1. From Relational Calculus to Relational Algebra
Tuple relational calculus,
domain relational calculus,
and relational algebra
5/15/2019
CS319 Theory of Databases

2. Domain Relational Calculus 1
.... reminder
Predicate Calculus query languages ...
a query = finding values satisfying predicate
Two kinds of predicate calculus language
primitive objects “tuples”  tuple relational calculus
primitive objects “domain values”
 domain relational calculus
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CS319 Theory of Databases

3. Domain Relational Calculus 2
Form of relational calculus expressions
Formula is built up using operators of FOPC from
atomic clauses of three types:
1. R(s1 s2 … sk), where R is relation name, and each si is domain variable
2. si q uj, where s and u are domain variables, and q is
an arithmetic comparison operator (such as <, = etc)
3. si q c, where s is domain variable, c is a constant
5/15/2019
CS319 Theory of Databases

4. Domain Relational Calculus 3
Semantics of the atomic clauses:
1. R(s1s2 …sk), where R is relation name,
and each si is domain variable
2. si q uj, where s and u are domain variables, and q is
an arithmetic comparison operator (such as <, = etc)
3. si q c, where s is domain variable, c is a constant
1. “s1s2… sk represents a tuple in R”
2. “domain value represented by si is in relationq to the domain value represented by uj”
3. “domain value represented by si is in relationq to the constant c”
5/15/2019
CS319 Theory of Databases

5. From tuple to domain relational calculus 1
The definition of safety is expressed in terms of constraints on component values of tuples ... so generalises directly to domain relational calculus
Theorem 2: For every safe tuple relational calculus expression there is an equivalent safe domain relational calculus expression.
Omit formal proof: essentially a syntactic transformation involving substitution for tuple variables
5/15/2019
CS319 Theory of Databases

6. From tuple to domain relational calculus 2
An illustrative example
Can express RCS in tuple relational calculus as:
{ w | ($u)($v)(R(u) Ù S(v) Ù f(u,v)) }
where f(u,v) º (u[2]=v[1] Ù w[1]=u[1] Ù w[2]=v[2])
in domain relational calculus, this becomes:
{ w1w2 | ($u1)($u2)($v1)($v2)
(R(u1u2) Ù S(v1v2) Ù F(u1,u2,v1,v2)) }
where F(u1,u2,v1,v2) º (u2=v1 Ù w1=u1 Ù w2=v2)
5/15/2019
CS319 Theory of Databases

7. From domain relational calculus to algebra 1
Theorem 3: For every safe expression in domain relational calculus, there is an equivalent relational algebra expression.
Proof (sketch only)
Use induction on number of operators in y to construct an algebraic expression for { t1t2 ... tn | y(t1, t2, ..., tn) }
To simplify the induction, begin with two lemmas:
there is a relational algebra exp to represent dom(y)
don’t need to consider  and  as independent cases
5/15/2019
CS319 Theory of Databases

8. From domain relational calculus to algebra 2
Use induction on number of operators in y to construct an algebraic expression for { t1t2 ... tn | y(t1, t2, ..., tn) }
By safety, enough to show:
for each subformula w of y of the form
{ t1t2 ... tm | w(t1, t2, ..., tm) }
$ a relational algebra expression E whose value is
dom(y)* Ç { t1t2 ... tm | w(t1, t2, ..., tm) }
where dom(y)* = set of tuples with compts in dom(y)
i.e. can restrict attention to tuples in dom(y)*
5/15/2019
CS319 Theory of Databases

9. From domain relational calculus to algebra 3
Lemma A: If y is any formula in domain relational calculus, there is a relational algebra expression to represent the unary relation dom(y)
Note: unary relation º set of 1-tuples º set
Proof: Suppose R has arity k. Let
D(R) º P1(R) È P2(R) È ... È Pk(R).
dom(y) is the union of all D(R)'s over relations R referred to in y together with the set of all constants {a1, a2, ..., an} referred to in y.
Thus can take D as an algebraic expression:
D = R referred to in y D(R) È {a1, a2, ..., an}
5/15/2019
CS319 Theory of Databases

10. From domain relational calculus to algebra 4
Lemma B: If y is any formula in domain relational
calculus, there is a formula y' in domain relational
calculus with no occurrences of Ù or " such that
{ t1t2 ... tn | y(t1, t2, ..., tn) } and { t1t2 ... tn | y'(t1, t2, ..., tn) }
are equivalent. This transformation respects safety.
Proof: Wherever the operators Ù and " appear in y:
replace f Ù r by (f  r).
replace ("v)(f(v)) by ($v)(f(v)).
Need to show that safety is preserved ...
5/15/2019
CS319 Theory of Databases

11. From domain relational calculus to algebra 5
Proof of Lemma B: Wherever the operators Ù and " appear in y:
replace f Ù r by (f  r).
replace ("v)(f(v)) by ($v)(f(v)).
To show that safety is preserved ...
Observe that dom(y) = dom(y'):
this takes care of the first safety condition.
Note also that if ("v)(f(v)) safe
v Ï dom(f)* Þ f(v) true Þ f(v) false
Hence ($v)(f(v)) is also safe
5/15/2019
CS319 Theory of Databases

12. From domain relational calculus to algebra 6
Proof of Theorem 3 (cont.)
Consider relation defined by { t1t2...tn | y(t1, t2, ..., tn) }
where y is a safe relational calculus expression
By Lemmata:
can assume neither Ù or " occurs in y
enough to show for each subformula w of y of form
{ t1t2...tm | w(t1, t2, ..., tm) }
$ a relational algebra expression E whose value is
dom(y)* Ç { t1t2...tm | w(t1, t2, ..., tm)}
where dom(y)* = set of tuples with compts in dom(y)
Prove this by induction on the number of operators in y.
5/15/2019
CS319 Theory of Databases

13. From domain relational calculus to algebra 7
I.e. prove for all subformulae w of y  in particular for y itself  by induction on number of operators N in w.
N=0: consider the relation defined by
dom(y)* Ç { t1t2 ... tm | w(t1, t2, ..., tm) }
where w is an atomic formula.
Let D be relational algebra expression for dom(y).
Two cases:
1. w(ti, tj) = ti q tj or w(ti) = ti q c
where q is an arithmetic comparison operator
2. w(t1, t2, ..., tm) = R(ti(1)ti(2) ... ti(k))
5/15/2019
CS319 Theory of Databases

14. From domain relational calculus to algebra 8
Proof of Theorem 3: Base of induction (cont.)
1. w(ti, tj) = ti q tj or w(ti) = ti q c
where q is an arithmetic comparison operator
2. w(t1, t2, ..., tm) = R(ti(1)ti(2) ... ti(k))
For case 1: use expression E º siqj(D´D).
For case 2: have w(t1, t2, ..., tm) = R (ti(1)ti(2) ... ti(k))
By safety, every index r, where 1rm, must be an
index i(j) for some j. Define algebraic expression
E º Õj(1), j(2), ..., j(m)(sC(R))
where C is conjunction of relations r=s over pairs (r,s)
such that i(r)=i(s) and j(r) is an index such that i(j(r))=r.
5/15/2019
CS319 Theory of Databases

15. From domain relational calculus to algebra 9
Illustrative example for Case 2:
Take domain relational calculus expression
{ t1t2t3 | R(t3t2t1t2) }
Consider indices j for which i(j) = r: this defines a pattern
j=1 j=2 j=3 j=4
r=1: •
r=2: • •
r=3: •
Suitable expression is E º Õ3, 4,1(s2=4(R))
5/15/2019
CS319 Theory of Databases

16. From domain relational calculus to algebra 10
Proof of Thm 3: The induction step
Three cases to consider in the induction step ...
Assume form of w(t1, t2, ..., tm) is
1. f(u1, u2, ..., up) Ú r(v1, v2, ..., vr)
2. f(t1, t2, ..., tm)
3. ($t) (f(t1, t2, ..., tm, t))
5/15/2019
CS319 Theory of Databases

17. From domain relational calculus to algebra 11
Proof of Thm 3: The induction step
Case 1: Assume form of w(t1, t2, ..., tm) is
f(u1, u2, ..., up) Ú r(v1, v2, ..., vr)
Can assume
(by safety) the variables u1, u2, ..., up, v1, v2, ..., vr
include all the variables t1, t2, ..., tm
variables in { u1, u2, ..., up } are distinct
variables in { v1, v2, ..., vr } are distinct
5/15/2019
CS319 Theory of Databases

18. From domain relational calculus to algebra 12
Proof of Thm 3: The induction step for operator Ú
Illustrative example shows principle
w(t1, t2, ..., tm)  f(u1, u2, ..., up) Ú r(v1, v2, ..., vr)
where, in particular case of m =4, p=3, r=2:
w(t1, t2, t3, t4) º f(t1, t3, t4) Ú r(t2, t4)
Let F and G be relational algebra expressions for
{ t1t2t3 | f(t1, t2, t3) } and { t1t2 | r(t1, t2) } respectively ....
Need to write down a relational algebra expression for
dom(y)* Ç { t1t2t3t4 | w(t1, t2, t3, t4) } which is also
dom(y)* Ç { t1t2t3t4 | f(t1, t3, t4) Ú r(t2, t4) }
... to do this, must use expression D for dom(y)
5/15/2019
CS319 Theory of Databases

19. From domain relational calculus to algebra 13
Proof of Thm 3: The induction step for operator Ú (cont.)
Need a relational algebra expression for
dom(y)* Ç { t1t2t3t4 | f(t1, t3, t4) Ú r(t2, t4) }
Set of tuples t1t2t3t4 satisfying f(t1, t3, t4) is constrained
so that t1t3t4 is a tuple in the relation defined by
algebraic expression F. If D is the algebraic expression
for dom(y) then F´D defines tuples t1t3t4t2 satisfying
f(t1, t3, t4) within dom(y)*. Hence Õ1, 4, 2, 3(F´D) defines
tuples t1t2t3t4 satisfying f(t1, t3, t4) within dom(y)*.
5/15/2019
CS319 Theory of Databases

20. From domain relational calculus to algebra 14
Proof of Thm 3: The induction step for operator Ú (cont.)
Need a relational algebra expression for
dom(y)* Ç { t1t2t3t4 | f(t1, t3, t4) Ú r(t2, t4) }
Õ1, 4, 2, 3(F´D) defines tuples t1t2t3t4 satisfying f(t1, t3, t4) in dom(y)*.
Similarly G ´ D ´ D defines tuples t2t4t1t3 satisfying
r(t2, t4) within dom(y)* and Õ3, 1, 4, 2(G ´ D ´ D) defines
tuples t1t2t3t4 satisfying r(t2, t4) within dom(y)*.
Hence can take E º Õ1, 4, 2, 3(F´D)ÈÕ3, 1, 4, 2(G´D´D)
5/15/2019
CS319 Theory of Databases

21. From domain relational calculus to algebra 15
Case 2: w(t1, t2, ..., tm) is f(t1, t2, ..., tm)
If F is an algebraic expression for
dom(y)* Ç { t1t2 ... tm | f(t1, t2, ..., tm) }
and D is an algebraic expression for dom(y) then
D ´ D ´ ... ´ D - F
m times
represents the relation
dom(y)* - { t1t2 ... tm | f(t1, t2, ..., tm) }
= dom(y)* - (dom(y)* - {t1t2 ... tm | f(t1, t2, ..., tm) })
= dom(y)* Ç { t1t2 ... tm | f(t1, t2, ..., tm) }
5/15/2019
CS319 Theory of Databases

22. From domain relational calculus to algebra 16
Case 3:
w(t1, t2, ..., tm) is (t) (f(t1, t2, ..., tm, t))
By induction, have an algebraic expression F for
dom(y)* Ç { t1t2 ... tmtm+1 | f(t1, t2, ..., tm, tm+1) }
Since y is safe:
t satisfies f(t1, t2, ..., tm, t) Þ t Î dom(y)*
Hence Õ1, 2, ..., m(F) represents the required relation:
dom(y)* Ç { t1t2 ... tm | (t) (f(t1, t2, ..., tm, t)) }
5/15/2019
CS319 Theory of Databases

23. From domain relational calculus to algebra 17
Have proved the equivalence of relational algebra and domain / tuple relational calculus ...
Theorems 1, 2 and 3 together prove
relational algebra
domain relational calculus
tuple relational calculus
all have the same expressive power. Thus
A query language is complete if and only if it has the expressive power of one of these formalisms.
5/15/2019
CS319 Theory of Databases

24. To follow … Mathematical foundations and features of SQL
5/15/2019
CS319 Theory of Databases