The problem of updating a database through a set of views consists in propagat-ing updates of the views to the base relations over which the view relations are defined, so that the changes to the database reflect exactly those to the views. This is a classical problem in database research, known as the view update prob

Feinerer, Ingo

Franconi, Enrico

Guagliardo, Paolo

Ingo Feinerer

Enrico Franconi

Paolo Guagliardo

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Lossless Selection Views under Constraints

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     Edinburgh Research Explorer                                      Lossless Selection Views under ConstraintsCitation for published version:Feinerer, I, Guagliardo, P & Franconi, E 2014, Lossless Selection Views under Constraints. in Proceedingsof the 8th Alberto Mendelzon Workshop on Foundations of Data Management, Cartagena de Indias,Colombia, June 4-6, 2014..Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Peer reviewed versionPublished In:Proceedings of the 8th Alberto Mendelzon Workshop on Foundations of Data Management, Cartagena deIndias, Colombia, June 4-6, 2014.General rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 05. Apr. 2019Lossless Selection Views under ConstraintsIngo Feinerer1, Enrico Franconi2, and Paolo Guagliardo21 Vienna University of Technologyfeinerer@logic.at2 Free University of Bozen-Bolzano{franconi,guagliardo}@inf.unibz.it1 IntroductionThe problem of updating a database through a set of views consists in propagat-ing updates of the views to the base relations over which the view relations aredefined, so that the changes to the database reflect exactly those to the views.This is a classical problem in database research, known as the view update prob-lem [1], which in recent years has received renewed attention.View updates can be consistently propagated in an unambiguous way underthe condition that the mapping between database and view relations is lossless,and that each database relation can be expressed in terms of the view relationsby means of a query, in much the same way the latter are defined from the former[3]. In such a context, database decompositions play an important role, in thattheir losslessness is associated with the existence of an explicit reconstructionoperator that prescribes how a database relation can be rebuilt from the piecesinto which it has been decomposed. In the case of horizontal decomposition, inwhich a relation is split into sub-relations containing each a subset of its rows,the reconstruction operator is the union.The study of horizontal decomposition in the literature has mostly focused onsettings where data values can only be compared for equality. However, most real-world applications make use of data values coming from domains with a richerstructure (e.g., ordering) on which a variety of other restrictions besides equalitycan be expressed (e.g., that of being within a range or above a threshold). It istherefore of practical interest to consider a scenario where some of the attributesin the database schema are interpreted over a specific domain, such as the realsor the integers, on which a set of special predicates (e.g., smaller/greater than)and functions (e.g., addition/subtraction) are defined, according to a first-orderlanguage C. We consider horizontal decomposition in such a setting and invest-igate its losslessness in the presence of integrity constraints. The only restrictionwe impose on the language C is that of being closed under negation.2 FundamentalsWe consider a source relation symbol R under so-called conditional domain con-straints (CDCs) of the form∀x, y .R(x, y) ∧ λ(x)→ δ(y) , (1)where λ(x) is a Boolean combination of equalities x = a between a variable fromx and a constant, and δ(y) ∈ C. The variables in x and y are associated withnon-interpreted and interpreted positions (that is, attributes) of R, respectively.By means of formulae in C, CDCs restrict the admissible values at interpretedpositions, whenever a certain condition holds on the non-interpreted ones.We consider a decomposed schema V consisting of view symbols having thesame arity as R, each of which is defined by a formula of the form∀x, y . V (x, y)↔ R(x, y) ∧ (λ(x) opσ(y)) , (2)where λ(x) is as in (1), σ(y) ∈ C, and op is ∧. A set Σ of formulae of the form(2), one for each V ∈ V, specifies a horizontal decomposition of R.3 LosslessnessLossless horizontal decomposition under CDCs can be characterised in terms ofunsatisfiability in C, which gives a decision procedure for losslessness wheneverthe satisfiability of sets of formulae in C is decidable. The main idea is as follows:1) With each equality between a variable and a constant a we associate a pro-positional variable pai , whose truth-value indicates whether the value at (non-interpreted) position i is a.2) For a given valuation α of such propositional variables, we consider the setconsisting of the C-formulae in the r.h.s. of all the CDCs that are “applicable”(i.e., the propositional representation of their l.h.s. is true) under α and thenegation of any C-formula δ(y) appearing in some selection in Σ where thepropositional representation of λ(x) is satisfied by α.3) Checking losslessness is equivalent to checking that there exists no valuationα for which the above set of C-formulae is satisfiable.The class of Unit Two-Variable Per Inequality formulae (UTVPIs) is a frag-ment of linear arithmetic over the integers for which the satisfiability problem isdecidable in polynomial time [4]. UTVPI have the form ay+by′ ≤ d, where y andy′ are integer variables, a, b ∈ {−1, 0, 1} and d ∈ Z. We refer to Boolean combin-ations of UTVPIs as BUTVPIs; the satisfiability problem for sets of BUTVPIs isNP-complete [5]. It can be shown that, when C is the language of either UTVPIsor BUTVPIs, the problem of deciding losslessness is co-NP-complete.4 SeparabilityWe also investigated lossless horizontal decomposition under CDCs in combina-tion with traditional integrity constraints, showing that functional dependencies(FDs) do not interact with CDCs and can therefore be allowed without any re-striction, whereas this is not the case for unary inclusion dependencies (UINDs).The main technical tool we employ in our study is the notion of separability : aclass of constraints is separable from CDCs if, after making explicit the result oftheir interaction, captured by a suitable finite set S of sound inference rules, wecan focus solely on CDCs and disregard the other constraints, as far as lossless2horizontal decomposition is concerned. Given a horizontal decomposition Σ anda combination ∆ of CDCs with other constraints that are separable from themw.r.t. S, we can check whether Σ is lossless under ∆ as follows:1) compute the deductive closure ∆∗ of ∆ w.r.t. S, which makes explicit theinteraction between CDCs and the other constraints in ∆ by adding entailedconstraints;2) by using the technique of Section 3, check whether Σ is lossless under theset cdc(∆∗) obtained from ∆∗ by discarding all of the constraints that arenot CDCs.A UIND R[i] ⊆ R[j] is satisfied if, for every source instance I, every value inthe i-th column of the extension RI of R under I appears also in the j-th columnof RI . Let yi and yj be the variables associated with the interpreted positions iand j of R, respectively; by means of the following domain propagation rule> → δ(yj) R[i] ⊆ R[j]> → δ(yi) , (dp)it is possible to derive a set of CDCs that fully captures the interaction betweena given set of UINDs and opportunely restricted CDCs w.r.t. lossless horizontaldecomposition, which makes possible to employ the general technique for decid-ing losslessness also in the presence of UINDs.5 DiscussionThe results on losslessness and separability briefly reported in this short paperare under revision for journal publication; a preliminary version appeared in [2].We remark that our technique for checking losslessness can also be applied whenop in (2) is ∨.We are currently investigating the possibility of generalising the separabilityresults for UINDs to arbitrary inclusion dependencies (INDs). We also stronglybelieve that equalities between non-interpreted variables (i.e., from x) in (1) and(2) can be allowed and our technique extended by representing such equalities bymeans of additional propositional variables and by adding suitable propositionalaxioms to handle transitivity and symmetry.References1. Bancilhon, F., Spyratos, N.: Update semantics of relational views. ACM Trans.Database Syst. 6(4), 557–575 (Dec 1981)2. Feinerer, I., Franconi, E., Guagliardo, P.: Lossless horizontal decomposition withdomain constraints on interpreted attributes. In: Big Data – Proc. of BNCOD 2013,LNCS, vol. 7968, pp. 77–91. Springer (2013)3. Franconi, E., Guagliardo, P.: Effectively updatable conjunctive views. In: Proc. ofAMW 2013. CEUR Workshop Proceedings, vol. 1087. CEUR-WS.org (2013)4. Schutt, A., Stuckey, P.J.: Incremental satisfiability and implication for UTVPI con-straints. INFORMS Journal on Computing 22(4), 514–527 (2010)5. Seshia, S.A., Subramani, K., Bryant, R.E.: On solving boolean combinations ofUTVPI constraints. JSAT 3(1-2), 67–90 (2007)3

Lossless Selection Views under Constraints

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