Chapter 5: Other Relational Languages

Chapter 5: Other Relational Languages

اسلاید 1: Chapter 5: Other Relational Languages

اسلاید 2: Chapter 5: Other Relational LanguagesTuple Relational CalculusDomain Relational CalculusQuery-by-Example (QBE)Datalog

اسلاید 3: Tuple Relational CalculusA nonprocedural query language, where each query is of the form{t | P (t ) }It is the set of all tuples t such that predicate P is true for tt is a tuple variable, t [A ] denotes the value of tuple t on attribute At  r denotes that tuple t is in relation rP is a formula similar to that of the predicate calculus

اسلاید 4: Predicate Calculus Formula1.Set of attributes and constants2.Set of comparison operators: (e.g., , , , , , )3.Set of connectives: and (), or (v)‚ not ()4.Implication (): x  y, if x if true, then y is truex  y x v y5.Set of quantifiers:t r (Q (t )) ”there exists” a tuple in t in relation r such that predicate Q (t ) is truet r (Q (t )) Q is true “for all” tuples t in relation r

اسلاید 5: Banking Examplebranch (branch_name, branch_city, assets ) customer (customer_name, customer_street, customer_city ) account (account_number, branch_name, balance ) loan (loan_number, branch_name, amount )depositor (customer_name, account_number )borrower (customer_name, loan_number )

اسلاید 6: Example QueriesFind the loan_number, branch_name, and amount for loans of over $1200 Find the loan number for each loan of an amount greater than $1200 {t |  s loan (t [loan_number ] = s [loan_number ]  s [amount ]  1200)} Notice that a relation on schema [loan_number ] is implicitly defined by the query{t | t  loan  t [amount ]  1200}

اسلاید 7: Example QueriesFind the names of all customers having a loan, an account, or both at the bank{t | s  borrower ( t [customer_name ] = s [customer_name ])  u  depositor ( t [customer_name ] = u [customer_name] ) Find the names of all customers who have a loan and an account at the bank{t | s  borrower ( t [customer_name ] = s [customer_name ])  u  depositor ( t [customer_name ] = u [customer_name ])

اسلاید 8: Example QueriesFind the names of all customers having a loan at the Perryridge branch{t | s  borrower (t [customer_name ] = s [customer_name ]  u  loan (u [branch_name ] = “Perryridge”  u [loan_number ] = s [loan_number ]))  not v  depositor (v [customer_name ] = t [customer_name ])} Find the names of all customers who have a loan at the Perryridge branch, but no account at any branch of the bank{t | s  borrower (t [customer_name ] = s [customer_name ]  u  loan (u [branch_name ] = “Perryridge”  u [loan_number ] = s [loan_number ]))}

اسلاید 9: Example QueriesFind the names of all customers having a loan from the Perryridge branch, and the cities in which they live{t | s  loan (s [branch_name ] = “Perryridge”  u  borrower (u [loan_number ] = s [loan_number ]  t [customer_name ] = u [customer_name ])   v  customer (u [customer_name ] = v [customer_name ]  t [customer_city ] = v [customer_city ])))}

اسلاید 10: Example QueriesFind the names of all customers who have an account at all branches located in Brooklyn:{t |  r  customer (t [customer_name ] = r [customer_name ])  (  u  branch (u [branch_city ] = “Brooklyn”   s  depositor (t [customer_name ] = s [customer_name ]   w  account ( w[account_number ] = s [account_number ]  ( w [branch_name ] = u [branch_name ]))))}

اسلاید 11: Safety of ExpressionsIt is possible to write tuple calculus expressions that generate infinite relations.For example, { t |  t r } results in an infinite relation if the domain of any attribute of relation r is infiniteTo guard against the problem, we restrict the set of allowable expressions to safe expressions.An expression {t | P (t )} in the tuple relational calculus is safe if every component of t appears in one of the relations, tuples, or constants that appear in PNOTE: this is more than just a syntax condition. E.g. { t | t [A] = 5  true } is not safe --- it defines an infinite set with attribute values that do not appear in any relation or tuples or constants in P.

اسلاید 12: Domain Relational CalculusA nonprocedural query language equivalent in power to the tuple relational calculusEach query is an expression of the form:{  x1, x2, …, xn  | P (x1, x2, …, xn)} x1, x2, …, xn represent domain variablesP represents a formula similar to that of the predicate calculus

اسلاید 13: Example QueriesFind the loan_number, branch_name, and amount for loans of over $1200 Find the names of all customers who have a loan from the Perryridge branch and the loan amount:{ c, a  |  l ( c, l   borrower  b ( l, b, a   loan  b = “Perryridge”))}{ c, a  |  l ( c, l   borrower   l, “ Perryridge”, a   loan)} { c  |  l, b, a ( c, l   borrower   l, b, a   loan  a > 1200)}Find the names of all customers who have a loan of over $1200 { l, b, a  |  l, b, a   loan  a > 1200}

اسلاید 14: Example QueriesFind the names of all customers having a loan, an account, or both at the Perryridge branch:{ c  |  s,n ( c, s, n   customer)   x,y,z ( x, y, z   branch  y = “Brooklyn”)   a,b ( x, y, z   account   c,a   depositor)} Find the names of all customers who have an account at all branches located in Brooklyn:{ c  |  l (  c, l   borrower   b,a ( l, b, a   loan  b = “Perryridge”))   a ( c, a   depositor   b,n ( a, b, n   account  b = “Perryridge”))}

اسلاید 15: Safety of ExpressionsThe expression:{  x1, x2, …, xn  | P (x1, x2, …, xn )} is safe if all of the following hold:All values that appear in tuples of the expression are values from dom (P ) (that is, the values appear either in P or in a tuple of a relation mentioned in P ).For every “there exists” subformula of the form  x (P1(x )), the subformula is true if and only if there is a value of x in dom (P1)such that P1(x ) is true.For every “for all” subformula of the form x (P1 (x )), the subformula is true if and only if P1(x ) is true for all values x from dom (P1).

اسلاید 16: Query-by-Example (QBE)Basic StructureQueries on One RelationQueries on Several RelationsThe Condition BoxThe Result RelationOrdering the Display of TuplesAggregate Operations Modification of the Database

اسلاید 17: QBE — Basic StructureA graphical query language which is based (roughly) on the domain relational calculusTwo dimensional syntax – system creates templates of relations that are requested by usersQueries are expressed “by example”

اسلاید 18: QBE Skeleton Tables for the Bank Example

اسلاید 19: QBE Skeleton Tables (Cont.)

اسلاید 20: Queries on One RelationFind all loan numbers at the Perryridge branch. _x is a variable (optional; can be omitted in above query) P. means print (display) duplicates are removed by default To retain duplicates use P.ALL

اسلاید 21: Queries on One Relation (Cont.)Display full details of all loansP._xP._yP._zMethod 1:Method 2: Shorthand notation

اسلاید 22: Queries on One Relation (Cont.)Find names of all branches that are not located in Brooklyn Find the loan number of all loans with a loan amount of more than $700

اسلاید 23: Queries on One Relation (Cont.)Find the loan numbers of all loans made jointly to Smith and Jones.Find all customers who live in the same city as Jones

اسلاید 24: Queries on Several RelationsFind the names of all customers who have a loan from the Perryridge branch.

اسلاید 25: Queries on Several Relations (Cont.)Find the names of all customers who have both an account and a loan at the bank.

اسلاید 26: Negation in QBEFind the names of all customers who have an account at the bank, but do not have a loan from the bank.¬ means “there does not exist”

اسلاید 27: Negation in QBE (Cont.)Find all customers who have at least two accounts.¬ means “not equal to”

اسلاید 28: The Condition BoxAllows the expression of constraints on domain variables that are either inconvenient or impossible to express within the skeleton tables.Complex conditions can be used in condition boxesExample: Find the loan numbers of all loans made to Smith, to Jones, or to both jointly

اسلاید 29: Condition Box (Cont.)QBE supports an interesting syntax for expressing alternative values

اسلاید 30: Condition Box (Cont.)Find all account numbers with a balance greater than $1,300 and less than $1,500 Find all account numbers with a balance greater than $1,300 and less than $2,000 but not exactly $1,500.

اسلاید 31: Condition Box (Cont.)Find all branches that have assets greater than those of at least one branch located in Brooklyn

اسلاید 32: The Result RelationFind the customer_name, account_number, and balance for all customers who have an account at the Perryridge branch.We need to:Join depositor and account.Project customer_name, account_number and balance.To accomplish this we:Create a skeleton table, called result, with attributes customer_name, account_number, and balance.Write the query.

اسلاید 33: The Result Relation (Cont.)The resulting query is:

اسلاید 34: Ordering the Display of TuplesAO = ascending order; DO = descending order.Example: list in ascending alphabetical order all customers who have an account at the bank When sorting on multiple attributes, the sorting order is specified by including with each sort operator (AO or DO) an integer surrounded by parentheses.Example: List all account numbers at the Perryridge branch in ascending alphabetic order with their respective account balances in descending order.

اسلاید 35: Aggregate OperationsThe aggregate operators are AVG, MAX, MIN, SUM, and CNTThe above operators must be postfixed with “ALL” (e.g., SUM.ALL. or AVG.ALL._x) to ensure that duplicates are not eliminated.Example: Find the total balance of all the accounts maintained at the Perryridge branch.

اسلاید 36: Aggregate Operations (Cont.)UNQ is used to specify that we want to eliminate duplicates Find the total number of customers having an account at the bank.

اسلاید 37: Query ExamplesFind the average balance at each branch.The “G” in “P.G” is analogous to SQL’s group by construct The “ALL” in the “P.AVG.ALL” entry in the balance column ensures that all balances are consideredTo find the average account balance at only those branches where the average account balance is more than $1,200, we simply add the condition box:

اسلاید 38: Query ExampleFind all customers who have an account at all branches located in Brooklyn. Approach: for each customer, find the number of branches in Brooklyn at which they have accounts, and compare with total number of branches in BrooklynQBE does not provide subquery functionality, so both above tasks have to be combined in a single query. Can be done for this query, but there are queries that require subqueries and cannot always be expressed in QBE.In the query on the next pageCNT.UNQ.ALL._w specifies the number of distinct branches in Brooklyn. Note: The variable _w is not connected to other variables in the queryCNT.UNQ.ALL._z specifies the number of distinct branches in Brooklyn at which customer x has an account.

اسلاید 39: Query Example (Cont.)

اسلاید 40: Modification of the Database – DeletionDeletion of tuples from a relation is expressed by use of a D. command. In the case where we delete information in only some of the columns, null values, specified by –, are inserted.Delete customer SmithDelete the branch_city value of the branch whose name is “Perryridge”.

اسلاید 41: Deletion Query ExamplesDelete all loans with a loan amount greater than $1300 and less than $1500.For consistency, we have to delete information from loan and borrower tables

اسلاید 42: Deletion Query Examples (Cont.)Delete all accounts at branches located in Brooklyn.

اسلاید 43: Modification of the Database – InsertionInsertion is done by placing the I. operator in the query expression. Insert the fact that account A-9732 at the Perryridge branch has a balance of $700.

اسلاید 44: Modification of the Database – Insertion (Cont.)Provide as a gift for all loan customers of the Perryridge branch, a new $200 savings account for every loan account they have, with the loan number serving as the account number for the new savings account.

اسلاید 45: Modification of the Database – UpdatesUse the U. operator to change a value in a tuple without changing all values in the tuple. QBE does not allow users to update the primary key fields.Update the asset value of the Perryridge branch to $10,000,000. Increase all balances by 5 percent.

اسلاید 46: Microsoft Access QBEMicrosoft Access supports a variant of QBE called Graphical Query By Example (GQBE)GQBE differs from QBE in the following waysAttributes of relations are listed vertically, one below the other, instead of horizontallyInstead of using variables, lines (links) between attributes are used to specify that their values should be the same.Links are added automatically on the basis of attribute name, and the user can then add or delete linksBy default, a link specifies an inner join, but can be modified to specify outer joins.Conditions, values to be printed, as well as group by attributes are all specified together in a box called the design grid

اسلاید 47: An Example Query in Microsoft Access QBEExample query: Find the customer_name, account_number and balance for all accounts at the Perryridge branch

اسلاید 48: An Aggregation Query in Access QBEFind the name, street and city of all customers who have more than one account at the bank

اسلاید 49: Aggregation in Access QBEThe row labeled Total specifies which attributes are group by attributeswhich attributes are to be aggregated upon (and the aggregate function). For attributes that are neither group by nor aggregated, we can still specify conditions by selecting where in the Total row and listing the conditions belowAs in SQL, if group by is used, only group by attributes and aggregate results can be output

اسلاید 50: DatalogBasic Structure Syntax of Datalog RulesSemantics of Nonrecursive DatalogSafety Relational Operations in DatalogRecursion in DatalogThe Power of Recursion

اسلاید 51: Basic StructureProlog-like logic-based language that allows recursive queries; based on first-order logic.A Datalog program consists of a set of rules that define views. Example: define a view relation v1 containing account numbers and balances for accounts at the Perryridge branch with a balance of over $700.v1 (A, B ) :– account (A, “Perryridge”, B ), B > 700.Retrieve the balance of account number “A-217” in the view relation v1.? v1 (“A-217”, B ).To find account number and balance of all accounts in v1 that have a balance greater than 800 ? v1 (A,B ), B > 800

اسلاید 52: Example QueriesEach rule defines a set of tuples that a view relation must contain.E.g. v1 (A, B ) :– account (A, “ Perryridge”, B ), B > 700 is read as for all A, B if (A, “Perryridge”, B )  account and B > 700 then (A, B )  v1The set of tuples in a view relation is then defined as the union of all the sets of tuples defined by the rules for the view relation.Example:interest_rate (A, 5) :– account (A, N, B ) , B < 10000 interest_rate (A, 6) :– account (A, N, B ), B >= 10000

اسلاید 53: Negation in DatalogDefine a view relation c that contains the names of all customers who have a deposit but no loan at the bank:c(N) :– depositor (N, A), not is_borrower (N). is_borrower (N) :–borrower (N,L).NOTE: using not borrower (N, L) in the first rule results in a different meaning, namely there is some loan L for which N is not a borrower. To prevent such confusion, we require all variables in negated “predicate” to also be present in non-negated predicates

اسلاید 54: Named Attribute NotationDatalog rules use a positional notation that is convenient for relations with a small number of attributesIt is easy to extend Datalog to support named attributes. E.g., v1 can be defined using named attributes as v1 (account_number A, balance B ) :– account (account_number A, branch_name “ Perryridge”, balance B ), B > 700.

اسلاید 55: Formal Syntax and Semantics of DatalogWe formally define the syntax and semantics (meaning) of Datalog programs, in the following stepsWe define the syntax of predicates, and then the syntax of rulesWe define the semantics of individual rulesWe define the semantics of non-recursive programs, based on a layering of rulesIt is possible to write rules that can generate an infinite number of tuples in the view relation. To prevent this, we define what rules are “safe”. Non-recursive programs containing only safe rules can only generate a finite number of answers.It is possible to write recursive programs whose meaning is unclear. We define what recursive programs are acceptable, and define their meaning.

اسلاید 56: Syntax of Datalog RulesA positive literal has the formp (t1, t2 ..., tn )p is the name of a relation with n attributeseach ti is either a constant or variableA negative literal has the form not p (t1, t2 ..., tn )Comparison operations are treated as positive predicates E.g. X > Y is treated as a predicate >(X,Y )“>” is conceptually an (infinite) relation that contains all pairs of values such that the first value is greater than the second valueArithmetic operations are also treated as predicatesE.g. A = B + C is treated as +(B, C, A), where the relation “+” contains all triples such that the third value is the sum of the first two

اسلاید 57: Syntax of Datalog Rules (Cont.)Rules are built out of literals and have the form:p (t1, t2, ..., tn ) :– L1, L2, ..., Lm. head bodyeach Li is a literalhead – the literal p (t1, t2, ..., tn ) body – the rest of the literalsA fact is a rule with an empty body, written in the form:p (v1, v2, ..., vn ).indicates tuple (v1, v2, ..., vn ) is in relation pA Datalog program is a set of rules

اسلاید 58: Semantics of a RuleA ground instantiation of a rule (or simply instantiation) is the result of replacing each variable in the rule by some constant.Eg. Rule defining v1 v1(A,B) :– account (A,“Perryridge”, B ), B > 700.An instantiation above rule: v1 (“A-217”, 750) :–account ( “A-217”, “Perryridge”, 750), 750 > 700.The body of rule instantiation R’ is satisfied in a set of facts (database instance) l if1.For each positive literal qi (vi,1, ..., vi,ni ) in the body of R’, l contains the fact qi (vi,1, ..., vi,ni ).2.For each negative literal not qj (vj,1, ..., vj,nj ) in the body of R’, l does not contain the fact qj (vj,1, ..., vj,nj ).

اسلاید 59: Semantics of a Rule (Cont.)We define the set of facts that can be inferred from a given set of facts l using rule R as:infer(R, l) = { p (t1, ..., tn) | there is a ground instantiation R’ of R where p (t1, ..., tn ) is the head of R’, and the body of R’ is satisfied in l }Given an set of rules  = {R1, R2, ..., Rn}, we defineinfer(, l) = infer (R1, l )  infer (R2, l )  ...  infer (Rn, l )

اسلاید 60: Layering of RulesDefine the interest on each account in Perryridgeinterest(A, l) :– perryridge_account (A,B), interest_rate(A,R), l = B * R/100. perryridge_account(A,B) :– account (A, “Perryridge”, B). interest_rate (A,5) :– account (N, A, B), B < 10000. interest_rate (A,6) :– account (N, A, B), B >= 10000.Layering of the view relations

اسلاید 61: Layering Rules (Cont.)A relation is a layer 1 if all relations used in the bodies of rules defining it are stored in the database.A relation is a layer 2 if all relations used in the bodies of rules defining it are either stored in the database, or are in layer 1.A relation p is in layer i + 1 if it is not in layers 1, 2, ..., iall relations used in the bodies of rules defining a p are either stored in the database, or are in layers 1, 2, ..., iFormally:

اسلاید 62: Semantics of a ProgramDefine I0 = set of facts stored in the database.Recursively define li+1 = li  infer (i+1, li )The set of facts in the view relations defined by the program (also called the semantics of the program) is given by the set of facts ln corresponding to the highest layer n.Let the layers in a given program be 1, 2, ..., n. Let i denote theset of all rules defining view relations in layer i.Note: Can instead define semantics using view expansion likein relational algebra, but above definition is better for handlingextensions such as recursion.

اسلاید 63: SafetyIt is possible to write rules that generate an infinite number of answers.gt(X, Y) :– X > Y not_in_loan (B, L) :– not loan (B, L)To avoid this possibility Datalog rules must satisfy the following conditions.Every variable that appears in the head of the rule also appears in a non-arithmetic positive literal in the body of the rule.This condition can be weakened in special cases based on the semantics of arithmetic predicates, for example to permit the rule p (A ) :- q (B ), A = B + 1Every variable appearing in a negative literal in the body of the rule also appears in some positive literal in the body of the rule.

اسلاید 64: Relational Operations in DatalogProject out attribute account_name from account.query (A) :–account (A, N, B ).Cartesian product of relations r1 and r2.query (X1, X2, ..., Xn, Y1, Y1, Y2, ..., Ym ) :– r1 (X1, X2, ..., Xn ), r2 (Y1, Y2, ..., Ym ).Union of relations r1 and r2. query (X1, X2, ..., Xn ) :–r1 (X1, X2, ..., Xn ), query (X1, X2, ..., Xn ) :–r2 (X1, X2, ..., Xn ), Set difference of r1 and r2.query (X1, X2, ..., Xn ) :–r1(X1, X2, ..., Xn ), not r2 (X1, X2, ..., Xn ),

اسلاید 65: Recursion in DatalogSuppose we are given a relation manager (X, Y ) containing pairs of names X, Y such that Y is a manager of X (or equivalently, X is a direct employee of Y). Each manager may have direct employees, as well as indirect employeesIndirect employees of a manager, say Jones, are employees of people who are direct employees of Jones, or recursively, employees of people who are indirect employees of JonesSuppose we wish to find all (direct and indirect) employees of manager Jones. We can write a recursive Datalog program. empl_jones (X ) :- manager (X, Jones ). empl_jones (X ) :- manager (X, Y ), empl_jones (Y ).

اسلاید 66: Semantics of Recursion in DatalogAssumption (for now): program contains no negative literalsThe view relations of a recursive program containing a set of rules  are defined to contain exactly the set of facts l computed by the iterative procedure Datalog-Fixpointprocedure Datalog-Fixpoint l = set of facts in the database repeat Old_l = l l = l  infer (, l )until l = Old_lAt the end of the procedure, infer (, l )  lInfer (, l ) = l if we consider the database to be a set of facts that are part of the programl is called a fixed point of the program.

اسلاید 67: Example of Datalog-FixPoint Iteration

اسلاید 68: A More General ViewCreate a view relation empl that contains every tuple (X, Y ) such that X is directly or indirectly managed by Y.empl (X, Y ) :– manager (X, Y ). empl (X, Y ) :– manager (X, Z ), empl (Z, Y ) Find the direct and indirect employees of Jones.? empl (X, “Jones”).Can define the view empl in another way too:empl (X, Y ) :– manager (X, Y ). empl (X, Y ) :– empl (X, Z ), manager (Z, Y ).

اسلاید 69: The Power of RecursionRecursive views make it possible to write queries, such as transitive closure queries, that cannot be written without recursion or iteration.Intuition: Without recursion, a non-recursive non-iterative program can perform only a fixed number of joins of manager with itselfThis can give only a fixed number of levels of managersGiven a program we can construct a database with a greater number of levels of managers on which the program will not work

اسلاید 70: Recursion in SQLStarting with SQL:1999, SQL permits recursive view definitionE.g. query to find all employee-manager pairs with recursive empl (emp, mgr ) as ( select emp, mgr from manager union select manager.emp, empl.mgr from manager, empl where manager.mgr = empl.emp ) select * from empl

اسلاید 71: Monotonicity A view V is said to be monotonic if given any two sets of facts I1 and I2 such that l1  I2, then Ev (I1)  Ev (I2 ), where Ev is the expression used to define V.A set of rules R is said to be monotonic if l1  I2 implies infer (R, I1 )  infer (R, I2 ), Relational algebra views defined using only the operations:  , |X|  and  (as well as operations like natural join defined in terms of these operations) are monotonic. Relational algebra views defined using set difference (–) may not be monotonic.Similarly, Datalog programs without negation are monotonic, but Datalog programs with negation may not be monotonic.

اسلاید 72: Non-MonotonicityProcedure Datalog-Fixpoint is sound provided the rules in the program are monotonic.Otherwise, it may make some inferences in an iteration that cannot be made in a later iteration. E.g. given the rules a :- not b. b :- c. c. Then a can be inferred initially, before b is inferred, but not later.We can extend the procedure to handle negation so long as the program is “stratified”: intuitively, so long as negation is not mixed with recursion

اسلاید 73: Non-Monotonicity (Cont.)There are useful queries that cannot be expressed by a stratified programExample: given information about the number of each subpart in each part, in a part-subpart hierarchy, find the total number of subparts of each part.A program to compute the above query would have to mix aggregation with recursionHowever, so long as the underlying data (part-subpart) has no cycles, it is possible to write a program that mixes aggregation with recursion, yet has a clear meaningThere are ways to evaluate some such classes of non-stratified programs

اسلاید 74: End of Chapter 5

اسلاید 75: Figure 5.1

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اسلاید 79: Figure 5.9

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اسلاید 83: Figure in-5-31

اسلاید 84: Figure in-5.36