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Lecture 5:Relational calculus

www.cl.cam.ac.uk/Teaching/current/Databases/

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Relational calculus

• There are two versions of the relational calculus:– Tuple relational calculus (TRC)– Domain relational calculus (DRC)

• Both TRC and DRC are simple subsets of first-order logic

• The difference is the level at which variables are used: for fields (domains) or for tuples

• The calculus is non-procedural (‘declarative’) compared to the relational algebra

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Domain relational calculus

• Queries have the form

{<x1,…,xn>| F(x1,…,xn)}

where x1,…,xn are domain variables and F is a formula with free variables {x1,…,xn}

• Answer: all tuples <v1,…,vn> that make F(v1,…,vn) true

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Example

Find all sailors with a rating above 7

{<I,N,R,A> | <I,N,R,A>Sailors R>7}

• The condition <I,N,R,A>Sailors ensures that the domain variables are bound to the appropriate fields of the Sailors tuple

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Example

• Simple projection:

{<I,N> | R,A.<I,N,R,A>Sailors}

• Simple projection and selection:

{<I,N> | R,A.<I,N,R,A>Sailors N=`Julia’}

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DRC formulae

• Atomic formulae: a ::=– <x1,…,xn>R

– xi binop xj, xi binop c, c binop xj, unop c, unop xi

• DRC Formulae: P, Q ::=– a P, PQ, PQ x.P x.P

• Recall that x and x are binders for x

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Example

Find the names of sailors rated >7 who’ve reserved boat 103

{<N> | I,A,R.<I,N,R,A>Sailors R>7

SI,BI,D.(<SI,BI,D>Reserves I=SI BI=103)}

• Note the use of and = to ‘simulate’ join

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Example

Find the names of sailors rated >7 who’ve reserved a red boat

{<N> | I,A,R.<I,N,R,A>Sailors R>7

SI,BI,D. (<SI,BI,D>Reserves SI=I B,C. (<B,C>Boats B=BI C=‘red’))}

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Example

Find the names of sailors who have reserved at least two boats

{<N> | I,R,A. <I,N,R,A>Sailors BI1,BI2,D1,D2.<I,BI1,D1>Reserves <I,BI2,D2>Reserves BI1BI2 }

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Example

Find names of sailors who’ve reserved all boats

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Example

Find names of sailors who’ve reserved all boats

{<N> | I,R,A. <I,N,R,A>Sailors B,C. ((<B,C>Boats) (<SI,BI,D>Reserves. I=SI BI=B)) }

{<N> | I,R,A. <I,N,R,A>Sailors <B,C>Boats. <SI,BI,D>Reserves. I=SI BI=B)) }

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Tuple relational calculus

• Similar to DRC except that variables range over tuples rather than field values

• For example, the query “Find all sailors with rating above 7” is represented in TRC as follows:

{S | SSailors S.rating>7}

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Semantics of TRC queries

• In general a TRC query is of the form

{t | P}

where FV(P)={t}

• The answer to such a query is the set of all tuples T for which P[T/t] is true

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Example

Find names and ages of sailors with a rating above 7

{P | SSailors. S.rating>7 P.sname=S.sname P.age=S.age}

Recall P rangesover tuple values

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Example

Find the names of sailors who have reserved at least two boats

{ P | SSailors. R1Reserves. R2Reserves. S.sid=R1.sid R1.sid=R2.sid R1.bid R2.bid P.sname=S.sname}

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Equivalence with relational algebra

• This equivalence was first considered by Codd in 1972

• Codd introduced the notion of relational completeness– A language is relationally complete if it can

express all the queries expressible in the relational algebra.

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Encoding relational algebra

• Let’s consider the first direction of the equivalence: can the relational algebra be coded up in the (domain) relational calculus?

• This translation can be done systematically, we define a translation function [-]

• Simple case:

[R] = {<x1,…,xn> | <x1,…,xn>R}

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Encoding selection

• Assume

[e] = {<x1,…,xn> | F }

• Then

[c(e)] = {<x1,…,xn> | F C’}

where C’ is obtained from C by replacing each attribute with the corresponding variable

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Encoding relational calculus

• Can we code up the relational calculus in the relational algebra?

• At the moment, NO!• Given our syntax we can define

‘problematic’ queries such as

{S | (SSailors)}• This (presumably) means the set of all

tuples that are not sailors, which is an infinite set…

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Safe queries

• A query is said to be safe if no matter how we instantiate the relations, it always produces a finite answer

• Unfortunately, safety (a semantic condition) is undecidable – That is, given a arbitrary query, no program can decide if it is

safe

• Fortunately, we can define a restricted syntactic class of queries which are guaranteed to be safe

• Safe queries can be encoded in the relational algebra

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Summary

You should now understand

• The relational calculus– Tuple relational calculus– Domain relational calculus

• Translation from relational algebra to relational calculus

• Safe queries and relational completeness

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