Relational model

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The relational model for database management is a data model based on predicate logic and set theory, it was invented by Edgar Codd.

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The model

The fundamental assumption of the relational model is that all data are represented as mathematical n-ary relations, an n-ary relation being a subset of the Cartesian product of n sets. In the mathematical model, reasoning about such data is done in two-valued predicate logic, meaning there are two possible evaluations for each proposition: either true or false (and in particular no third value such as unknown, often associated with the concept of NULL). Data are operated upon by means of a relational calculus or algebra, these being equivalent in expressive power.

The relational model of data permits the database designer to create a consistent, logical representation of information. Consistency is achieved by including declared constraints in the database design, which is usually referred to as the logical schema. The theory includes a process of database normalization whereby a design with certain desirable properties can be selected from a set of logically equivalent alternatives. The access plans and other implementation and operation details are handled by the DBMS engine, and are not be reflected in the logical model. This contrasts with common practice for SQL DBMSs in which performance tuning often requires changes to the logical model.

The basic relational building block is the domain or data type, usually abbreviated nowadays to type. A tuple is an unordered set of attribute values. An attribute is an ordered pair of attribute name and type name. An attribute value is an ordered pair of attribute and value, the value being of the type of the attribute.

A relation consists of a heading and a body. A heading is a set of attributes. A body (of an n-ary relation) is a set of n-tuples. The heading of the relation is also the heading of each of its tuples.

The concept of the relation as a set of unordered n-tuples is the distinguishing feature of the relational model. In mathematics the relation is a set of ordered n-tuples. It was one of E.F. Codd's great insights that using attribute names instead of an ordering would be so much more convenient, in general, in a computer language based on relations. An immediate and important consequence of this distinguishing feature is that in the relational model the Cartesian product becomes commutative.

A table is an accepted visual representation of a relation; a tuple is similar to the concept of row, but note that in the database language SQL the columns of a table are ordered.

A relvar (relation variable) is a named variable of some specific relation type, to which at all times some relation of that type is assigned.

The basic principle of the relational model is the Information Principle: all information is represented by data values in relations. In accordance with this Principle, a relational database is a set of relvars and the result of every query is presented as a relation.

The consistency of a relational database is enforced, not by rules built into the applications that use it, but rather by constraints, declared as part of the logical schema and enforced by the DBMS for all applications. In general constraints are expressed using relational comparison operators, of which just one, "is subset of" (⊆), is theoretically sufficient. In practice, several useful shorthands are expected to be available, of which the most important are candidate key (really, superkey) and foreign key constraints.

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Interpretation

To fully appreciate the relational model of data it is essential to understand the intended interpretation of a relation.

The body of a relation is sometimes called its extension. This is because it is to be interpreted as a representation of the extension of some predicate, this being the set of true propositions that can be formed by replacing each free variable in that predicate by a name (a term that designates something).

There is a one-to-one correspondence between the free variables of the predicate and the attribute names of the relation heading. Each tuple of the relation body provides attribute values to instantiate the predicate by substituting each of its free variables. The result is a proposition that is deemed, on account of the appearance of the tuple in the relation body, to be true. Contrariwise, every tuple whose heading conforms to that of the relation but which does not appear in the body is deemed to be false.

The assumption underlying that last sentence is known as the closed world assumption, and the relational model depends on it.

For a formal exposition of these ideas, see the section Set Theory Formulation, below.

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Competition

Other models are the hierarchical model and network model. Some systems using these older architectures are still in use today in data centers with high data volume needs or where existing systems are so complex and abstract it would be cost prohibitive to migrate to systems employing the relational model; also of note are newer object-oriented databases, even though many of them are DBMS-construction kits, rather than proper DBMSs.

The relational model was the first formal database model. After it was defined, informal models were made to describe hierarchical databases (the hierarchical model) and network databases (the network model). Hierarchical and network databases existed before relational databases, but were only described as models after the relational model was defined, in order to establish a basis for comparison.

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History

The relational model was invented by E.F. (Ted) Codd as a general model of data, and subsequently maintained and developed by Chris Date and Hugh Darwen among others. In The Third Manifesto (first published in 1995) Date and Darwen show how the relational model can accommodate certain desired object-oriented features without compromising its fundamental principles.

The foundation for the relational model included important works published by Georg Cantor (1874) and D.L Childs (1968). Cantor was a 19th century German mathematician who published a number of articles and was the principal creator of set theory. Childs is an American mathematician whose "Description of a Set-Theoretic Data Structure" was cited by Codd in his seminal 1970 paper "A Relational Model of Data for Large Shared Data Banks". Childs' STDS uses set theory as the basis for querying data using set operations such as union, intersection, domain, range, restriction, cardinality and Cartesian product. The use of sets and set operations provided independence from physical data structures, a pioneering concept at the time.

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Misimplementation

SQL, initially pushed as the standard language for relational databases, was actually always in violation of it. SQL DBMS's are thus not actually RDBMS's, and the current ISO SQL standard doesn't mention the relational model or use relational terms or concepts. (disputed — see [[: talk:Relational model#{{{1|Disputed}}}|talk page]])

The following deviations from the relational model have been noted in SQL:

  • Duplicate rows: The same row can appear more than once in an SQL table. The same tuple cannot appear more than once in a relation.
  • Anonymous columns: A column in an SQL table can be unnamed and thus unable to be referenced in expressions. The relational model requires every attribute to be named and referenceable.
  • Duplicate column names: Two or more columns of the same SQL table can have the same name and are thus unable to be referenced, on account of the obvious ambiguity. The relational model requires every attribute to be referenceable.
  • Column order significance: The order of columns in an SQL table is defined and significant, one consequence being that SQL's implementations of Cartesian product and union are both noncommutative. The relational model requires there to be no significance to any ordering of the attributes of a relation.
  • Views without CHECK OPTION: Updates to a view defined without CHECK OPTION can be accepted but the resulting update to the database does not necessarily have the expressed effect on its target. For example, an invocation of INSERT can be accepted but the inserted rows might not all appear in the view, or an invocation of UPDATE can result in rows disappearing from the view. The relational model requires updates to a view to have the same effect as if the view were a base relvar.
  • Columnless tables unrecognized: SQL requires every table to have at least one column, but there are two relations of degree zero (of cardinality one and zero) and they are needed to represent extensions of predicates that contain no free variables.
  • NULL: This special mark can appear instead of a value wherever a value can appear in SQL, in particular in place of a column value in some row. The deviation from the relational model arises from the fact that the implementation of this ad hoc concept in SQL involves the use of three-valued logic, under which the comparison of NULL with itself does not yield true but instead yields the third truth value, unknown; similarly the comparison NULL with something other than itself does not yield false but instead yields unknown. It is because of this behaviour in comparisons that NULL is described as a mark rather than a value. The relational model depends on the law of excluded middle under which anything that is not true is false and anything that is not false is true; it also requires every tuple in a relation body to have a value for every attribute of that relation. This particular deviation is disputed by some if only because E.F. Codd himself eventually advocated the use of special marks and a 4-valued logic, but this was based on his observation that there are two distinct reasons why one might want to use a special mark in place of a value, which led opponents of the use of such logics to discover more distinct reasons and at least as many as 19 have been noted, which would require a 21-valued logic. SQL itself uses NULL for several purposes other than to represent "value unknown". For example, the sum of the empty set is NULL, meaning zero, the average of the empty set is NULL, meaning undefined, and NULL appearing in the result of a LEFT JOIN can mean "no value because there is no matching row in the right-hand operand".
  • Concepts: SQL uses concepts "table", "column", "row" instead of "relvar", "attribute", "tuple". These are not merely differences in terminology. For example, a "table" may contain duplicate rows, whereas a relation in a relvar may not.
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Implementation

There have been several attempts to produce a true implementation of the relational database model as originally defined by Codd and explained by Date, Darwen and others, but none have been popular successes so far. Rel is one of the more recent attempts to do this.

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Controversies

Codd himself, some years after publication of his 1970 model, proposed a three-valued logic version of it in order to deal with missing information, and in his The Relational Model for Database Management Version 2 (1990) he went a step further with a four-valued logic version. But these have never been implemented, presumably because of attending complexity. SQL's NULL construct was intended to be part of a three-valued logic system, but fell short of that due to logical errors in the standard and in its implementations. See the section Misimplementation, above.

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Design

Database normalization is usually performed when designing a relational database, to improve the logical consistency of the database design and the transactional performance.

There are two commonly used systems of diagramming to aid in the visual representation of the relational model: the entity-relationship diagram (ERD), and the related IDEF diagram used in the IDEF1X method created by the U.S. Air Force based on ERDs.

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Example database

An idealized, very simple example of a description of some relvars and their attributes:

  • Customer(Customer ID, Tax ID, Name, Address, City, State, Zip, Phone)
  • Order(Order No, Customer ID, Invoice No, Date Placed, Date Promised, Terms, Status)
  • Order Line(Order No, Order Line No, Product Code, Qty)
  • Invoice(Invoice No, Customer ID, Order No, Date, Status)
  • Invoice Line(Invoice No, Line No, Product Code, Qty Shipped)
  • Product(Product Code, Product Description)

In this design we have six relvars: Customer, Order, Order Line, Invoice, Invoice Line and Product. The bold, underlined attributes are candidate keys. The non-bold, underlined attributes are foreign keys.

Usually one candidate key is arbitrarily chosen to be called the primary key and used in preference over the other candidate keys, which are then called alternate keys.

A candidate key is a unique identifier enforcing that no tuple will be duplicated; this would make the relation into something else, namely a bag, by violating the basic definition of a set. A key can be composite, that is, can be composed of several attributes. Below is a tabular depiction of a relation of our example Customer relvar; a relation can be thought of as a value that can be attributed to a relvar.

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Set Theory Formulation

Basic notions in the relational model are relation names and attribute names. We will represent these as strings such as "Person" and "name" and we will usually use the variables r, s, t, ... and a, b, c to range over them. Another basic notion is the set of atomic values that contains values such as numbers and strings.

Our first definition concerns the notion of tuple, which formalizes the notion of row or record in a table:

Def. A tuple is a partial function from attribute names to atomic values.
Def. A header is a finite set of attribute names.
Def.- The projection of a tuple t on a finite set of attributes A is t[A] = { (a, v) : (a, v) ∈ t, a ∈ A }.

The next definition defines relation which formalizes the contents of a table as it is defined in the relational model.

Def. A relation is a tuple (H, B) with H, the header, and B, the body, a set of tuples that all have the domain H.

Such a relation closely corresponds to what is usually called the extension of a predicate in first-order logic except that here we identify the places in the predicate with attribute names. Usually in the relational model a database schema is said to consist of a set of relation names, the headers that are associated with these names and the constraints that should hold for every instance of the database schema.

Def. A relation universe U over a header H is a non-empty set of relations with header H.
Def. A relation schema (H, C) consists of a header H and a predicate C(R) that is defined for all relations R with header H.
Def. A relation satisfies the relation schema (H, C) if it has header H and satisfies C.
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Key constraints and functional dependencies

One of the simplest and most important types of relation constraints is the key constraint. It tells us that in every instance of a certain relational schema the tuples can be identified by their values for certain attributes.

Def. A superkey is written as a finite set of attribute names.
Def. A superkey K holds in a relation (H, B) if K ⊆ H and there are no two distinct tuples t1 and t2 in B such that t1[K] = t2[K].
Def. A superkey holds in a relation universe U over a header H if it holds in all relations in U.
Def. - A superkey K holds as a candidate key for a relation universe U over H if it holds as a superkey for U and there is no proper subset of K that also holds as a superkey for U.
Def. A functional dependency (or FD for short) is written as X->Y with X and Y finite sets of attribute names.
Def. A functional dependency X->Y holds in a relation (H, B) if X and Y are subsets of H and for all tuples t1 and t2 in B it holds that if t1[X] = t2[X] then t1[Y] = t2[Y]
Def. A functional dependency X->Y holds in a relation universe U over a header H if it holds in all relations in U.
Def. A functional dependency is trivial under a header H if it holds in all relation universes over H.
Theorem A FD X->Y is trivial under a header H iff Y ⊆ X ⊆ H.
Theorem A superkey K holds in a relation universe U over H iff K ⊆ H and K->H holds in U.
Def. (Armstrong's rules) Let S be a set of FDs then the closure of S under a header H, written as S+, is the smallest superset of S such that:
(reflexivity) if Y ⊆ X ⊆ H then X->Y in S+
(transitivity) if X->Y in S+ and Y->Z in S+ then X->Z in S+
(augmentation) if X->Y in S+ and Z ⊆ H then X∪Z -> Y∪Z in S+
Theorem Armstrong's rules are sound and complete, i.e., given a header H and a set S of FDs that only contain subsets of H then the FD X->Y is in S+ iff it holds in all relation universes over H in which all FDs in S hold.
Def. If X is a finite set of attributes and S a finite set of FDs then the completion of X under S, written as X+, is the smallest superset of X such that:
if Y->Z in S and Y ⊆ X+ then Z ⊆ X+

The completion of an attribute set can be used to compute if a certain dependency is in the closure of a set of FDs.

Theorem Given a header H and a set S of FDs that only contain subsets of H it holds that X->Y is in S+ iff Y ⊆ X+.
Algorithm (deriving candidate keys from FDs) 
      INPUT: a set S of FDs that contain only subsets of a header H
      OUTPUT: the set C of superkeys that hold as candidate keys in
              all relation universes over H in which all FDs in S hold
      begin
        C := ∅;          // found candidate keys
        Q := { H };      // superkeys that contain candidate keys
        while Q <> ∅ do
          let K be some element from Q;
          Q := Q - { K };  
          minimal := true;
          for each X->Y in S do 
            K' := (K - Y) ∪ X;   // derive new superkey
            if K' ⊂ K then
              minimal := false;
              Q := Q ∪ { K' };
            end if
          end for
          if minimal and there is not a subset of K in C then
            remove all supersets of K from C;
            C := C ∪ { K };
          end if
        end while
      end
Def. Given a header H and a set of FDs S that only contain subsets of H an irreducible cover of S is a set T of FDs such that
  1. S+ = T+
  2. there is no proper subset U of T such that S+ = U+,
  3. if X->Y in T then Y is a singleton set and
  4. if X->Y in T and Z a proper subset of X then Z->Y is not in S+.
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See also

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References

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External links

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