1. Database Management Systems,1 Relational Calculus

2. Database Management Systems,2 Tuple Relational Calculus(TRC)  Tuple variable is a variable that takes on the tuples of a particular relation schema as values  It has the form {T|p(T)} –Where T is the tuple variable and p(.) is the formula that describe T. –The result of the query is the set of all tuples for which the formula is true…  The formula is constructed using the tuples and relational operators: >, <, =, not equal, less than or equal, greater than or equal.  The formulas may be atomic: composed of tuple variables, operators, and constants  Formula may also be recursively defined in terms of other sub-formulas.

3. Database Management Systems,3 Examples  Find all sailors with rating above 7: (Q11) {S|S  Sailors  S.rating>7} –S is instantiated for each tuple of Sailor and the test is applied to it. The answer is the set of those instances passing the test.  Find the names and ages of the sailors with rating above 7: (Q12) {P|S  Sailors (S.rating>7  P.sname=S.sname  P.age=S.age)}  Find the sailor names, boat id, and reservation dates for each reservation: (Q13) {P|  R  Reserves  S  Sailors (R.sid=S.sid  P.bid=R.bid  P.day=R.day  P.sname=S.sname)}

4. Database Management Systems,4 Examples (cont.)  Find the name of the sailors who have reserved boat 103 :(Q1) {P|  S  Sailors  R  Reserves (R.sid=S.sid  R.bid=103  P.sname=S.sname )}  Find the name of the sailors who have reserved a red boat: (Q2) {P|  S  Sailors  R  Reserves   B  Boats (R.sid=S.sid  B.bid=R.bid  B.color=‘red’  P.sname=S.sname)}  Find the name of the sailors who have reserved a at least two boats :(Q7) {P|  S  Sailors  R1  Reserves   R2  Reserves (S.sid=R1.sid  R1.sid=R2.sid  R1.bid  R2.bid  P.sname=S.sname)}

5. Database Management Systems,5 Examples (cont.)  Find the name of the sailors who have reserved all boats :(Q9) {P|  S  Sailors  B  Boats  R  Reserves(S.sid=R.sid  R.bid=B.bid  P.sname=S.sname)} –English statement for this query is “find sailors S such that for all boats B there is a Reserves tuple…”  Find the name of the sailors who have reserved all red boats :(Q14) {P|  S  Sailors   B  Boats (B.color =‘red’  (  R  Reserves(S.sid=R.sid  R.bid=B.bid  P.sname=S.sname)))} or replacing implication by NOT and OR: {P|  S  Sailors   B  Boats (B.color  ‘red’  (  R  Reserves(S.sid=R.sid  R.bid=B.bid  P.sname=S.sname)))} –Find sailors S such that for all boats B, either the boat is not red or the sailor S has reserved boat B.

6. Database Management Systems,6 Aggregate Functions  Aggregate functions and grouping are not included as basic algebraic operations. However, they are useful, like division and join operation. Count and Average, Minimum, Maximum, and Sum are aggregate functions.  Example for Count and Average:  Number of sailors reserving each boat: bid ₣ count sid (Reserves)  Total number of students and the average rating: ₣ count sid, Average rating (Sailors)

7. Database Management Systems,7 Review  Major problem with calculus: unsafe queries (P|  S  Sailors) is syntactically correct but it is an unsafe query.  If a query language can express all the queries that can be expresses in relational algebra it is called a relationally complete language  Relational algebra query describe a procedure to compute output relation from input relations, using RelAlg operators  Relational calculus describe the tuples in output relation, using a subset of first order predicate logic  All relational algebra queries can be expresses in relational calculus, the converse is true only for restricted relational calculus