copyright © 2003-2008 curt hill the relational calculus another way to do queries

Copyright © 2003-2008 Curt Hill

The Relational Calculus

Another way to do queries


Calculus• Declarative way to form a query• Two types

– Tuple Relational Calculus– Domain Relational Calculus

• Subsets of first order logic• Formulas are the main unit

– Used to construct the tables that satisfy the query


Formulas• Operators• Quantifiers• Constants• Comparisons• All tuples that makes the formula

true are in the answer table


TRC and DRC• Tuple Relational Calculus considers

a tuple as variable• Domain Relational Calculus

considers a Domain a variable– A domain is math for the field or

attribute values


Atomic Formulas• A simple statement connecting a

tuple or domain to some data• We may compare any tuple value

or domain value with constants• Since tables are sets they are

expressed as set expressions:– {C|CCourses}– The set of all tuples C such that C is

an element of the Courses relation• You may also compare a tuple

element with a value


Formulas• Atomic formulas need to be

combined to be useful• They are combined with AND ()

OR () and modified with NOT ()• {C|CCourses C.dept=CS}

– This should be easy to convert into a relational algebra selection

– This is declarative not procedural


Query By Example• QBE is a graphical way to specify this• {C|CCourses C.dept=CS}• Click a table to indicater• Fill in the desired values

Dept Number Crhr Title

CS


Quantifiers• Two common quantifiers

– There existsT(predicate(T))– For all T(predicate(T))

• These bind a formula to a variable• This variable may then be used in

a larger formula Courses(Courses.number>299)

– True if there is one course that is so numbered


Quantified queries• Any student taking any upper level

course• {S | SStudents

GGrades (G.number>299 S.naid=G.naid)}

• Any student taking only upper level courses

• {S | SStudents GGrades (G.number>299 S.naid=G.naid)}


Unsafe queries• Any query with an infinite number

of answers– Legal syntax, illegal results

• {C|CCourses}• This possibility is not present in the

algebra– The algebra always deals with finite

relations


Equivalence and completeness• Every series of relational algebra

expressions may be expressed as a safe calculus formula

• Every safe calculus formula may be expressed as a series of relational algebra operations– A must for implementation

• Any language that can express any relational algebra expression is said to be relationally complete– SQL is relationally complete


Domain Relational Calculus

• Most of the previous examples were of the Tuple Relational Calculus

• In the DRC we show the construction of the tuple using domain values or only show the domain values

• Different syntax, but equivalent

Domain Relational Constructions

• In the DRC we construct tuples from fields– Not rely on them as variables

• Then we determine the values we want these fields to have

• The notation for this construction is:<f1,f2,f3…fn>where the fs are fields


DRC Example• Consider this formula

{<a, n>| m>299 d, m, x, n (<d,m,x,n> Grades) n, a, r(<n,a,r> Students)}

• This says construct a table of tuples– Two elements in each tuple, a and m

• Student name is a, naid is n– There must exist a grade tuple and a

student tuple with these characteristics• Same naid and course number > 299



DRC and TRC example• In TRC we might say:

{S | SStudents GGrades (G.number>299 S.naid=G.naid)}

• This gives approximately the same table as the previous except there are more fields in each row

• The DRC makes it easy to construct tuples that do not exist in any table


Queries using relational calculus

• Consider the college schema tables:– Course– Students– Grade– Faculty– Faculty_teach– Department– Division


Find student grades• DRC

{<naid,addr,dp,cr,sc>| sc(<dp,cr,naid,sc> Grades) name(<naid,name,addr> Students)}

• TRC{T| T2 Grades T.naid = T2.naid T3 Students T3.naid = T.naid T.addr = T3.addr T.dp = T2.dp T.cr = T2.cr T.sc = T2.sc}


Find all the courses taught by faculty members (include

credit hours)• {<name,dept,number,crhr>|

naid,dp,dg,ar,title(<naid,name,dp,dg,ar>Faculty) <dept,number,crhr,title>Course <dept,number,naid>Faculty_Teach)}

• Implied equality in naid, dept, number, crhr


Access Choose Tables


Access Choose Fields


Some others• Find the departmental chairs

for each faculty member• Find all the students who got a B or

better in any CS class• Find all the students that each

faculty member has• Find all the students who got

an A in Calculus and an A in CIS 385

copyright © 2003-2008 curt hill the relational calculus another way to do queries

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