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CPSC 504: Data Management Review of Relational Model 1/2 Laks V.S. Lakshmanan Dept. of CS UBC The Basic Model • A DB = a (finite) set of (finite) relations. • A relation has a well-defined schema – its set of attributes, each with its domain: e.g., dom(songName), dom(Artist), dom(AlbumNo), dom(fileFormat), … • A relation – a subset of the Cartesian (aka cross) product of its attribute domains: • e.g., myfavs Í dom(AlbumNo) x dom(songName) x dom(Artist) x dom(fileFormat). How do you get there? • In practice, you often use the ER (for entity-relationship) model as an informal spec of a design of a relational database schema. • Schema = structure or table template. • Relation = table instance. Example ER schema A schema in the ER model is specified using an ER diagram. Actor Movie Stars in Name DOB Addr Title Year Dir Role Some constraints • Each relation has a key: candidate keys, primary keys, superkeys, …? • Functional dependency (FD): formalization and generalization of key. • r(R), X, Y Í R. r |= XàY iff any pair of tuples in r that agree on X, necessarily agree on Y. • e.g.: AlbumNoàArtist, AlbumNo, songNameàAlbumNo, …, fileFormat. • {AlbumNo, songName} – a key, but not AlbumNo. • Note: ER diagrams can express a rich class of (but not all) constraints. More on FDs • FDs help determine keys: there are efficient algorithms as well as an assortment of (computationally) hard problems. • FDs are pointers to redundancies in data: e.g., same Artist (value) will appear many times, once for each song in an album of that artist. • Why is redundancy (in representation) bad? • $efficient algorithms for redundancy removal – the so -called normal forms. • There is a full assortment of them – 1NF, 2NF, 3NF, BCNF, 4NF, … • FDs/keys help with indexing, with query optimization, with data exchange, etc. More on Normal Forms • 1NF really says tuples must be “flat”. • Other NFs really seek to capture different degrees of redundancy (and eliminate them). • 3NF, BCNF – some of the most important ones in practice. • Normalization algorithms: start from U, the set of all attributes, and figure out which attr sets should stay together, which ones should split up. Some more on FDs • Constraints such as FDs are called integrity constraints. Why? • ICs are statements in predicate logic. • What does an FD really say? • Need to reason with ICs, so: – Can determine keys, – Can check if certain ICs are enforced. – Can understand interaction of ICs. Last bits on FDs • FDs admit a sound and complete axiom system: – Reflexivity: if X Í Y, then YàX. – Augmentation: if WàZ and X Í Y, then WYàZX. – Transitivity: if XàY and YàZ, then XàZ. • Not all classes of ICs admit a S&C axiom system. • Some consequences of the above axiom system for FDs. ICs, Tableaux, and Queries • An FD can also be stated in a “pictorial” form: Schema/column A B C D headers x1 x2 _ _ Pattern rows. x1 x3 _ _ x2 = x3 Summary row/ Consequent/ Conclusion. ICs, Tableuax, and Queries • Tableaux are general; can express what conditions must hold when certain patterns are present in data. • Can express equality generating dependencies (egd’s) and tuple generating dependencies (tgd’s). • Can also express queries (by using summary row for saying what you want in the output). ICs, Tableaux, and Queries • Tableaux can also be expressed in the form of rules, also known as “Horn clauses”. • Examples: X2 = X3 ß r(X1,X2,_,_), r(X1,X3,_,_). Which really is saying … X2 = X3 ß r(X1,X2,Y3,Y4), r(X1,X3,Y5,Y6). Logically speaking, what are these rules saying? p(X,Y) ß a(X,Z), b(Z,Y), a(Y,W), W>100. Some more on NFs – 3NF • Def. of 3NF: r(R), given set of applicable FDs F, is in 3NF iff for every XàA that holds for r, X is a superkey for r according to F OR A belongs to some key. • e.g.: r(AlbumNo, Artist, songName, fileFormat) is not in 3NF, while r1(AlbumNo, Artist), r2(AlbumNo, songName, fileFormat) are. Why? Some more on NFs – BCNF • Def. of BCNF: r(R), given applicable FDs F, is in BCNF iff for every XàA that holds for r, X is a superkey for r according to F. • Clearly stronger than 3NF. • e.g.: {S(treet), (s)T(ate), Z(ip)}, with STàZ, ZàT. – it’s in 3NF. • Not in BCNF. What’s the redundancy here? • {ZT, ZS} is in BCNF. • Can you anticipate any problems with the design {ZT, ZS}? • Majority of 3NF designs in practice tend to be in BCNF. More on ICs • Inclusion depencency (IND): r1(Emp, Dept, Sal), r2(Mgr, Dept, Budget). r2[Mgr] Í r1[Emp]: every manager is an employee. • INDs are a generalization of referential integrity constraints. • FDs, INDs. – most fundamental. • What can we say about reasoning with INDs, or with INDs + FDs? More on INDs • Good news -- $a complete axiomatization. • Reflexivity: r[X]Ír[X]. • Projection+permutation: if r[A1,…,Am] Ís[B1…Bm], then r[Ai1…Aik] Ís[Bi1…Bik]. • Transitivity: if r[X] Ís[Y] and s[Y] Ít[Z], then r[X] Ít[Z]. • But, while testing whether F |= f can be done in linear time, testing I |= i is PSPACE- complete! Story of FDs+INDs • Recall: FDs admit complete axiomatization, as do INDs. • But, put together, there is no complete (k-ary) axiomatization for INDs+FDs! • We have seen some (very) positive results and some negative results. • What approaches can you think of for tackling the negative situations? Last bits on ICs • There is a rich landscape of lot more classes of ICs. • Fundamentally split into two syntactic classes: – Equality generating dependencies (egds), e.g., FDs. – Tuple generating dependencies (tgds), e.g., INDs. • The value is NOT that these fancier ICs arise as is in practice. • Rather, they arise in solving various problems: e.g., integrating data from diverse sources. • E.g. tgd: (\forall M,D,B) [r2(M, D, B) à (\exists S) r1(M, D, S)]. • What is it saying, logically speaking? More Reading • J.D. Ullman: Principles of Database and Knowledge-Base Systems, vol. I and II. CS Press, 1988. • R. Ramakrishnan & J. Gehrke. Database Management Sytems. Mc-Graw Hill, latest edition. • S. Abiteboul, R. Hull, and V. Vianu. Foundations of Databases. 1995. Available online from DBLP. Some Resources for Research Literature • DBLP • Citeseer • ACM digital library • If you aren’t already a member, consider becoming ACM student member: really a great deal! Some top conferences/journals for DMM Research • ACM SIGMOD/PODS • VLDB / PVLDB • IEEE ICDE • ICDT / EDBT • KDD, PKDD, ICDM, SDM, … • CIKM • WWW, WSDM, ICWSM, • TODS, IS, VLDBJ, TKDE, TCS, JACM, JCSS, I&C, …

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