4 - relational calculus-2

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CS 222

Spring 2010-11

Lecture 2

orra a ya a uDepartment of Computer Science

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Relational Calculus RA and RC are formal Languages of Relational model

RA is Procedural and RC is Declarative (non procedural) RC expression specifies what data to be retrieved rather than how it is to be retrieved

RC comes in two flavors Tuple Relation Calculus (TRC), Domain Relation Calculus (DRC)

Calculus has variables, constants, comparison operators, logicalconnectives and quantifiers. TRC: Variables range over (i.e., get bound to) tuples

DRC: Variables range over domain elements (= field values)

o an are s mp e su se s o rs r er re ca e og c

Both TRC and DRC are simple subsets of First Order Predicate Logic

In first-order logic (predicate calculus), predicate is a truth-valued function withar uments

When we substitute values for the arguments, function yields an expression, called aproposition, which can be either true or false

Expressions in the calculus are called formulas. An answer tuple isessentially an assignment of constants to variables that make the

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ExampleSno Sname Rating Age

22 Dustine 7 45.0

Sid bid day

22 101 10/10/08

bid bname color

101 Interlake Blue

29 Brutus 1 33.0

31 Lubber 8 55.5

32 And 8 25.5

22 102 10/10/08

22 103 10/08/08

22 104 10/07/08

102 Interlake Red103 Clipper Green

58 Rusty 10 35.0

64 Horatio 7 35.0

31 102 10/10/08

31 103 11/06/08

Instance of Boats

.

74 Horatio 9 35.0

85 Art 3 25.5

64 101 09/05/08

64 102 09/08/08

o .

Instance of Sailors Instance of Reserves

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Tuple Relational Calculus (TRC)

Interested in finding tuples for which a predicate is true.Based on the use of tuple variables

Query: {T|P(T)} T is tuple variable

P(T) is a formula (predicate) that describes T

Tuple variable is a variable that ranges over a named

relation: i.e, variable whose only permitted values aretu les of the relation

Specify range of a tuple variable T as the employeerelation as: Employee (T)

u , u r w v uTrue. Find all sailors with a rating above 7.

.

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Al ebra Vs Calculus condition (formula)(R) = { T/ R (T) formula }

1,2,..,k(R) = { tk/ (u) (R(u) t[1] = u[i1] .. t[k] = u[ik] }

=

R S = { t/ R(t) S(t) }

RX S = {tr+s/ (ur) (vs) (R(u) S(v) t[1] = u[i1] .. t[r] = u[r] t[r+1] = v[1] .. t[r+s] = v[s] }

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TRC Examples

Find all sailors with a rating above 7.

}7.|{ >

orratingSSailorsSS

}7.)(|{ > ratingSSSailorsS

r u r r u u r . wras:

..

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Tuple Relational Calculus

Can use two quantifiers to tell how many instancesthe predicate applies to: x s en a uan er ere ex s s

Universal Quantifier (for all)

Tuple variables qualified by or are called bound

variables, otherwise called freevariables

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Existential quantifier used in formulae that must betrue for at least one instance, such as:

{T/Staff(T) (B) (Branch(B) (B.branchNO=S.branchNo) B.city = Rourkela)}

branchNo as the branchNo of the current Staff tuple,T, and is located in Rourkela.

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Universal quantifier used in statements about every,

(B) (B.city Delhi) Means For all Branch tuples, the address is not in

.

One can also use ~ B B.cit = Delhi whichmeans there are no branches with an address inDelhi

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Formulae should be unambiguous and make sense

A (well-formed) formula is made out of atoms R(S

i

), where Si

is a tuple variable and R is a relation

R.a op S.b , op is one of

R.a op constant< > = , , , , ,

Can recursively build up formulae from atoms: If F1 and F2 are formulae, so are their conjunction, F1 F2;

disjunction, F1 F2 ; negation, ~ F1; and implication F1 F2 If F is a formula with free variable X then X F and X

(F) are also formulae

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Example Tuple Relational Calculus

Find the names and ages of sailors with a rating above 7

)}....7.(|{ ageSagePnameSnamePratingSSailorsSP ==>

BoatsBai orsP

......Re snameSsnamePbidBbidRsidRsidSservesR ===

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uple Relational Calculus Expressions can generate an infinite set.It is possible

have an infinite number of answers! Such queriesare called unsafe.

{ S/~Staff(S) }

,must be values in the domain of the expression.

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Uses variables that make values from domains

.

If F d d d stands for a formula com osed of , ,atoms and d1,d2,,dn represent domain variables,

then:1, 2,, n 1, 2,, n r

relational calculus expression

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Domain Relational Calculus Find all sailors with a rating above 7

{/ < I,N,T,A > Sailors T > 7 } he condition < I,N,T,A > Sailors ensures that the

domain variables I, N, T and A are bound to fields ofthe same Sailors tuple.

Modify this query to answer:

rating under 9, and are called Joe.

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Domain Relational Calculus Find sailors rated > 7 whove reserved boat #103

I N T A I N T A Sailors T, , , | , , , >

7

== 103Re,,,, BrIIrservesDBrIrDBrIr

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Find sailors rated > 7 whove reserved a redboat

I N T A I N T A Sailors T, , , | , , ,

> 7

=

= =

B BN C B BN C Boats B Br C red, , , , ' '

Observe how the parentheses control the

scope o eac quan er s n ng.

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I N T A I N T A Sailors, , , , , ,

B BN C Boats, ,

= =

Ir Br D serves I Ir Br B, , Re

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Summary

Relational calculus is non-operational,and users define ueries in terms ofwhat they want, not in terms of how to

com ute it. Declarativeness. Algebra and safe calculus have same

,of relational completeness.

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