The relational Algebra & calculus cont. Reference : UC Berkeley

Transcription

1 The relational Algebra & calculus cont Reference : UC Berkeley

2 Formal Relational Query Languages Two mathematical Query Languages form the basis for real languages (e.g. SQL), and for implementation: ❶ Relational Algebra: More operational, very useful for representing execution plans. ❷ Relational Calculus: Lets users describe what they want, rather than how to compute it. (Non-operational, declarative.) Understanding Algebra & Calculus is key to understanding SQL, query processing! Database Management Systems, R. Ramakrishnan and J. Gehrke 1

3 Review Why do we need Query Languages anyway? Two key advantages Less work for user asking query More opportunities for optimization Relational Algebra Theoretical foundation for SQL Higher level than programming language but still must specify steps to get desired result Relational Calculus Formal foundation for Query-by-Example A first-order logic description of desired result Only specify desired result, not how to get it

4 Relational Algebra Review Reserves Sailors Boats sid bid day /10/ /12/96 sid sname rating age 22 dustin lubber rusty bid bname color 101 Interlake Blue 102 Interlake Red 103 Clipper Green 104 Marine Red Basic operations: Selection ( σ ) Projection ( π ) Cross-product ( ) Set-difference ( ) Union ( ) Additional operations: Intersection () Join ( ) Division ( / ) : gives a subset of rows. : deletes unwanted columns. : combine two relations. : tuples in relation Prediction: 1, but These not 2 relational operators are going to look hauntingly familiar when we get to them! : tuples in relation 1 and 2. Query Optimization and Execution Relational Operators Files and Access Methods Buffer Management Disk Space Management :tuples in both relations. :like but only keep tuples where common fields are equal. :tuples from relation 1 with matches in relation 2 DB

5 Relational Algebra Review Reserves Sailors Boats sid bid day /10/ /12/96 Basic operations: Selection ( σ ) Projection ( π ) Cross-product ( ) Set-difference ( ) Union ( ) Additional operations: Intersection () Join ( ) Division ( / ) sid sname rating age 22 dustin lubber rusty bid bname color 101 Interlake Blue 102 Interlake Red 103 Clipper Green 104 Marine Red Find names of sailors who ve reserved a green boat π σ ( ( sname ( ( color= Green Boats) Reserves ) ) Sailors)

6 Relational Algebra Review Reserves Sailors Boats sid bid day /10/ /12/96 sid sname rating age 22 dustin lubber rusty bid bname color 101 Interlake Blue 102 Interlake Red 103 Clipper Green 104 Marine Red Or better yet: Find names of sailors who ve reserved a green boat ( sname π σ ( color= Green Boats) ( πbid ) ( Reserves) ( πsid ) ( Sailors) ) *Given the previous algebra, a query optimizer would replace it with this!

7 Find sailors who ve reserved a red or a green boat Can identify all red or green boats, then find sailors who ve reserved one of these boats: ( Tempboats, ( )) color ' red ' color ' green' Boats sname ( Tempboats Re serves Sailors)

8 Find sailors who ve reserved a red and a green boat Previous approach won t work! Must identify sailors who ve reserved red boats, sailors who ve reserved green boats, then find the intersection (note that sid is a key for Sailors): ( Tempred, (( sid color ' red ' ( Tempgreen, (( sid color ' green' Boats) Re serves)) Boats) Re serves)) sname (( Tempred Tempgreen) Sailors)

9 Intermission Some algebra exercises for you to practice which are out on the class web site Algebra and calculus exercises make for good exam questions!

10 Relational Calculus High-level, first-order logic description A formal definition of what you want from the database Comes in two flavours: Tuple relational calculus (TRC, like SQL) Domain relational calculus (DRC, like QBE). Calculus has variables, constants, comparison ops, logical connectives and quantifiers. Expressions in the calculus are called formulas. An answer tuple is essentially an assignment of constants to variables that make the formula evaluate to true.

11 Tuple The current values (relation instance) of a relation are specified by a table Tuple is represented by a row in a table customer_name customer_street customer_city attributes (or columns, fields) Jones Smith Curry Lindsay Main North North Park Harrison Rye Rye Pittsfield tuples (or rows) customer

12 Domain A domain is referred to in a relation schema by the domain name and has a set of associated values. Students(sid: string, name: string, login: string, age: integer, gpa: real) The set of values associated with domain string is the set of all character strings.

13 Example: Find all sailors with a rating above 7 Tuple R.C.: {S S Sailors S.rating > 7} sid sname rating age 28 yuppy lubber guppy rusty From all that is, find me the set of things that are tuples in the Sailors relation and whose rating field is greater than 7. Domain R.C.: {<S,N,R,A> <S,N,R,A> Sailors R > 7} From all that is, find me column values S, N, R, and A, where S is an integer, N is a string, R is an integer, A is a floating point number, such that <S, N, R, A> is a tuple in the Sailors relation and R is greater than 7.

14 Review of Relational Calculus Building Blocks Variables TRC: Variables are bound to tuples. DRC: Variables are bound to domain elements (= column values) Constants 7, Foo, , etc. Comparison operators (op) =, <>, <, >, etc. Logical connectives - not and - or - implies - is a member of Quantifiers X(p(X)): For every X, p(x) must be true X(p(X)): There exists at least one X such that p(x) is true

15 The advantage level of logical connectives Ops (<,>,=)are highest The quantifiers and are the second, is higher than. Then it is logical notation, the is higher than, is higher than When brackets are used, the one inside will be used first.

16 Tuple Relational Calculus Query form: {T p(t)} T is a tuple and p(t) denotes a formula in which tuple variable T appears. Answer: set of all tuples T for which the formula p(t) evaluates to true. Formula is recursively defined: Atomic formulas get tuples from relations or compare values Formulas built from other formulas using logical operators.

17 TRC Formulas An atomic formula is one of the following: R Rel R.a op S.b R.a op constant, where op is one of,,,,, A formula can be: an atomic formula p, p q, pq R( p( R)) R( p( R)) where p and q are formulas where variable R is a tuple variable where variable R is a tuple variable

18 Free and Bound Variables The use of quantifiers X and X in a formula is said to bind X in the formula. A variable that is not bound is free. Important restriction {T p(t)} The variable T that appears to the left of ` must be the only free variable in the formula p(t). In other words, all other tuple variables must be bound using a quantifier.

19 Free and Bound Variables The use of quantifiers X and X in a formula is said to bind X in the formula. A variable that is not bound is free. Important restriction {T p(t)} The variable T that appears to the left of ` must be the only free variable in the formula p(t). In other words, all other tuple variables must be bound using a quantifier.

20 Use of (For every) x (P(x)): only true if P(x) is true for every x in the universe: e.g. x ((x.color = Red ) means everything that exists is red Usually we are less grandiose in our assertions: x ( (x Boats) (x.color = Red ) is a logical implication a b means that if a is true, b must be true a b is the same as a b

21 a b is the same as a b a T F T T T b F T F If a is true, b must be true! If a is true and b is false, the expression evaluates to false. If a is not true, we don t care about b The expression is always true.

22 Quantifier Shortcuts x ((x Boats) (x.color = Red )) For every x in the Boats relation, the color must be Red. Can also be written as: x Boats(x.color = Red ) x ( (x Boats) (x.color = Red )) There exists a tuple x in the Boats relation whose color is Red. Can also be written as: x Boats (x.color = Red )

23 Selection and Projection Selection Find all sailors with rating above 8 {S S Sailors S.rating > 8} S1 S1 S1 S1 sid sname rating age 28 yuppy lubber guppy rusty Projection Find names and ages of sailors with rating above 8. {S S1 Sailors(S1.rating > 8 S.sname = S1.sname S.age = S1.age)} sname age S yuppy 35.0 S rusty 35.0 S is a tuple variable of 2 fields (i.e. {S} is a projection of Sailors)

24 Joins Find sailors rated > 7 who ve reserved boat #103 {S SSailors S.rating > 7 R(RReserves R.sid = S.sid R.bid = 103)} S S S R R sid sname rating age 22 dustin lubber rusty sid bid day /10/ /12/96 Note the use of to find a tuple in Reserves that `joins with the Sailors tuple under consideration. What if there was another tuple {58, 103, 12/13/96} in the Reserves relation?

25 Joins (continued) What does this expression compute? Find sailors rated > 7 who ve reserved a red boat {S SSailors S.rating > 7 R(RReserves R.sid = S.sid B(BBoats B.bid = R.bid B.color = red ))} Notice how the parentheses control the scope of each quantifier s binding.

26 Division Not supported as a primitive operator, but useful for expressing queries like: Find sailors who have reserved all boats. Let A have 2 fields, x and y; B have only field y: A/B = x x, y A y B i.e., A/B contains all x tuples (sailors) such that for every y tuple (boat) in B, there is an xy tuple in A. Or: If the set of y values (boats) associated with an x value (sailor) in A contains all y values in B, the x value is in A/B. In general, x and y can be any lists of fields; y is the list of fields in B, and xy is the list of fields of A. 10

27 Division Recall the algebra expression A/B A value x in A is disqualified if by attaching a y value from B, we obtain an xy tuple that is not in A. (e.g: only give me A tuples that have a match in B. In calculus, use the operator: e.g. Find sailors who ve reserved all boats: {S SSailors BBoats (RReserves (S.sid = R.sid B.bid = R.bid))} Find all sailors S such that For all tuples B in Boats There is at least one tuple in Reserves showing that sailor S has reserved B.

28 Division Algebra basic operators Disqualified x values: x (( x ( A) B) A) A/B: x ( A) Disqualified x values

29 Expressive Power (Theorem due to Codd): 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 languages (e.g., SQL) can express every query that is expressible in relational algebra/calculus. (actually, SQL is more powerful, as we will see )

30 Tuple Relational Calculus A tuple relational calculus query has the form { T p(t) }, where T is a tuple variable and p(t) denotes a formula that describes T. T[i] op Q[j], this presents the op relationship between the ith constituent of tuple T and the jth constituent of tuple Q.

31 Example 1 We know student relational model S(Sno, Sname, age, sex), use TRC to represent a query: find all male students name(sname) and student number(sno). {t ( u)(s(u) u[4]= male t[1]=u[2] t[2]=u[1])}

32 Example 2 We know relation SC(Sno,Cno,grade), use domain to represent the following query:find the students number and score whose course number is 1. First stetp: TRC {t ( u)(sc(u) u[2]= 1 t[1]=u[1] t[2]=u[3]} Second step:trc to DRC {t 1 t 2 ( u 1 )( u 2 )( u 3 )(SC(u 1 u 2 u 3 ) u 2 = 1 t 1 =u 1 t 2 =u 3 ))} Which can be simplified as: {t 1 t 2 SC(t 1, 1,t 2 )}

33 TRC 其他表达方法 2 原子公式 R (t): 表示 t 是关系 R 中的一个元组 t[i]u[j]: 表示 t 的第 i 个分量和 u 的第 j 个分量满足比较关系 t[i]c 或 C t[i] : 含义同上, 只不过 C 为常量 3 公式的递归定义 (1) 每个原子公式是一个公式 ; (2) 设 1 2 是公式, 则 也是公式 ; (3) 设 是公式,t 是元组变量, 则 ( t) ( t) 也是公式 ; (4) 除此之外没有其它形式的公式 数据库系统数据库系统 33

34 4 关系代数运算均可用关系演算来表示, 反之亦然 5 用关系演算来表达查询例 1, 求年龄大于或等于 20 的学生 : S20 = { t S(t) t[4]20 } 例 2, 求学生姓名及所在的系 : S: S# SN SD SA S1 A CS 20 S2 B CS 21 S3 C MA 19 S4 D CI 19 S5 E MA 20 S6 F CS 22 S1={ t (2) ( u )(S(u) t[1]=u[2] t[2]=u[3] )} S1 A CS 20 S2 B CS 21 S3 C MA 19 S4 D CI 19 S5 E MA 20 S6 F CS 数据库系统数据库系统 34

35 Domain Relational Calculus Query has the form: Database Management Systems, R. Ramakrishnan x1,x2,...,xn p Answer includes all tuples make the formula p x1,x2,...,xn x1,x2,...,xn x1,x2,...,xn that 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. 1

36 DRC Formulas Atomic formula: x1,x2,...,xn Rname, or X op Y, or X op constant op is one of,,,,, Formula: an atomic formula, or p, p q, pq, where p and q are formulas, or X ( p(x)), where variable X is free in p(x), or X ( p(x)), where variable X is free in p(x) The use of quantifiers Xand X is said to bind X. A variable that is not bound is free. Database Management Systems, R. Ramakrishnan 1

37 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: the variables x1,..., xn that appear to the left of ` must be the only free variables in the formula p(...). Database Management Systems, R. Ramakrishnan 1

38 Find all sailors with a rating above 7 I,N,T,A I,N,T,A Sailors T 7 The condition 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 I,N,T,A Sailors to the left of (which should be read as such that) says that every tuple that satisfies T>7 is in the answer. Modify this query to answer: I,N,T, A Find sailors who are older than 18 or have a rating under 9, and are called Joe. Database Management Systems, R. Ramakrishnan 1

39 练习题 若要查询 选修了课程号 C#='C2' 的学生的学生号, 正确的表达是 I. II. III. select S# from SC where C#='C2' A. 仅 I 和 II B. 仅 II 和 III C. 仅 I 和 III D. 都正确

40 (2011 计算机等级考试考题 ) 1. 设关系 R 和 S 具有相同的属性个数, 且相对应属性的值取自同一个域, 则 R- (R-S) 等价于 A. RUS B.R S C.RxS D. R-S

41 Relational Completeness means Theory Relational Calculus Practice Query Optimization and Execution Relational Operators Relational Algebra Files and Access Methods Buffer Management Relational Model Disk Space Management DB

42 Now we can study SQL! Practice Query Optimization and Execution SQL Relational Operators Files and Access Methods Buffer Management Disk Space Management DB