CS 319: Theory of Databases: C6
Dr. Alexandra I. Cristea
http://www.dcs.warwick.ac.uk/~acristea/
… previous
RA
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Content
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Generalities DB
Integrity constraints (FD revisited)
Relational Algebra (revisited)
Query optimisation
Tuple calculus
Domain calculus
Query equivalence
Temporal Data
No Nulls
The Askew Wall
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Tuple Calculus
• non-procedural query language
• A query in TC is expressed as:
{t|P(t)}
where:
t – set of tuples (also called free or range variable)
P – predicate that is true for t (also called formula)
Also:
t[A] – value of tuple t on attribute A
tr – means the tuple t is in relation r
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Example TC language
• QUEL (used by INGRES DBMS)
• SQL (to some extent)
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Examples TC
1. {t | t loan ^ t[amount] > 1200}
2. {t | s loan (t[loan-number] = s[loan-number]
^ s[amount] > 1200)}
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Satefy of expressions:
Tuple Calculus (& DC)
• A TC expression {t | P(t)} may give an infinite
relation, e.g.,
{ t |  (t  loan) }
• Define dom(P) as the set of all values
referenced by P (e.g. literals or relations)
• A TC expression {t | P(t)} is safe when all
values appearing in the result are values from
dom(P).
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Queries in TC & DC
Translate the following questions (for the bank
database) to the tuple & domain calculus:
1. Give the name of customers that have a loan
with a branch where they also have an account.
2. Give the name of customers who have a loan at
a branch where they do not have an account.
3. Give the name of customers who have a loan at
every branch where they have an account.
4. Give the name of customers who have loans
only at branches where they have an account.
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1. Give the name of customers that have a loan with
a branch where they also have an account.
TC: {t| bborrower(t[c_n]=b[c_n] ^
lloan(l[l_n]=b[l_n] ^
aaccount(a[b_n]=l[b_n] ^
ddepositor(a[a_n]=d[a_n] ^
d[c_n]=b[c_n] ))))}
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2. Give the name of customers who have a loan at a
branch where they do not have an account.
TC: {t| bborrower(t[c_n]=b[c_n] ^
lloan(l[l_n]=b[l_n] ^
aaccount(a[b_n]=l[b_n] ^
ddepositor(a[a_n]=d[a_n] ^
d[c_n]=b[c_n] ))))}
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3. Give the name of customers who have a loan at
every branch where they have an account.
This is necessary for
TC: {t| ddepositor(t[c_n]=d[c_n] ^
the safety of the
d2depositor((d2[c_n]=d[c_n] ^expression, to reduce
aaccount(a[a_n]=d2[a_n])) => the search space
( lloan(l[b_n]=a[b_n] ^
bborrow(b[l_n]=l[l_n] ^
b[c_n]=d2[c_n])))))}
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4. Give the name of customers who have loans only
at branches where they have an account.
TC: {t| dborrow(t[c_n]=d[c_n] ^
d2borrow((d2[c_n]=d[c_n] ^
aloan(a[l_n]=d2[l_n]) =>
( laccount(l[b_n]=a[b_n] ^
bdepositor(b[a_n]=l[a_n] ^
b[c_n]=d2[c_n]))))))}
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4. Give the name of customers who have loans only
at branches where they have an account.
TC: {t| bborrow(t[c_n]=b[c_n] ^
b2borrow((b2[c_n]=b[c_n] ^
lloan(l[l_n]=b2[l_n])) =>
( aaccount(a[b_n]=l[b_n] ^
ddepositor(d[a_n]=a[a_n] ^
d[c_n]=n2[c_n])))))}
With more
intuitive attrs
naming.
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Library TC/DC questions
1. Give the name and department of borrowers,
and the ISBN-number of the books of which
they were the first borrower.
2. Give the name and department of borrowers
who have never borrowed the most recently
purchased copy of a book.
3. Give the title of books that are “present” in
the library, reserved by someone, but not yet
borrowed (by anyone).
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1. Give the name and department of borrowers, and
the ISBN-number of the books of which they were
the first borrower.
TC: {t | bborrow( t[name]=b[name] ^
t[department]=b[department] ^
ccopy(b[barcode]=c[barcode] ^ t[ISBN]=c[ISBN] ^
c2copy(c2[ISBN]=c[ISBN] ^
b2borrow(b2[barcode]=c2[barcode] ^
b2[from]<b[from])))) }
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2. Give the name and department of borrowers who
have never borrowed the most recently purchased
copy of a book.
TC {t | bborrow( t[name]=b[name] ^
t[department]=b[department] ^
b2borrow((b2[name]=b[name] ^
b2[department]=b[department] ^ c2copy
(c2[barcode]=b2[barcode])) =>
(c3copy(c3[ISBN]=c2[ISBN] ^ c3[cpYear]>c2[cpYear] ^
b2[from]  c3[cpYear])))}
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3. Give the title of books that are “present” in the
library, reserved by someone, but not yet borrowed
(by anyone).
TC: {t| bbook(t[title]=b[title] ^
rreservation(r[ISBN]=b[ISBN] ^
ccopy(c[ISBN]=b[ISBN] ^ c[present] = “true” ^
c2copy(c2[ISBN]=c[ISBN] ^
bborrow(b[barcode]=c2[barcode])))))}
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Beer Database
• visits(drinker, bar)
• serves(bar, beer)
• likes(drinker, beer).
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Beer questions with a difference
1. Give all drinkers that visit bars that don’t serve any
beer they like
2. Give all drinkers that only visit bars that serve a
beer they like
3. Give all drinkers that only visit bars that serve no
beer they like
4. Give all drinkers that only visit bars that serve all
beers they like (and maybe other beers as well)
5. Give all drinkers that only visit bars that only serve
beers they like (and thus serve nothing else)
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Content
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Generalities DB
Integrity constraints (FD revisited)
Relational Algebra (revisited)
Query optimisation
Tuple calculus
Domain calculus
Query equivalence
Temporal Data
No Nulls
The Askew Wall
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Domain calculus
• Uses domain variables to take values from an
attributes domain, instead of tuple values
• is related to QBE language
• An expression in DC is of the form:
{<x1,x2,…,xn> | P(x1,x2,…,xn)}
where:
x1,…xn are domain variables
P is a formula
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Example DC language
• QBE
• Others: ILL, FQL, DEDUCE
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Example DC
1. {<l,b,a> | <l,b,a>  loan ^ a > 1200}
2. {<l> | b,a (<l, b, a>loan ^ a > 1200)}
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Queries in TC & DC
Translate the following questions (for the bank
database) to the tuple & domain calculus:
1. Give the name of customers that have a loan
with a branch where they also have an account.
2. Give the name of customers who have a loan at
a branch where they do not have an account.
3. Give the name of customers who have a loan at
every branch where they have an account.
4. Give the name of customers who have loans
only at branches where they have an account.
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1. Give the name of customers that have a loan with
a branch where they also have an account.
TC: {t| bborrower(t[c_n]=b[c_n] ^
lloan(l[l_n]=b[l_n] ^
aaccount(a[b_n]=l[b_n] ^
ddepositor(a[a_n]=d[a_n] ^
d[c_n]=b[c_n] ))))}
DC: {<cn>| ln (<cn,ln>borrower ^
bn,a(<ln,bn,a>loan ^
an,b(<an,bn,b>account ^
<cn,an>depositor)))}
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2. Give the name of customers who have a loan at a
branch where they do not have an account.
TC: {t| bborrower(t[c_n]=b[c_n] ^
lloan(l[l_n]=b[l_n] ^
aaccount(a[b_n]=l[b_n] ^
ddepositor(a[a_n]=d[a_n] ^
d[c_n]=b[c_n] ))))}
DC: {<cn>| ln (<cn,ln>borrower ^
bn,a(<ln,bn,a>loan ^
an,b(<an,bn,b>account ^
<cn,an>depositor)))}
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3. Give the name of customers who have a loan at
every branch where they have an account.
This is necessary for
TC: {t| ddepositor(t[c_n]=d[c_n] ^
the safety of the
d2depositor((d2[c_n]=d[c_n] ^expression, to reduce
aaccount(a[a_n]=d2[a_n]) => the search space
( lloan(l[b_n]=a[b_n] ^
bborrow(b[l_n]=l[l_n] ^
b[c_n]=d2[c_n]))))))}
DC: {<cn>| an(<cn,an>depositor ^
an2 (( <cn,an2>depositor ^
bn,b(<an2,bn,b>account) =>
(ln,a(<ln,bn,a>loan ^
<cn,ln>borrow)))))}
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4. Give the name of customers who have loans only
at branches where they have an account.
TC: {t| dborrow(t[c_n]=d[c_n] ^
d2borrow((d2[c_n]=d[c_n] ^
aloan(a[l_n]=d2[l_n]) =>
( laccount(l[b_n]=a[b_n] ^
bdepositor(b[a_n]=l[a_n] ^
b[c_n]=d2[c_n]))))))}
DC: {<cn>| an(<cn,an>borrow ^
an2 (( <cn,an2>borrow ^
bn,b(<an2,bn,b>loan) =>
(ln,a(<ln,bn,a>account ^
<cn,ln>depositor)))))}
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4. Give the name of customers who have loans only
at branches where they have an account.
TC: {t| bborrow(t[c_n]=b[c_n] ^
b2borrow((b2[c_n]=b[c_n] ^
lloan(l[l_n]=b2[l_n]) =>
( aaccount(a[b_n]=l[b_n] ^
ddepositor(d[a_n]=a[a_n] ^
d[c_n]=n2[c_n]))))))}
With more
intuitive attrs
naming.
DC: {<cn>| ln(<cn,ln>borrow ^
ln2 (( <cn,ln2>borrow ^
bn,a(<ln2,bn,a>loan) =>
(an,b(<an,bn,b>account ^
<cn,an>depositor)))))}
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Library TC/DC questions
1. Give the name and department of borrowers,
and the ISBN-number of the books of which
they were the first borrower.
2. Give the name and department of borrowers
who have never borrowed the most recently
purchased copy of a book.
3. Give the title of books that are “present” in
the library, reserved by someone, but not yet
borrowed (by anyone).
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1. Give the name and department of borrowers, and
the ISBN-number of the books of which they were
the first borrower.
TC: {t | bborrow( t[name]=b[name] ^
t[department]=b[department] ^
ccopy(b[barcode]=c[barcode] ^ t[ISBN]=c[ISBN] ^
c2copy(c2[ISBN]=c[ISBN] ^
b2borrow(b2[barcode]=c2[barcode] ^
b2[from]<b[from])))) }
DC: {<n,d,ISBN> | b,f,t( <n,b,d,f,t>borrow ^
Y,p,d1(<b,ISBN,d1,Y,p>copy ^
b2,d2,Y2,p2(<b2,ISBN,d2,Y2,p2>copy ^
n2,f2,t2,d3(<n2,b2,d3,f2,t2>borrow ^ f2<f)))) }
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2. Give the name and department of borrowers who
have never borrowed the most recently purchased
copy of a book.
TC {t | bborrow( t[name]=b[name] ^
t[department]=b[department] ^
b2borrow((b2[name]=b[name] ^
b2[department]=b[department] ^ c2copy
(c2[barcode]=b2[barcode])) =>
(c3copy(c3[ISBN]=c2[ISBN] ^ c3[cpYear]>c2[cpYear] ^
b2[from]  c3[cpYear])))}
DC {<n, d> | b,f,t (<n,b,d,f,t>borrow ^
b2,f2,t2((<n,b2,d,f2,t2>borrow ^
ISBN,d2,Y2,p2(<b2,ISBN,d2,Y2,p2>copy]) =>
(b3,d3,Y3,p3(<b3,ISBN,d3,Y3,p3>copy ^ Y3>Y2 ^ f2 
Y3)))))}
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3. Give the title of books that are “present” in the
library, reserved by someone, but not yet borrowed
(by anyone).
TC: {t| bbook(t[title]=b[title] ^
rreservation(r[ISBN]=b[ISBN] ^
ccopy(c[ISBN]=b[ISBN] ^ c[present] = “true” ^
c2copy(c2[ISBN]=c[ISBN] ^
bborrow(b[barcode]=c2[barcode])))))}
DC: {<title>| ISBN,p,y(<ISBN,title,p,y>book ^
n,d,date,c(<n,d,ISBN,date,c>reservation ^
b2,d2,Y2,p2(<b2,ISBN,d2,Y2,p2>copy ^ p2 = “true”
^ b3,d3,Y3,p3(<b3,ISBN,d3,Y3,p3>copy ^
n4,d4,f4,t4(<n4,b3,d4,f4,t4>borrow)))))}
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Summary
• We have learned TC & DC
• We have learned to perform simple and
more complex queries in TC & DC
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… to follow
Query equivalence
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