EECE 433 Chapter 4 Part 2, Spring 2012

Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke Relational Calculus Chapter 4, Part B Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke Relational Calculus  Comes in two flavors: Tuple relational calculus (TRC) and Domain relational calculus (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 logic.  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. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke Tuple Relational Calculus  Query has the form: { T | p(T) }  Answer includes all tuples T that make the formula p(T) 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. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke TRC Formulas  Atomic formula:   R.a op R.b  R.a op constant ; op is one of  Formula:  an atomic formula, or  , where p and q are formulas  , where variable X is free in p(X), or  , where variable X is free in p(X)  The use of quantifiers and is said to bind X. R Rel     , , , , ,    p p q p q , , X p X ( ( )) X p X ( ( )) X  X Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke TRC Formulas  Note that if a variable R does not appear in an atomic formula of the form , we follow the convention that the type of R is a tuple whose fields include only fields of R that appear in the formula.  Example:  Print names of Sailors: {T | (T.name = S.name)}  What would this give: {T | (T.name = S.name)} R Rel  SSailors  SSailors  TSailors  Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke Examples of TRC Queries  Assume Sailors (sid, sname, rating, age); Reservations(sid, bid, day); Boats(bid, bname, color)  Q12: Find the names and ages of Sailors with a rating above 7. (i.e. print the name and age of sailor each time you find a sailor with rating above 7).  Q2: Find the names of Sailors who have reserved a red boat  Q7: Find the names of Sailors who have reserved at least two boats.  Q10: Find the names of Sailors who have reserved all boats.  Q14: Find the sailors who have reserved all red boats (alternatively: For each sailor such that no matter what boat you pick, if that boat is red then the sailor has a reservation for it). Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke Find the names of Sailors who have reserved exactly two boats. ))} .1 .3 .2 .3 . .3 ( ) . . .2 .1 .2 . .1 (( . Re 3 Re 2 Re 1 | { R bid bid R R bid R bid S sid sid R S sname P sname R bid bid R sid R S sid R sid S sid serves R serves R serves R Sailors S P                      Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke Domain Relational Calculus  Query has the form: x x xn p x x xn 1 2 1 2 , ,..., | , ,...,                Answer includes all tuples that make the formula be true. x x xn 1 2 , ,..., p x x xn 1 2 , ,...,          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. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke DRC Formulas  Atomic formula:  , or X op Y, or X op constant  op is one of  Formula:  an atomic formula, or  , where p and q are formulas, or  , where variable X is free in p(X), or  , where variable X is free in p(X)  The use of quantifiers and is said to bind X.  A variable that is not bound is free. x x xn Rname 1 2 , ,...,     , , , , ,    p p q p q , , X p X ( ( )) X p X ( ( )) X  X Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1 Free and Bound Variables  The use of quantifiers and in a formula is said to bind X.  A variable that is not bound is free.  Let us revisit the definition of a query: X  X x x xn p x x xn 1 2 1 2 , ,..., | , ,...,                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 3ed, R. Ramakrishnan and J. Gehrke 1 Q11: Find all sailors with a rating above 7  The condition ensures that the domain variables I, N, T and A are bound to fields of the same Sailors tuple.  The term 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:  Find sailors who are older than 18 or have a rating under 9, and are called ‘Joe’. I N T A I N T A Sailors T , , , | , , ,         7 I N T A Sailors , , ,  I N T A , , , I N T A , , , Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1 Find sailors rated > 7 who have reserved boat #103  We have used as a shorthand for  Note the use of to find a tuple in Reserves that `joins with’ the Sailors tuple under consideration. I N T A I N T A Sailors T , , , | , , ,        7                  Ir Br D Ir Br D serves Ir I Br , , , , Re 103   Ir Br D , , ...          Ir Br D ...  Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1 Q1: Find the names of Sailors who have reserved boat 103  In-class example… Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1 Find sailors rated > 7 who’ve reserved a red boat  Observe how the parentheses control the scope of each quantifier’s binding. I N T A I N T A Sailors T , , , | , , ,        7         Ir Br D Ir Br D serves Ir I , , , , Re                      B BN C B BN C Boats B Br C red , , , , ' ' Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1 Q9: Find sailors who’ve reserved all boats  To find sailors who’ve reserved all red boats: I N T A I N T A Sailors , , , | , , ,       B BN C  Boats , ,                         Ir Br D serves I Ir Br B , , Re C red Ir Br D serves I Ir Br B                           ' ' , , Re ..... Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1 Q14: Find names of sailors who’ve reserved all red boats  In-class example… Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1 Unsafe Queries, Expressive Power  It is possible to write syntactically correct calculus queries that have an infinite number of answers! Such queries are called unsafe.  e.g.,  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. S S Sailors |                 Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1 Summary  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.