Introduction to Data Management. Lecture #10 (Relational Calculus, Continued)

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1 Introduction to Data Management Lecture #10 (Relational Calculus, Continued) Instructor: Mike Carey Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1 Announcements v Key deliverables and dates: HW #3: Last chance to turn it in! Midterm is next Tuesday (bring cheat-sheet) v Today s plan: Finish up relational calculus Take any review questions Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 2

2 Ex: Wisconsin Sailing Club Database Sailors Reserves Boats sid sname rating age sid bid date bid bname color 22 Dustin Brutus Lubber Andy Rusty Horatio Zorba Horatio Art Bob /10/ /10/ /8/ /7/ /10/ /6/ /12/ /5/ /8/ /8/ Interlake blue 102 Interlake red 103 Clipper green 104 Marine red Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 3 Tuple Relational Calculus v Query in TRC has the form: { t(a1,a2,...) P(t) } v Answer includes all possible tuples t with the specified schema that make formula P(t) true. v Formula is recursively defined, starting with simple atomic formulas and building up bigger formulas using logical connectives. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 4

3 Review: Find sailors with a rating > 7 v Simplest answer, if all Sailors attributes desired: { s s Sailors s.rating > 7 } v Also equivalent to a more general formulation: { t(sid, sname, rating, age) s Sailors ( t.sid = s.sid t.sname = s.sname t.rating=s.rating t.age = s.age s.rating > 7 ) } (Notice how each specifies the answer s schema and values.) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 5 Ex: TRC Query Semantics Sailors sid sname rating age 22 Dustin Brutus Lubber sid bid date /10/98 bid bname color 101 Interlake blue sid sname rating age 32 Andy Rusty Horatio Dustin nid nname nvalue 71 Zorba Horatio Art Bob Pi sid sname rating age 58 Rusty Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 6

4 Review: Find ids of sailors who are older than 30.0 or who have a rating under 8 and are named Horatio { t(sid) s Sailors ( (s.age > 30.0 (s.rating < 8 s.sname = Horatio )) t.sid = s.sid ) } Again, how result schema and values are specified Use of Boolean formula to specify the query constraints Highly declarative nature of this form of query language! Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 7 Unsafe Queries and Expressive Power v 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 v 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. v Relational Completeness: Query language (e.g., SQL) can express every query that is expressible in relational algebra/calculus. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 8

5 Find names of sailors who ve reserved a red boat { t(sname) s Sailors (t.sname = s.sname r Reserves (r.sid = s.sid b Boats (b.bid = r.bid b.color = red ))) } Again, how result schema and values are specified How joins appear here as value-matching predicates Highly declarative nature of this form of query language! Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 9 Find ids of sailors who ve reserved a red or a green boat { t(sid) s Sailors (t.sid = s.sid r Reserves (r.sid = s.sid b Boats (b.bid = r.bid (b.color = red b.color = green )))) } Just had to add a disjunctive test to the previous query! Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 10

6 Find ids of sailors who ve reserved a red and a green boat { t(sid) s Sailors (t.sid = s.sid r1 Reserves (r1.sid = s.sid b1 Boats (b1.bid = r1.bid b1.color = red )) r2 Reserves (r2.sid = s.sid b2 Boats (b2.bid = r2.bid b2.color = green )))} This required several more variables! (Q: Why?) Q: Could we have done this with just s, r, b1, and b2? (And why?) Think of tuple variables as fingers pointing at the tables rows Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 11 Example: Tuple Variable Bindings Sailors sid sname rating age 22 Dustin Brutus Lubber Andy Rusty Horatio Zorba (Bindings at one point in time J ) s Reserves sid bid date /10/ /10/ /8/ /7/98 Boats 74 Horatio /10/98 bid bname color 85 Art /6/ Interlake blue 95 Bob /12/ Interlake red /5/ Clipper green /8/ Marine red /8/93 Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 12 r1 r2 b1 b2

7 Find the names of sailors who ve reserved all boats { t(sname) s Sailors (t.sname = s.sname b Boats ( r Reserves (r.sid = s.sid b.bid = r.bid) ) ) } How universal quantification addresses the all query use case Highly declarative nature of this form of query language! Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 13 Find the names of sailors who ve reserved all Interlake boats { t(sname) s Sailors (t.sname = s.sname b Boats (b.bname = Interlake ( r Reserves (r.sid = s.sid b.bid = r.bid) ) ) ) } v Or, if you prefer: { t(sname) s Sailors (t.sname = s.sname b Boats (b.bname Interlake ( r Reserves (r.sid = s.sid b.bid = r.bid) ) ) ) } Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 14

8 Relational Calculus Summary v Relational calculus is non-operational, and users define queries in terms of what they want, not in terms of how to compute it. (Declarativeness: What, not how! ) v Algebra and safe calculus subset have same expressive power, leading to the notion of relational completeness for query languages. v Two calculus variants: TRC (tuple relational calculus, which we ve studied) and DRC (domain relational calculus). Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 15 What Have We Learned So Far? v First we do conceptual modeling (E-R schema) eid name Employee salary mgr emp 1 N M pcttime WorksIn Manages budget v Then we develop a good logical schema (relational) Employee(eid, name, salary, mgrid) PK(eid), FK(mgrid) REFS Employee WorksIn(empid, deptid, pcttime) PK(empid, deptid), FK(empid) REFS Employee, FK(deptid) REFS Department Department(did, dname, budget) PK(did) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 16 N did dname Department

9 What Have We Learned So Far? Employee(eid, name, salary, mgrid) WorksIn(empid, deptid, pcttime) Department(did, dname, budget) v We can double-check the schema using relational DB design theory (dependency theory & normal forms) v Queries are then written against the relational schema Relational algebra π eid, salary esal, msal ((σ salary > msal (Employee mgrid = mid (π eid mid, salary msal (Employee))) Tuple relational calculus { t(eid, esal, msal) e Employee (t.eid = e.eid t.esal = e.salary m Employee (t.msal = m.salary e.mgrid = m.eid e.salary > m.salary) )} Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 17 Questions Before We Proceed? v Lingering E-R questions? E.g., key cardinality and/or inheritance mapping? v FD/normalization questions? v Relational algebra questions? v Relational calculus questions? v Okay, then: You re ready for the Midterm Exam! (You may bring a 2-sided 8.5x11 cheat sheet.) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 18