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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
Next lecture: Basic SQL
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