Libkin, Leonid

Sirangelo, Cristina

English

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Open and Closed World Assumptionsin Data ExchangeLeonid Libkin1 and Cristina Sirangelo21 School of Informatics, University of Edinburgh2 LSV, ENS–Cachan, CNRS and INRIAData exchange is the problem of finding an instance of a target schema,given an instance of a source schema and a specification of a mapping betweenthe source and the target schemas, and answering queries over target instances.A schema mapping is a condition, often expressed in a fragment of first-orderlogic, that relates instances of source and target schemas. Most commonly it isa conjunction of sentences of the form∀x¯ ϕ(x¯) → ∃z¯ ψ(x¯, z¯), (1)where ϕ and ψ are both conjunctions of atomic formulae. For example, if wehave a source database S(·, ·) that lists employees and their salaries, and a targetdatabase T (·, ·) that lists people and their children, then a natural mapping is∀x∀y (S(x, y) → ∃z T (x, z)).The study of both data exchange and schema mappings has been activelypursued recently (see, e.g., recent surveys [3, 11, 4]). Existing implementationshave been incorporated into major database products.Theoretical foundations of data exchange were developed in [5, 6]. For asource instance S and a schema mapping M , a target instance T is a solu-tion for S if S and T together satisfy the conditions of M . For instance, if wehave a mapping given by a conjunction of sentences of the form (1), it meansthat for each tuple a¯ making ϕ(a¯) true in the source, one can find a tuple b¯ ofvalues in the target database so that the conjunction ψ(a¯, b¯) is true in the target.This semantics implies that for the same source instance one can have manydifferent target solutions. In fact this semantics of data exchange follows the socalled open world assumption (OWA), as target solutions may contain arbitraryfacts, some of which are not directly implied by the source data and the schemamapping. An alternative approach, developed in [12] adopts the so called closedworld assumption (CWA) [14], for defining the semantics of data exchange. Hereintuitively only facts which are a direct “translation” of source data are allowedin the target.If a query Q is posed against target instances, then it must be answered in away that is consistent with the source. A common approach is to look for certainanswers:certainM (Q,S) =⋂{Q(T ) | T is a solution}. (2)A desirable feature in data exchange is to find a particular solution to be materi-alized in the target, which allows to answer queries over the target (with certainanswer semantics), with no access to source data. In other words, we want aspecific target instance T0 so that certainM (Q,S) = Q′(T0), for some query Q′.Data exchange literature identified two particularly nice solutions, called thecanonical solution [5] and the core [6]. It was shown in [5, 6] that, under theOWA, certain answers (2) can be found in polynomial time if Q is a union ofconjunctive queries (i.e., in the (∃,∧,∨)-fragment of FO). In fact, in that caseit is easy to construct a query Q′ so that certainM (Q,S) = Q′(T0) if T0 is eitherthe canonical solution, or the core.While this looks reasonable, the definition of query answering is actuallyunderspecified. We know well how to answer queries on complete databases; incontrast, databases arising in data exchange are often incomplete as they containnull values (for example, it is common for target schemas to have attributes thatdo not correspond to any attributes in the source schema; these are populatedwith nulls). Thus, the notions of query answering over incomplete databases areessential in data exchange, and have to be incorporated into the algorithms foranswering queries in data exchange.Let us briefly recall the standard approach to query answering over databaseswith incomplete information [10]. Such a database is formally a relational struc-ture over the disjoint union of two domains: D of data values, and V of variables(or nulls, usually denoted by the symbol ⊥ with subscripts). For such a databaseR, we let Rep(R) denote the set of all complete databases (over D) that it rep-resents. This can be interpreted again under two assumptions: under the closedworld assumption, Rep(R) consists of all databases obtained by applying map-pings V → D to R; under the open world assumption, Rep(R) is the set of alldatabases that contain v(R), for some mapping v : V → D.The answers to a query Q over R are defined as2Q(R) =⋂{Q(R′) | R′ ∈ Rep(R)}.That is, 2Q(R) contains answers independent of the interpretation of nulls. Oneeasy case of finding such answers is known: if Q is a union of conjunctive queries,then 2Q(R), under both closed and open world assumptions, is obtained by astraightforward evaluation of Q on R and then discarding tuples that containnulls (elements of V ). This is referred to as naive evaluation. Clearly, naiveevaluation is done in PTIME; adding negation to queries (e.g., going to the fullFO) makes the problem coNP-complete under the closed world assumption [1]and undecidable under the OWA.The theory of incomplete information explains why the OWA works well foranswering unions of conjunctive queries in data exchange. In fact the canonicalsolution in data exchange, viewed as an open-world incomplete database, turnsout to represent the set of all data exchange solutions under the OWA (propertyalready referred to as universality in [5]). It follows that, if Tc is the canonicalsolution, 2Q(Tc) computes data exchange certain answers certainM (Q,S) underthe OWA. If Q is a union of conjunctive query, all one needs to do is to computethe canonical solution, and naively evaluate the query over it – then results of[10] mentioned above guarantee that the process computes certain answers.However, going beyond unions of conjunctive queries poses problems. Theyappear even in the simplest mappings of the form ∀x¯ R(x¯) → R′(x¯) that copycontents of each relation R into a new relation R′. Under the OWA, even FOquery answering in data exchange is undecidable, and for every Boolean query,either its certain answers are false for all instances, or the certain answers for itsnegation are false for all instances [2]. There are other examples of anomalousbehavior of query answering in data exchange.The reason behind such anomalies is that our intuition tells us that themapping ∀x¯ R(x¯) → R′(x¯) forces R′ to be a copy of R; however, under the OWAsemantics, R′ is allowed to be an arbitrary extension of R. It seems natural thento adopt the CWA that would not permit adding arbitrary facts to R′ and wouldonly put there what the constraints dictate – that is, a copy of R.This approach was developed in [12]. It introduced the notion of CWA-solutions based on three assumptions. First, every fact put in a target instancemust be justified. Second, every justification for putting a tuple into the tar-get should not generate multiple nulls. And third, every fact true in the targetinstance must be derivable from the source instance and constraints in the map-pings. When these are properly formalized, the CWA-solutions can be character-ized as homomorphic images of the canonical solution that have a homomorphismback into the canonical solution. It follows that every CWA-solution contains acopy of the core [12].When the definition of certainM (Q,S) is applied to CWA-solutions, a propernotion of query answers Q(T ) has to be adopted for CWA-solutions T . The mostnatural semantics is 2Q(T ), where T is viewed as a closed-world incompletedatabase. This approach naturally leads to a new semantics of query answering:a tuple t is in certainM (Q,S) under the CWA if, for every CWA-solution T andevery R ∈ Rep(T ), this tuple is in Q(R). That is,certainCWAM(Q,S) =⋂{Q(R) | R ∈ Rep(T ), T is a CWA-solution.}It was shown in [12] that the canonical solution is also a CWA-solution and,viewed as a closed-world incomplete database, represents all databases that arein Rep(T ) for some CWA-solution T . It follows that, as in the OWA case,the certain answers can be computed over the canonical solution Tc: namely,certainCWAM(Q,S) = 2Q(Tc). Hence, the problem of answering queries in dataexchange is reduced to the well-studied problems of answering queries in closed-world databases with nulls. In particular, from [1, 10] one derives that comput-ing certainCWAM(Q,S) is coNP-complete for arbitrary FO queries and tractablefor unions of conjunctive queries (for which the results coincide under both theOWA and the CWA).Extensions of this approach to mappings that involve constraints over targetschemas were developed in [9].But fully open or fully closed mappings, being two extreme cases, are boundto have their shortcomings. Consider again the example of a source databaseS(·, ·) that lists employees and their salaries, and a target database T (·, ·) thatlists people and their children. The mapping is given by ∀x∀y (S(x, y) →∃z T (x, z)). Under the CWA, if we assume that each employee has one salary,in the target each employee will have exactly one value for the ‘child’ attribute.This example indicates that it is natural to declare some target attributes asopen, and some as closed. In the above example, we would formulate the map-ping as ∀x∀y (S(x, y) → ∃z T (xcl, zop)), saying that only employees from thesource are moved to the target, but the number of values of the ‘child’ attributeare arbitrary.Such mixed mappings were studied in [13]. Data exchange solutions undermixed mappings are incomplete databases where domain elements are annotatedas either open (op) or closed (cl). Such an approach for databases with nulls wasused back in the 80s, see [8]. If T is such an annotated instance, RepA(T ) is de-fined so as to apply the OWA on values annotated as open, and the CWA onvalues annotated as closed. In particular, to get an instance in RepA(T ), afterapplying a valuation v to T , any tuple in v(T ) can be replicated arbitrarily manytimes by changing the values of attributes annotated as open, but keeping thesame value of attributes annotated as closed. For example, RepA({(acl,⊥op)})contains all relations R whose projection on the first attribute is {a}. SimilarlyRepA({(acl,⊥op1,⊥cl2)}) contains all relations whose first projection is {a} andwhose third projection is a singleton. On the contrary, RepA({(acl,⊥cl)}) con-tains all one-tuple relations {(a, b)} for some constant b. If all annotations in Tare open (closed, resp), clearly RepA(T ) coincides with Rep(T ) under the OWA(CWA, resp.).Annotated data exchange solutions have been defined along the lines of CWA-solutions. Each tuple put in the target still needs to be justified by some sourcedata and the schema mapping. The tuple annotation (determined by the schemamapping) will then give either and open-world or closed-world semantics to thetuple attributes. The restriction that all facts true in the target must be im-plied by the source and the schema mapping is enforced only limited to closedattributes.A particular annotated solution, the annotated canonical solution, has theusual property that it represents the semantics of all the solutions. That is, if Tcis the annotated canonical solution, RepA(Tc) contains precisely all the instancesR that are in RepA(T ) for some annotated solution T . When annotation isremoved from Tc, this is the usual canonical solution of [5] and [12], obtained bydisregarding annotation in the schema mapping. This also shows, as proved in[13], that if a mapping has all open (resp, all closed) annotation, the semanticsof annotated solutions coincides with the semantics of the OWA solutions of [5](respectively CWA-solutions of [12]).The semantics of query answering is standard: for an annotated schema map-ping Mα a tuple t is certainMα(Q,S) if, for every annotated solution T and ev-ery R ∈ RepA(T ), this tuple is in Q(R). Since the semantics of the annotatedcanonical solution Tc captures the semantics of all solutions, the usual equationcertainMα(Q,S) = 2Q(Tc) holds also under annotated mappings. Thus the com-plexity of computing certain answers for arbitrary FO queries, must range fromcoNP-complete (in the case of all ‘closed’ annotation) to undecidable (in the caseof all ‘open’ annotation).In particular it was shown that for query answering, there is a complexitytrichotomy that depends on the number of target attributes declared as openper each atom in a rule in the mapping. If the number is 0 (i.e., we are underthe CWA), then the complexity is in general coNP-complete. If the number is 2or greater, then, as for the OWA, the problem is undecidable. But if the numberis 1, the problem is decidable, albeit with a high complexity: coNEXPTIME-complete. Note that this means that each atom in a rule has at most one openannotation; the rule itself (or the mapping) can have arbitrarily many openannotations. We prohibit atoms such as T (xop, yop) but can have, for example,T1(xop, ycl) ∧ T1(yop, zcl) ∧ T2(xop, zcl).In the case of unions of conjunctive queries, all the cases collapse and theusual naive evaluation over the canonical solution produces the answer in poly-nomial time.Finally, [13] considered the issue of composing schema mappings, and showedthat the OWA approach of [7] extends to CWAmappings, but does not guaranteecomposition in general for mixed mappings.Acknowledgments Supported by EPSRC grants E005039 and F028288, and theFET-Open Project FoX (grant agreement 233599).References1. S. Abiteboul, P. Kanellakis, G. Grahne. On the representation and querying ofsets of possible worlds. TCS 78 (1991), 158–187.2. M. Arenas, P. Barcelo´, R. Fagin, L. Libkin. Locally consistent transformations andquery answering in data exchange. In PODS 2004, pages 229–240.3. P. Barcelo´. Logical foundations of relational data exchange. SIGMOD Record 38(2009).4. P. Bernstein, S. Melnik. Model management 2.0: manipulating richer mappings.SIGMOD’07, pages 1–12.5. R. Fagin, Ph. Kolaitis, R. Miller, L. Popa. Data exchange: semantics and queryanswering. Theor. Comput. Sci. 336(1): 89–124 (2005).6. R. Fagin, Ph. Kolaitis, L. Popa. Data exchange: getting to the core. ACM TODS30(1): 174–210 (2005).7. R. Fagin, Ph. Kolaitis, L. Popa, W.C. Tan. Composing schema mappings: second-order dependencies to the rescue. ACM TODS 30(4) 994–1055 (2005).8. G. Gottlob, R. Zicari. Closed world databases opened through null values. InVLDB’88, pages 50–61.9. A. Hernich, N. Schweikardt. CWA-solutions for data exchange settings with targetdependencies. In PODS’07, pages 113–122.10. T. Imielinski, W. Lipski. Incomplete information in relational databases. J. ACM31 (1984), 761–791.11. Ph. Kolaitis. Schema mappings, data exchange, and metadata management. InPODS 2005.12. L. Libkin. Data exchange and incomplete information. In PODS’06, pages 60–69.13. L. Libkin and C. Sirangelo. Data exchange and schema mappings in open andclosed worlds In PODS’08, pages 139–148.14. R. Reiter. On closed world databases. In Logic and Databases, Plenum Press, 1978,pages 55–76.

Open and Closed World Assumptions in Data Exchange

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