lt18-preservefd

7/29/2019 lt18-preservefd

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Database Systems&ApplicationsLecture 18Preserving FD's &Tuple Calculus


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A decomposition of a relation R, on which the dependencies F hold,into relations R1and R2 is dependency preserving if:

(FR1 FR2)+ = F+

Where FR1 is the set of dependencies that apply between attributesonly in R1 and

similarly FR2 is the set of dependencies that apply between attributes

only in R2.


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Preserving FD's

Consider a relation (Project,Project-Budget,Department,Dept-Address) with following functional dependencies:

Project-Budget

Project Department

Dept-Address


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We decompose the PROJECTS relation into two relations on whichtwo corresponding sets of FDs held:

(Project,Project-Budget,Department)

(Department, Dept-Address)

Is it preserving FDs ?

Preserving FD's


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Relational Calculus


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Introduction

Procedural Query language

query specification involves giving a step bystep process of obtaining the query resulte.g., relational algebra

usage calls for detailed knowledge of the

operators involved

difficult for the use of non-experts


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Declarative Query language

query specification involves giving the logicalconditions the results are required to satisfy

easy for the use of non-experts


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Tuple relational Calculus ..aDQL

Tuple relational calculus

Calculus has variables, constants, comparison ops,

logical connectives and quantifiers.TRC: Variables range over (i.e., get bound to) tuples.

Like SQL.


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TRC is simple subset a of first-order logic.

Expressions in the calculus are called formulas.

Answer : tuple is an assignment of constants to

variables that make the formula evaluate to true.


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First-Order Predicate Logic

Predicate: is a feature of language which you can useto make a statement about something, e.g., toattribute a property to that thing.

Peter is tall. We predicated tallness of peter or attributedtallness to peter.

A predicate may be thought of as a kind of function whichapplies to individuals and yields a proposition.

Proposition logic is concerned only with sententialconnectives such as and, or, not.

Predicate Logic, where a logic is concerned not only

with the sentential connectives but also with the


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First order predicate logic

First-order predicate logic, first-order says we considerpredicates on the one hand, and individuals on theother; that atomic sentences are constricted byapplying the former to the latter; and that quantificationis permitted only over the individuals.

First-order logic permits reasoning about propositionalconnectives and also about quantification.

All men are motal

Peter is a man

Peter is mortal


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Tuple Relational Calculus

Queryhas the form: {T|p(T)}

p(T)denotes a formula in which tuple variable Tappears.

Answer is the set of all tuples T for which the

formula p(T)evaluates to true.

Formula is recursively defined:

ystart with simple atomic formulas (get tuples from

relations or make comparisons of values)ybuild bigger and better formulas using the logical

connectives.


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TRC FormulasAn Atomic formula is one of the following:

R RelR.aopS.b

R.aopconstant

constantop R.aop is one of

Aformula can be:

an atomic formula

where p and q are formulas

where variable R is a tuple variable

where variable R is a tuple variable

,>,=, , ,

p,pq,pq

R

( p(R )

)

R ( p (R ) )


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Free and Bound Variables

The use of quantifiers and in a formulais said to bindX in the formula. A variable that is not bound is free.

Let us revisit the definition of a query: {T|p(T)}

$XX

There is an important restriction

the variable Tthat appears to the left of `|must be the onlyfree variable in the formulap(T).

in other words, all other tuple variables mustbe bound using a quantifier.


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bid bname color

101 Interlake blue

102 Interlake red

103 Clipper green

104 Marine red

sid sname rating age

22 Vinod 7 35.0

29 Deepu 1 33.031 Silvia 8 35.5

32 Jeffrey 8 45.5

58 Kim 10 45.0

64 Kingsley 7 35.0

71 Samulya 10 16.0

74 Victor 9 45.0

85 Kalyan 3 33.5

95 Praveen 3 33.5

Boats Sid Bid Day

22 101 10/10/11

22 102 10/10/11

22 103 10/8/11

22 104 10/7/11

31 102 11/10/11

31 103 11/6/11

31 104 11/12/98

64 101 9/5/11

64 102 9/8/11

74 103 9/8/11

ReservesSailors


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Selection and Projection

Find all sailors with rating above 7

{S |S Sailors S.rating > 7}


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Find names and ages of sailors with ratingabove 7.

{P | S1 Sailors(S1.rating > 7P.sname = S1.snameP.age = S1.age)}

Note, here P is a tuple variable of 2 fields (i.e. {P} is

a projection ofsailors), since only 2 fields are evermentioned and P is never used to range over anyrelations in the query.


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Find sailors rated > 7 whove reserved boat #103

{S | SSailors S.rating > 7

R(RReserves R.sid= S.sid

R.bid = 103)}

Joins

Note the use of

to find atuple in Reserves that `joinswith the Sailors tuple underconsideration.


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Joins (continued)

d sailors rated > 7 whove reserved a red b

{S | S

Sailors

S.rating > 7R(RReserves R.sid = S.sid

B(BBoats B.bid = R.bid B.color = red))}


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Division (makes more sense here???)

Find all sailors S such that for each tuple B in Boatsthere is a tuple in Reserves showing that sailor Shas reserved it.

Find sailors whove reserved all boats

(hint, use )

S | SSailors

BBoats (RReserves(S.sid = R.sid B.bid = R.bid))}


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Division a trickier example

{S | SSailors B Boats ( B.color = red R(RReserves S.sid = R.sid

B.bid = R.bid))}

Find sailors whove reserved all Red boats

{S | SSailors

B Boats ( B.color red R(RReserves S.sid = R.sid

B.bid = R.bid))}

Alternatively


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a b is the same as a b

If a is true, b must be true for theimplication to be true. If a istrue and b is false, theimplication evaluates to false.

If a is not true, we dont careabout b, the expression isalways true.

a

T

F

T F

b

T

T T

F

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