This document discusses the relational calculus, including the tuple relational calculus (TRC) and domain relational calculus (DRC). The TRC uses variables for tuples, while the DRC uses variables for fields/domains. Both can express the same queries as the relational algebra through translations. While the relational calculus can express some "unsafe" infinite queries, a subset of "safe" queries can be guaranteed to produce finite answers and are equivalent to the relational algebra.

1. 1
Lecture 5:
Relational calculus
www.cl.cam.ac.uk/Teaching/current/Databases/

2. 2
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

3. 3
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

4. 4
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

5. 5
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’}

6. 6
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, P∧Q, P∨Q
– ∃x.P
– ∀x.P
• Recall that ∃x and ∀x are binders for x

7. 7
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

8. 8
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’))}

9. 9
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 ∧
BI1≠BI2 }

10. 10
Example
Find names of sailors who’ve reserved all
boats

11. 11
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)) }

12. 12
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 | S∈Sailors ∧ S.rating>7}

13. 13
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

14. 14
Example
Find names and ages of sailors with a rating
above 7
{P | ∃S∈Sailors. S.rating>7∧
P.sname=S.sname∧
P.age=S.age}
Recall P ranges
over tuple values

15. 15
Example
Find the names of sailors who have
reserved at least two boats
{ P | ∃S∈Sailors.
∃R1∈Reserves. ∃R2∈Reserves.
S.sid=R1.sid ∧ R1.sid=R2.sid ∧
R1.bid ≠ R2.bid ∧
P.sname=S.sname}

16. 17
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.

17. 18
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}

18. 19
Encoding selection
• Assume
[e] = {<x1,…,xn> | F }
• Then
[σc(e)] = {<x1,…,xn> | F ∧ C’}
where C’ is obtained from C by replacing
each attribute with the corresponding
variable

19. 20
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 | ¬ (S∈Sailors)}
• This (presumably) means the set of all
tuples that are not sailors, which is an
infinite set… 

20. 21
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

21. 22
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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