Operations of Relational Algebra

Transcription

1 ITCS 3160 DATA BASE DESIGN AND IMPLEMENTATION JING YANG 2010 FALL Class 11: The Relational Algebra and Relational Calculus (2) 2 Operations of Relational Algebra 1

2 3 Operations of Relational Algebra (cont d.) Notation for Query Trees 4 Query tree Represents the input relations of query as leaf nodes of the tree Represents the relational algebra operations as internal nodes 2

3 5 Additional Relational Operations 6 Generalized projection Allows functions of attributes to be included d in the projection list Aggregate functions and grouping Common functions applied to collections of numeric values Include SUM, AVERAGE, MAXIMUM, and MINIMUM 3

4 7 Additional Relational Operations (cont d.) Group tuples by the value of some of their attributes tes Apply aggregate function independently to each group Recursive Closure Operations 8 Operation applied to a recursive relationship between tuples of same type Question: what is the purpose of the following expression? What if we want to find all employees supervised by James Borg? 4

5 OUTER JOIN Operations 9 Outer joins Keep all tuples in R, or all those in S, or all those in both relations regardless of whether or not they have matching tuples in the other relation Types LEFT OUTER JOIN, RIGHT OUTER JOIN, FULL OUTER JOIN OUTER JOIN Operations 10 Example: 5

6 The OUTER UNION Operation 11 Take union of tuples from two relations that have some common attributestes Not union (type) compatible Partially compatible All tuples from both relations included in the result Tuples with the same value combination will appear only once Summary 12 The relational model has rigorously defined query languages that are simple and powerful. Relational algebra is more operational; useful as internal representation for query evaluation plans. Several ways of expressing a given query; a query optimizer should choose the most efficient version. 6

7 RELATIONAL CALCULUS Relational Calculus 14 Declarative expression Specify a retrieval request; nonprocedural language Any retrieval that can be specified in basic relational algebra Can also be specified in relational calculus 7

8 First-Order Predicate Logic 15 Propositional logic is concerned only with sentential connectives such as and, or, not. Proposition: a statement that affirms or denies something Peter is tall First order predicate logic additionally covers predicates and quantification Predicates 16 Example: {x x is a positive integer less than 4} is the set {1,2,3}. An element of the set {x P(x)}, is an object t for which the statement P(t) is true. With such statements, P(x) is referred to as the Predicate making x the subject of the proposition. 8

9 Quantification 17 All known human languages make use of quantification. For example, in English: Every student in my class is smart. There was somebody in my class that was able to correctly answer every one of the questions I gave. Most of the people I talked to didn't have a clue who the candidates were. Relational Calculus 18 Comes in two flavors: Tuple relational calculus (TRC) and Domain relational l calculus l (DRC). Calculus has variables, constants, comparison ops, logical connectives and quantifiers. TRC: Variables range over (i.e., get bound to) tuples. DRC: Variables range over domain elements (= field values). Both TRC and DRC are simple subsets of first-order predicate logic. Expressions (predicates) in the calculus are called formulas. An answer tuple is essentially an assignment of constants to variables that make the formula evaluate to true. 9

10 Tuple Relational Calculus 19 Query: {T P(T)} T is tuple variable P(T) is a formula that describes T Result, the set of all tuples t for which P(t) evaluates True. Find all sailors with a rating above 7. { S S Sailors S. rating > 7} S Sailors in our book: Sailors(S) specifies that the range relation of tuple variable S is Sailors (s may take as its value any individual tuple from Sailors) Tuple Relational Calculus 20 Atomic formula R Rel eg. S Sil Sailors in our book: Sailors(S) Rel: range relation of R R.a op S.b, op is one of <, >, =,,, R.a op constant eg. S. rating > 7 10

11 21 TRC Formula Any atomic formula p, p q, (in our book: NOT(p), p AND q, p OR q ) Existential quantifiers R( p( R)) Universal quantifiers Example p q R( p( R)) Find the names and ages of sailors with a rating above 7 { P S Sailors ( S. rating > 7 P. name = S. name P. age = S. age)} Free and Bound Variables 22 The use of quantifiers X and X in a formula is said to bind X. A variable that is not bound is free. Revisit Query: {T P(T)} T is tuple variable P(T) is a formula that describes T Th i i i i h i bl There is an important restriction: the variables T that appear to the left of ` must be the only free variables in the formula p(...). 11

12 23 Sample Queries in Tuple Relational Calculus Practice Retrieve the birth date and address of the employee whose first name is John 2. Retrieve all employees whose salary is higher than For each employee, retrieve the employee s first and last name and the first and last name of his/her immediate supervisor 4. List the name of employees who have at least one dependent 12

13 Domain Relational Calculus 25 Query has the form: x1, x2,..., xn p x1, x2,..., xn Answer includes all tuples x1, x2,..., xn that make the formula px1, x2,..., xn be true. Formula is recursively defined, starting with simple atomic formulas (getting tuples from relations or making comparisons of values), and building bigger and better formulas using the logical connectives. Free and Bound Variables The use of quantifiers X and X in a formula is said to bind X. A variable that is not bound is free. Let us revisit the definition of a query: x1, x2,..., xn p x1, x2,..., xn There is an important restriction: i the variables x1,..., xn that appear to the left of ` must be the only free variables in the formula p(...). 13

14 27 DRC Formulas Atomic formula: x 1, x 2,..., xn Rname, or X op Y, or X op constant op is one of <>=,,,,, Formula: an atomic formula, or p, p q, p q, where p and q are formulas, or XX ( p ( X )), where variable X is free in p(x), or X ( p( X)), where variable X is free in p(x) Find all sailors with a rating above 7 28 INTA,,, INTA,,, Sailors T> 7 The condition I, N, T, A Sailors ensures that the domain variables I, N, T and A are bound to fields of the same Sailors tuple. The term I, N, T, A to the left of ` (which should be I, N, T, A read as such that) says that every tuple that satisfies T>7 is in the answer. Modify this query to answer: Find sailors who are older than 18 14

15 Find sailors rated > 7 who ve reserved boat # I, N, T, A I, N, T, A Sailors T > 7 IrBrD,, IrBrD,, Reserves Ir= I Br= 103 We have used shorthand for ( ) Ir, Br, D... ( ( ( ))) as a Ir Br D... Note the use of to find a tuple in Reserves that `joins with the Sailors tuple under consideration. 30 Find sailors rated > 7 who ve reserved a red boat I, N, T, A I, N, T, A Sailors T > 7 IrBrD,, IrBrD,, Reserves Ir= I BBNC,, BBNC,, Boats B= Br C= ' red' Observe how the parentheses control the scope of each quantifier s binding. This may look cumbersome, but with a good user interface, it is very intuitive. 15

16 31 Find sailors who ve reserved all boats I, N, T, A I, N, T, A Sailors BBNC,, BBNC,, Boats IrBrD,, IrBrD,, Reserves I= Ir Br= B 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. 32 Find sailors who ve reserved all boats (again!) I, N, T, A I, N, T, A Sailors BBN,, C Boats Ir, Br, D Reserves I = Ir Br = B Simpler notation, same query. (Much clearer!) To find sailors who ve reserved all red boats:... C ' red ' Ir, Br, D Reserves I = Ir Br = B 16

17 Unsafe Queries, Expressive Power 33 It is possible to write syntactically correct calculus queries that have an infinite number of answers! Such queries are called unsafe. e.g., S S Sailors It is known that every query that can be expressed in relational algebra can be expressed as a safe query in DRC / TRC; the converse is also true. Relational Completeness: Query language (e.g., SQL) can express every query that is expressible in relational algebra/calculus. Summary 34 Relational calculus is non-operational, and users define queries in terms of what they want, not in terms of how to compute it. (Declarativeness.) Algebra and safe calculus have same expressive power, leading to the notion of relational completeness. 17

18 Reference 35 Raghu Ramakrishnan: Database Management Systems 18