Revision of Midterm 2 Prof. Sin-Min Lee Department of Computer Science

Relational Calculus Important features: –Declarative formal query languages for relational model –Based on the branch mathematical logic known as predicate calculus –Two types of RC: 1) tuple relational calculus 2) domain relational calculus –A single statement can be used to perform a query

Tuple Relational Calculus based on specifying a number of tuple variables a tuple variable refers to any tuple

Generic Form {t | COND (t)} –where – t is a tuple variable and –COND(t) is Boolean expression involving t

Simple example 1 To find all employees whose salary is greater than $50,000 –{t| EMPLOYEE(t) and t.Salary>5000} where EMPLOYEE(t) specifies the range of tuple variable t –The above operation selects all the attributes

Simple example 2 To find only the names of employees whose salary is greater than $50,000 –{t.FNAME, t.NAME| EMPLOYEE(t) and t.Salary>5000} The above is equivalent to SELECT T.FNAME, T.LNAME FROM EMPLOYEE T WHERE T.SALARY > 5000

Elements of a tuple calculus In general, we need to specify the following in a tuple calculus expression: –Range Relation (I.e, R(t)) = FROM –Selected combination= WHERE –Requested attributes= SELECT

More Example:Q0 Retrieve the birthrate and address of the employee(s) whose name is ‘John B. Smith’ {t.BDATE, t.ADDRESS| EMPLOYEE(t) AND t.FNAME=‘John’ AND t.MINIT=‘B” AND t.LNAME=‘Smith}

Formal Specification of tuple Relational Calculus A general format: {t 1.A 1, t 2.A 2,…,t n.A n |COND ( t 1,t 2,…, t n, t n+1, t n+2,…,t n+m )} –where –t 1,…,t n+m are tuple var –A i : attribute  R(t i ) –COND (formula) Where COND corresponds to statement about the world, which can be True or False

Elements of formula A formula is made of Predicate Calculus atoms: – an atom of the from R(ti) –t i.A op t j.B op  {=,,..} –F1 And F2 where F1 and F2 are formulas –F1 OR F2 –Not (F1) –F’=(  t) (F) or F’= (  t) (F)  Y friends (Y, John)  X likes(X, ICE_CREAM)

Example Queries Using the Existential Quantifier Retrieve the name and address of all employees who work for the ‘ Research ’ department {t.FNAME, t.LNAME, t.ADDRESS| EMPLOYEE(t) AND (  d) (DEPARTMENT (d) AND d.DNAME=‘Research’ AND d.DNUMBER=t.DNO)}

More Example For every project located in ‘Stafford’, retrieve the project number, the controlling department number, and the last name, birthrate, and address of the manger of that department.

Cont. {p.PNUMBER,p.DNUM,m.LNAME,m.BD ATE, m.ADDRESS|PROJECT(p) and EMPLOYEE(M) and P.PLOCATION=‘Stafford’ and (  d) (DEPARTMENT(D) AND P.DNUM=d.DNUMBER and d.MGRSSN=m.SSN))}

Safe Expressions A safe expression R.C: –An expression that is guaranteed to generate a finite number of rows (tuples) Example: –{t | not EMPLOYESS(t))} results values not being in its domain (I.e., EMPLOYEE)

Domain Relational Calculus (DRC) Another type of formal predicate calculus- based language QBE is based on DRC The language shares a lot of similarities with the tuple calculus

DRC The only difference is the type of variables: –variables range over singles values from domains of attributes An expression of DRC is: –{x 1, x 2,…,x n |COND(x 1,x 2,…,x n, x n+2,…,x n+m )} where x 1,x 2,…,x n+m are domain var range over attributers COND is a condition (or formula)

Examples Retrieve the birthdates and address of the employee whose name is ‘John B. Smith’ {uv| (  q)(  r)(  s) (EMPLOYEE(qrstuvwxyz) and q=‘John’ and r=‘B’ and s=‘Smith’

Alternative notation Ssign the constants ‘John’, ‘B’, and ‘Smith’ directly {uv|EMPLOYEE (‘John’, ’B’, ’Smith’,t,u,v,x,y,z)}

More example Retrieve the name and address of all employees who work for the ‘Reseach’ department {qsv | (  z) EMPLOYEE(qrstuvwxyz) and (  l) (  m) (DEPARTMENT (lmno) and l=‘Research’ and m=z))}

More example List the names of managers who have at least on e dependent {sq| (  t) EMPLOYEE(qrstuvwxyz) and ((  j)( DEPARTMENT (hijk) and ((  l) | (DEPENTENT (lmnop) and t=j and t=l))))}

QBE Query-By-Example –Supports graphical query language based on DRC –Implemented in commercial db such as Access/Paradox –Query can be specified by filling in templates of relations –Fig 9.5

Summary It can be shown that any query that can be expressed in the relational algebra, it can also be expressed in domain and tuple relational calculus

Quiz In what sense doe R.C differ from R.A, and in what sense are they similar?

Relational Algebra Relational algebra operations operate on relations and produce relations ( “ closure ” ) f: Relation -> Relationf: Relation x Relation -> Relation Six basic operations: –Projection   (R) –Selection   (R) –UnionR 1 [ R 2 –DifferenceR 1 – R 2 –ProductR 1 £ R 2 –(Rename)   (R)

Example Data Instance sidname 1Jill 2Qun 3Nitin 4Marty fidname 1Ives 2Saul 8Roth sidexp-gradecid 1A A A 3C C cidsubjsem DBF AIS ArchF03 fidcid STUDENT Takes COURSE PROFESSOR Teaches

Natural Join and Intersection Natural join: special case of join where  is implicit – attributes with same name must be equal: STUDENT ⋈ Takes ´ STUDENT ⋈ STUDENT.sid = Takes.sid Takes Intersection: as with set operations, derivable from difference A-B B-A A B A  B

Division A somewhat messy operation that can be expressed in terms of the operations we have already defined Used to express queries such as “ The fid's of faculty who have taught all subjects ” Paraphrased: “ The fid ’ s of professors for which there does not exist a subject that they haven ’ t taught ”

Division Using Our Existing Operators All possible teaching assignments: Allpairs: NotTaught, all (fid,subj) pairs for which professor fid has not taught subj: Answer is all faculty not in NotTaught:  fid,subj (PROFESSOR £  subj (COURSE)) Allpairs -  fid,subj (Teaches ⋈ COURSE)  fid (PROFESSOR) -  fid (NotTaught) ´  fid (PROFESSOR) -  fid (  fid,subj (PROFESSOR £  subj (COURSE)) -  fid,subj (Teaches ⋈ COURSE))

Division: R 1  R 2 Requirement: schema(R 1 ) ¾ schema(R 2 ) Result schema: schema(R 1 ) – schema(R 2 ) “ Professors who have taught all courses ” : What about “ Courses that have been taught by all faculty ” ?  fid (  fid,subj ( Teaches ⋈ COURSE)   subj (COURSE))

The Big Picture: SQL to Algebra to Query Plan to Web Page SELECT * FROM STUDENT, Takes, COURSE WHERE STUDENT.sid = Takes.sID AND Takes.cID = cid STUDENT Takes COURSE Merge Hash by cid Optimizer Execution Engine Storage Subsystem Web Server / UI / etc Query Plan – an operator tree

Relational Calculus: A Logical Way of Expressing Query Operations First-order logic (FOL) can also be thought of as a query language, and can be used in two ways: –Tuple relational calculus –Domain relational calculus –Difference is the level at which variables are used: for attributes (domains) or for tuples The calculus is non-procedural (declarative) as compared to the algebra –More like what we ’ ll see in SQL –More convenient to express certain things

Domain Relational Calculus Queries have form: { | p} Predicate: boolean expression over x 1,x 2, …, x n –Precise operations depend on the domain and query language – may include special functions, etc. –Assume the following at minimum:  RX op Y X op constconst op X where op is , , , , ,  x i,x j, … are domain variables domain variables predicate

More Complex Predicates Starting with these atomic predicates, build up new predicates by the following rules: –Logical connectives: If p and q are predicates, then so are p  q, p  q,  p, and p  q (x>2)  (x<4) (x>2)   (x>0) –Existential quantification: If p is a predicate, then so is  x.p  x. (x>2)  (x<4) –Universal quantification: If p is a predicate, then so is  x.p  x.x>2  x.  y.y>x

Some Examples Faculty ids Course names for courses with students expecting a “ C ” Courses taken by Jill

Logical Equivalences There are two logical equivalences that will be heavily used: –p  q   p  q (Whenever p is true, q must also be true.) –  x. p(x)   x.  p(x) (p is true for all x) The second can be a lot easier to check!

Free and Bound Variables A variable v is bound in a predicate p when p is of the form  v … or  v … A variable occurs free in p if it occurs in a position where it is not bound by an enclosing  or  Examples: –x is free in x>2 –x is bound in  x.x>y

Can Rename Bound Variables Only When a variable is bound one can replace it with some other variable without altering the meaning of the expression, providing there are no name clashes Example:  x.x>2 is equivalent to  y.y>2 Otherwise, the variable is defined outside our “ scope ”…

Safety Pitfall in what we have done so far – how do we interpret: { |   STUDENT} –Set of all binary tuples that are not students: an infinite set (and unsafe query) A query is safe if no matter how we instantiate the relations, it always produces a finite answer –Domain independent: answer is the same regardless of the domain in which it is evaluated –Unfortunately, both this definition of safety and domain independence are semantic conditions, and are undecidable

Safety and Termination Guarantees There are syntactic conditions that are used to guarantee “ safe ” formulas –The definition is complicated, and we won ’ t discuss it; you can find it in Ullman ’ s Principles of Database and Knowledge-Base Systems –The formulas that are expressible in real query languages based on relational calculus are all “ safe ” Many DB languages include additional features, like recursion, that must be restricted in certain ways to guarantee termination and consistent answers

Mini-Quiz How do you write: –Which students have taken more than one course from the same professor? –What is the highest course number offered?

Translating from RA to DRC Core of relational algebra: , , , x, - We need to work our way through the structure of an RA expression, translating each possible form. –Let TR[e] be the translation of RA expression e into DRC. Relation names: For the RA expression R, the DRC expression is { |  R}

Selection: TR[   R] Suppose we have   (e ’ ), where e ’ is another RA expression that translates as: TR[e ’ ]= { | p} Then the translation of  c (e ’ ) is { | p  ’ } where  ’ is obtained from  by replacing each attribute with the corresponding variable Example: TR[  #1=#2  #4>2.5 R] (if R has arity 4) is { |  R  x 1 =x 2  x 4 >2.5}

Projection: TR[  i 1, …,i m (e)] If TR[e]= { | p} then TR[  i 1,i 2, …,i m (e)]= { |  x j 1,x j 2, …, x j k.p}, where x j 1,x j 2, …, x j k are variables in x 1,x 2, …, x n that are not in x i 1,x i 2, …, x i m Example: With R as before,  #1,#3 (R)={ |  x 2,x 4.  R}

Union: TR[R 1  R 2 ] R 1 and R 2 must have the same arity For e 1  e 2, where e 1, e 2 are algebra expressions TR[e 1 ]={ |p} and TR[e 2 ]={ |q} Relabel the variables in the second: TR[e 2 ]={ |q ’ } This may involve relabeling bound variables in q to avoid clashes TR[e 1  e 2 ]={ |p  q ’ }. Example: TR[R 1  R 2 ] = { |  R 1   R 2

Other Binary Operators Difference: The same conditions hold as for union If TR[e 1 ]={ |p} and TR[e 2 ]={ |q} Then TR[e 1 - e 2 ]= { |p  q} Product: If TR[e 1 ]={ |p} and TR[e 2 ]={ |q} Then TR[e 1  e 2 ]= { | p  q} Example: TR[R  S]= { |  R   S }

Summary Can translate relational algebra into (domain) relational calculus. Given syntactic restrictions that guarantee safety of DRC query, can translate back to relational algebra These are the principles behind initial development of relational databases –SQL is close to calculus; query plan is close to algebra –Great example of theory leading to practice!

Limitations of the Relational Algebra / Calculus Can ’ t do: –Aggregate operations –Recursive queries –Complex (non-tabular) structures Most of these are expressible in SQL, OQL,