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Database Systems and

Applications

Lecture 19

Domain Calculus

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RelationalCalculus

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Domain Relational Calculus

Queryhas the form:

{x1 , x2 , . . . , xnp(x1 , x2 , . . . , xn)}Answerincludes all tuples that

make the formula be true.x1 , x2 , . . . , xn

p (x1, x2 , . . . , xn )Formula is recursively defined, starting with

simple atomic formulas (getting tuples fromrelations or making comparisons of values),and building bigger and better formulas usingthe logical connectives.

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DRC Formulas Atomic formula:

, or X op Y, or X opconstant

op is one of

Formula:

an atomic formula, or

, where p and q are formulas, or

, where variable X is freein p(X), or , where variable X is free in p(X)

x1 , x2 , . . . , xn Rname ,>,=,, ,

p , pq , pq$ X( p ( X))X \( p \( X \) \) }} {}

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Free and Bound Variables

The use ofquantifiers and in aformula is said to bind X. A variable that is not bound is free.

Let us revisit the definition of a query:

$ XX}} {}

{x1 , x2 , . . . , xnp(x1 , x2 , . . . , xn)}

There is an important restriction: thevariables x1, ..., xn that appear to the leftof `| must be the onlyfree variables inthe formula p(...).

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Find all sailors with a ratingabove 7

The condition ensuresthat the domain variables I, N, TandA are boundto fields of the same Sailors tuple.

The term to the left of `| (whichshould 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 SailorsT>7}I , N ,T , A Sailors

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

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Find sailors rated > 7 whove reservedboat #103

We have used as ashorthand for

Note the use of to find a tuple inReserves that `joins with the Sailors tupleunder consideration.

{I , N ,T , A I , N ,T , A SailorsT>7

$Ir ,Br ,D (Ir , Br ,DReserves

Ir=I Br=103

)}$ Ir , Br , D ( . .. )$ Ir ($ Br ( $ D ( .. . ) ))

$

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Find sailors rated > 7 whove reserveda red boat

Observe how the parentheses control the scope ofeach quantifiers binding.

{I , N ,T , A I , N ,T , A SailorsT>7

$Ir ,Br ,D(Ir , Br , DReserves Ir=I$B,BN,C(B,BN,CBoatsB=BrC='red ') )}

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Find sailors whove reservedall boats

Find all sailors I such that for each 3-tuple either it is not a tuple in Boats or there is a tuple

in Reserves showing that sailor I has reserved it.

Find sailors whove reserved all redboats ?

{I , N , T , A I , N ,T , A SailorsB, ital BN,C\( left ( langle B, ital BN,C rangle ital Boats right )}} {}

(

Ir ,Br ,

D

(Ir , Br ,D

Re serves

I=Ir

Br=B )

))

}

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{ | I,T,A ( Sailors

Br1,Br2,D1,D2( Reserves

reserves Br1 Br2)) }

Find the sailors who reserved at least twoboats.

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Unsafe Queries, ExpressivePower

syntactically correct calculusqueries that have an infinite number

of answers! Unsafe queries. e.g., TRC

Solution???? Dont do that!

{S( S Sailors ) }

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Summary The relational model has rigorously defined query

languages simple and powerful. Relational algebra is more operational

useful as internal representation for query evaluationplans.

Relational calculus is non-operational users define queries in terms of what they want, not in

terms of how to compute it. (Declarative)

Several ways of expressing a given query

a queryoptimizershould choose the most efficientversion.

Algebra and safe calculus have same expressivepower

leads to the notion ofrelational completeness.