1. Copyright © 2003-2008 Curt Hill The Relational Calculus Another way to do queries

2. Copyright © 2003-2008 Curt Hill 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

3. Copyright © 2003-2008 Curt Hill Formulas Operators Quantifiers Constants Comparisons All tuples that makes the formula true are in the answer table

4. Copyright © 2003-2008 Curt Hill 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

5. Copyright © 2003-2008 Curt Hill 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|C  Courses} –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

6. Copyright © 2003-2008 Curt Hill Formulas Atomic formulas need to be combined to be useful They are combined with AND (  ) OR (  ) and modified with NOT (  ) {C|C  Courses  C.dept=CS} –This should be easy to convert into a relational algebra selection –This is declarative not procedural

7. Copyright © 2003-2008 Curt Hill Query By Example QBE is a graphical way to specify this {C|C  Courses  C.dept=CS} Click a table to indicater Fill in the desired values DeptNumberCrhrTitle CS

8. Copyright © 2003-2008 Curt Hill Quantifiers Two common quantifiers –There exists  T(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

9. Copyright © 2003-2008 Curt Hill Quantified queries Any student taking any upper level course {S | S  Students   G  Grades (G.number>299  S.naid=G.naid)} Any student taking only upper level courses {S | S  Students   G  Grades (G.number>299  S.naid=G.naid)}

10. Copyright © 2003-2008 Curt Hill Unsafe queries Any query with an infinite number of answers –Legal syntax, illegal results {C|  C  Courses} This possibility is not present in the algebra –The algebra always deals with finite relations

11. Copyright © 2003-2008 Curt Hill 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

12. Copyright © 2003-2008 Curt Hill 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

13. 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: where the fs are fields Copyright © 2003-2008 Curt Hill

14. DRC Example Consider this formula { | m>299   d, m, x, n (  Grades)   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 Copyright © 2003-2008 Curt Hill

15. DRC and TRC example In TRC we might say: {S | S  Students   G  Grades (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

16. Copyright © 2003-2008 Curt Hill Queries using relational calculus Consider the college schema tables: –Course –Students –Grade –Faculty –Faculty_teach –Department –Division

17. Copyright © 2003-2008 Curt Hill Find student grades DRC { |  sc(  Grades)   name(  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}

18. Copyright © 2003-2008 Curt Hill Find all the courses taught by faculty members (include credit hours) { |  naid,dp,dg,ar,title(  Faculty)   Course   Faculty_Teach )} Implied equality in naid, dept, number, crhr

19. Copyright © 2003-2008 Curt Hill Access Choose Tables

20. Copyright © 2003-2008 Curt Hill Access Choose Fields

21. Copyright © 2003-2008 Curt Hill 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