This chapter introduces the mathematical concepts behind relational databases. It is not required reading, so if you bog down or want to get straight to some simple examples feel free to jump ahead to the next chapter and come back when you have more time and patience. This stuff is supposed to be fun! This material originally appeared as a part of Stefan Simkovics' Master's Thesis (). SQL has become the most popular relational query language. The name SQL is an abbreviation for Structured Query Language. In 1974 Donald Chamberlin and others defined the language SEQUEL (Structured English Query Language) at IBM Research. This language was first implemented in an IBM prototype called SEQUEL-XRM in 1974-75. In 1976-77 a revised version of SEQUEL called SEQUEL/2 was defined and the name was changed to SQL subsequently. A new prototype called System R was developed by IBM in 1977. System R implemented a large subset of SEQUEL/2 (now SQL) and a number of changes were made to SQL during the project. System R was installed in a number of user sites, both internal IBM sites and also some selected customer sites. Thanks to the success and acceptance of System R at those user sites IBM started to develop commercial products that implemented the SQL language based on the System R technology. Over the next years IBM and also a number of other vendors announced SQL products such as SQL/DS (IBM), DB2 (IBM), ORACLE (Oracle Corp.), DG/SQL (Data General Corp.), and SYBASE (Sybase Inc.). SQL is also an official standard now. In 1982 the American National Standards Institute (ANSI) chartered its Database Committee X3H2 to develop a proposal for a standard relational language. This proposal was ratified in 1986 and consisted essentially of the IBM dialect of SQL. In 1987 this ANSI standard was also accepted as an international standard by the International Organization for Standardization (ISO). This original standard version of SQL is often referred to, informally, as SQL/86. In 1989 the original standard was extended and this new standard is often, again informally, referred to as SQL/89. Also in 1989, a related standard called Database Language Embedded SQL (ESQL) was developed. The ISO and ANSI committees have been working for many years on the definition of a greatly expanded version of the original standard, referred to informally as SQL2 or SQL/92. This version became a ratified standard - International Standard ISO/IEC 9075:1992, Database Language SQL - in late 1992. SQL/92 is the version normally meant when people refer to the SQL standard. A detailed description of SQL/92 is given in . At the time of writing this document a new standard informally referred to as SQL3 is under development. It is planned to make SQL a Turing-complete language, i.e., all computable queries (e.g., recursive queries) will be possible. This has now been completed as SQL:2003. As mentioned before, SQL is a relational language. That means it is based on the relational data model first published by E.F. Codd in 1970. We will give a formal description of the relational model later (in ) but first we want to have a look at it from a more intuitive point of view. A relational database is a database that is perceived by its users as a collection of tables (and nothing else but tables). A table consists of rows and columns where each row represents a record and each column represents an attribute of the records contained in the table. shows an example of a database consisting of three tables: SUPPLIER is a table storing the number (SNO), the name (SNAME) and the city (CITY) of a supplier. PART is a table storing the number (PNO) the name (PNAME) and the price (PRICE) of a part. SELLS stores information about which part (PNO) is sold by which supplier (SNO). It serves in a sense to connect the other two tables together. SUPPLIER: SELLS: SNO | SNAME | CITY SNO | PNO ----+---------+-------- -----+----- 1 | Smith | London 1 | 1 2 | Jones | Paris 1 | 2 3 | Adams | Vienna 2 | 4 4 | Blake | Rome 3 | 1 3 | 3 4 | 2 PART: 4 | 3 PNO | PNAME | PRICE 4 | 4 ----+---------+--------- 1 | Screw | 10 2 | Nut | 8 3 | Bolt | 15 4 | Cam | 25 The tables PART and SUPPLIER can be regarded as entities and SELLS can be regarded as a relationship between a particular part and a particular supplier. As we will see later, SQL operates on tables like the ones just defined but before that we will study the theory of the relational model. The mathematical concept underlying the relational model is the set-theoretic relation which is a subset of the Cartesian product of a list of domains. This set-theoretic relation gives the model its name (do not confuse it with the relationship from the Entity-Relationship model). Formally a domain is simply a set of values. For example the set of integers is a domain. Also the set of character strings of length 20 and the real numbers are examples of domains. The Cartesian product of domains D1, D2, ... Dk, written D1 × D2 × ... × Dk is the set of all k-tuples v1, v2, ... vk, such that v1 ∈ D1, v2 ∈ D2, ... vk ∈ Dk. For example, when we have k=2, D1={0,1} and D2={a,b,c} then D1 × D2 is {(0,a),(0,b),(0,c),(1,a),(1,b),(1,c)}. A Relation is any subset of the Cartesian product of one or more domains: R ⊆ D1 × D2 × ... × Dk. For example {(0,a),(0,b),(1,a)} is a relation; it is in fact a subset of D1 × D2 mentioned above. The members of a relation are called tuples. Each relation of some Cartesian product D1 × D2 × ... × Dk is said to have arity k and is therefore a set of k-tuples. A relation can be viewed as a table (as we already did, remember where every tuple is represented by a row and every column corresponds to one component of a tuple. Giving names (called attributes) to the columns leads to the definition of a relation scheme. A relation scheme R is a finite set of attributes A1, A2, ... Ak. There is a domain Di, for each attribute Ai, 1 <= i <= k, where the values of the attributes are taken from. We often write a relation scheme as R(A1, A2, ... Ak). A relation scheme is just a kind of template whereas a relation is an instance of a relation scheme. The relation consists of tuples (and can therefore be viewed as a table); not so the relation scheme. We often talked about domains in the last section. Recall that a domain is, formally, just a set of values (e.g., the set of integers or the real numbers). In terms of database systems we often talk of data types instead of domains. When we define a table we have to make a decision about which attributes to include. Additionally we have to decide which kind of data is going to be stored as attribute values. For example the values of SNAME from the table SUPPLIER will be character strings, whereas SNO will store integers. We define this by assigning a data type to each attribute. The type of SNAME will be VARCHAR(20) (this is the SQL type for character strings of length <= 20), the type of SNO will be INTEGER. With the assignment of a data type we also have selected a domain for an attribute. The domain of SNAME is the set of all character strings of length <= 20, the domain of SNO is the set of all integer numbers. In the previous section () we defined the mathematical notion of the relational model. Now we know how the data can be stored using a relational data model but we do not know what to do with all these tables to retrieve something from the database yet. For example somebody could ask for the names of all suppliers that sell the part 'Screw'. Therefore two rather different kinds of notations for expressing operations on relations have been defined: The Relational Algebra which is an algebraic notation, where queries are expressed by applying specialized operators to the relations. The Relational Calculus which is a logical notation, where queries are expressed by formulating some logical restrictions that the tuples in the answer must satisfy. The Relational Algebra was introduced by E. F. Codd in 1972. It consists of a set of operations on relations: SELECT (σ): extracts tuples from a relation that satisfy a given restriction. Let R be a table that contains an attribute A. σA=a(R) = {t ∈ R ∣ t(A) = a} where t denotes a tuple of R and t(A) denotes the value of attribute A of tuple t. PROJECT (π): extracts specified attributes (columns) from a relation. Let R be a relation that contains an attribute X. πX(R) = {t(X) ∣ t ∈ R}, where t(X) denotes the value of attribute X of tuple t. PRODUCT (×): builds the Cartesian product of two relations. Let R be a table with arity k1 and let S be a table with arity k2. R × S is the set of all k1 + k2-tuples whose first k1 components form a tuple in R and whose last k2 components form a tuple in S. UNION (∪): builds the set-theoretic union of two tables. Given the tables R and S (both must have the same arity), the union R ∪ S is the set of tuples that are in R or S or both. INTERSECT (∩): builds the set-theoretic intersection of two tables. Given the tables R and S, R ∩ S is the set of tuples that are in R and in S. We again require that R and S have the same arity. DIFFERENCE (− or ∖): builds the set difference of two tables. Let R and S again be two tables with the same arity. R - S is the set of tuples in R but not in S. JOIN (∏): connects two tables by their common attributes. Let R be a table with the attributes A,B and C and let S be a table with the attributes C,D and E. There is one attribute common to both relations, the attribute C. R ∏ S = πR.A,R.B,R.C,S.D,S.E(σR.C=S.C(R × S)). What are we doing here? We first calculate the Cartesian product R × S. Then we select those tuples whose values for the common attribute C are equal (σR.C = S.C). Now we have a table that contains the attribute C two times and we correct this by projecting out the duplicate column. Let's have a look at the tables that are produced by evaluating the steps necessary for a join. Let the following two tables be given: R: S: A | B | C C | D | E ---+---+--- ---+---+--- 1 | 2 | 3 3 | a | b 4 | 5 | 6 6 | c | d 7 | 8 | 9 First we calculate the Cartesian product R × S and get: R x S: A | B | R.C | S.C | D | E ---+---+-----+-----+---+--- 1 | 2 | 3 | 3 | a | b 1 | 2 | 3 | 6 | c | d 4 | 5 | 6 | 3 | a | b 4 | 5 | 6 | 6 | c | d 7 | 8 | 9 | 3 | a | b 7 | 8 | 9 | 6 | c | d After the selection σR.C=S.C(R × S) we get: A | B | R.C | S.C | D | E ---+---+-----+-----+---+--- 1 | 2 | 3 | 3 | a | b 4 | 5 | 6 | 6 | c | d To remove the duplicate column S.C we project it out by the following operation: πR.A,R.B,R.C,S.D,S.E(σR.C=S.C(R × S)) and get: A | B | C | D | E ---+---+---+---+--- 1 | 2 | 3 | a | b 4 | 5 | 6 | c | d DIVIDE (÷): Let R be a table with the attributes A, B, C, and D and let S be a table with the attributes C and D. Then we define the division as: R ÷ S = {t ∣ ∀ ts ∈ S ∃ tr ∈ R such that tr(A,B)=t∧tr(C,D)=ts} where tr(x,y) denotes a tuple of table R that consists only of the components x and y. Note that the tuple t only consists of the components A and B of relation R. Given the following tables R: S: A | B | C | D C | D ---+---+---+--- ---+--- a | b | c | d c | d a | b | e | f e | f b | c | e | f e | d | c | d e | d | e | f a | b | d | e R ÷ S is derived as A | B ---+--- a | b e | d For a more detailed description and definition of the relational algebra refer to [] or []. Recall that we formulated all those relational operators to be able to retrieve data from the database. Let's return to our example from the previous section () where someone wanted to know the names of all suppliers that sell the part Screw. This question can be answered using relational algebra by the following operation: πSUPPLIER.SNAME(σPART.PNAME='Screw'(SUPPLIER ∏ SELLS ∏ PART)) We call such an operation a query. If we evaluate the above query against the our example tables () we will obtain the following result: SNAME ------- Smith Adams The relational calculus is based on the first order logic. There are two variants of the relational calculus: The Domain Relational Calculus (DRC), where variables stand for components (attributes) of the tuples. The Tuple Relational Calculus (TRC), where variables stand for tuples. We want to discuss the tuple relational calculus only because it is the one underlying the most relational languages. For a detailed discussion on DRC (and also TRC) see or . The queries used in TRC are of the following form: x(A) ∣ F(x) where x is a tuple variable A is a set of attributes and F is a formula. The resulting relation consists of all tuples t(A) that satisfy F(t). If we want to answer the question from example using TRC we formulate the following query: {x(SNAME) ∣ x ∈ SUPPLIER ∧ ∃ y ∈ SELLS ∃ z ∈ PART (y(SNO)=x(SNO) ∧ z(PNO)=y(PNO) ∧ z(PNAME)='Screw')} Evaluating the query against the tables from again leads to the same result as in . The relational algebra and the relational calculus have the same expressive power; i.e., all queries that can be formulated using relational algebra can also be formulated using the relational calculus and vice versa. This was first proved by E. F. Codd in 1972. This proof is based on an algorithm (Codd's reduction algorithm) by which an arbitrary expression of the relational calculus can be reduced to a semantically equivalent expression of relational algebra. For a more detailed discussion on that refer to and . It is sometimes said that languages based on the relational calculus are higher level or more declarative than languages based on relational algebra because the algebra (partially) specifies the order of operations while the calculus leaves it to a compiler or interpreter to determine the most efficient order of evaluation. As is the case with most modern relational languages, SQL is based on the tuple relational calculus. As a result every query that can be formulated using the tuple relational calculus (or equivalently, relational algebra) can also be formulated using SQL. There are, however, capabilities beyond the scope of relational algebra or calculus. Here is a list of some additional features provided by SQL that are not part of relational algebra or calculus: Commands for insertion, deletion or modification of data. Arithmetic capability: In SQL it is possible to involve arithmetic operations as well as comparisons, e.g.: A < B + 3. Note that + or other arithmetic operators appear neither in relational algebra nor in relational calculus. Assignment and Print Commands: It is possible to print a relation constructed by a query and to assign a computed relation to a relation name. Aggregate Functions: Operations such as average, sum, max, etc. can be applied to columns of a relation to obtain a single quantity. The most often used command in SQL is the SELECT statement, used to retrieve data. The syntax is: SELECT [ ALL | DISTINCT [ ON ( expression [, ...] ) ] ] * | expression [ [ AS ] output_name ] [, ...] [ INTO [ TEMPORARY | TEMP ] [ TABLE ] new_table ] [ FROM from_item [, ...] ] [ WHERE condition ] [ GROUP BY expression [, ...] ] [ HAVING condition [, ...] ] [ { UNION | INTERSECT | EXCEPT } [ ALL ] select ] [ ORDER BY expression [ ASC | DESC | USING operator ] [ NULLS { FIRST | LAST } ] [, ...] ] [ LIMIT { count | ALL } ] [ OFFSET start ] [ FOR { UPDATE | SHARE } [ OF table_name [, ...] ] [ NOWAIT ] [...] ] Now we will illustrate the complex syntax of the SELECT statement with various examples. The tables used for the examples are defined in . Here are some simple examples using a SELECT statement: To retrieve all tuples from table PART where the attribute PRICE is greater than 10 we formulate the following query: SELECT * FROM PART WHERE PRICE > 10; and get the table: PNO | PNAME | PRICE -----+---------+-------- 3 | Bolt | 15 4 | Cam | 25 Using * in the SELECT statement will deliver all attributes from the table. If we want to retrieve only the attributes PNAME and PRICE from table PART we use the statement: SELECT PNAME, PRICE FROM PART WHERE PRICE > 10; In this case the result is: PNAME | PRICE --------+-------- Bolt | 15 Cam | 25 Note that the SQL SELECT corresponds to the projection in relational algebra not to the selection (see for more details). The qualifications in the WHERE clause can also be logically connected using the keywords OR, AND, and NOT: SELECT PNAME, PRICE FROM PART WHERE PNAME = 'Bolt' AND (PRICE = 0 OR PRICE <= 15); will lead to the result: PNAME | PRICE --------+-------- Bolt | 15 Arithmetic operations can be used in the target list and in the WHERE clause. For example if we want to know how much it would cost if we take two pieces of a part we could use the following query: SELECT PNAME, PRICE * 2 AS DOUBLE FROM PART WHERE PRICE * 2 < 50; and we get: PNAME | DOUBLE --------+--------- Screw | 20 Nut | 16 Bolt | 30 Note that the word DOUBLE after the keyword AS is the new title of the second column. This technique can be used for every element of the target list to assign a new title to the resulting column. This new title is often referred to as alias. The alias cannot be used throughout the rest of the query. The following example shows how joins are realized in SQL. To join the three tables SUPPLIER, PART and SELLS over their common attributes we formulate the following statement: SELECT S.SNAME, P.PNAME FROM SUPPLIER S, PART P, SELLS SE WHERE S.SNO = SE.SNO AND P.PNO = SE.PNO; and get the following table as a result: SNAME | PNAME -------+------- Smith | Screw Smith | Nut Jones | Cam Adams | Screw Adams | Bolt Blake | Nut Blake | Bolt Blake | Cam In the FROM clause we introduced an alias name for every relation because there are common named attributes (SNO and PNO) among the relations. Now we can distinguish between the common named attributes by simply prefixing the attribute name with the alias name followed by a dot. The join is calculated in the same way as shown in . First the Cartesian product SUPPLIER × PART × SELLS is derived. Now only those tuples satisfying the conditions given in the WHERE clause are selected (i.e., the common named attributes have to be equal). Finally we project out all columns but S.SNAME and P.PNAME. Another way to perform joins is to use the SQL JOIN syntax as follows: SELECT sname, pname from supplier JOIN sells USING (sno) JOIN part USING (pno); giving again: sname | pname -------+------- Smith | Screw Adams | Screw Smith | Nut Blake | Nut Adams | Bolt Blake | Bolt Jones | Cam Blake | Cam (8 rows) A joined table, created using JOIN syntax, is a table reference list item that occurs in a FROM clause and before any WHERE, GROUP BY, or HAVING clause. Other table references, including table names or other JOIN clauses, can be included in the FROM clause if separated by commas. JOINed tables are logically like any other table listed in the FROM clause. SQL JOINs come in two main types, CROSS JOINs (unqualified joins) and qualified JOINs. Qualified joins can be further subdivided based on the way in which the join condition is specified (ON, USING, or NATURAL) and the way in which it is applied (INNER or OUTER join). CROSS JOIN T1 CROSS JOIN T2 A cross join takes two tables T1 and T2 having N and M rows respectively, and returns a joined table containing all N*M possible joined rows. For each row R1 of T1, each row R2 of T2 is joined with R1 to yield a joined table row JR consisting of all fields in R1 and R2. A CROSS JOIN is equivalent to an INNER JOIN ON TRUE. Qualified JOINs T1 NATURAL INNER LEFT RIGHT FULL OUTER JOIN T2 ON search condition USING ( join column list ) A qualified JOIN must specify its join condition by providing one (and only one) of NATURAL, ON, or USING. The ON clause takes a search condition, which is the same as in a WHERE clause. The USING clause takes a comma-separated list of column names, which the joined tables must have in common, and joins the tables on equality of those columns. NATURAL is shorthand for a USING clause that lists all the common column names of the two tables. A side-effect of both USING and NATURAL is that only one copy of each joined column is emitted into the result table (compare the relational-algebra definition of JOIN, shown earlier). INNER JOIN For each row R1 of T1, the joined table has a row for each row in T2 that satisfies the join condition with R1. The words INNER and OUTER are optional for all JOINs. INNER is the default. LEFT, RIGHT, and FULL imply an OUTER JOIN. LEFT OUTER JOIN First, an INNER JOIN is performed. Then, for each row in T1 that does not satisfy the join condition with any row in T2, an additional joined row is returned with null fields in the columns from T2. The joined table unconditionally has a row for each row in T1. RIGHT OUTER JOIN First, an INNER JOIN is performed. Then, for each row in T2 that does not satisfy the join condition with any row in T1, an additional joined row is returned with null fields in the columns from T1. The joined table unconditionally has a row for each row in T2. FULL OUTER JOIN First, an INNER JOIN is performed. Then, for each row in T1 that does not satisfy the join condition with any row in T2, an additional joined row is returned with null fields in the columns from T2. Also, for each row in T2 that does not satisfy the join condition with any row in T1, an additional joined row is returned with null fields in the columns from T1. The joined table unconditionally has a row for every row of T1 and a row for every row of T2. JOINs of all types can be chained together or nested where either or both of T1 and T2 can be JOINed tables. Parenthesis can be used around JOIN clauses to control the order of JOINs which are otherwise processed left to right. SQL provides aggregate functions such as AVG, COUNT, SUM, MIN, and MAX. The argument(s) of an aggregate function are evaluated at each row that satisfies the WHERE clause, and the aggregate function is calculated over this set of input values. Normally, an aggregate delivers a single result for a whole SELECT statement. But if grouping is specified in the query, then a separate calculation is done over the rows of each group, and an aggregate result is delivered per group (see next section). If we want to know the average cost of all parts in table PART we use the following query: SELECT AVG(PRICE) AS AVG_PRICE FROM PART; The result is: AVG_PRICE ----------- 14.5 If we want to know how many parts are defined in table PART we use the statement: SELECT COUNT(PNO) FROM PART; and get: COUNT ------- 4 SQL allows one to partition the tuples of a table into groups. Then the aggregate functions described above can be applied to the groups — i.e., the value of the aggregate function is no longer calculated over all the values of the specified column but over all values of a group. Thus the aggregate function is evaluated separately for every group. The partitioning of the tuples into groups is done by using the keywords GROUP BY followed by a list of attributes that define the groups. If we have GROUP BY A1, ⃛, Ak we partition the relation into groups, such that two tuples are in the same group if and only if they agree on all the attributes A1, ⃛, Ak. If we want to know how many parts are sold by every supplier we formulate the query: SELECT S.SNO, S.SNAME, COUNT(SE.PNO) FROM SUPPLIER S, SELLS SE WHERE S.SNO = SE.SNO GROUP BY S.SNO, S.SNAME; and get: SNO | SNAME | COUNT -----+-------+------- 1 | Smith | 2 2 | Jones | 1 3 | Adams | 2 4 | Blake | 3 Now let's have a look of what is happening here. First the join of the tables SUPPLIER and SELLS is derived: S.SNO | S.SNAME | SE.PNO -------+---------+-------- 1 | Smith | 1 1 | Smith | 2 2 | Jones | 4 3 | Adams | 1 3 | Adams | 3 4 | Blake | 2 4 | Blake | 3 4 | Blake | 4 Next we partition the tuples into groups by putting all tuples together that agree on both attributes S.SNO and S.SNAME: S.SNO | S.SNAME | SE.PNO -------+---------+-------- 1 | Smith | 1 | 2 -------------------------- 2 | Jones | 4 -------------------------- 3 | Adams | 1 | 3 -------------------------- 4 | Blake | 2 | 3 | 4 In our example we got four groups and now we can apply the aggregate function COUNT to every group leading to the final result of the query given above. Note that for a query using GROUP BY and aggregate functions to make sense, the target list can only refer directly to the attributes being grouped by. Other attributes can only be used inside the arguments of aggregate functions. Otherwise there would not be a unique value to associate with the other attributes. Also observe that it makes no sense to ask for an aggregate of an aggregate, e.g., AVG(MAX(sno)), because a SELECT only does one pass of grouping and aggregation. You can get a result of this kind by using a temporary table or a sub-SELECT in the FROM clause to do the first level of aggregation. The HAVING clause works much like the WHERE clause and is used to consider only those groups satisfying the qualification given in the HAVING clause. Essentially, WHERE filters out unwanted input rows before grouping and aggregation are done, whereas HAVING filters out unwanted group rows post-GROUP. Therefore, WHERE cannot refer to the results of aggregate functions. On the other hand, there's no point in writing a HAVING condition that doesn't involve an aggregate function! If your condition doesn't involve aggregates, you might as well write it in WHERE, and thereby avoid the computation of aggregates for groups that you're just going to throw away anyway. If we want only those suppliers selling more than one part we use the query: SELECT S.SNO, S.SNAME, COUNT(SE.PNO) FROM SUPPLIER S, SELLS SE WHERE S.SNO = SE.SNO GROUP BY S.SNO, S.SNAME HAVING COUNT(SE.PNO) > 1; and get: SNO | SNAME | COUNT -----+-------+------- 1 | Smith | 2 3 | Adams | 2 4 | Blake | 3 In the WHERE and HAVING clauses the use of subqueries (subselects) is allowed in every place where a value is expected. In this case the value must be derived by evaluating the subquery first. The usage of subqueries extends the expressive power of SQL. If we want to know all parts having a greater price than the part named 'Screw' we use the query: SELECT * FROM PART WHERE PRICE > (SELECT PRICE FROM PART WHERE PNAME='Screw'); The result is: PNO | PNAME | PRICE -----+---------+-------- 3 | Bolt | 15 4 | Cam | 25 When we look at the above query we can see the keyword SELECT two times. The first one at the beginning of the query - we will refer to it as outer SELECT - and the one in the WHERE clause which begins a nested query - we will refer to it as inner SELECT. For every tuple of the outer SELECT the inner SELECT has to be evaluated. After every evaluation we know the price of the tuple named 'Screw' and we can check if the price of the actual tuple is greater. (Actually, in this example the inner query need only be evaluated once, since it does not depend on the state of the outer query.) If we want to know all suppliers that do not sell any part (e.g., to be able to remove these suppliers from the database) we use: SELECT * FROM SUPPLIER S WHERE NOT EXISTS (SELECT * FROM SELLS SE WHERE SE.SNO = S.SNO); In our example the result will be empty because every supplier sells at least one part. Note that we use S.SNO from the outer SELECT within the WHERE clause of the inner SELECT. Here the subquery must be evaluated afresh for each tuple from the outer query, i.e., the value for S.SNO is always taken from the current tuple of the outer SELECT. A somewhat different way of using subqueries is to put them in the FROM clause. This is a useful feature because a subquery of this kind can output multiple columns and rows, whereas a subquery used in an expression must deliver just a single result. It also lets us get more than one round of grouping/aggregation without resorting to a temporary table. If we want to know the highest average part price among all our suppliers, we cannot write MAX(AVG(PRICE)), but we can write: SELECT MAX(subtable.avgprice) FROM (SELECT AVG(P.PRICE) AS avgprice FROM SUPPLIER S, PART P, SELLS SE WHERE S.SNO = SE.SNO AND P.PNO = SE.PNO GROUP BY S.SNO) subtable; The subquery returns one row per supplier (because of its GROUP BY) and then we aggregate over those rows in the outer query. These operations calculate the union, intersection and set theoretic difference of the tuples derived by two subqueries. The following query is an example for UNION: SELECT S.SNO, S.SNAME, S.CITY FROM SUPPLIER S WHERE S.SNAME = 'Jones' UNION SELECT S.SNO, S.SNAME, S.CITY FROM SUPPLIER S WHERE S.SNAME = 'Adams'; gives the result: SNO | SNAME | CITY -----+-------+-------- 2 | Jones | Paris 3 | Adams | Vienna Here is an example for INTERSECT: SELECT S.SNO, S.SNAME, S.CITY FROM SUPPLIER S WHERE S.SNO > 1 INTERSECT SELECT S.SNO, S.SNAME, S.CITY FROM SUPPLIER S WHERE S.SNO < 3; gives the result: SNO | SNAME | CITY -----+-------+-------- 2 | Jones | Paris The only tuple returned by both parts of the query is the one having SNO=2. Finally an example for EXCEPT: SELECT S.SNO, S.SNAME, S.CITY FROM SUPPLIER S WHERE S.SNO > 1 EXCEPT SELECT S.SNO, S.SNAME, S.CITY FROM SUPPLIER S WHERE S.SNO > 3; gives the result: SNO | SNAME | CITY -----+-------+-------- 2 | Jones | Paris 3 | Adams | Vienna There is a set of commands used for data definition included in the SQL language. The most fundamental command for data definition is the one that creates a new relation (a new table). The syntax of the CREATE TABLE command is: CREATE TABLE table_name (name_of_attr_1 type_of_attr_1 [, name_of_attr_2 type_of_attr_2 [, ...]]); To create the tables defined in the following SQL statements are used: CREATE TABLE SUPPLIER (SNO INTEGER, SNAME VARCHAR(20), CITY VARCHAR(20)); CREATE TABLE PART (PNO INTEGER, PNAME VARCHAR(20), PRICE DECIMAL(4 , 2)); CREATE TABLE SELLS (SNO INTEGER, PNO INTEGER); The following is a list of some data types that are supported by SQL: INTEGER: signed fullword binary integer (31 bits precision). SMALLINT: signed halfword binary integer (15 bits precision). DECIMAL (p[,q]): signed packed decimal number of up to p digits, with q digits to the right of the decimal point. If q is omitted it is assumed to be 0. FLOAT: signed doubleword floating point number. VARCHAR(n): varying length character string of maximum length n. CHAR(n): fixed length character string of length n. Indexes are used to speed up access to a relation. If a relation R has an index on attribute A then we can retrieve all tuples t having t(A) = a in time roughly proportional to the number of such tuples t rather than in time proportional to the size of R. To create an index in SQL the CREATE INDEX command is used. The syntax is: CREATE INDEX index_name ON table_name ( name_of_attribute ); To create an index named I on attribute SNAME of relation SUPPLIER we use the following statement: CREATE INDEX I ON SUPPLIER (SNAME); The created index is maintained automatically, i.e., whenever a new tuple is inserted into the relation SUPPLIER the index I is adapted. Note that the only changes a user can perceive when an index is present are increased speed for SELECT and decreases in speed of updates. A view can be regarded as a virtual table, i.e., a table that does not physically exist in the database but looks to the user as if it does. By contrast, when we talk of a base table there is really a physically stored counterpart of each row of the table somewhere in the physical storage. Views do not have their own, physically separate, distinguishable stored data. Instead, the system stores the definition of the view (i.e., the rules about how to access physically stored base tables in order to materialize the view) somewhere in the system catalogs (see ). For a discussion on different techniques to implement views refer to SIM98. In SQL the CREATE VIEW command is used to define a view. The syntax is: CREATE VIEW view_name AS select_stmt where select_stmt is a valid select statement as defined in . Note that select_stmt is not executed when the view is created. It is just stored in the system catalogs and is executed whenever a query against the view is made. Let the following view definition be given (we use the tables from again): CREATE VIEW London_Suppliers AS SELECT S.SNAME, P.PNAME FROM SUPPLIER S, PART P, SELLS SE WHERE S.SNO = SE.SNO AND P.PNO = SE.PNO AND S.CITY = 'London'; Now we can use this virtual relation London_Suppliers as if it were another base table: SELECT * FROM London_Suppliers WHERE PNAME = 'Screw'; which will return the following table: SNAME | PNAME -------+------- Smith | Screw To calculate this result the database system has to do a hidden access to the base tables SUPPLIER, SELLS and PART first. It does so by executing the query given in the view definition against those base tables. After that the additional qualifications (given in the query against the view) can be applied to obtain the resulting table. To destroy a table (including all tuples stored in that table) the DROP TABLE command is used: DROP TABLE table_name; To destroy the SUPPLIER table use the following statement: DROP TABLE SUPPLIER; The DROP INDEX command is used to destroy an index: DROP INDEX index_name; Finally to destroy a given view use the command DROP VIEW: DROP VIEW view_name; Once a table is created (see ), it can be filled with tuples using the command INSERT INTO. The syntax is: INSERT INTO table_name (name_of_attr_1 [, name_of_attr_2 [, ...]]) VALUES (val_attr_1 [, val_attr_2 [, ...]]); To insert the first tuple into the relation SUPPLIER (from ) we use the following statement: INSERT INTO SUPPLIER (SNO, SNAME, CITY) VALUES (1, 'Smith', 'London'); To insert the first tuple into the relation SELLS we use: INSERT INTO SELLS (SNO, PNO) VALUES (1, 1); To change one or more attribute values of tuples in a relation the UPDATE command is used. The syntax is: UPDATE table_name SET name_of_attr_1 = value_1 [, ... [, name_of_attr_k = value_k]] WHERE condition; To change the value of attribute PRICE of the part 'Screw' in the relation PART we use: UPDATE PART SET PRICE = 15 WHERE PNAME = 'Screw'; The new value of attribute PRICE of the tuple whose name is 'Screw' is now 15. To delete a tuple from a particular table use the command DELETE FROM. The syntax is: DELETE FROM table_name WHERE condition; To delete the supplier called 'Smith' of the table SUPPLIER the following statement is used: DELETE FROM SUPPLIER WHERE SNAME = 'Smith'; In every SQL database system system catalogs are used to keep track of which tables, views indexes etc. are defined in the database. These system catalogs can be queried as if they were normal relations. For example there is one catalog used for the definition of views. This catalog stores the query from the view definition. Whenever a query against a view is made, the system first gets the view definition query out of the catalog and materializes the view before proceeding with the user query (see for a more detailed description). For more information about system catalogs refer to . In this section we will sketch how SQL can be embedded into a host language (e.g., C). There are two main reasons why we want to use SQL from a host language: There are queries that cannot be formulated using pure SQL (i.e., recursive queries). To be able to perform such queries we need a host language with a greater expressive power than SQL. We simply want to access a database from some application that is written in the host language (e.g., a ticket reservation system with a graphical user interface is written in C and the information about which tickets are still left is stored in a database that can be accessed using embedded SQL). A program using embedded SQL in a host language consists of statements of the host language and of embedded SQL (ESQL) statements. Every ESQL statement begins with the keywords EXEC SQL. The ESQL statements are transformed to statements of the host language by a precompiler (which usually inserts calls to library routines that perform the various SQL commands). When we look at the examples throughout we realize that the result of the queries is very often a set of tuples. Most host languages are not designed to operate on sets so we need a mechanism to access every single tuple of the set of tuples returned by a SELECT statement. This mechanism can be provided by declaring a cursor. After that we can use the FETCH command to retrieve a tuple and set the cursor to the next tuple. For a detailed discussion on embedded SQL refer to , , or .