1. 1 Relational Calculus Chapter 4 – Part II

2. 2 Formal Relational Query Languages  Two mathematical Query Languages form the basis for “real” languages (e.g. SQL), and for implementation:  Relational Algebra : how to compute it, operational, Representing execution plans (inside) · Relational Calculus : what you want, declarative. Influent SQL design (outside)

3. 3 Relational Calculus  Calculus has variables, constants, comparison ops (,=, , .  ), logical connectives ( ,  ) Quantifiers ( ,  )  Comes in two flavours: Tuple relational calculus (TRC) : Variables range over tuples. (SQL) Domain relational calculus (DRC). Variables range over domain elements. (QBE) Both formula in TRC and DRC are simple subsets of first- order logic.

4. 4 Tuple Relational Calculus (TRC)  Tuple variable: variable that takes on tuples of a relation schema as values.  Query form: T = tuple variable p(T) = formula that describes T  Result = set of all tuples t for which p(t) with T=t evaluates to be true  Formula P(T) specified in First Order Logic

5. 5 Find all sailors with a rating above 7  Variable S bound to a tuple in relation Sailors

6. 6 TRC Formulas  Formula:  an atomic formula R  Rel, R.a op S.b, R.a op constant, costant op R.a  Or recursively defined  p, p  q, p  q, p  q, where p, q are formulas  R(p(R)), where R is a tuple variable  R(p(R)), where R is a tuple variable  Terms.  bind --- the quantifier  and  are bind to variable R.  free --- a variable is free if no quantifier binds to it in a formula

7. 7 Semantics of TRC Queries  F is an atomic formula R  Rel, and R is assigned a tuple in the instance of relation Rel  {S|S  Sailors}  F is a comparison R.a op S.b, R.a op constant, or constant op R.a, and the tuples assigned to R and S have field names R.a and S.b that make the comparison true.  F is of the form  p, and p is not true; the form p  q, and both p and q are true ; the form p  q, and one of them is true ; the form p  q, and q is true whenever p is true.  F is of the form  R(p( R )), and there is some assignment of tuples to the free variables in p ( R ), including the variable R that make the formula p( R ) true ;  F is of the form  R (p ( R )), and there is some assignment of tuples to the free variables in p( R ) that makes the formula p( R ) true no matter what tuple is assigned to R.

8. 8 SailorsReserves TRC Example

9. 9 Find the names and ages of sailors with rating > 7 P is a Tuple variable with two fields: name and age. - the only fields mentioned in P - P does not range over any relation in the query. { P |  S  Sailors (S.rating > 7  P.name = S.sname  P.age = S.age)} nameage Lubber55.0 rusty35.0 Sailors

10. 10 Find the names of the sailors who reserved boat 103 { P |  S  Sailors  R  Reserves (R.sid = S.sid  R.bid = 103  P.sname = S.sname)} Retrieve all sailor tuples for which there exists a tuple in Reserves having the same value in the sid field and with bid =103 Question: How is the answer tuple looks like? (columns? Rows?) What is difference? { S |  S  Sailors   R  Reserves (R.sid = S.sid  R.bid = 103)}

11. 11 Find the names of the sailors who have reserved a red boat { P |  S  Sailors  R  Reserves (R.sid = S.sid  P.sname = S.sname   B  Boats(B.bid = R.bid  B.color ='red'))} Retrieve all sailor tuples S for which there exists tuples R in Reserves and B in Boats such that R.sid = S.sid  B.bid = R.bid  B.color ='red' { P |  S  Sailors  R  Reserves  B  Boats (R.sid = S.sid  B.bid = R.bid  B.color ='red'  P.sname = S.sname)}

12. 12 Find the names of the sailors who have reserved all boat Find sailors S such that for all boats B there is a Reserves tuple showing that sailor S has reserved boat B. { P |  S  Sailors  B  Boats (  R  Reserves (S.sid = R.sid  R.bid = B.bid  P.sname = S.sname))}

13. 13 Find the sailors who have reserved all red boat { P |  S  Sailors   B  Boats (B.color = 'red'  (  R  Reserves (S.sid = R.sid  R.bid = B.bid))} Question: How to change the algebraic representation?

14. 14 Find the sailors who have reserved all red boat { P |  S  Sailors   B  Boats (B.color = 'red'  (  R  Reserves (S.sid = R.sid  R.bid = B.bid))}  Logically p  q is equivalent to  p  q (Why?) { P |  S  Sailors   B  Boats (B.color  'red'  (  R  Reserves (S.sid = R.sid  R.bid = B.bid))}  is rewritten as:  to restrict attention to those red boat.

15. 15 Domain Relational Calculus  Query has the form: v Answer includes all tuples that make the formula be true.

16. 16 DRC Formulas  Atomic formula:   Rel, where Rel is a relation with n variables  X op Y  X op constant  op is one of comparison ops (,=, , .  ),  Or recursively defined  p, p  q, p  q, p  q, where p, q are formulas  X(p(X)), where X is a domain variable  X(p(X)), where X is a domain variable

17. 17 Free and Bound Variables  The use of quantifiers and in a formula is said to bind X.  A variable that is not bound is free.  Let us revisit the definition of a query: v There is an important restriction: the variables x1,..., xn that appear to the left of `|’ must be the only free variables in the formula p(...).

18. 18 Find all sailors with a rating above 7  the domain variables I, N, T and A are bound to fields of the same Sailors tuple.  that every tuple that satisfies T> 7 is in the answer.  Question: Modify to answer:  Find sailors who are older than 18 or have a rating under 9, and are called ‘Joe’. Sailors

19. 19 Find sailors rating > 7 who’ve reserved boat #103  We have used as a shorthand for  Note the use of to find a tuple in Reserves that `joins with’ the Sailors tuple under consideration. Reserves Sailors bidbnamecolor 101interlakered 103marinegreen Boats { |  Sailors  T>7   Ir,Br,D (  Reserves  Ir =I )}

20. 20 Find sailors rated > 7 who’ve reserved a red boat  Observe how the parentheses control the scope of each quantifier’s binding.  Rewrite using 'red' as constant BN = red Reserves Sailors bidbnamecolor 101interlakered 103marinegreen Boats

21. 21 Find sailors who’ve reserved all boats  Find all sailors I such that for each 3-tuple { B, BN, C } either it is not a tuple in Boats or there is a tuple in Reserves showing that sailor I has reserved it. Reserves Sailors bidbnamecolor 101interlakered 103marinegreen Boats

22. 22 Algebra vs. Calculus  Two formal query languages are equal in the power of expressivness?  Algebra  Calculus?YES!  Calculus  Algebra? Unsafe query {S |  (S  Sailors)}Correct??? Safe TRC --- Dom(Q, I) -answers for Q contains only values that are in Dom(Q, I) -  R(p(R)), tuple r contains only constant in Dom(Q,I) -  R(p(R)), tuple r contains a constant that is not in Dom(Q,I), then p(r) is true.

23. 23 Summary  Algebra and safe calculus have same expressive power  Relationally complete --- if a query language can express all the queries that can be expressed in relational algebra.  Relational calculus is non-operational, and users define queries in terms of what they want, not in terms of how to compute it. (Declarativeness.)