DOCSLIB.ORG

(BCNF) Decomposition of a Relation Scheme Example (Same

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
(BCNF) Decomposition of a Relation Scheme Example (Same Functional Dependencies (Review) • A functional dependency X → Y holds over relation Schema Refinement schema R if, for every allowable instance r of R: and Normalization t1 ∈ r, t2 ∈ r, π (t1) = π (t2) πX πX implies Y (t1) = Y (t2) (where t1 and t2 are tuples;X and Y are sets of attributes) CS 186, Fall 2002, Lecture 6 R&G Chapter 15 • In other words: X → Y means Given any two tuples in r, if the X values are the same, then the Y values must also be the same. (but not vice versa) Nobody realizes that some people • Can read “→” as “determines” expend tremendous energy merely to be normal. Albert Camus Normal Forms Boyce-Codd Normal Form (BCNF) • Back to schema refinement… • Reln R with FDs F is in BCNF if, for all X → A in F+ • Q1: is any refinement is needed??! –A ∈ X (called a trivial FD), or • If a relation is in a normal form (BCNF, 3NF etc.): – X is a superkey for R. – we know that certain problems are avoided/minimized. • In other words: “R is in BCNF if the only non-trivial FDs – helps decide whether decomposing a relation is useful. over R are key constraints.” • Role of FDs in detecting redundancy: • If R in BCNF, then every field of every tuple records – Consider a relation R with 3 attributes, ABC. information that cannot be inferred using FDs alone. • No (non-trivial) FDs hold: There is no redundancy here. – Say we know FD X → A holds this example relation: X Y A •Given A → B: If A is not a key, then several tuples could have the same A value, and if so, they’ll all have the same B value! • Can you guess the value of the x y1 a • 1st Normal Form – all attributes are atomic missing attribute? x y2 ? st nd rd • 1 ⊃2 (of historical interest) ⊃ 3 ⊃ Boyce-Codd ⊃ … •Yes, so relation is not in BCNF Decomposition of a Relation Scheme Example (same as before) S N L R W H • If a relation is not in a desired normal form, it can be 123-22-3666 Attishoo 48 8 10 40 decomposed into multiple relations that each are in that normal form. 231-31-5368 Smiley 22 8 10 30 131-24-3650 Smethurst 35 5 7 30 • Suppose that relation R contains attributes A1 ... An. A Hourly_Emps decomposition of R consists of replacing R by two or more 434-26-3751 Guldu 35 5 7 32 relations such that: 612-67-4134 Madayan 35 8 10 40 – Each new relation scheme contains a subset of the attributes of R, and •SNLRWH hasFDs S → SNLRWH and R → W – Every attribute of R appears as an attribute of at least one of the new relations. • Q: Is this relation in BCNF? No, The second FD causes a violation; W values repeatedly associated with R values. 1 Decomposing a Relation Problems with Decompositions • Easiest fix is to create a relation RW to store these associations, and to remove W from the main • There are three potential problems to consider: schema: 1) May be impossible to reconstruct the original relation! S N L R H (Lossiness) 123-22-3666 Attishoo 48 8 40 • Fortunately, not in the SNLRWH example. R W 231-31-5368 Smiley 22 8 30 2) Dependency checking may require joins. 810 131-24-3650 Smethurst 35 5 30 • Fortunately, not in the SNLRWH example. 57 434-26-3751 Guldu 35 5 32 3) Some queries become more expensive. 612-67-4134 Madayan 35 8 40 Wages • e.g., How much does Guldu earn? Hourly_Emps2 Tradeoff: Must consider these issues vs. •Q: Are both of these relations are now in BCNF? redundancy. •Decompositions should be used only when needed. –Q: potential problems of decomposition? An Aside – Natural Join Lossless Decomposition (example) S N L R H • Natural Join is a fundamental operator of relational 123-22-3666 Attishoo 48 8 40 algebra R W 231-31-5368 Smiley 22 8 30 ba 810 131-24-3650 Smethurst 35 5 30 ba • Semantics of R ba S are: 57 – Compute Cartesian Product of R and S 434-26-3751 Guldu 35 5 32 – Select out those tuples where the common attributes of R 612-67-4134 Madayan 35 8 40 and S have the same values – Keep all unique attributes of these tuples plus one copy of S N L R W H each of the common attributes. 123-22-3666 Attishoo 48 8 10 40 • More on this in the next lecture 231-31-5368 Smiley 22 8 10 30 = 131-24-3650 Smethurst 35 5 7 30 434-26-3751 Guldu 35 5 7 32 612-67-4134 Madayan 35 8 10 40 Lossy Decomposition (example) Lossless Join Decompositions A B C A B B C 1 2 3 12 2 3 • Decomposition of R into X and Y is lossless-join w.r.t. 4 5 6 a set of FDs F if, for every instance r that satisfies F: 45 5 6 π π 7 2 8 X(r) Y (r) = r 2 8 72 ⊆ π π • It is always true that r X (r) Y (r) A → B; C → B – In general, the other direction does not hold! If it does, the decomposition is lossless-join. A B C A B B C • Definition extended to decomposition into 3 or more 123 relations in a straightforward way. 12 2 3 456 ba 5 6 = • It is essential that all decompositions used to deal with 45 728 redundancy be lossless! (Avoids Problem #1) 2 8 72 1 2 8 7 2 3 2 More on Lossless Decomposition Lossless Decomposition (example) • The decomposition of R into X and Y is lossless with respect to F if and only if the A B C A C B C closure of F contains: 1 2 3 1 3 2 3 4 5 6 5 6 X ∩ Y → X, or 4 6 7 2 8 7 8 2 8 X ∩ Y → Y in example: decomposing ABC into AB and BC is A → B; C → B lossy, because intersection (i.e., “B”) is not a key of either resulting relation. A C B C A B C 1 3 1 2 3 • Useful result: If W → Z holds over R and W ∩ Z is 2 3 4 6 ba 4 5 6 empty, then decomposition of R into R-Z and WZ is 5 6 = 7 8 7 2 8 2 8 loss-less. But, now we can’t check A → B without doing a join! Dependency Preserving Decomposition Dependency Preserving Decompositions (Contd.) • Dependency preserving decomposition (Intuitive): • Decomposition of R into X and Y is dependency – If R is decomposed into X, Y and Z, and we ∪ + + preserving if (FX FY ) = F enforce the FDs that hold individually on X, on Y – i.e., if we consider only dependencies in the closure F + that and on Z, then all FDs that were given to hold can be checked in X without considering Y, and in Y without on R must also hold. (Avoids Problem #2 on considering X, these imply all dependencies in F +. our list.) • Important to consider F + in this definition: → → → • Projection of set of FDs F : If R is decomposed into –ABC, A B, B C, C A, decomposed into AB and BC. – Is this dependency preserving? Is C → A preserved????? X and Y the projection of F on X (denoted FX ) is the set of FDs U → V in F+ (closure of F , not just F ) such •note: F+ contains F ∪ {A → C, B → A, C → B}, so… that all of the attributes U, V are in X. (same holds → → → → for Y of course) •FAB contains A B and B A; FBC contains B C and C B •So, (FAB ∪ FBC)+ contains C → A Decomposition into BCNF • Consider relation R with FDs F. If X → Y violates BCNF, BCNF and Dependency Preservation decompose R into R - Y and XY (guaranteed to be loss-less). – Repeated application of this idea will give us a collection of relations • In general, there may not be a dependency preserving that are in BCNF; lossless join decomposition, and guaranteed to decomposition into BCNF. terminate. – e.g., CSZ, CS → Z, Z → C → → → – e.g., CSJDPQV, key C, JP C, SD P, J S – Can’t decompose while preserving 1st FD; not in BCNF. – {contractid, supplierid, projectid,deptid,partid, qty, value} • Similarly, decomposition of CSJDPQV into SDP, JS and –To deal with SD → P, decompose into SDP, CSJDQV. CJDQV is not dependency preserving (w.r.t. the FDs –To deal with J → S, decompose CSJDQV into JS and CJDQV JP → C, SD → P and J → S). – So we end up with: SDP, JS, and CJDQV • {contractid, supplierid, projectid,deptid,partid, qty, value} • Note: several dependencies may cause violation of BCNF. The – However, it is a lossless join decomposition. order in which we ``deal with’’ them could lead to very different sets of relations! – In this case, adding JPC to the collection of relations gives us a dependency preserving decomposition. • but JPC tuples are stored only for checking the f.d. (Redundancy!) 3 Third Normal Form (3NF) What Does 3NF Achieve? → • Reln R with FDs F is in 3NF if, for all X → A in F+ • If 3NF violated by X A, one of the following holds: – X is a subset of some key K (“partial dependency”) A ∈ X (called a trivial FD), or • We store (X, A) pairs redundantly. X is a superkey of R, or • e.g.
Load more

Recommended publications
  • Functional Dependencies
    Basics of FDs Manipulating FDs Closures and Keys Minimal Bases Functional Dependencies T. M. Murali October 19, 21 2009 T. M. Murali October 19, 21 2009 CS 4604: Functional Dependencies Students(Id, Name) Advisors(Id, Name) Advises(StudentId, AdvisorId) Favourite(StudentId, AdvisorId) I Suppose we perversely decide to convert Students, Advises, and Favourite into one relation. Students(Id, Name, AdvisorId, AdvisorName, FavouriteAdvisorId) Basics of FDs Manipulating FDs Closures and Keys Minimal Bases Example Name Id Name Id Students Advises Advisors Favourite I Convert to relations: T. M. Murali October 19, 21 2009 CS 4604: Functional Dependencies I Suppose we perversely decide to convert Students, Advises, and Favourite into one relation. Students(Id, Name, AdvisorId, AdvisorName, FavouriteAdvisorId) Basics of FDs Manipulating FDs Closures and Keys Minimal Bases Example Name Id Name Id Students Advises Advisors Favourite I Convert to relations: Students(Id, Name) Advisors(Id, Name) Advises(StudentId, AdvisorId) Favourite(StudentId, AdvisorId) T. M. Murali October 19, 21 2009 CS 4604: Functional Dependencies Basics of FDs Manipulating FDs Closures and Keys Minimal Bases Example Name Id Name Id Students Advises Advisors Favourite I Convert to relations: Students(Id, Name) Advisors(Id, Name) Advises(StudentId, AdvisorId) Favourite(StudentId, AdvisorId) I Suppose we perversely decide to convert Students, Advises, and Favourite into one relation. Students(Id, Name, AdvisorId, AdvisorName, FavouriteAdvisorId) T. M. Murali October 19, 21 2009 CS 4604: Functional Dependencies Name and FavouriteAdvisorId. Id ! Name Id ! FavouriteAdvisorId AdvisorId ! AdvisorName I Can we say Id ! AdvisorId? NO! Id is not a key for the Students relation. I What is the key for the Students? fId, AdvisorIdg.
    [Show full text]
  • Relational Database Design Chapter 7
    Chapter 7: Relational Database Design Chapter 7: Relational Database Design First Normal Form Pitfalls in Relational Database Design Functional Dependencies Decomposition Boyce-Codd Normal Form Third Normal Form Multivalued Dependencies and Fourth Normal Form Overall Database Design Process Database System Concepts 7.2 ©Silberschatz, Korth and Sudarshan 1 First Normal Form Domain is atomic if its elements are considered to be indivisible units + Examples of non-atomic domains: Set of names, composite attributes Identification numbers like CS101 that can be broken up into parts A relational schema R is in first normal form if the domains of all attributes of R are atomic Non-atomic values complicate storage and encourage redundant (repeated) storage of data + E.g. Set of accounts stored with each customer, and set of owners stored with each account + We assume all relations are in first normal form (revisit this in Chapter 9 on Object Relational Databases) Database System Concepts 7.3 ©Silberschatz, Korth and Sudarshan First Normal Form (Contd.) Atomicity is actually a property of how the elements of the domain are used. + E.g. Strings would normally be considered indivisible + Suppose that students are given roll numbers which are strings of the form CS0012 or EE1127 + If the first two characters are extracted to find the department, the domain of roll numbers is not atomic. + Doing so is a bad idea: leads to encoding of information in application program rather than in the database. Database System Concepts 7.4 ©Silberschatz, Korth and Sudarshan 2 Pitfalls in Relational Database Design Relational database design requires that we find a “good” collection of relation schemas.
    [Show full text]
  • Fundamentals of Database Systems [Normalization – II]
    Outline First Normal Form Second Normal Form Third Normal Form Boyce-Codd Normal Form Fundamentals of Database Systems [Normalization { II] Malay Bhattacharyya Assistant Professor Machine Intelligence Unit Indian Statistical Institute, Kolkata October, 2019 Outline First Normal Form Second Normal Form Third Normal Form Boyce-Codd Normal Form 1 First Normal Form 2 Second Normal Form 3 Third Normal Form 4 Boyce-Codd Normal Form Outline First Normal Form Second Normal Form Third Normal Form Boyce-Codd Normal Form First normal form The domain (or value set) of an attribute defines the set of values it might contain. A domain is atomic if elements of the domain are considered to be indivisible units. Company Make Company Make Maruti WagonR, Ertiga Maruti WagonR, Ertiga Honda City Honda City Tesla RAV4 Tesla, Toyota RAV4 Toyota RAV4 BMW X1 BMW X1 Only Company has atomic domain None of the attributes have atomic domains Outline First Normal Form Second Normal Form Third Normal Form Boyce-Codd Normal Form First normal form Definition (First normal form (1NF)) A relational schema R is in 1NF iff the domains of all attributes in R are atomic. The advantages of 1NF are as follows: It eliminates redundancy It eliminates repeating groups. Note: In practice, 1NF includes a few more practical constraints like each attribute must be unique, no tuples are duplicated, and no columns are duplicated. Outline First Normal Form Second Normal Form Third Normal Form Boyce-Codd Normal Form First normal form The following relation is not in 1NF because the attribute Model is not atomic. Company Country Make Model Distributor Maruti India WagonR LXI, VXI Carwala Maruti India WagonR LXI Bhalla Maruti India Ertiga VXI Bhalla Honda Japan City SV Bhalla Tesla USA RAV4 EV CarTrade Toyota Japan RAV4 EV CarTrade BMW Germany X1 Expedition CarTrade We can convert this relation into 1NF in two ways!!! Outline First Normal Form Second Normal Form Third Normal Form Boyce-Codd Normal Form First normal form Approach 1: Break the tuples containing non-atomic values into multiple tuples.
    [Show full text]
  • A Hybrid Approach to Functional Dependency Discovery
    A Hybrid Approach to Functional Dependency Discovery Thorsten Papenbrock Felix Naumann Hasso Plattner Institute (HPI) Hasso Plattner Institute (HPI) 14482 Potsdam, Germany 14482 Potsdam, Germany [email protected] [email protected] ABSTRACT concrete metadata. One reason is that they depend not only Functional dependencies are structural metadata that can on a schema but also on a concrete relational instance. For be used for schema normalization, data integration, data example, child ! teacher might hold for kindergarten chil- cleansing, and many other data management tasks. Despite dren, but it does not hold for high-school children. Conse- their importance, the functional dependencies of a specific quently, functional dependencies also change over time when dataset are usually unknown and almost impossible to dis- data is extended, altered, or merged with other datasets. cover manually. For this reason, database research has pro- Therefore, discovery algorithms are needed that reveal all posed various algorithms for functional dependency discov- functional dependencies of a given dataset. ery. None, however, are able to process datasets of typical Due to the importance of functional dependencies, many real-world size, e.g., datasets with more than 50 attributes discovery algorithms have already been proposed. Unfor- and a million records. tunately, none of them is able to process datasets of real- world size, i.e., datasets with more than 50 columns and We present a hybrid discovery algorithm called HyFD, which combines fast approximation techniques with efficient a million rows, as a recent study showed [20]. Because validation techniques in order to find all minimal functional the need for normalization, cleansing, and query optimiza- dependencies in a given dataset.
    [Show full text]
  • Database Normalization
    Outline Data Redundancy Normalization and Denormalization Normal Forms Database Management Systems Database Normalization Malay Bhattacharyya Assistant Professor Machine Intelligence Unit and Centre for Artificial Intelligence and Machine Learning Indian Statistical Institute, Kolkata February, 2020 Malay Bhattacharyya Database Management Systems Outline Data Redundancy Normalization and Denormalization Normal Forms 1 Data Redundancy 2 Normalization and Denormalization 3 Normal Forms First Normal Form Second Normal Form Third Normal Form Boyce-Codd Normal Form Elementary Key Normal Form Fourth Normal Form Fifth Normal Form Domain Key Normal Form Sixth Normal Form Malay Bhattacharyya Database Management Systems These issues can be addressed by decomposing the database { normalization forces this!!! Outline Data Redundancy Normalization and Denormalization Normal Forms Redundancy in databases Redundancy in a database denotes the repetition of stored data Redundancy might cause various anomalies and problems pertaining to storage requirements: Insertion anomalies: It may be impossible to store certain information without storing some other, unrelated information. Deletion anomalies: It may be impossible to delete certain information without losing some other, unrelated information. Update anomalies: If one copy of such repeated data is updated, all copies need to be updated to prevent inconsistency. Increasing storage requirements: The storage requirements may increase over time. Malay Bhattacharyya Database Management Systems Outline Data Redundancy Normalization and Denormalization Normal Forms Redundancy in databases Redundancy in a database denotes the repetition of stored data Redundancy might cause various anomalies and problems pertaining to storage requirements: Insertion anomalies: It may be impossible to store certain information without storing some other, unrelated information. Deletion anomalies: It may be impossible to delete certain information without losing some other, unrelated information.
    [Show full text]
  • The Theory of Functional and Template Dependencs[Es*
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 17 (1982) 317-331 317 North-Holland Publishing Company THE THEORY OF FUNCTIONAL AND TEMPLATE DEPENDENCS[ES* Fereidoon SADRI** Department of Computer Science, Princeton University, Princeton, NJ 08540, U.S.A. Jeffrey D. ULLMAN*** Department of ComprrtzrScience, Stanfoc:;lUniversity, Stanford, CA 94305, U.S.A. Communicated by M. Nivat Received June 1980 Revised December 1980 Abstract. Template dependencies were introduced by Sadri and Ullman [ 171 to generuize existing forms of data dependencies. It was hoped that by studyin;g a 4arge and natural class of dependen- cies, we could solve the inference problem for these dependencies, while that problem ‘was elusive for restricted subsets of the template dependencjes, such as embedded multivalued dependencies. At about the same time, other generalizations of known clependency forms were developed, such as the implicational dependencies of Fagin [l 1j and the algebraic dependencies of Vannakakis and Papadimitriou [20]. Unlike the template dependencies, the latter forms include the functional dependencies as special cases. In this paper we show that no nontrivial functional dependency follows from template dependencies, and we characterize those template dependencies that follow from functional dependencies. We then give a complete set of axioms for reasoning about combinations of functional and template dependencies. As a result, template dependencies augmented1 by functional dependencies can serve as a sub!;tltute for the more genera1 implicational or algebraic dependenc:ies, providing the same ability to rl:present those dependencies that appear ‘in nature’,, while providing a somewhat simpler notation and set of axioms than the more genera4 classes.
    [Show full text]
  • SEMESTER-II Functional Dependencies
    For SEMESTER-II Paper II : Database Management Systems(CODE: CSC-RC-2016) Topics: Functional dependencies, Normalization and Normal forms up to 3NF Functional Dependencies A functional dependency (FD) is a relationship between two attributes, typically between the primary key and other non-key attributes within a table. A functional dependency denoted by X Y , is an association between two sets of attribute X and Y. Here, X is called the determinant, and Y is called the dependent. For example, SIN ———-> Name, Address, Birthdate Here, SIN determines Name, Address and Birthdate.So, SIN is the determinant and Name, Address and Birthdate are the dependents. Rules of Functional Dependencies 1. Reflexive rule : If Y is a subset of X, then X determines Y . 2. Augmentation rule: It is also known as a partial dependency, says if X determines Y, then XZ determines YZ for any Z 3. Transitivity rule: Transitivity says if X determines Y, and Y determines Z, then X must also determine Z Types of functional dependency The following are types functional dependency in DBMS: 1. Fully-Functional Dependency 2. Partial Dependency 3. Transitive Dependency 4. Trivial Dependency 5. Multivalued Dependency Full functional Dependency A functional dependency XY is said to be a full functional dependency, if removal of any attribute A from X, the dependency does not hold any more. i.e. Y is fully functional dependent on X, if it is Functionally Dependent on X and not on any of the proper subset of X. For example, {Emp_num,Proj_num} Hour Is a full functional dependency. Here, Hour is the working time by an employee in a project.
    [Show full text]
  • Chapter 7+8+9
    7+8+9: Functional Dependencies and Normalization how “good” how do we know the is our data database design won’t model cause any problems? design? what do we grouping of know about attributes so far has the quality been intuitive - how of the do we validate the logical design? schema? 8 issue of quality of design an illustration - figure 7.1b STOCK (Store, Product, Price, Quantity, Location, Discount, Sq_ft, Manager) are there data redundancies? for price? location? quantity? discount? repeated appearances of a data value ≠ data redundancy unneeded repetition that does what is data not add new meaning = data redundancy? redundancy data redundancy → modification anomalies are there data redundancies? yes - for price, location, and discount insertion anomaly - adding a washing machine with a price (cannot add without store number) update anomaly - store 11 moves to Cincinnati (requires updating multiple rows) deletion anomaly - store 17 is closed (pricing data about vacuum cleaner is lost) decomposition of the STOCK instance no data redundancies/ modification anomalies in STORE or PRODUCT data redundancies/ modification anomalies persist in INVENTORY normalized free from data redundancies/modification anomalies relational schema reverse-engineered from tables design-specific ER diagram reverse-engineered from relational schema how do we systematically identify data redundancies? how do we know how to decompose the base relation schema under investigation? how we know that the data decomposition is correct and complete? the seeds of data redundancy are undesirable functional dependencies. functional dependencies relationship between attributes that if we are given the value of one of the attributes we can look up the value FD of the other the building block of normalization principles functional dependencies an attribute A (atomic or composite) in a relation schema R functionally determines another attribute B (atomic or composite) in R if: ! for a given value a1 of A there is a single, specific value b1 of B in every relation state ri of R.
    [Show full text]
  • Functional Dependencies and Normalization Juliana Freire
    Design Theory for Relational Databases: Functional Dependencies and Normalization Juliana Freire Some slides adapted from L. Delcambre, R. Ramakrishnan, G. Lindstrom, J. Ullman and Silberschatz, Korth and Sudarshan Relational Database Design • Use the Entity Relationship (ER) model to reason about your data---structure and relationships, then translate model into a relational schema (more on this later) • Specify relational schema directly – Like what you do when you design the data structures for a program Pitfalls in Relational Database Design • Relational database design requires that we find a “good” collection of relation schemas • A bad design may lead to – Repetition of Information – Inability to represent certain information • Design Goals: – Avoid redundant data – Ensure that relationships among attributes are represented – Facilitate the checking of updates for violation of database integrity constraints Design Choices: Small vs. Large Schemas Which design do you like better? Why? EMPLOYEE(ENAME, SSN, ADDRESS, PNUMBER) PROJECT(PNAME, PNUMBER, PMGRSSN) EMP_PROJ(ENAME, SSN, ADDRESS, PNUMBER, PNAME, PMGRSSN) An employee can be assigned to at most one project, many employees participate in a project What’s wrong? EMP(ENAME, SSN, ADDRESS, PNUM, PNAME, PMGRSSN)" The description of the project (the name" and the manager of the project) is repeated" for every employee that works in that department." Redundancy! " The project is described redundantly." This leads to update anomalies." EMP(ENAME, SSN, ADDRESS, PNUM, PNAME, PMGRSSN)" Update
    [Show full text]
  • Functional Dependency and Normalization for Relational Databases
    Functional Dependency and Normalization for Relational Databases Introduction: Relational database design ultimately produces a set of relations. The implicit goals of the design activity are: information preservation and minimum redundancy. Informal Design Guidelines for Relation Schemas Four informal guidelines that may be used as measures to determine the quality of relation schema design: Making sure that the semantics of the attributes is clear in the schema Reducing the redundant information in tuples Reducing the NULL values in tuples Disallowing the possibility of generating spurious tuples Imparting Clear Semantics to Attributes in Relations The semantics of a relation refers to its meaning resulting from the interpretation of attribute values in a tuple. The relational schema design should have a clear meaning. Guideline 1 1. Design a relation schema so that it is easy to explain. 2. Do not combine attributes from multiple entity types and relationship types into a single relation. Redundant Information in Tuples and Update Anomalies One goal of schema design is to minimize the storage space used by the base relations (and hence the corresponding files). Grouping attributes into relation schemas has a significant effect on storage space Storing natural joins of base relations leads to an additional problem referred to as update anomalies. These are: insertion anomalies, deletion anomalies, and modification anomalies. Insertion Anomalies happen: when insertion of a new tuple is not done properly and will therefore can make the database become inconsistent. When the insertion of a new tuple introduces a NULL value (for example a department in which no employee works as of yet).
    [Show full text]
  • Chapter 4: Functional Dependencies
    Chapter 4 FUNCTIONAL DEPENDENCIES Two primary purposes of databases are to attenuate data redundancy and enhance data reliability. Any a priorz’ knowledge of restrictions or constraints on permissible sets of data has considerable usefulness in reaching these goals, as we shall see. Data dependencies are one way to formulate such ad- vance knowledge. In this chapter we shall cover one type of data dependency, the functional dependency. In Chapter 7 we cover two other types of data dependencies, the multivalued and join dependencies. Other general classes of data dependencies are treated in Chapter 14. 4.1 DEFINITIONS We discussed keys in Chapter 1. Functional dependencies are a generaliza- tion. Table 4.1 depicts the relation assign(PILOT FLIGHT DATE DEPARTS). Assign tells which pilot flies a given flight on a given day, and what time the flight leaves. Not every combination of pilots, flights, dates, and times is allowable in assign. The following restrictions appIy, among others. 1. For each flight there is exactly one time. 2. For any given pilot, date, and time, there is only one flight. 3. For a given flight and date, there is only one pilot. These restrictions are examples of functional dependencies. Informally, a functional dependency occurs when the values of a tuple on one set of at- tributes uniquely determine the values on another set of attributes. Our restrictions can be phrased as 1. TIME functionally depends on FLIGHT, 2. FLIGHT functionally depends on {PILOT, DATE, TIME}, and 3. PILOT functionally depends on {FLIGHT, DATE). 42 Definitions 43 Table 4.1 The relation assign(PILOT FLIGHT DATE DEPARTS).
    [Show full text]
  • Functional Dependencies and Normalization
    Functional Dependencies and Normalization Definition 1 Let R be a relation scheme and let X ⊆ R and Y ⊆ R. We say that a relation instance r(R) satisfies a functional dependency X ! Y if for every pair of tuples t1 2 r and t2 2 r, if t1[X] = t2[X] then t1[Y ] = t2[Y ]. Definition 2 An instance r of relation scheme R is called a legal instance if it is a true reflection of the mini-world facts it represents (i.e. it satisfies all constraints imposed on it in the real world). Definition 3 Let R be a relation scheme and let X ⊆ R and Y ⊆ R. Then X ! Y , a functional dependency on scheme R, is valid if every legal instance r(R) satisfies X ! Y . Mathematical Properties of Functional Dependencies Definition 4 Let F be a set of FDs on scheme R and f be another FD on R. Then, F implies f, denoted by F j= f, if every relation instance r(R) that satisfies all FDs in F also satisfies f. Definition 5 Let F be a set of FDs on scheme R. Then, the closure of F , denoted by F +, is the set of all FDs implied by F . Definition 6 Let F and G be sets of FDs on scheme R. Then, F and G are equivalent, denoted by F ≡ G, if F j= G and G j= F . Inference rules for FDs (Armstrong's Axioms) Reflexivity: If Y ⊆ X then, X ! Y . Such FDs are called trivial FDs. Augmentation: If X ! Y , then XZ ! Y Z.
    [Show full text]