Cristian Molinaro

Irina Trubitsyna

Marco Calautti

Sergio Greco

English

AIR Universita degli studi di Milano

Rewriting-based Check of Chase TerminationMarco Calautti, Sergio Greco, Cristian Molinaro, and Irina Trubitsyna{calautti,greco,cmolinaro,trubitsyna}@dimes.unical.itDIMES, Università della Calabria, 87036 Rende (CS), ItalyKeywords: Chase, Data Dependencies, Data Exchange, Data Integration.1 IntroductionThe Chase is a fixpoint algorithm enforcing satisfaction of data dependencies(also called constraints) in databases. It has been proposed more than thirtyyears ago [2,18] and has seen a revival of interest in recent years in both databasetheory and practical applications. Indeed, the availability of data coming fromdifferent sources easily results in inconsistent or incomplete data (i.e., data notsatisfying data dependencies) and, therefore, techniques for fixing inconsistenciesare crucial [1,3,5,7,8,13,17].The chase algorithm is used, directly or indirectly, on an everyday basis bypeople who design databases, and it is used in commercial systems to reasonabout the consistency and correctness of a data design. New applications ofthe chase in meta-data management, ontological reasoning, data exchange, datacleaning, and query optimization have been proposed as well [6,9].The chase algorithm solves possible violations of constraints by inserting newtuples, possibly containing null values [4]. The following example shows a casewhere a given database does not satisfy a set of tuple generating dependencies(TGDs) and the application of the chase algorithm produces a new consistentdatabase by adding tuples with nulls.Example 1. Consider the following set of constraints Σ1 describing departmentsand their employees:∀x ∀y Department(x) ∧Managed(x, y) → Employee(y)∀x Employee(x) → ∃ y WorksFor(x, y)∀x ∀y WorksFor(x, y) → ∃ z Managed(y, z)Consider the database D = {Department(d),Managed(d,m)}. Since the firstconstraint is not satisfied, the tuple Employee(m) is inserted. This update oper-ation fires the second constraint to insert the tuple WorksFor(m, η1), which inturn fires the third constraint so that the tuple Managed(η1, η2) is added to thedatabase (η1 and η2 are new labeled nulls). At this point, the chase terminatessince the database is consistent, that is, all dependencies are satisfied. 2Unfortunately, the chase algorithm may not terminate. For instance, in Exam-ple 1 if we delete from the first constraint the atom Department(x), the chasenever terminates and adds an infinite number of tuples to the database. It hasbeen formally proved in [12] that the problem of deciding whether the chaseprocedure terminates is semi-decidable. The first and basic effort concerning theformalization of a (decidable) sufficient condition guaranteeing chase termina-tion is weak acyclicity [11]. Informally, it checks whether the constraints do notallow for nulls to cyclically propagate inside predicates’ positions. Consideringthe example above, we have that a value in the second position of predicateManaged is copied to Employee (denoted by M2 → E1). This forces the in-troduction of a new null value in the second position of WorksFor, (denotedas E1 →∗ WF2); this value is then copied in the first position of Managed(WF2 → M1) and it also forces the introduction of a new null value in the se-cond position of Managed (WF2 →∗ M2). Since it is possible to reach positionM2 from itself through a connection of the form →∗ , an infinite number of nullscould be introduced during the chase procedure.Different extensions of weak acyclicity have been proposed. Safety [20] andsuper-weak acyclicity [19] identify the positions in which null values can be prop-agated. Stratification-based approaches [10,20,16] analyse whether dependenciesmay fire each other and thus propagate null values from one to another. See[14] for a comprehensive survey on this topic. Nevertheless, despite the previ-ously mentioned results, there are still important classes of terminating datadependencies which are not identified by any of the previously mentioned cri-teria: Example 1 showed one such a case. To overcome such limitations, rewrit-ing techniques have been proposed [15,16]. In the following section we give anoverview on them and show how the constraints of Example 1 can be rewrittenby using predicate adornments in order to allow simple termination conditions(even the simplest one, weak-acyclicity) to understand that the chase procedureterminates. Issues regarding the extension of these techniques to managing alsoequality generating dependencies (EGDs) are discussed in Section 3.2 Constraint RewritingWe start by introducing the basic idea of the Adn technique [15], which canbe used in conjunction with current termination criteria, enabling us to detectmore sets of constraints as terminating. The technique consists of rewriting a setof TGDs Σ into a new set Σα which is “better” than the original one for thepurpose of checking termination. Rather than applying a termination criterion toΣ, the new set Σα is used and if Σα satisfies the criterion then chase terminationfor Σ is guaranteed. This allows us to recognize larger classes of constraints forwhich chase termination is guaranteed: if Σ satisfies chase termination criterionC, then the rewritten set Σα satisfies C as well, but the vice versa is not true,that is, there are significant classes of constraints for which Σα satisfies C andΣ does not.Example 2. Consider again the set of TGDs Σ1. The Adn technique first rewritesTGDs by associating strings of b symbols to body atoms and to head positionscontaining universally quantified variables. Then, f symbols are associated forexistentially quantified variables. This new set of TGDs is denoted by Base(Σ1):∀x∀y Departmentb(x) ∧Managedbb(x, y) → Employeeb(y)∀x Employeeb(x) → ∃ y WorksForbf (x, y)∀x∀y WorksForbb(x, y) → ∃ z Managedbf (y, z)Subsequently, because of the presence of atomsWorksForbf (x, y) andManagedbf (x, y)in Base(Σ1), the rewriting continues by producing the following set of TGDsDerived(Σ1):∀x∀y WorksForbf (x, y) → ∃ z Managedff (y, z)∀x∀y Departmentb(x) ∧Managedbf (x, y) → Employeef (y)∀x Employeef (x) → ∃ y WorksForff (x, y)∀x∀y WorksForff (x, y) → ∃ z Managedff (y, z)At this point, the generation of Derived(Σ1) terminates, since the atomDepartmentb(x) cannot be joined with Managedff (x, y) to produce a new ador-ned TGD. The rewritten set of TGDs Adn(Σ1) is weakly-acyclic, whereas theoriginal set Σ1 is not recognized by any chase termination criteria. 2Rewriting Algorithm Improvement. The rewriting algorithm Adn has been fur-ther improved into the Adn+ algorithm [16] by using different adornments foreach existentially quantified variable and by considering how TGDs may fire eachother in the generation of adorned atoms. During the rewriting process, this al-gorithm also performs a basic cyclicity check, allowing to eventually determinethe termination of the chase, without necessarily relying on other criteria. Thenew criterion is called Acyclicity. To the best of our knowledge, the class ofTGDs recognized by this criterion is the most general class known so far.3 Adding EGDsIn the previous sections, we have considered the case where all constraints areTGDs. In this section, we show how the chase termination problem radicallychanges when we allow also EGDs.Given a set of TGDs Σ for which the chase does not terminate, we can showthat the addition of EGDs to Σ may allow to have a terminating chase sequence.On the other hand, if the chase always terminates for Σ, adding EGDs to Σ maymake the chase of Σ non-terminating.Example 3. Consider the following two sets of constraints Σ3 (left) and Σ′3(right):r1 : ∀x A(x) → ∃y N(y) r′1 : ∀x N(x) → ∃y ∃z S(x, y, z)r2 : ∀x N(x) → ∃y E(x, y) r′2 : ∀x ∀y ∀z S(x, y, y) → N(y)r3 : ∀x∀y E(x, y) → N(y) r′3 : ∀x ∀y ∀z S(x, y, z) → T (x, y, z)r4 : ∀x∀y E(x, y) → x = y r′4 : ∀x ∀y ∀z T (x, y, z) → y = zand the database D = {N(a)}. The chase applied to the database D and the sub-set of TGDs {r1, r2} of Σ3 is not terminating as it introduces an infinite numberof tuples E(η1, η2), E(η3, η1), ... The introduction of the EGD r3 allows to havea terminating sequence, which produces the universal solution {N(a), E(a, a)}.The subset of TGDs {r′1, r′2, r′3} of Σ′3 is terminating for all database in-stances as recognized by several criteria (e.g., super-weak acyclicity). However,the chase fixpoint applied to Σ′3 and the database D is non-terminating as it intro-duces an infinite number of tuples S(a, η1, η1), T (a, η1, η1), N(η1), S(η1, η2, η2),T (η1, η2, η2), N(η2), .... 2As shown in the previous example, when for a set of dependencies it is notthe case that every chase sequence is terminating, the existence of at least oneterminating chase sequence, for every database, might still be guaranteed. Thus,one could extend rewriting techniques such as Adn+ to sets of TGDs and EGDs,in order to find whether there exists, for every database, at least one terminatingchase sequence. In order to cope with the aforementioned issues, algorithm Adn+can be extended in such a way that some adornments generated by rewritingTGDs are changed in order to satisfy the head equalities of EGDs. Specifically,the algorithm first tries to adorn as many EGDs as possible, and then considerthe rewriting of a single TGD. The basic idea is illustrated in the followingexample.Example 4. Consider the set of dependencies Σ3 of Example 3. As initially EGDr4 cannot be adorned, TGD r1 is rewritten into:∀x Ab(x) → ∃y Nf1(y)and r2 is rewritten into:∀x N b(x) → ∃y Ebf2(x, y)Now, EGD r4 can be used to “merge” distinct symbols. This is accomplishedby constructing the following adorned version of r4 using the atom Ebf2(x, y):∀x ∀y Ebf2(x, y) → x = yThis indicates that every occurrence of the symbol f2 in the obtained adorneddependencies has to be replaced with b, thereby obtaining:∀x Ab(x) → ∃y Nf1(y)∀x ∀y N b(x) → ∃y Ebb(x, y)∀x ∀y Ebb(x, y) → x = yThen, TGD r3 is adorned, obtaining:∀x∀y Ebb(x, y) → N b(y)Then, TGD r2 is adorned using atom Nf1(x), obtaining:∀x∀y Nf1(x) → ∃y Ef1f3(x, y)Again, EGD r4 is used, getting:∀x∀y Ef1f3(x, y) → x = yConsequently, f3 is replaced with f1 and we get:∀x Ab(x) → ∃y Nf1(y)∀x∀y N b(x) → ∃y Ebb(x, y)∀x∀y Ebb(x, y) → N b(y)∀x∀y Nf1(x) → ∃y Ef1f1(x, y)∀x∀y Ebb(x, y) → x = y∀x∀y Ef1f1(x, y) → x = yFinally, atom Ef1f1(x, y) is used to adorn r3, obtaining:∀x∀y Ef1f1(x, y) → Nf1(y)At this point, the rewriting stops, since no new adorned dependency can be con-structed. Intuitively, the algorithm identifies the existence of a terminating chasesequence because symbols f1, f2, f3, which represent nulls constructed w.r.t. thesymbols occurring in the body of the TGD, are not “cyclic” in the followingsense. Symbol f1 “depends on” symbol b in body(r1), f2 depends on symbol bin body(r2), and f3 depends on symbol f1. Since no pair of symbols fi, fj in thefinal set of dependency is such that fi depends on fj and vice versa, the set Σ3has a terminating chase sequence. 2References1. F. N. Afrati and P. G. Kolaitis. Repair checking in inconsistent databases: algo-rithms and complexity. In ICDT, pages 31–41, 2009.2. A. V. Aho, C. Beeri, and J. D. Ullman. The theory of joins in relational databases.ACM Trans. Database Syst., 4(3):297–314, 1979.3. M. Arenas, L. E. Bertossi, and J. Chomicki. Consistent query answers in inconsis-tent databases. In PODS, pages 68–79, 1999.4. C. Beeri and M. Y. Vardi. Formal systems for tuple and equality generatingdependencies. 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Rewriting-based Check of Chase Termination

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Rewriting-based Check of Chase Termination

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