Relational Calculus Zachary G. Ives November 15, 2018 University of Pennsylvania CIS 550 – Database & Information Systems November 15, 2018 Some slide content courtesy of Susan Davidson & Raghu Ramakrishnan

Administrivia Reminder: Homework 1 due 9/28 (next Monday) Upcoming research talks of possible interest: Amol Deshpande, U Maryland, Monday 10/19, 3PM – adding probabilities to databases

The Relational Algebra as “Virtual Machine” Instructions Six basic operations: Projection  (R) Selection  (R) Union R1 [ R2 Difference R1 – R2 Product R1 £ R2 (Rename) ->b (R) And some other useful ones: Join R1 ⋈ R2 Intersection R1 Å R2 STUDENT Takes COURSE Calculus SELECT * FROM STUDENT, Takes, COURSE WHERE STUDENT.sid = Takes.sID AND Takes.cID = cid

Our Example Data Instance STUDENT Takes COURSE sid name 1 Jill 2 Qun 3 Nitin 4 Marty sid exp-grade cid 1 A 550-0109 520-1009 3 C 501-0109 4 cid subj sem 550-0109 DB F09 520-1009 AI S09 501-0109 Arch PROFESSOR Teaches fid name 11 Ives 12 Taskar 18 Martin fid cid 11 550-0109 12 520-1009 18 501-0109

Relational Calculus, Variant I: Domain Relational Calculus (DRC) Queries have form: {<x1,x2, …, xn>| p} Predicate: Boolean expression over x1,x2, …, xn Precise operations depend on the domain and query language – may include special functions, etc. Assume the following at minimum: <xi,xj,…>  R X op Y X op const const op X where op is , , , , ,  xi,xj,… are domain variables domain variables predicate

Complex Predicates in the Calculus 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 Subjects for courses with students expecting a “C” All course numbers for which there exists a smaller course number

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! Example: The highest course number offered

Terminology: 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: {<sid,name>| <sid,name>  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?

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 {<x1,x2, …, xn>| <x1,x2, …, xn>  R}

Selection: TR[ R] Suppose we have (e’), where e’ is another RA expression that translates as: TR[e’]= {<x1,x2, …, xn>| p} Then the translation of c(e’) is {<x1,x2, …, xn>| p’} where ’ is obtained from  by replacing each attribute with the corresponding variable Example: TR[#1=#2 #4>2.5R] (if R has arity 4) is {<x1,x2, x3, x4>| < x1,x2, x3, x4>  R  x1=x2  x4>2.5}

Projection: TR[i1,…,im(e)] If TR[e]= {<x1,x2, …, xn>| p} then TR[i1,i2,…,im(e)]= {<x i1,x i2, …, x im >|  xj1,xj2, …, xjk.p}, where xj1,xj2, …, xjk are variables in x1,x2, …, xn that are not in x i1,x i2, …, x im Example: With R as before, #1,#3 (R)={<x1,x3>| x2,x4. <x1,x2, x3,x4> R}

Union: TR[R1  R2] R1 and R2 must have the same arity For e1  e2, where e1, e2 are algebra expressions TR[e1]={<x1,…,xn>|p} and TR[e2]={<y1,…yn>|q} Relabel the variables in the second: TR[e2]={< x1,…,xn>|q’} This may involve relabeling bound variables in q to avoid clashes TR[e1e2]={<x1,…,xn>|pq’}. Example: TR[R1  R2] = {< x1,x2, x3,x4>| <x1,x2, x3,x4>R1  <x1,x2, x3,x4>R2

Other Binary Operators Difference: The same conditions hold as for union If TR[e1]={<x1,…,xn>|p} and TR[e2]={< x1,…,xn>|q} Then TR[e1- e2]= {<x1,…,xn>|pq} Product: If TR[e1]={<x1,…,xn>|p} and TR[e2]={< y1,…,ym>|q} Then TR[e1 e2]= {<x1,…,xn, y1,…,ym >| pq} Example: TR[RS]= {<x1,…,xn, y1,…,ym >| <x1,…,xn> R  <y1,…,ym > S }

What about the Tuple Relational Calculus (TRC)? We’ve been looking at the Domain Relational Calculus The Tuple Relational Calculus is nearly the same, but variables are at the level of a tuple, not an attribute {Q | 9 S  COURSES, 9 T 2 Takes (S.cid = T.cid Æ Q.cid = S.cid Æ Q.exp-grade = T.exp-grade)} The use of the “output” variable is a bit unintuitive, but otherwise it should be very similar

Limitations of the Relational Algebra / Calculus Can’t do: Aggregate operations (average, sum, count) Track the number of duplicate elements (bag semantics) Recursive queries Complex (non-tabular) structures Most of these are expressible in SQL, OQL, XQuery – using other special operators Sometimes we even need the power of a Turing-complete programming language

Summary Can translate relational algebra into relational calculus DRC and TRC are slightly different syntaxes but equivalent 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!