Temple University – CIS Dept. CIS616– Principles of Database Systems V. Megalooikonomou Relational Model (based on notes by Silberchatz,Korth, and Sudarshan.

Temple University – CIS Dept. CIS616– Principles of Database Systems V. Megalooikonomou Relational Model (based on notes by Silberchatz,Korth, and Sudarshan and notes by C. Faloutsos at CMU)

Overview history concepts Formal query languages relational algebra rel. tuple calculus rel. domain calculus

History before: records, pointers, sets etc introduced by E.F. Codd (1923-2003) in 1970 revolutionary!!! first systems: 1977-8 (System R; Ingres) Turing award in 1981

Concepts Database: a set of relations (= tables) rows: tuples columns: attributes (or keys) superkey, candidate key, primary key

Example Database:

Example: cont’d Database: rel. schema (attr+domains) tuple k-th attribute (Dk domain)

Example: cont’d rel. schema (attr+domains) instance

Example: cont’d rel. schema (attr+domains) instance Di: the domain of the I-th attribute (eg., char(10) Formally: an instance is a subset of (D1 x D2 x …x Dn)

Example: cont’d superkey (eg., ‘ssn, name’): determines record cand. key (eg., ‘ssn’, or ‘st#’): minimal superkey (no subset of it is a superkey) primary key: one of the cand. keys

Another example Example: if Customer-name = {Jones, Smith, Curry, Lindsay} Customer-street = {Main, North, Park} Customer-city = {Harrison, Rye, Pittsfield} Then r = { (Jones, Main, Harrison), (Smith, North, Rye), (Curry, North, Rye), (Lindsay, Park, Pittsfield)} is a relation over Customer-name x Customer-street x Customer-city

Relations, tuples Relation Schema: A 1, A 2, …, A n are attributes R = (A 1, A 2, …, A n ) is a relation schema E.g. Customer-schema = (customer-name, customer-street, customer-city) r(R) is a relation on the relation schema R E.g.customer (Customer-schema) Relations are unordered Order of tuples is irrelevant (tuples may be stored in an arbitrary order)

Database A database consists of multiple relations Information about an enterprise is broken up into parts, with each relation storing one part of the information E.g.: account : stores information about accounts depositor : stores information about which customer owns which account customer : stores information about customers Storing all information as a single relation such as bank(account-number, balance, customer-name,..) results in repetition of information (e.g. two customers own an account) the need for null values (e.g. represent a customer without an account) Normalization theory (discuss later) deals with how to design relational schemas

Formal query languages How do we collect information? Eg., find ssn’s of people in cis331 (recall: everything is a set!) One solution: Relational algebra, i.e., set operators (procedural language) Q1: Which operators?? Q2: What is a minimal set of operators?

. set union U set difference ‘-’ Relational operators

Example: Q: find all students (part or full time) A: PT-STUDENT union FT-STUDENT

Observations: two tables are ‘union compatible’ if they have the same attributes (i.e., same arity: number of attributes and same ‘domains’) Q: how about intersection ? U

Observations: A: redundant: STUDENT intersection STAFF = STUDENT - (STUDENT - STAFF) STUDENT STAFF

. set union set difference ‘-’ Relational operators U

Other operators? E.g., find all students on ‘Main street’ A: ‘selection’

Other operators? Notice: selection (and rest of operators) expect tables, and produce tables --> can be cascaded!! For selection, in general:

Selection - examples Find all ‘Smiths’ on ‘Forbes Ave’ ‘condition’ can be any boolean combination of ‘=‘, ‘>’, ‘>=‘,...

selection. set union set difference R - S Relational operators R U S

selection picks rows - how about columns? A: ‘projection’ - eg.: finds all the ‘ssn’ - removing duplicates Relational operators

Cascading: ‘find ssn of students on ‘forbes ave’ Relational operators

selection projection. set union set difference R - S Relational operators R U S

Are we done yet? Q: Give a query we can not answer yet! Relational operators

A: any query across two or more tables, eg., ‘find names of students in cis351’ Q: what extra operator do we need?? A: surprisingly, the cartesian product is enough! Relational operators TAKES SSNc-idgrade 123cis331A 234cis331B

Cartesian product E.g., dog-breeding: MALE x FEMALE gives all possible couples x =

so what? Eg., how do we find names of students taking cis351? TAKES SSNc-idgrade 123cis331A 234cis331B

Cartesian product A: SsnNameAddressssncidgrade 123smithmain str123cis331A 234jonesforbes ave123cis331A 123smithmain str234cis331B 234jonesforbes ave234cis331B

Cartesian product SsnNameAddressssncidgrade 123smithmain str123cis331A 234jonesforbes ave123cis331A 123smithmain str234cis331B 234jonesforbes ave234cis331B

SsnNameAddressssncidgrade 123smithmain str123cis331A 234jonesforbes ave123cis331A 123smithmain str234cis331B 234jonesforbes ave234cis331B

selection projection cartesian product MALE x FEMALE set union set difference R - S FUNDAMENTAL Relational operators R U S

Relational ops Surprisingly, they are enough, to help us answer almost any query we want!! derived operators, for convenience set intersection join (theta join, equi-join, natural join) ‘rename’ operator division

Joins Equijoin:

Cartesian product A: SsnNameAddressssncidgrade 123smithmain str123cis331A 234jonesforbes ave123cis331A 123smithmain str234cis331B 234jonesforbes ave234cis331B

Joins Equijoin: theta-joins: generalization of equi-join - any condition

Joins very popular: natural join: R S like equi-join, but it drops duplicate columns: STUDENT(ssn, name, address) TAKES(ssn, cid, grade)

Joins nat. join has 5 attributes SsnNameAddressssncidgrade 123smithmain str123cis331A 234jonesforbes ave123cis331A 123smithmain str234cis331B 234jonesforbes ave234cis331B equi-join: 6

Natural Joins - nit-picking if no attributes in common between R, S: nat. join -> cartesian product:

Overview - rel. algebra fundamental operators derived operators joins etc rename division examples

rename op. Q: why? A: Shorthand (BEFORE can be a relational algebra expression) self-joins; … for example, find the grand-parents of ‘Tom’, given PC(parent-id, child-id)

rename op. PC(parent-id, child-id)

rename op. first, WRONG attempt: (why? how many columns?) Second WRONG attempt:

rename op. we clearly need two different names for the same table - hence, the ‘rename’ op.

Division Rarely used, but powerful. Suited for queries that include the phrase “for all” Example: find suspicious suppliers, i.e., suppliers that supplied all the parts in A_BOMB

Division

Observations: ~reverse of cartesian product It can be derived from the 5 fundamental operators (!!) How?

Division Answer: gives those tuples t in such that for some tuple u in S, tu not in R.

Sample schema CLASS c-idc-nameunits cis331d.b.2 cis321o.s.2 TAKES SSNc-idgrade 123cis331A 234cis331B find names of students that take cis351

Examples find names of students that take cis351

Sample schema find course names of ‘smith’ CLASS c-idc-nameunits cis331d.b.2 cis321o.s.2 TAKES SSNc-idgrade 123cis331A 234cis331B

Examples find course names of ‘smith’

Examples find ssn of ‘overworked’ students, ie., that take cis331, cis342, cis350

Examples find ssn of ‘overworked’ students, ie., that take cis331, cis342, cis350: almost correct answer: )( )( )( 350 342 331 TAKES namec c c        

Examples find ssn of ‘overworked’ students, ie., that take cis331, cis342, cis350 - Correct answer: )]([ ([ ([ 350 342 331 TAKES namecssn namecssn namecssn        

Examples find ssn of students that work at least as hard as ssn=123 (ie., they take all the courses of ssn=123, and maybe more)

Sample schema CLASS c-idc-nameunits cis331d.b.2 cis321o.s.2 TAKES SSNc-idgrade 123cis331A 234cis331B

Examples find ssn of students that work at least as hard as ssn=123 (ie., they take all the courses of ssn=123, and maybe more

Conclusions Relational model: only tables (‘relations’) relational algebra: powerful, minimal: 5 operators can handle almost any query! most non-trivial op.: join

The bank database

Temple University – CIS Dept. CIS616– Principles of Database Systems V. Megalooikonomou Relational Model II (based on notes by Silberchatz,Korth, and Sudarshan and notes by C. Faloutsos at CMU)

Extended Relational-Algebra- Operations Generalized Projection Outer Join Aggregate Functions

Generalized Projection Extends the projection operation by allowing arithmetic functions to be used in the projection list  F1, F2, …, Fn (E) E is any relational-algebra expression Each of F 1, F 2, …, F n are arithmetic expressions involving constants and attributes in the schema of E. Given relation credit-info(customer-name, limit, credit-balance), find how much more each person can spend:  customer-name, limit – credit-balance (credit-info)

Aggregate Functions and Operations Aggregation function takes a collection of values and returns a single value as a result ?

Aggregate Functions and Operations Aggregation function takes a collection of values and returns a single value as a result avg: average value min: minimum value max: maximum value sum: sum of values count: number of values Aggregate operation in relational algebra G 1, G 2, …, G n g F 1 ( A 1), F 2 ( A 2),…, F n ( A n) (E) E is any relational-algebra expression G 1, G 2 …, G n is a list of attributes on which to group (can be empty) Each F i is an aggregate function Each A i is an attribute name

Aggregate Operation – Example Relation r : AB   C 7 3 10 g sum(c) (r) sum-C 27

Aggregate Operation – Example Relation account grouped by branch-name: branch-name g sum(balance) (account) branch-nameaccount-numberbalance Perryridge Brighton Redwood A-102 A-201 A-217 A-215 A-222 400 900 750 700 branch-namebalance Perryridge Brighton Redwood 1300 1500 700

Aggregate Functions (Cont.) Result of aggregation does not have a name Use rename operation For convenience, we permit renaming as part of aggregate operation branch-name g sum(balance) as sum-balance (account)

Outer Join An extension of the join operation that avoids loss of information Computes the join and then adds tuples from one relation that does not match tuples in the other relation to the result of the join Uses null values: null signifies that the value is unknown or does not exist All comparisons involving null are (roughly speaking) false by definition precise meaning of comparisons with nulls later

Outer Join – Example Relation loan loan-numberamount L-170 L-230 L-260 3000 4000 1700 Relation borrower customer-nameloan-number Jones Smith Hayes L-170 L-230 L-155 branch-name Downtown Redwood Perryridge

Outer Join – Example Inner Join loan borrower loan borrower Left Outer Join loan-numberamount L-170 L-230 3000 4000 customer-name Jones Smith branch-name Downtown Redwood loan-numberamount L-170 L-230 L-260 3000 4000 1700 customer-name Jones Smith null branch-name Downtown Redwood Perryridge

Outer Join – Example Right Outer Join loan borrower Full Outer Join loan-numberamount L-170 L-230 L-155 3000 4000 null customer-name Jones Smith Hayes branch-name Downtown Redwood null loan-numberamount L-170 L-230 L-260 L-155 3000 4000 1700 null customer-name Jones Smith null Hayes branch-name Downtown Redwood Perryridge null

Null Values It is possible for tuples to have a null value, denoted by null, for some of their attributes null signifies unknown value or a value that does not exist The result of any arithmetic expression involving null is null Aggregate functions ignore null values An arbitrary decision. Could have returned null as result instead Follow semantics of SQL in its handling of null values For duplicate elimination and grouping, null is treated like any other value, and two nulls are assumed to be the same Alternative: assume each null is different from each other Both arbitrary decisions, we follow SQL

Null Values Comparisons with null values return the special truth value unknown If false was used instead of unknown, then not (A = 5 Three-valued logic using the truth value unknown: OR: (unknown or true) = true, (unknown or false) = unknown (unknown or unknown) = unknown AND: (true and unknown) = unknown, (false and unknown) = false, (unknown and unknown) = unknown NOT: (not unknown) = unknown In SQL “P is unknown” evaluates to true if predicate P evaluates to unknown Result of select predicate is treated as false if it evaluates to unknown

Modification of the Database The content of the database may be modified using the following operations: Deletion Insertion Updating All these operations are expressed using the assignment operator

The bank database

Deletion Expressed similarly to a query Instead of displaying tuples, the selected tuples are removed from the database Can delete only whole tuples cannot delete values on only particular attributes A deletion is expressed in relational algebra by: r  r – E where r is a relation and E is a relational algebra query.

Deletion Examples Delete all account records in the Perryridge branch account  account –  branch-name = “Perryridge” (account) Delete all loan records with amount in the range of 0 to 50 loan  loan –   amount  0  and amount  50 (loan) Delete all accounts at branches located in Needham r 1    branch-city = “Needham” (account branch) r 2   branch-name, account-number, balance (r 1 ) r 3   customer-name, account-number (r 2 depositor) account  account – r 2 depositor  depositor – r 3

Insertion To insert data into a relation, we either: specify a tuple to be inserted write a query whose result is a set of tuples to be inserted in relational algebra, an insertion is expressed by?

Insertion To insert data into a relation, we either: specify a tuple to be inserted write a query whose result is a set of tuples to be inserted in relational algebra, an insertion is expressed by: r  r  E where r is a relation and E is a relational algebra expression Insertion of a single tuple: let E be a constant relation containing one tuple

Insertion Examples Insert information in the database specifying that Smith has $1200 in account A-973 at the Perryridge branch. account  account  {(“Perryridge”, A-973, 1200)} depositor  depositor  {(“Smith”, A-973)} Provide as a gift for all loan customers in the Perryridge branch, a $200 savings account. Let the loan number serve as the account number for the new savings account. r 1  (  branch-name = “Perryridge” (borrower loan)) account  account  (  branch-name, loan-number (r 1 )) x {(200)}) OR account  account   branch-name, account-number,200 (r 1 ) depositor  depositor   customer-name, loan-number,(r 1 )

Updating A mechanism to change a value in a tuple without charging all values in the tuple Use the generalized projection operator r   F1, F2, …, FI, (r) Each F, is either the i-th attribute of r, if the i-th attribute is not updated, or, if the attribute is to be updated F i is an expression, involving only constants and the attributes of r, which gives the new value for the attribute

Update Examples Make interest payments by increasing all balances by 5 percent account   AN, BN, BAL * 1.05 (account) where AN, BN and BAL stand for account-number, branch- name and balance, respectively Pay all accounts with balances over $10,000 6 percent interest and pay all others 5 percent account   AN, BN, BAL * 1.06 (  BAL  10000 (account))   AN, BN, BAL * 1.05 (  BAL  10000 (account))

Views … if it is not desirable for all users to see the entire logical model (i.e., all the relations stored in the database) A person who needs to know a customer’s loan number but has no need to see the loan amount can see the relation:  customer-name, loan-number (borrower loan) Any relation that is not of the conceptual model but is made visible to a user as a “virtual relation” is called a view.

View Definition A view is defined using the create view statement which has the form create view v as where is any legal relational algebra query expression. The view name, v, can be used to refer to the virtual relation View definition  creating a new relation by evaluating the query expression. Rather, a view definition causes the saving of an expression to be substituted into queries using the view Materialized views

View Examples Consider the view (named all-customer) consisting of branches and their customers create view all-customer as  branch-name, customer-name (depositor account)   branch-name, customer-name (borrower loan) We can find all customers of the Perryridge branch by writing:  customer-name (  branch-name = “Perryridge” (all-customer))

Updates Through View Database modifications expressed as views must be translated to modifications of the actual relations in the database E.g., a person who needs to see all loan data in the loan relation except amount. The view is defined as: create view branch-loan as  branch-name, loan-number (loan) Since we allow a view name to appear wherever a relation name is allowed, the person may write: branch-loan  branch-loan  {(“Perryridge”, L-37)}

Updates Through Views (Cont’d.) The previous insertion must be represented by an insertion into the actual relation loan from which the view branch- loan is constructed An insertion into loan requires a value for amount. The insertion can be dealt with by either: rejecting the insertion and returning an error message to the user inserting a tuple (“L-37”, “Perryridge”, null ) into the loan relation Some updates through views are impossible to translate into database relation updates create view v as  branch-name = “Perryridge” (account)) v  v  (L-99, Downtown, 23) Others cannot be translated uniquely all-customer  all-customer  (Perryridge, John) Have to choose loan or account, and create a new loan/account number!

Views Defined Using Other Views One view may be used in the expression defining another view A view relation v 1 is said to depend directly on a view relation v 2 if v 2 is used in the expression defining v 1 A view relation v 1 is said to depend on view relation v 2 if either v 1 depends directly to v 2 or there is a path of dependencies from v 1 to v 2 A view relation v is said to be recursive if it depends on itself

View Expansion Define the meaning of views in terms of other views… Let view v 1 be defined by an expression e 1 that may itself contain uses of view relations View expansion of an expression repeats the following replacement step: repeat Find any view relation v i in e 1 Replace the view relation v i by the expression defining v i until no more view relations are present in e 1 This loop will terminate as long as …?

View Expansion Define the meaning of views in terms of other views… Let view v 1 be defined by an expression e 1 that may itself contain uses of view relations View expansion of an expression repeats the following replacement step: repeat Find any view relation v i in e 1 Replace the view relation v i by the expression defining v i until no more view relations are present in e 1 This loop will terminate as long as the view definitions are not recursive

General Overview - rel. model history concepts Formal query languages relational algebra rel. tuple calculus rel. domain calculus

Overview - detailed Relational tuple calculus Why do we need it? Details Examples Equivalence with rel. algebra More examples; ‘safety’ of expressions Rel. domain calculus + QBE

Motivation Q: weakness of rel. algebra? A: procedural describes the steps (i.e., ‘how’) … still useful, for query optimization though

Solution: rel. calculus describes what we want two equivalent flavors: ‘tuple’ calculus and ‘domain’ calculus basis for SQL and QBE, resp.

Rel. tuple calculus (RTC) first order logic ‘Give me tuples ‘t’, satisfying predicate P – e.g.:

..or Tuple Relational Calculus A 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 t t is a tuple variable, t [A] denotes the value of tuple t on attribute A t  r denotes that tuple t is in relation r P is a formula similar to that of the predicate calculus

Details symbols allowed: quantifiers: “for all”, “there exists” Atom:

Specifically Formula: Atom if P1, P2 are formulas, so are if P(s) is a formula, so are

Specifically Reminders: DeMorgan implication: double negation: ‘every human is mortal : no human is immortal’

Reminder: our Mini-U db CLASS c-idc-nameunits cis331d.b.2 cis321o.s.2 TAKES SSNc-idgrade 123cis331A 234cis331B

Examples find all student records output tuple of type ‘STUDENT’

Examples (selection) find student record with ssn=123

Examples (projection) find name of student with ssn=123

Examples (projection) find name of student with ssn=123 ‘t’ has only one column

‘Tracing’ s t

Examples cont’d (union) get records of both PT and FT students

Examples difference: find students that are not staff (assuming that STUDENT and STAFF are union-compatible)

Examples difference: find students that are not staff

Cartesian product e.g., dog-breeding: MALE x FEMALE gives all possible couples x =

Cartesian product find all the pairs of (male, female)

‘Proof’ of equivalence rel. algebra rel. tuple calculus

Overview - detailed rel. tuple calculus why? details examples equivalence with rel. algebra more examples; ‘safety’ of expressions re. domain calculus + QBE

More examples join: find names of students taking cis351

More examples join: find names of students taking cis351 projection selection join

More examples 3-way join: find names of students taking a 2-unit course

Reminder: our Mini-U db

More examples 3-way join: find names of students taking a 2-unit course selection projection join

More examples 3-way join: find names of students taking a 2-unit course - in rel. algebra??

Even more examples: self -joins: find Tom’s grandparent(s)

Hard examples: DIVISION find suppliers that shipped all the ABOMB parts

General pattern three equivalent versions: 1) if it’s bad, he shipped it 2)either it was good, or he shipped it 3) there is no bad shipment that he missed

More on division find (SSNs of) students that take all the courses that ssn=123 does (and maybe even more) find students ‘s’ so that if 123 takes a course => so does ‘s’

More on division find students that take all the courses that ssn=123 does (and maybe even more)

Safety of expressions FORBIDDEN: It has infinite output!! Instead, always use

Safety of expressions Possible to write tuple calculus expressions that generate infinite relations, e.g., {t |  t  r } results in an infinite relation if the domain of any attribute of relation r is infinite To 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 P

More examples: Banking example branch (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)

Example Queries Find the loan-number, branch-name, and amount for loans of over $1200 {t | t  loan  t [amount]  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

Example Queries Find 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])

Example Queries Find 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]))} 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]))  not  v  depositor (v[customer-name] = t[customer-name]) }

Example Queries Find the names of all customers having a loan from the Perryridge branch, and the cities they live in {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])))}

Example Queries Find the names of all customers who have an account at all branches located in Brooklyn: {t |  c  customer (t[customer.name] = c[customer-name])   s  branch(s[branch-city] = “Brooklyn”   u  account ( s[branch-name] = u[branch-name]   s  depositor ( t[customer-name] = s[customer-name]  s[account-number] = u[account-number] )) )}

Temple University – CIS Dept. CIS616– Principles of Database Systems V. Megalooikonomou Relational Model III (based on notes by Silberchatz,Korth, and Sudarshan and notes by C. Faloutsos at CMU)

General Overview - rel. model history concepts Formal query languages relational algebra rel. tuple calculus rel. domain calculus

Overview - detailed rel. tuple calculus why? details examples equivalence with rel. algebra more examples; ‘safety’ of expressions rel. domain calculus + QBE