Dr. Alexandra I. Cristea

http://www.dcs.warwick.ac.uk/~acristea/

CS 319: Theory of Databases: C6

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Content1. Generalities DB2. Integrity constraints (FD revisited)3. Relational Algebra (revisited)4. Query optimisation5. Tuple calculus6. Domain calculus7. Query equivalence8. Temporal Data9. No Nulls10. 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 tr – 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 & DCTranslate 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| bborrower(t[c_n]=b[c_n] ^ lloan(l[l_n]=b[l_n] ^ aaccount(a[b_n]=l[b_n] ^ ddepositor(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| bborrower(t[c_n]=b[c_n] ^ lloan(l[l_n]=b[l_n] ^ aaccount(a[b_n]=l[b_n] ^ ddepositor(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.

TC: {t| ddepositor(t[c_n]=d[c_n] ^ d2depositor((d2[c_n]=d[c_n] ^ aaccount(a[a_n]=d2[a_n])) => ( lloan(l[b_n]=a[b_n] ^ bborrow(b[l_n]=l[l_n] ^ b[c_n]=d2[c_n])))))}

This is necessary for the safety of the expression, to reduce the search space

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4. Give the name of customers who have loans only at branches where they have an account.

TC: {t| dborrow(t[c_n]=d[c_n] ^ d2borrow((d2[c_n]=d[c_n] ^ aloan(a[l_n]=d2[l_n]) => ( laccount(l[b_n]=a[b_n] ^

bdepositor(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| bborrow(t[c_n]=b[c_n] ^ b2borrow((b2[c_n]=b[c_n] ^ lloan(l[l_n]=b2[l_n])) => ( aaccount(a[b_n]=l[b_n] ^

ddepositor(d[a_n]=a[a_n] ^ d[c_n]=n2[c_n])))))}

With more intuitive attrs naming.

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Library TC/DC questions1. 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 | bborrow( t[name]=b[name] ^ t[department]=b[department] ^

ccopy(b[barcode]=c[barcode] ^ t[ISBN]=c[ISBN] ^ c2copy(c2[ISBN]=c[ISBN] ^

b2borrow(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 | bborrow( t[name]=b[name] ^ t[department]=b[department] ^ b2borrow((b2[name]=b[name] ^ b2[department]=b[department] ^ c2copy (c2[barcode]=b2[barcode])) =>

(c3copy(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| bbook(t[title]=b[title] ^ rreservation(r[ISBN]=b[ISBN] ^ ccopy(c[ISBN]=b[ISBN] ^ c[present] = “true” ^ c2copy(c2[ISBN]=c[ISBN] ^

bborrow(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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Content1. Generalities DB2. Integrity constraints (FD revisited)3. Relational Algebra (revisited)4. Query optimisation5. Tuple calculus6. Domain calculus7. Query equivalence8. Temporal Data9. No Nulls10. 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 & DCTranslate 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| bborrower(t[c_n]=b[c_n] ^ lloan(l[l_n]=b[l_n] ^ aaccount(a[b_n]=l[b_n] ^ ddepositor(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| bborrower(t[c_n]=b[c_n] ^ lloan(l[l_n]=b[l_n] ^ aaccount(a[b_n]=l[b_n] ^ ddepositor(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.

TC: {t| ddepositor(t[c_n]=d[c_n] ^ d2depositor((d2[c_n]=d[c_n] ^ aaccount(a[a_n]=d2[a_n]) => ( lloan(l[b_n]=a[b_n] ^ bborrow(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)))))}

This is necessary for the safety of the expression, to reduce the search space

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4. Give the name of customers who have loans only at branches where they have an account.

TC: {t| dborrow(t[c_n]=d[c_n] ^ d2borrow((d2[c_n]=d[c_n] ^ aloan(a[l_n]=d2[l_n]) => ( laccount(l[b_n]=a[b_n] ^

bdepositor(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| bborrow(t[c_n]=b[c_n] ^ b2borrow((b2[c_n]=b[c_n] ^ lloan(l[l_n]=b2[l_n]) => ( aaccount(a[b_n]=l[b_n] ^

ddepositor(d[a_n]=a[a_n] ^ d[c_n]=n2[c_n]))))))}

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)))))}

With more intuitive attrs naming.

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Library TC/DC questions1. 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 | bborrow( t[name]=b[name] ^ t[department]=b[department] ^

ccopy(b[barcode]=c[barcode] ^ t[ISBN]=c[ISBN] ^ c2copy(c2[ISBN]=c[ISBN] ^

b2borrow(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 | bborrow( t[name]=b[name] ^ t[department]=b[department] ^ b2borrow((b2[name]=b[name] ^ b2[department]=b[department] ^ c2copy (c2[barcode]=b2[barcode])) =>

(c3copy(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| bbook(t[title]=b[title] ^ rreservation(r[ISBN]=b[ISBN] ^ ccopy(c[ISBN]=b[ISBN] ^ c[present] = “true” ^ c2copy(c2[ISBN]=c[ISBN] ^

bborrow(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