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Chapter: Fundamentals of Database Systems : The Relational Data Model and SQL : The Relational Algebra and Relational Calculus

The Tuple Relational Calculus

1. Tuple Variables and Range Relations 2. Expressions and Formulas in Tuple Relational Calculus 3. The Existential and Universal Quantifiers 4. Sample Queries in Tuple Relational Calculus 5. Notation for Query Graphs 6. Transforming the Universal and Existential Quantifiers 7. Using the Universal Quantifier in Queries 8. Safe Expressions

The Tuple Relational Calculus


In this and the next section, we introduce another formal query language for the relational model called relational calculus. This section introduces the language known as tuple relational calculus, and Section 6.7 introduces a variation called domain relational calculus. In both variations of relational calculus, we write one declarative expression to specify a retrieval request; hence, there is no description of how, or in what order, to evaluate a query. A calculus expression specifies what is to be retrieved rather than how to retrieve it. Therefore, the relational calculus is considered to be a nonprocedural language. This differs from relational algebra, where we must write a sequence of operations to specify a retrieval request in a particular order of applying the operations; thus, it can be considered as a procedural way of stating a query. It is possible to nest algebra operations to form a single expression; however, a certain order among the operations is always explicitly specified in a relational algebra expression. This order also influences the strategy for evaluating the query. A calculus expression may be written in different ways, but the way it is writ-ten has no bearing on how a query should be evaluated.


It has been shown that any retrieval that can be specified in the basic relational algebra can also be specified in relational calculus, and vice versa; in other words, the expressive power of the languages is identical. This led to the definition of the concept of a relationally complete language. A relational query language L is considered relationally complete if we can express in L any query that can be expressed in relational calculus. Relational completeness has become an important basis for comparing the expressive power of high-level query languages. However, as we saw in Section 6.4, certain frequently required queries in database applications cannot be expressed in basic relational algebra or calculus. Most relational query languages are relationally complete but have more expressive power than relational algebra or relational calculus because of additional operations such as aggregate functions, grouping, and ordering. As we mentioned in the introduction to this chapter, the relational calculus is important for two reasons. First, it has a firm basis in mathematical logic. Second, the standard query language (SQL) for RDBMSs has some of its foundations in the tuple relational calculus.

Our examples refer to the database shown in Figures 3.6 and 3.7. We will use the same queries that were used in Section 6.5. Sections 6.6.6, 6.6.7, and 6.6.8 discuss dealing with universal quantifiers and safety of expression issues. (Students inter-ested in a basic introduction to tuple relational calculus may skip these sections.)


1. Tuple Variables and Range Relations


The tuple relational calculus is based on specifying a number of tuple variables. Each tuple variable usually ranges over a particular database relation, meaning that the variable may take as its value any individual tuple from that relation. A simple tuple relational calculus query is of the form:


{t | COND(t)}


where t is a tuple variable and COND(t) is a conditional (Boolean) expression involving t that evaluates to either TRUE or FALSE for different assignments of tuples to the variable t. The result of such a query is the set of all tuples t that evalu-ate COND(t) to TRUE. These tuples are said to satisfy COND(t). For example, to find all employees whose salary is above $50,000, we can write the following tuple calcu-lus expression:


{t | EMPLOYEE(t) AND t.Salary>50000}


The condition EMPLOYEE(t) specifies that the range relation of tuple variable t is EMPLOYEE. Each EMPLOYEE tuple t that satisfies the condition t.Salary>50000 will be retrieved. Notice that t.Salary references attribute Salary of tuple variable t; this notation resembles how attribute names are qualified with relation names or aliases in SQL, as we saw in Chapter 4. In the notation of Chapter 3, t.Salary is the same as writing t[Salary].


The above query retrieves all attribute values for each selected EMPLOYEE tuple t. To retrieve only some of the attributes—say, the first and last names—we write


{t.Fname, t.Lname | EMPLOYEE(t) AND t.Salary>50000}


Informally, we need to specify the following information in a tuple relational calcu-lus expression:


              For each tuple variable t, the range relation R of t. This value is specified by a condition of the form R(t). If we do not specify a range relation, then the variable t will range over all possible tuples “in the universe” as it is not restricted to any one relation.


              A condition to select particular combinations of tuples. As tuple variables range over their respective range relations, the condition is evaluated for every possible combination of tuples to identify the selected combinations for which the condition evaluates to TRUE.


                    A set of attributes to be retrieved, the requested attributes. The values of these attributes are retrieved for each selected combination of tuples. 

Before we discuss the formal syntax of tuple relational calculus, consider another query.

Query 0. Retrieve the birth date and address of the employee (or employees) whose name is John B. Smith.


Q0: {t.Bdate, t.Address | EMPLOYEE(t) AND t.Fname=‘John’ AND t.Minit=‘B’

AND t.Lname=‘Smith’}


In tuple relational calculus, we first specify the requested attributes t.Bdate and t.Address for each selected tuple t. Then we specify the condition for selecting a tuple following the bar (|)—namely, that t be a tuple of the EMPLOYEE relation whose Fname, Minit, and Lname attribute values are ‘John’, ‘B’, and ‘Smith’, respectively.


2. Expressions and Formulas in Tuple Relational Calculus


A general expression of the tuple relational calculus is of the form


{t1.Aj, t2.Ak, ..., tn.Am | COND(t1, t2, ..., tn, tn+1, tn+2, ..., tn+m)}

where t1, t2, ..., tn, tn+1, ..., tn+m are tuple variables, each Ai is an attribute of the rela-tion on which ti ranges, and COND is a condition or formula.13 of the tuple rela-tional calculus. A formula is made up of predicate calculus atoms, which can be one of the following:


                                                    An atom of the form R(ti), where R is a relation name and ti is a tuple vari-able. This atom identifies the range of the tuple variable ti as the relation whose name is R. It evaluates to TRUE if ti is a tuple in the relation R, and evaluates to FALSE otherwise.


                                                    An atom of the form ti.A op tj.B, where op is one of the comparison opera-tors in the set {=, <, , >, , }, ti and tj are tuple variables, A is an attribute of the relation on which ti ranges, and B is an attribute of the relation on which tj ranges.


                                                    An atom of the form ti.A op c or c op tj.B, where op is one of the compari-son operators in the set {=, <, , >, , }, ti and tj are tuple variables, A is an attribute of the relation on which ti ranges, B is an attribute of the relation on which tj ranges, and c is a constant value.


Each of the preceding atoms evaluates to either TRUE or FALSE for a specific combi-nation of tuples; this is called the truth value of an atom. In general, a tuple variable t ranges over all possible tuples in the universe. For atoms of the form R(t), if t is assigned to a tuple that is a member of the specified relation R, the atom is TRUE; oth-erwise, it is FALSE. In atoms of types 2 and 3, if the tuple variables are assigned to tuples such that the values of the specified attributes of the tuples satisfy the condi-tion, then the atom is TRUE.


A formula (Boolean condition) is made up of one or more atoms connected via the logical operators AND, OR, and NOT and is defined recursively by Rules 1 and 2 as follows:


       Rule 1: Every atom is a formula.

              Rule 2: If F1 and F2 are formulas, then so are (F1 AND F2), (F1 OR F2), NOT (F1), and NOT (F2). The truth values of these formulas are derived from their component formulas F1 and F2 as follows:


                    (F1 AND F2) is TRUE if both F1 and F2 are TRUE; otherwise, it is FALSE.


                    (F1 OR F2) is FALSE if both F1 and F2 are FALSE; otherwise, it is TRUE.


                    NOT (F1) is TRUE if F1 is FALSE; it is FALSE if F1 is TRUE.

                    NOT (F2) is TRUE if F2 is FALSE; it is FALSE if F2 is TRUE.


3. The Existential and Universal Quantifiers


In addition, two special symbols called quantifiers can appear in formulas; these are the universal quantifier () and the existential quantifier (). Truth values for formulas with quantifiers are described in Rules 3 and 4 below; first, however, we need to define the concepts of free and bound tuple variables in a formula. Informally, a tuple variable t is bound if it is quantified, meaning that it appears in an (t) or (t) clause; otherwise, it is free. Formally, we define a tuple variable in a formula as free or bound according to the following rules:


              An occurrence of a tuple variable in a formula F that is an atom is free in F.


              An occurrence of a tuple variable t is free or bound in a formula made up of logical connectives—(F1 AND F2), (F1 OR F2), NOT(F1), and NOT(F2)— depending on whether it is free or bound in F1 or F2 (if it occurs in either). Notice that in a formula of the form F = (F1 AND F2) or F = (F1 OR F2), a tuple variable may be free in F1 and bound in F2, or vice versa; in this case, one occurrence of the tuple variable is bound and the other is free in F.


              All free occurrences of a tuple variable t in F are bound in a formula F of the form F = ( t)(F) or F = (t)(F). The tuple variable is bound to the quanti-fier specified in F . For example, consider the following formulas:


F1 : d.Dname=‘Research’


F2 : ( t)(d.Dnumber=t.Dno)

F3 : (d)(d.Mgr_ssn=‘333445555’)


The tuple variable d is free in both F1 and F2, whereas it is bound to the () quan-tifier in F3. Variable t is bound to the () quantifier in F2.


We can now give Rules 3 and 4 for the definition of a formula we started earlier:


              Rule 3: If F is a formula, then so is (t)(F), where t is a tuple variable. The formula (t)(F) is TRUE if the formula F evaluates to TRUE for some (at least one) tuple assigned to free occurrences of t in F; otherwise, (t)(F) is FALSE.


              Rule 4: If F is a formula, then so is (t)(F), where t is a tuple variable. The formula (t)(F) is TRUE if the formula F evaluates to TRUE for every tuple

(in the universe) assigned to free occurrences of t in F; otherwise, (t)(F) is FALSE.


The () quantifier is called an existential quantifier because a formula (t)(F) is TRUE if there exists some tuple that makes F TRUE. For the universal quantifier,

(t)(F) is TRUE if every possible tuple that can be assigned to free occurrences of t in F is substituted for t, and F is TRUE for every such substitution. It is called the uni-versal or for all quantifier because every tuple in the universe of tuples must make F TRUE to make the quantified formula TRUE.


4. Sample Queries in Tuple Relational Calculus


We will use some of the same queries from Section 6.5 to give a flavor of how the same queries are specified in relational algebra and in relational calculus. Notice that some queries are easier to specify in the relational algebra than in the relational calculus, and vice versa.


Query 1. List the name and address of all employees who work for the ‘Research’ department.


Q1: {t.Fname, t.Lname, t.Address | EMPLOYEE(t) AND (d)(DEPARTMENT(d)

AND d.Dname=‘Research’ AND d.Dnumber=t.Dno)}


The only free tuple variables in a tuple relational calculus expression should be those that appear to the left of the bar (|). In Q1, t is the only free variable; it is then bound successively to each tuple. If a tuple satisfies the conditions specified after the bar in Q1, the attributes Fname, Lname, and Address are retrieved for each such tuple. The conditions EMPLOYEE(t) and DEPARTMENT(d) specify the range relations for t and d. The condition d.Dname = ‘Research’ is a selection condition and corresponds to a SELECT operation in the relational algebra, whereas the condition d.Dnumber = t.Dno is a join condition and is similar in purpose to the (INNER) JOIN operation (see Section 6.3).


Query 2. For every project located in ‘Stafford’, list the project number, the controlling department number, and the department manager’s last name, birth date, and address.


Q2: {p.Pnumber, p.Dnum, m.Lname, m.Bdate, m.Address | PROJECT(p) AND

EMPLOYEE(m) AND p.Plocation=‘Stafford’ AND ((d)(DEPARTMENT(d)

AND p.Dnum=d.Dnumber AND d.Mgr_ssn=m.Ssn))}


In Q2 there are two free tuple variables, p and m. Tuple variable d is bound to the existential quantifier. The query condition is evaluated for every combination of tuples assigned to p and m, and out of all possible combinations of tuples to which p and m are bound, only the combinations that satisfy the condition are selected.


Several tuple variables in a query can range over the same relation. For example, to specify Q8—for each employee, retrieve the employee’s first and last name and the first and last name of his or her immediate supervisor—we specify two tuple variables e and s that both range over the EMPLOYEE relation:


Q8: {e.Fname, e.Lname, s.Fname, s.Lname | EMPLOYEE(e) AND EMPLOYEE(s)

AND e.Super_ssn=s.Ssn}


Query 3 . List the name of each employee who works on some project con-trolled by department number 5. This is a variation of Q3 in which all is changed to some. In this case we need two join conditions and two existential quantifiers.


Q0 : {e.Lname, e.Fname | EMPLOYEE(e) AND ((x)(w)(PROJECT(x) AND

WORKS_ON(w) AND x.Dnum=5 AND w.Essn=e.Ssn AND



Query 4. Make a list of project numbers for projects that involve an employee whose last name is ‘Smith’, either as a worker or as manager of the controlling department for the project.


Q4: { p.Pnumber | PROJECT(p) AND (((e)(w)(EMPLOYEE(e)

AND WORKS_ON(w) AND w.Pno=p.Pnumber

AND e.Lname=‘Smith’ AND e.Ssn=w.Essn) )




AND p.Dnum=d.Dnumber AND d.Mgr_ssn=m.Ssn

AND m.Lname=‘Smith’)))}


Compare this with the relational algebra version of this query in Section 6.5. The UNION operation in relational algebra can usually be substituted with an OR connective in relational calculus.


5. Notation for Query Graphs


In this section we describe a notation that has been proposed to represent relational calculus queries that do not involve complex quantification in a graphical form. These types of queries are known as select-project-join queries, because they only involve these three relational algebra operations. The notation may be expanded to more general queries, but we do not discuss these extensions here. This graphical representation of a query is called a query graph. Figure 6.13 shows the query graph for Q2. Relations in the query are represented by relation nodes, which are dis-played as single circles. Constant values, typically from the query selection conditions, are represented by constant nodes, which are displayed as double circles or ovals. Selection and join conditions are represented by the graph edges (the lines that connect the nodes), as shown in Figure 6.13. Finally, the attributes to be retrieved from each relation are displayed in square brackets above each relation.

The query graph representation does not indicate a particular order to specify which operations to perform first, and is hence a more neutral representation of a select-project-join query than the query tree representation (see Section 6.3.5), where the order of execution is implicitly specified. There is only a single query graph corresponding to each query. Although some query optimization techniques were based on query graphs, it is now generally accepted that query trees are prefer-able because, in practice, the query optimizer needs to show the order of operations for query execution, which is not possible in query graphs.


In the next section we discuss the relationship between the universal and existential quantifiers and show how one can be transformed into the other.


6. Transforming the Universal and Existential Quantifiers


We now introduce some well-known transformations from mathematical logic that relate the universal and existential quantifiers. It is possible to transform a universal quantifier into an existential quantifier, and vice versa, to get an equivalent expression. One general transformation can be described informally as follows: Transform one type of quantifier into the other with negation (preceded by NOT); AND and OR replace one another; a negated formula becomes unnegated; and an unnegated formula becomes negated. Some special cases of this transformation can be stated as follows, where the symbol stands for equivalent to:


(x) (P(x)) NOT (x) (NOT (P(x)))

(x) (P(x)) NOT (x) (NOT (P(x)))

(x) (P(x) AND Q(x)) NOT (x) (NOT (P(x)) OR NOT (Q(x)))

(x) (P(x) OR Q(x)) NOT (x) (NOT (P(x)) AND NOT (Q(x)))

(x) (P(x)) OR Q(x)) NOT (x) (NOT (P(x)) AND NOT (Q(x)))

(x) (P(x) AND Q(x)) NOT (x) (NOT (P(x)) OR NOT (Q(x)))


Notice also that the following is TRUE, where the symbol stands for implies:


(x)(P(x)) (x)(P(x))

NOT (x)(P(x)) NOT (x)(P(x))


7. Using the Universal Quantifier in Queries


Whenever we use a universal quantifier, it is quite judicious to follow a few rules to ensure that our expression makes sense. We discuss these rules with respect to the query Q3.


Query 3. List the names of employees who work on all the projects controlled by department number 5. One way to specify this query is to use the universal quantifier as shown:


Q3: {e.Lname, e.Fname | EMPLOYEE(e) AND ((x)(NOT(PROJECT(x)) OR NOT (x.Dnum=5) OR ((w)(WORKS_ON(w) AND w.Essn=e.Ssn AND x.Pnumber=w.Pno))))}

We can break up Q3 into its basic components as follows:


Q3: {e.Lname, e.Fname | EMPLOYEE(e) AND F }

F          = ((x)(NOT(PROJECT(x)) OR F1))

F1 = NOT(x.Dnum=5) OR F2

F2 =         ((w)(WORKS_ON(wAND  w.Essn=e.Ssn

AND x.Pnumber=w.Pno))


We want to make sure that a selected employee e works on all the projects controlled by department 5, but the definition of universal quantifier says that to make the quantified formula TRUE, the inner formula must be TRUE for all tuples in the uni-verse. The trick is to exclude from the universal quantification all tuples that we are not interested in by making the condition TRUE for all such tuples. This is necessary because a universally quantified tuple variable, such as x in Q3, must evaluate to TRUE for every possible tuple assigned to it to make the quantified formula TRUE.


The first tuples to exclude (by making them evaluate automatically to TRUE) are those that are not in the relation R of interest. In Q3, using the expression NOT(PROJECT(x)) inside the universally quantified formula evaluates to TRUE all tuples x that are not in the PROJECT relation. Then we exclude the tuples we are not interested in from R itself. In Q3, using the expression NOT(x.Dnum=5) evaluates to TRUE all tuples x that are in the PROJECT relation but are not controlled by depart-ment 5. Finally, we specify a condition F2 that must hold on all the remaining tuples in R. Hence, we can explain Q3 as follows:


              For the formula F  = (x)(F) to be TRUE, we must have the formula F be


TRUE for all tuples in the universe that can be assigned to x. However, in Q3 we are only interested in F being TRUE for all tuples of the PROJECT relation


that are controlled by department 5. Hence, the formula F is of the form (NOT(PROJECT(x)) OR F1). The ‘NOT (PROJECT(x)) OR ...’ condition is TRUE for all tuples not in the PROJECT relation and has the effect of eliminating these tuples from consideration in the truth value of F1. For every tuple in the PROJECT relation, F1 must be TRUE if F is to be TRUE.


              Using the same line of reasoning, we do not want to consider tuples in the PROJECT relation that are not controlled by department number 5, since we are only interested in PROJECT tuples whose Dnum=5. Therefore, we can write:


IF (x.Dnum=5) THEN F2


which is equivalent to


(NOT (x.Dnum=5) OR F2)

              Formula F1, hence, is of the form NOT(x.Dnum=5) OR F2. In the context of Q3, this means that, for a tuple x in the PROJECT relation, either its Dnum5 or it must satisfy F2.


Finally, F2 gives the condition that we want to hold for a selected EMPLOYEE tuple: that the employee works on every PROJECT tuple that has not been excluded yet. Such employee tuples are selected by the query.

In English, Q3 gives the following condition for selecting an EMPLOYEE tuple e: For every tuple x in the PROJECT relation with x.Dnum=5, there must exist a tuple w in WORKS_ON such that w.Essn=e.Ssn and w.Pno=x.Pnumber. This is equivalent to saying that EMPLOYEE e works on every PROJECT x in DEPARTMENT number 5. (Whew!)


Using the general transformation from universal to existential quantifiers given in Section 6.6.6, we can rephrase the query in Q3 as shown in Q3A, which uses a negated existential quantifier instead of the universal quantifier:


Q3A:                             {e.Lname, e.Fname | EMPLOYEE(e) AND (NOT (x) (PROJECT(x) AND

(x.Dnum=5) AND (NOT (w)(WORKS_ON(wAND  w.Essn=e.Ssn

AND x.Pnumber=w.Pno))))}


We now give some additional examples of queries that use quantifiers.


Query 6. List the names of employees who have no dependents.


Q6:                               {e.Fname, e.Lname | EMPLOYEE(e) AND (NOT (d)(DEPENDENT(d)

AND e.Ssn=d.Essn))}


Using the general transformation rule, we can rephrase Q6 as follows:


Q6A:                             {e.Fname, e.Lname | EMPLOYEE(e) AND ((d)(NOT(DEPENDENT(d))

OR NOT(e.Ssn=d.Essn)))}


Query 7. List the names of managers who have at least one dependent.


Q7:                               {e.Fname, e.Lname | EMPLOYEE(e) AND ((d)(ρ)(DEPARTMENT(d)

AND DEPENDENT(ρ) AND e.Ssn=d.Mgr_ssn AND ρ.Essn=e.Ssn))}


This query is handled by interpreting managers who have at least one dependent as managers for whom there exists some dependent.



8. Safe Expressions


Whenever we use universal quantifiers, existential quantifiers, or negation of predicates in a calculus expression, we must make sure that the resulting expression makes sense. A safe expression in relational calculus is one that is guaranteed to yield a finite number of tuples as its result; otherwise, the expression is called unsafe. For example, the expression


{t | NOT (EMPLOYEE(t))}


is unsafe because it yields all tuples in the universe that are not EMPLOYEE tuples, which are infinitely numerous. If we follow the rules for Q3 discussed earlier, we will get a safe expression when using universal quantifiers. We can define safe expressions more precisely by introducing the concept of the domain of a tuple relational calculus expression: This is the set of all values that either appear as constant values in the expression or exist in any tuple in the relations referenced in the expression. For example, the domain of {t | NOT(EMPLOYEE(t))} is the set of all attribute values appearing in some tuple of the EMPLOYEE relation (for any attribute). The domain of the expression Q3A would include all values appearing in EMPLOYEE, PROJECT, and WORKS_ON (unioned with the value 5 appearing in the query itself).


An expression is said to be safe if all values in its result are from the domain of the expression. Notice that the result of {t | NOT(EMPLOYEE(t))} is unsafe, since it will, in general, include tuples (and hence values) from outside the EMPLOYEE relation; such values are not in the domain of the expression. All of our other examples are safe expressions.

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