1. Relational Calculus Ameetinder Singh CS 157A

2. Tuple Relational Calculus  non-procedural query language as compared to relational algebra that is procedural.  only requires the user to specify what information is required and not how that information should be obtained.  A query in tuple relational calculus expressed as {t | P(t)} {t | P(t)}  i.e. set of all tuples t such that P is true for t

3. Query Examples  To find the branch-name, loan-number, and amount for loans over $1000: {t | t  loan  t[amount] > 1000} {t | t  loan  t[amount] > 1000}  To find only the loan-number attribute, rather than all the attributes of the loan relation: {t |  s  loan ( t[loan-number] = s[loan- number]  s[amount] > 1000)} {t |  s  loan ( t[loan-number] = s[loan- number]  s[amount] > 1000)} This query will retrieve those tuples in loan- number such that there is a tuple in loan with the amount attribute > 1000 This query will retrieve those tuples in loan- number such that there is a tuple in loan with the amount attribute > 1000

4. Examples (Cont’d)  To find the names of all customers who borrowed a loan from the Downtown branch. {t |  s  borrower ( t[customer-name] = s[customer- name]   u  loan (u[loan-number] = s[loan-number]  u[branch-name] = “Downtown”))} {t |  s  borrower ( t[customer-name] = s[customer- name]   u  loan (u[loan-number] = s[loan-number]  u[branch-name] = “Downtown”))} requires two “there exists” clauses since it involves two relations requires two “there exists” clauses since it involves two relations retrieves the set of all (customer-name) tuples for which the customer has a loan from the Downtown branch retrieves the set of all (customer-name) tuples for which the customer has a loan from the Downtown branch

5. Examples (Cont’d)  To find customers who have a loan, an account, or both at the bank {t |  s  borrower ( t[customer-name] = s[customer-name])   u  depositor (t[customer-name] = u[customer-name] )} {t |  s  borrower ( t[customer-name] = s[customer-name])   u  depositor (t[customer-name] = u[customer-name] )} If a customer has both account and a loan from the bank, the customer-name only appears once If a customer has both account and a loan from the bank, the customer-name only appears once Changing (  ) to (  ) in the previous query will generate results with the set of customer-name tuples who have both an account and a loan at the bank. Changing (  ) to (  ) in the previous query will generate results with the set of customer-name tuples who have both an account and a loan at the bank.

6. Examples (Cont’d)  To find all the customers who have an account at the bank but do not have a loan from the bank, we use the not (  ) symbol in the previous expression {t |   s  borrower ( t[customer-name] = s[customer-name])   u  depositor ( t[customer-name] = u[customer-name] )}

7. Examples (Cont’d) Some queries require the use of implication (  ). P  Q means, “if P is true, then Q must be true.” To find all customers who have an account at all branches located in Brooklyn, we write a query using “for all” (  ) construct as follows:

8. Examples (Cont’d) {t |  r  customer ( r[customer-name] = t[customer- name])  (  u  branch (u[branch-city] = “Brooklyn”   s  depositor (t[customer-name] = s[customer- name]   w  account (w[account-number] = s[account-number]  w[branch-name] = s[branch- name]))))} this is interpreted as “the set of all customers such that, for all tuples u in the branch relation, if the value of u on attribute branch-city is Brooklyn, then the customer has an account at the branch whose name appears in the branch-name attribute of u.”

9. Formal Definition A tuple-relational-calculus expression is in the form {t | P(t)} {t | P(t)} where P is a formula where P is a formula The formula in tuple-relational-calculus is built up out of atoms.

10. Formal Definition (Cont’d) An atom can have one of the following forms: An atom can have one of the following forms: s  r, where s is a tuple variable in relation rs  r, where s is a tuple variable in relation r s[x]  u[y], where s and u are tuple variables, x is an attribute on which s is defined, y is an attribute on which u is defined, and  is a comparison operator ( , , , , ,  ); it is required that x and y have domains whose members can be compared by .s[x]  u[y], where s and u are tuple variables, x is an attribute on which s is defined, y is an attribute on which u is defined, and  is a comparison operator ( , , , , ,  ); it is required that x and y have domains whose members can be compared by . s[x]  c, where s is a tuple variable, x is and attribute on which s is defined,  is a comparison operator, and c is a constant in the domain of attribute x.s[x]  c, where s is a tuple variable, x is and attribute on which s is defined,  is a comparison operator, and c is a constant in the domain of attribute x.

11. Formal Definition (Cont’d) Formulae can be built up from atoms by using the following rules: an atom is a formulaan atom is a formula If P1 is a formula, then so are  P1 and (P1)If P1 is a formula, then so are  P1 and (P1) If P1 and P2 are formulae, then so are P1  P2, P1  P2, and P1  P2If P1 and P2 are formulae, then so are P1  P2, P1  P2, and P1  P2

12. Safety of Expressions A tuple-relational-calculus expression may generate an infinite relation. {t |  ( t  loan)} {t |  ( t  loan)} there are infinitely many tuples that are not in loan. Domain of P is the set of all values referenced by P. Domain of P is the set of all values referenced by P. the expression {t | P(t)} is safe if all values in the result are in the domain of P

13. Domain Relational Calculus Closely related to tuple-relational- calculus Uses domain variables that take on values from an attributes domain, rather than values for an entire tuple

14. Formal Definition An expression in the domain relational calculus is in the form { | P(x1, x2,……. xn)} { | P(x1, x2,……. xn)} where x1, x2,……. xn represent domain variables. P represents formula composed of atoms.

15. Formal Definition (Cont’d) An atom in the domain relational calculus has one of the following forms:  r, where r is a relation on n attributes and x1, x2,…xn are domain variables.  r, where r is a relation on n attributes and x1, x2,…xn are domain variables. x  y, where x and y are domain variables and  is a comparison operator ( , , , , ,  ); it is required that x and y have domains that can be compared by x  y, where x and y are domain variables and  is a comparison operator ( , , , , ,  ); it is required that x and y have domains that can be compared by  x  c, where x is a domain variable,  is a comparison operator, and c is a constant in the domain of attribute for which x is a domain variablex  c, where x is a domain variable,  is a comparison operator, and c is a constant in the domain of attribute for which x is a domain variable

16. Formal Definition (Cont’d) Formulae can be built up from atoms by using the following rules:  an atom is a formula.  If P1 is a formula, then so are  P1 and (P1)  If P1 and P2 are formulae, then so are P1  P2, P1  P2, and P1  P2 Same as tuple-relational-calculus

17. Example queries  To find the branch-name, loan-number, and amount for loans over $1000: { |  loan  a > 1000} { |  loan  a > 1000}  To find only the loan-number attribute, rather than all the attributes of the loan relation: { |  b,a (  loan  a > 1000)} { |  b,a (  loan  a > 1000)}

18. Examples (Cont’d)  To find the names of all customers who borrowed a loan from the Downtown branch and find the loan amount: { |  l (  borrower   b (  loan  b = “Downtown”))}  To find all customers who have an account at all branches located in Brooklyn: { |  n (  customer)   x,y,z(  branch  y = “Brooklyn”  a,b (  account   depositor)}

19. Examples (Cont’d) interpreted the set of all (customer name) tuples c such that, for all (branch-name, branch-city, assets) tuples x, y, z, if the branch- city is Brooklyn, then the following conditions hold:  There exists a tuple in the relation account with account number a and branch name x  There exists a tuple in the relation depositor with customer c and account number a

20. Safety of expressions Expressions may generate an infinite relation To overcome this problem we define safety of domain relational calculus expressions. An expression such as { |   loan )} { |   loan )} generates result with values that are not in the domain of the expression.

21. Safety of expressions (Cont’d) An expression { | P(x1, x2,…. xn)} is safe if all of the following conditions hold:  all values that appear in tuples of the expression are values from domain of P.  for every “there exists” sub formula of the form  x (P1 (x)), the subformula is true if and only if there is a value x in the domain P1 such that P1 (x) is true.  for every “for all” sub formula of the form  x (P1 (x)), the subformula is true if and only if P1 (x) is true for all values x from domain of P1