schema will unnecessarily favor home address and phone over office address and phone or vice versa. A system with M2 will incorrectly favor answers that return a person’s home address together with office phone number. A system with M5 will also return a person’s home address together with office phone, and does not distinguish such answers from answers with correct correlations. A probabilistic mediated schema will avoid this problem. Consider a probabilistic mediated schema M that includes M3 and M4 , each with probability 0.5. For each of them and each source schema, we generate a probabilistic mapping (Section 3). For example, the set of probabilistic mappings pM for S1 is shown in Figure 1.5(a) and (b). Now consider an instance of S1 with a tuple (’Alice’, ’123-4567’, ’123, A Ave.’, ’765-4321’, ’456, B Ave.’) and a query SELECT name, phone, address FROM People The answer generated by our system with respect to M and pM is shown in Figure 1.5(c). (As we describe in detail in the following sections, we allow users to compose queries using any attribute in the source.) Compared with using one of M2 to M5 as a mediated schema, our method generates better query results in that (1) it treats answers with home address and home phone and answers with office address and office phone equally, and (2) it favors answers with the correct correlation between address and phone number. 

4.2

Probabilistic Mediated Schema

Consider a set of source schemas {S1 , . . . , Sn }. We denote the attributes in schema Si , i ∈ [1, n], by attr(Si ), and the set of all source attributes as A. That is, A = attr(S1 ) ∪ . . . ∪ attr(Sn ). We denote a mediated schema for the set of sources {S1 , . . . , Sn } by M = {A1 , . . . , Am }, where each of the Ai ’s is called a mediated attribute. The mediated attributes are sets of attributes from the sources, i.e., Ai ⊆ A; for each i, j ∈ [1, m], i 6= j ⇒ Ai ∩ Aj = ∅. Note that whereas in a traditional mediated schema an attribute has a name, we do not deal with naming of an attribute in our mediated schema and allow users to use any source attribute in their queries. (In practice, we can use the most frequent source attribute to represent a mediated attribute when exposing the mediated schema to users.) If a query contains an attribute a ∈ Ai , i ∈ [1, m], then when answering the query we replace a everywhere with Ai .

26 A probabilistic mediated schema consists of a set of mediated schemas, each with a probability indicating the likelihood that the schema correctly describes the domain of the sources. We formally define probabilistic mediated schemas as follows.

Definition 1.25 (Probabilistic Mediated Schema) Let {S1 , . . . , Sn } be a set of schemas. A probabilistic mediated schema (p-med-schema) for {S1 , . . . , Sn } is a set M = {(M1 , P r(M1 )), . . . , (Ml , P r(Ml ))} where for each i ∈ [1, l], Mi is a mediated schema for S1 , . . . , Sn , and for each i, j ∈ [1, l], i 6= j, Mi and Mj correspond to different clusterings of the source attributes; P r(Mi ) ∈ (0, 1], and Σli=1 P r(Mi ) = 1.  Semantics of queries: Next we define the semantics of query answering with respect to a p-med-schema and a set of p-mappings for each mediated schema in the p-med-schema. Answering queries with respect to p-mappings returns a set of answer tuples, each with a probability indicating the likelihood that the tuple occurs as an answer. We consider by-table semantics here. Given a query Q, we compute answers by first answering Q with respect to each possible mapping, and then for each answer tuple t summing up the probabilities of the mappings with respect to which t is generated. We now extend this notion for query answering that takes p-med-schema into consideration. Intuitively, we compute query answers by first answering the query with respect to each possible mediated schema, and then for each answer tuple taking the sum of its probabilities weighted by the probabilities of the mediated schemas.

Definition 1.26 (Query Answer) Let S be a source schema and M = {(M1 , P r(M1 )), . . . , (Ml , P r(Ml ))} be a p-med-schema. Let pM ={pM (M1 ), . . ., pM (Ml )} be a set of p-mappings where pM (Mi ) is the p-mapping between S and Mi . Let D be an instance of S and Q be a query. Let t be a tuple. Let P r(t|Mi ), i ∈ [1, l], be the probability of t in the answer of Q with respect to Mi and pM (Mi ). Let p = Σli=1 P r(t|Mi ) ∗ P r(Mi ). If p > 0, then we say (t, p) is a by-table answer with respect to M and pM. We denote all by-table answers by QM,pM (D).  We say that query answers A1 and A2 are equal (denoted A1 = A2 ) if A1 and A2 contain exactly the same set of tuples with the same probability assignments.

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Expressive power: A natural question to ask at this point is whether probabilistic mediated schemas provide any added expressive power compared to deterministic ones. Theorem 1.27 shows that if we consider one-to-many schema mappings, where one source attribute can be mapped to multiple mediated attributes, then any combination of a p-med-schema and p-mappings can be equivalently represented using a deterministic mediated schema with p-mappings, but may not be represented using a p-med-schema with deterministic schema mappings. Note that we can easily extend the definition of query answers to one-to-many mappings as one mediated attribute can correspond to no more than one source attribute.

Theorem 1.27 (Subsumption) The following two claims hold. 1 Given a source schema S, a p-med-schema M, and a set of p-mappings pM between S and possible mediated schemas in M, there exists a deterministic mediated schema T and a p-mapping pM between S and T , such that ∀D, Q : QM,pM (D) = QT,pM (D). 2 There exists a source schema S, a mediated schema T , a p-mapping pM between S and T , and an instance D of S, such that for any pmed-schema M and any set m of deterministic mappings between S and possible mediated schemas in M, there exists a query Q such that QM,m (D) 6= QT,pM (D).  In contrast, Theorem 1.28 shows that if we restrict our attention to one-to-one mappings, then a probabilistic mediated schema does add expressive power.

Theorem 1.28 There exists a source schema S, a p-med-schema M, a set of one-to-one p-mappings pM between S and possible mediated schemas in M, and an instance D of S, such that for any deterministic mediated schema T and any one-to-one p-mapping pM between S and T , there exists a query Q such that, QM,pM (D) 6= QT,pM (D).  Constructing one-to-many p-mappings in practice is much harder than constructing one-to-one p-mappings. And, when we are restricted to one-to-one p-mappings, p-med-schemas grant us more expressive power while keeping the process of mapping generation feasible.

4.3

P-med-schema Creation

We now show how to create a probabilistic mediated schema M. Given source tables S1 , . . . , Sn , we first construct the multiple schemas M1 , . . . , Mp in M, and then assign each of them a probability. We exploit two pieces of information available in the source tables: (1) pairwise similarity of source attributes; and (2) statistical co-occurrence properties

28 of source attributes. The former will be used for creating multiple mediated schemas, and the latter for assigning probabilities on each of the mediated schemas. The first piece of information tells us when two attributes are likely to be similar, and is generated by a collection of schema matching modules. This information is typically given by some pairwise attribute similarity measure, say s. The similarity s(ai , aj ) between two source attributes ai and aj depicts how closely the two attributes represent the same real-world concept. The second piece of information tells us when two attributes are likely to be different. Consider for example, source table schemas S1: (name,address,email-address) S2: (name,home-address) Pairwise string similarity would indicate that attribute address can be similar to both email-address and home-address. However, since the first source table contains address and email-address together, they cannot refer to the same concept. Hence, the first table suggests address is different from emailaddress, making it more likely that address refers to home-address. Creating Multiple Mediated Schemas: The creation of the multiple mediated schemas constituting the p-med-schema can be divided conceptually into three steps. First, we remove infrequent attributes from the set of all source attributes; that is, attribute names that do not appear in a large fraction of source tables. This step ensures that our mediated schema contains only information that is relevant and central to the domain. In the second step we construct a weighted graph whose nodes are the attributes that survived the filter of the first step. An edge in the graph is labeled with the pairwise similarity between the two nodes it connects. Finally, several possible clusterings of nodes in the resulting weighted graph give the various mediated schemas. Algorithm 1 describes the various steps in detail. The input is the set of source schemas creating S1 , . . . , Sn and a pairwise similarity function s, and the output is the multiple mediated schemas in M. Steps 1–3 of the algorithm find the attributes that occur frequently in the sources. Steps 4 and 5 construct the graph of these high-frequency attributes. We allow an error ǫ on the threshold τ for edge weights. We thus have two kinds of edges: certain edges, having weight at least τ + ǫ, and uncertain edges, having weight between τ − ǫ and τ + ǫ. Steps 6-8 describe the process of obtaining multiple mediated schemas. Specifically, a mediated schema in M is created for every subset of the uncertain edges. For every subset, we consider the graph resulting from omitting that subset from the graph. The mediated schema includes a mediated attribute for each connected component in the resulting graph. Since, in the worst case,

Uncertainty in Data Integration 0: 1: 2:

3: 4:

5: 6:

7:

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Input: Source schemas S1 , . . . , Sn . Output: A set of possible mediated schemas. Compute A = {a1 , . . . , am }, the set of all source attributes; for each (j ∈ [1, m]) |{i∈[1,n]|aj ∈Si }| ; Compute frequency f (aj ) = n Set A = {aj |j ∈ [1, m], f (aj ) ≥ θ}; //θ is a threshold Construct a weighted graph G(V, E), where (1) V = A, and (2) for each aj , ak ∈ A, s(aj , ak ) ≥ τ − ǫ, there is an edge (aj , ak ) with weight s(aj , ak ); Mark all edges with weight less than τ + ǫ as uncertain; for each (uncertain edge e = (a1 , a2 ) ∈ E) Remove e from E if (1) a1 and a2 are connected by a path with only certain edges, or (2) there exists a3 ∈ V , such that a2 and a3 are connected by a path with only certain edges and there is an uncertain edge (a1 , a3 ); for each (subset of uncertain edges) Omit the edges in the subset and compute a mediated schema where each connected component in the graph corresponds to an attribute in the schema; return distinct mediated schemas. Algorithm 1: Generate all possible mediated schemas.

the number of resulting graphs is exponential in the number of uncertain edges, the parameter ǫ needs to be chosen carefully. In addition, Step 6 removes uncertain edges that when omitted will not lead to different mediated schemas. Specifically, we remove edges that connect two nodes already connected by certain edges. Also, we consider only one among a set of uncertain edges that connect a particular node with a set of nodes that are connected by certain edges. Probability Assignment: The next step is to compute probabilities for possible mediated schemas that we have generated. As a basis for the probability assignment, we first define when a mediated schema is consistent with a source schema. The probability of a mediated schema in M will be the proportion of the number of sources with which it is consistent.

Definition 1.29 (Consistency) Let M be a mediated schema for sources S1 , . . . , Sn . We say M is consistent with a source schema Si , i ∈ [1, n], if there is no pair of attributes in Si that appear in the same cluster in M . Intuitively, a mediated schema is consistent with a source only if it does not group distinct attributes in the source (and hence distinct real-world concepts)

30 0:

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Input: Possible mediated schemas M1 , . . . , Ml and source schemas S 1 , . . . , Sn . Output: P r(M1 ), . . . , P r(Ml ). for each (i ∈ [1, l]) Count the number of source schemas that are consistent with Mi , denoted as ci ; for each (i ∈ [1, l]) Set P r(Mi ) = P lci . i=1 ci

Algorithm 2: Assign probabilities to possible mediated schemas. 0: 1: 2: 3: 4: 5: 6:

Input: Mediated schemas M1 , . . . , Ml . Output: A consolidated single mediated schema T . Set T = M1 . for (i = 2, . . . , l) modify T as follows: for each (attribute A′ in Mi ) for each (attribute A in T ) Divide A into A ∩ A′ and A − A′ ; return T . Algorithm 3: Consolidate a p-med-schema.

into a single cluster. Algorithm 2 shows how to use the notion of consistency to assign probabilities on the p-med-schema.

4.4

Consolidation

To complete the fully automatic setup of the data integration system, we consider the problem of consolidating a probabilistic mediated schema into a single mediated schema and creating p-mappings to the consolidated schema. We require that the answers to queries over the consolidated schema be equivalent to the ones over the probabilistic mediated schema. The main reason to consolidate the probabilistic mediated schema into a single one is that the user expects to see a single schema. In addition, consolidating to a single schema has the advantage of more efficient query answering: queries now need to be rewritten and answered based on only one mediated schema. We note that in some contexts, it may be more appropriate to show the application builder a set of mediated schemas and let her select one of them (possibly improving on it later on). Consolidating a p-med-schema: Consider a p-med-schema M = {(M1 , P r(M1 )), . . . , (Ml , P r(Ml ))}. We consolidate M into a single mediated schema T . Intuitively, our algorithm (see Algorithm 3) generates the “coarsest refinement” of the possible mediated schemas in M such that every cluster in any of the Mi ’s is equal to the union of a set of clusters in T . Hence,

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any two attributes ai and aj will be together in a cluster in T if and only if they are together in every mediated schema of M. The algorithm initializes T to M1 and then modifies each cluster of T based on clusters from M2 to Ml .

Example 1.30 Consider a p-med-schema M = {M1 , M2 }, where M1 contains three attributes {a1 , a2 , a3 }, {a4 }, and {a5 , a6 }, and M2 contains two attributes {a2 , a3 , a4 } and {a1 , a5 , a6 }. The target schema T would then contain four attributes: {a1 }, {a2 , a3 }, {a4 }, and {a5 , a6 }.  Note that in practice the consolidated mediated schema is the same as the mediated schema that corresponds to the weighted graph with only certain edges. Here we show the general algorithm for consolidation, which can be applied even if we do not know the specific pairwise similarities between attributes. Consolidating p-mappings: Next, we consider consolidating p-mappings specified w.r.t. M1 , . . . , Ml to a p-mapping w.r.t. the consolidated mediated schema T . Consider a source S with p-mappings pM1 , . . . , pMl for M1 , . . . , Ml respectively. We generate a single p-mapping pM between S and T in three steps. First, we modify each p-mapping pMi , i ∈ [1, l], between S and Mi to a p-mapping pMi′ between S and T . Second, we modify the probabilities in each pMi′ . Third, we consolidate all possible mappings in pMi′ ’s to obtain pM . The details are as follows. 1. For each i ∈ [1, l], modify p-mapping pMi : Do the following for every possible mapping m in pMi : For every correspondence (a, A) ∈ m between source attribute a and mediated attribute A in Mi , proceed as follows. (1) Find the set of all mediated attributes B in T such that B ⊂ A. Call this set B. (2) Replace (a, A) in m with the set of all (a, B)’s, where B ∈ B. Call the resulting p-mapping pMi′ . 2. For each i ∈ [1, l], modify probabilities in pMi′ : Multiply the probability of every schema mapping in pMi′ by P r(Mi ), which is the probability of Mi in the p-med-schema. (Note that after this step the sum of probabilities of all mappings in pMi′ is not 1.) 3. Consolidate pMi′ ’s: Initialize pM to be an empty p-mapping (i.e., with no mappings). For each i ∈ [1, l], add pMi′ to pM as follows: For each schema mapping m in pMi′ with probability p: if m is in pM , with probability p′ , modify the probability of m in pM to (p + p′ ); if m is not in pM , then add m to pM with probability p. The resulting p-mapping, pM , is the final consolidated p-mapping. The probabilities of all mappings in pM add to 1.

32 Note that Step 2 can map one source attribute to multiple mediated attributes; thus, the mappings in the result pM are one-to-many mappings, and so typically different from the p-mapping generated directly on the consolidated schema. The following theorem shows that the consolidated mediated schema and the consolidated p-mapping are equivalent to the original p-med-schema and p-mappings.

Theorem 1.31 (Merge Equivalence) For all queries Q, the answers obtained by posing Q over a p-med-schema M = {M1 , . . . , Ml } with pmappings pM1 , . . . , pMl is equal to the answers obtained by posing Q over the consolidated mediated schema T with consolidated p-mapping pM . 

4.5

Other approaches

He and Chang [21] considered the problem of generating a mediated schema for a set of web sources. Their approach was to create a mediated schema that is statistically maximally consistent with the source schemas. To do so, they assume that the source schemas are created by a generative model applied to some mediated schema, which can be thought of as a probabilistic mediated schema. The probabilistic mediated schema we described in this chapter has several advantages in capturing heterogeneity and uncertainty in the domain. We can express a wider class of attribute clusterings, and in particular clusterings that capture attribute correlations. Moreover, we are able to combine attribute matching and co-occurrence properties for the creation of the probabilistic mediated schema, allowing for instance two attributes from one source to have a nonzero probability of being grouped together in the mediated schema. Also, the approach for p-med-schema creation described in this chapter is independent of a specific schema-matching technique, whereas the approach in [21] is tuned for constructing generative models and hence must rely on statistical properties of source schemas. Magnani et al. [30] proposed generating a set of alternative mediated schemas based on probabilistic relationships between relations (such as an Instructor relation intersects with a Teacher relation but is disjoint with a Student relation) obtained by sampling the overlapping of data instances. Here we focus on matching attributes within relations. In addition, our approach allows exploring various types of evidence to improve matching and we assign probabilities to the mediated schemas we generate. Chiticariu et. al. [5] studied the generation of multiple mediated schemas for an existing set of data sources. They consider multi-table data sources, not considered in this chapter, but explore interactive techniques that aid humans in arriving at the mediated schemas. There has been quite a bit of work on automatically creating mediated schemas that focused on the theoretical analysis of the semantics of merging

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schemas and the choices that need to be made in the process [2, 4, 23, 25, 31, 33]. The goal of these work was to make as many decisions automatically as possible, but where some ambiguity arises, refer to input from a designer.

5.

Future Directions

The investigation of data integration with uncertainty is only beginning. This chapter described some of the fundamental concepts on which such systems will be built, but there is a lot more to do. The main challenge is to build actual data integration systems that incorporate uncertainty and thereby uncover a new set of challenges, such as efficiency and understanding what are the common types of uncertainty that arise in data integration applications, so techniques can be tailored for these cases. The work we described showed how to create p-mediated schemas and schema mappings automatically. This is only a way to bootstrap a pay-asyou-go integration system. The next challenge is to find methods to improve it over time (see [24] for a first work on doing so). We would also like to incorporate multi-table sources, rather than only single-table ones as we described so far. Finally, when we have many data sources, the sources tend to be redundant and contain dependencies (and therefore not offer independent sources of evidence). An important line of work is to discover these dependencies and use them to provide more precise answers to queries. We are currently exploring how the formalism and techniques from this chapter can be extended to consider uncertain and interdependent data sources, and how query answering can be performed efficiently even in the presence of dependencies.

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