Lu Chaojun, SJTU 1 Extended Relational Algebra

Bag Semantics A relation (in SQL, at least) is really a bag (or multiset). –It may contain the same tuple more than once –There is no specified order (unlike a list). Select, project, and join work for bags as well as sets. –Just work on a tuple-by-tuple basis, and don't eliminate duplicates. Lu Chaojun, SJTU 2

Why Bags? Efficient implementation –e.g. projection, union –Q: How to eliminate duplicates? Some queries use bags –e.g. Aggregate Find the average grades Lu Chaojun, SJTU 3

Bag Union R  S: Sum the times an element appears in the two bags, i.e. if t appears n/m times in R/S, then t appears n+m times in R  S. Example { 1,2, 1}  { 1,2, 3} = { 1,1,1,2,2, 3}. 4 Lu Chaojun, SJTU

Bag Intersection R  S: Take the minimum of the number of occurrences in each bag, i.e. t appears min(n,m) times in R  S. Example { 1,2, 1}  { 1,2, 3,3} = { 1,2 }. 5 Lu Chaojun, SJTU

Bag Difference R  S: Proper-subtract the number of occurrences in the two bags, i.e. t appears max(0, n  m) times in R  S. Example { 1,2, 1}  { 1,2, 3,3} = { 1 }. 6 Lu Chaojun, SJTU

Other Operators on Bags Projection, selection, product, join –No duplicate elimination 7 Lu Chaojun, SJTU

Extensions to Relational Model Not a part of the formal relational model, but appear in real query languages like SQL. –Modification: insert, delete, update. –Aggregation: count, sum, average –Views –Null values 8 Lu Chaojun, SJTU

Extended RA Duplicate-elimination operator Sorting operator Extended projection Grouping-and-aggregation operator Outerjoin operator 9 Lu Chaojun, SJTU

Duplicate Elimination  ( R) = relation with one copy of each tuple that appears one or more times in R. 10 Lu Chaojun, SJTU

Aggregation Operators These are not relational operators; rather they summarize a column in some way. Five standard operators: Sum, Average, Count, Min, and Max. 11 Lu Chaojun, SJTU

Grouping Operator  L (R), where L is a list of elements that are either –Individual ( grouping) attributes or –Of the form  (A), where  is an aggregation operator and A the attribute to which it is applied. Example  sno,AVG(grade) (SC) 12 Lu Chaojun, SJTU

Grouping Operator(cont.)  L (R) is computed by: 1. Group R according to all the grouping attributes on list L. 2. Within each group, compute  (A), for each element  (A) on list L. 3. Result is the relation that consists of one tuple for each group. The components of that tuple are the values associated with each element of L for that group. 13 Lu Chaojun, SJTU

Extended Projection Allow the columns in the projection to be functions of one or more columns in the argument relation. Example  name,2011  age (Student) 14 Lu Chaojun, SJTU

Sorting  L (R) = list of tuples of R, ordered according to attributes on list L. Note that result type is outside the normal types (set or bag) for relational algebra. –Consequence:  cannot be followed by other relational operators. 15 Lu Chaojun, SJTU

Outerjoin The normal join can lose information, because a tuple that doesn't join with any from the other relation becomes dangling. The null value can be used to pad dangling tuples so they appear in the join. Outerjoin operator: o Variations: theta-outerjoin, left- and right- outerjoin (pad only dangling tuples from the left (resp., right). 16 Lu Chaojun, SJTU

A Logic for Relations Datalog Lu Chaojun, SJTU 17

Introduction A query language for relational model may be based on –Algebra: relational algebra –Logic: relational calculus e.g. Datalog More natural for recursive queries 18 Lu Chaojun, SJTU

Predicates and Atoms RDB vs. Datalog RDB Datalog relation R( ) predicate R( ) attributes(tuples) arguments x schema R(X) (relational)atom R(x) tuple t  R R(t) is TRUE –R(x) is a boolean-valued function if x contains variables; proposition otherwise. 19 Lu Chaojun, SJTU

Arithmetic Atoms Comparison between two arithmetic expressions exp1  exp2 –Predicate  (exp1,exp2) –infinite and unchanging relation 20 Lu Chaojun, SJTU

Datalog Rules Example Happy(sno)  S(sno,n,a,d) AND SC(sno,cno,g) AND g>=95 AND C(cno,cn) AND cn=‘Database’ Rules: Head  Body –Head: relational atom –Body: AND of subgoals Subgoal: atom or NOT atom Atom: P(arg), P is relation name or arithmetic predicate; arg may be variable or constant –  : if Or :- 21 Lu Chaojun, SJTU

Datalog Rules (cont.) Query: a collection of one or more rules Result: a relation appearing in rule heads –Designate the intended answer when there are more than one relation in rule heads 22 Lu Chaojun, SJTU

Meaning of Datalog Rules Meaning I: –Assign possible values to variables in the rule –If the assignment makes all the subgoals TRUE, then it forms a tuple of the result relation. Meaning II: –Consider consistent assignment of tuples for each nonnegated, relational subgoals. (see safety) –Then consider the negated, relational subgoals and the arithmetic subgoals, to see if the assignment of values to variables makes them all TRUE. If yes, a tuple is added to the result relation. 23 Lu Chaojun, SJTU

Example: Meaning I S(x,y)  R(x,z) AND R(z,y) AND NOT R(x,y) Consider all possible assignments: R: A B 1. x=1, z=2 make R(x,z) TRUE 1 2 y=3 make R(z,y) TRUE 2 3 NOT R(x,y) TRUE thus add (1,3) to S; S: C D 2. x=2, z=3 make R(x,z) TRUE 1 3 no y make R(z,y) TRUE 24 Lu Chaojun, SJTU

Example: Meaning II S(x,y)  R(x,z) AND R(z,y) AND NOT R(x,y) Consider consistent assignment of tuples: R: A B 1. t 1 for R(x,z), t 1 for R(z,y) t t 1 for R(x,z), t 2 for R(z,y) t t 2 for R(x,z), t 1 for R(z,y) 4. t 2 for R(x,z), t 2 for R(z,y) S: C D 1 3 only case 2 is a consistent assignment 25 Lu Chaojun, SJTU

Safety Every variable in the rule must appear in some nonnegated relational subgoal. To make the result a finite relation. Example: safety violation 1. S(x)  R(y) x not in subgoal 2. S(x)  NOT R(x) x not in nonnegated subgoal 3. S(x)  R(y) AND x < y x not in relational subgoal 26 Lu Chaojun, SJTU

Datalog Program -- Query A collection of rules Predicates/Relations are divided into two classes: –Extensional Relations/Predicates: stored in DB –Intensional Relations/Predicates: defined by rules EDB predicates can’t appear in the head, only in body; IDB predicates can appear in head, body, or both. 27 Lu Chaojun, SJTU

Datalog Rules Applied to Bags When there are no negated relational subgoals: –Meaning I for evaluating Datalog rules applies to bags as well as sets –But for bags, Meaning II is simpler for evaluating. When there are negated relational subgoals: –There is not a clearly defined meaning under the bag model. 28 Lu Chaojun, SJTU

From RA to Datalog R  S I(x)  R(x) AND S(x) R  S I(x)  R(x) I(x)  S(x) R  S I(x)  R(x) AND NOT S(x)  A (R) I(a)  R(a,b) 29 Lu Chaojun, SJTU

From RA to Datalog(cont.)  F (R) I(x)  R(x) AND F  C1 AND C2 (R) I(x)  R(x) AND C1 AND C2  C1 OR C2 (R) I(x)  R(x) AND C1 I(x)  R(x) AND C2 R  S I(x,y)  R(x) AND S(y) R S I(x,y,z)  R(x,y) AND S(y,z) 30 Lu Chaojun, SJTU

Multiple Operations in Datalog Create IDB predicates for intermediate relations Example A(x,y,z)  R(x,y,z) AND x > 10 B(x,y,z)  R(x,y,z) AND y = ‘ok’ C(x,y,z)  A(x,y,z) AND B(x,y,z) D(x,z)  C(x,y,z) 31 Lu Chaojun, SJTU

Expressive Power of Datalog Non-recursive Datalog = RA Datalog simulates SQL SELECT-FROM- WHERE without aggregation and grouping Recursive Datalog is more powerful than RA and SQL None is full in expressive power (Turing completeness) 32 Lu Chaojun, SJTU

End