database systems {week 03a}

Database Systems{week 03a}

Rensselaer Polytechnic InstituteCSCI-4380 – Database SystemsDavid Goldschmidt, Ph.D.

Functional dependencies (FDs) A functional dependency on relation

R is a logical expression of the form X Y X and Y are sets of attributes of R▪ i.e. X = { A1, A2, ..., An } and Y = { B1, B2, ...,

Bm } where n = m or n <> m

X Y means that whenever any pair of tuplesin R have the same values for attributes in X, then they must also have the same values for attributes in Y

Functional dependencies (FDs) For each X Y defined on relation R,

it means that X functionally determines Y Or more specifically, the attributes of X

functionally determine the attributes of Y

More generally, a functional dependency adds meaning to attributes of R In some cases, the occurrence of duplicate

tuples does not make semantic sense

Rules about FDs

For a given relation R, we look at the set of all functional dependencies to tell us what tuples we can (and should) store

We can also reason by applying simple inference rules to the tuples e.g. transitivity, splitting/combining, etc.

Trivial functional dependencies A constraint of any kind on relation R

is said to be trivial if it holds for every instance of R If Y X, then X Y is true for all

relations In other words, a trivial functional

dependency has a right-hand side (Y) that is a subset of its left-hand side (X)

e.g. name artist name e.g. name name

What’s the point?We can remove trivial FDs!

Reflexivity rule

Trivial-dependency rule

The functional dependency X Y is equivalent to X Z where attributesof Z are all those attributes of Y thatare not also attributes of X In other words, some of the attributes on

the right-hand side (of X Y) are also on the left (X)

We can simplify this by removing attributes from the right-hand side that also appear on the left

Augmentation

Given functional dependency X Y

We can always add a set Z of attribute(s) XZ YZ

This is called augmentation

Splitting

Given functional dependency X Y as A1, A2, ..., An B1, B2, ..., Bm

We can split it into multiple functional dependencies (singletons) as follows: A1, A2, ..., An B1 A1, A2, ..., An B2 ... A1, A2, ..., An Bm

Combining

Given functional dependencies as follows: A1, A2, ..., An B1 A1, A2, ..., An B2 ... A1, A2, ..., An Bm

We can combine attributes on the right-hand side to form functional dependency A1, A2, ..., An B1, B2, ..., Bm

Transitivity

Given functional dependencies X Y and Y Z

We can unequivocally conclude that X Z

And if some attributes of Z are also attributes of X, we can eliminate them from the right-hand side (trivial-dependency rule)

Keys

For a given relation R, we look at the set of functional dependencies to identify which attribute(s) imply all the rest These attribute(s) form a key on R

Set K = { A1, A2, ..., An } is a key on R if: K functionally determine all other

attributes of R No proper subset of K functionally

determinesall other attributes of R

Uniqueness of keys

By definition, a key must be unique A key K must functionally determine

all other attributes of relation R e.g. Student( id, name, address ) The key is the id attribute

Minimality of keys

By definition, a key must be minimal No proper subset of key K can

functionally determine all other attributes of relation R e.g. Student( id, name, address ) Even though id and name together

might be unique, the id attribute is minimal

Superkeys

A set of attributes that contains a keyis called a superkey (a superset of a key) The uniqueness constraint must be

satisfied The minimality constraint need not be

satisfied Every key is a superkey e.g. Student( id, name, address ) Attribute id is both a key and a superkey Attributes (id, name) form a superkey

Exercises (part one)

Model a US Census relation Name, SSN, address, city, state, zip,

area code, phone number, etc. Use only a single relation Describe functional dependencies Identify keys and superkeys

Inference

Given relation R with attributes A, B, C, D, E and A BC, CD E, BE C What does AE functionally determine

(infer)? In other words, AE _____?

Closure

Given a set of attributes X, theclosure X+ is the set of attributes functionally determined by X

Given a relation R and a set F of functional dependencies, we need a way to find whether a functional dependency X Y is true with respect to F

Closure example

Given relation R with attributes A, B, C, D, E and A BC, CD E, BE C AE _____? From reflexivity, AE+ = { A, E } From A BC, AE+ = { A, B, C, E } No other rules are applicable or add to

AE+ We conclude that AE ABCE or simply

AE BC Or AE A, AE B, AE C, and AE E

Closure of a set of attributes Given a set F of functional

dependencies, the closure X+ of a set of attributes X is determined by the following algorithm: Initialize X+ to X Repeat until X+ does not change:▪ Find any unapplied functional dependency Y

Zin F such that Y X+▪ Set X+ = X+ Z

Closure

A set F of functional dependencies implies a functional dependency X Y if Y X+

In other words, if Y is in the closure of X, then functional dependency X Y is true

Keys revisited

A key K for a given relation R is a minimal set of attributes A1, A2, ..., An such that closure {A1, A2, ..., An}+ is the set of all attributes of R MusicGroup(name, artist, genre, dateformed,

datefirstjoined) name genre dateformed name artist datefirstjoined K must be (name, artist) because K+ = {name,

artist, genre, dateformed, datefirstjoined}

Exercises (part two)

Review the US Census relation What other functional dependencies

can you infer? Pick pairs of attributes (e.g. name and

state) and identify the resulting closure In other words, what is the set of

attributes X+ functionally determined by set X (the pair)?