Database System Concepts

Chapter 6: Formal Relational Query Languages

10/6/2020

Relational Calculus

Relational Calculus

• A relational calculus expression creates a new relation, which is specified in terms of variables that range over

– rows of the stored database relations (in tuple calculus)

• {S |S Sailors S.rating > 7}

– columns of the stored relations (in domain calculus)

• {<S,N,R,A>| <S,N,R,A> Sailors R > 7}

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Relational Calculus

• In a calculus expression, there is no order of operations to specify how to retrieve the query result

– a calculus expression specifies only what information the result should contain

– this is the main distinguishing feature between relational algebra and relational calculus.

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Relational Calculus

• Relational calculus is considered to be a nonprocedural or declarative language.

– this differs from relational algebra, where we must write a sequence of operations to specify a retrieval request;

– relational algebra can be considered as a procedural way of stating a query

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Relational Calculus Building Blocks

• VariablesTRC: Variables are bound to tuples.DRC: Variables are bound to domain elements (= column values)

• Constants7, “Foo”, 3.14159, etc.

• Comparison operators=, <>, <, >, etc.

• Logical connectives - not – and - or - implies - is a member of

• QuantifiersX(p(X)): For every X, p(X) must be trueX(p(X)): There exists at least one X such that p(X) is true

Relational Calculus

• English example: Find all sailors with a rating above 7

– Tuple R.C.:{S |S Sailors S.rating > 7}

“From all that is, find me the set of things that are tuples in the Sailors relation and whose rating field is greater than 7.”

– Domain R.C.: {<S,N,R,A>| <S,N,R,A> Sailors R > 7}

“From all that is, find me column values S, N, R, and A, where S is an integer, N is a string, R is an integer, A is a floating point number, such that <S, N, R, A> is a tuple in the Sailors relation and R is greater than 7.”

sid sname rating age

28 yuppy 9 35.0

31 lubber 8 55.5

44 guppy 5 35.0

58 rusty 10 35.0

Tuple Relational Calculus

• Query form: {T | p(T)}

– T is a tuple and p(T) denotes a formula in which tuple variable T appears.

• Answer:

– set of all tuples T for which the formula p(T) evaluates to true.

• Formula is recursively defined:

– Atomic formulas get tuples from relations or compare values

– Formulas built from other formulas using logical operators.

• An atomic formula is one of the following:

R Rel

R.a op S.b

R.a op constant, where

op is one of

• A formula can be:

– an atomic formula

– where p and q are formulas

– where variable R is a tuple variable

– where variable R is a tuple variable

TRC Formulas

= , , , , ,

qpqpp ,,))(( RpR

))(( RpR

Free and Bound Variables

• The use of quantifiers X and X in a formula is said to

bind X in the formula.

– A variable that is not bound is free.

• Important restriction

{T | p(T)}

– The variable T that appears to the left of `|’ must be the only free variable in the formula p(T).

– In other words, all other tuple variables must be bound using a quantifier.

Use of (For every)

• x (P(x)):

only true if P(x) is true for every x in the universe:

e.g. x ((x.color = “Red”) )

➢means everything that exists is red

• Usually we are less grandiose in our assertions:

x ( (x Boats) (x.color = “Red”))

• is a logical implication

a b means that if a is true, b must be true

a b is the same as a b

a b is the same as a b

• If a is true, b must be true!

– If a is true and b is false, the expression evaluates to false.

• If a is not true, we don’t care about b

– The expression is always true.

aT

F

T F

b

T

T T

F

Quantifier Shortcuts

• x ((x Boats) (x.color = “Red”))

“For every x in the Boats relation, the color must be Red.”

Can also be written as:

x Boats(x.color = “Red”)

• x ( (x Boats) (x.color = “Red”))

“There exists a tuple x in the Boats relation whose color is Red.”

Can also be written as:

x Boats (x.color = “Red”)

Selection and Projection

• Selection

Find all sailors with rating above 8

{S |S Sailors S.rating > 8}

{S | S1 Sailors(S1.rating > 8 S.sname = S1.sname S.age = S1.age)}

S is a tuple variable of 2 fields (i.e. {S} is a projection of Sailors)

sid sname rating age

28 yuppy 9 35.0

31 lubber 8 55.5

44 guppy 5 35.0

58 rusty 10 35.0

sname age

• Projection

Find names and ages of sailors with rating above 8.

S

S1

yuppy 35.0

S1

S1

S1

S rusty 35.0

Sailors

Note the use of to find a tuple in Reserves that `joins with’ the Sailors tuple under consideration.

{S | SSailors S.rating > 7 R(RReserves R.sid = S.sid

R.bid = 103)}

Joins

Find sailors rated > 7 who’ve reserved

boat #103

sid sname rating age

22 dustin 7 45.0

31 lubber 8 55.5

58 rusty 10 35.0

sid bid day

22 101 10/10/96

58 103 11/12/96

S

S

S

R

R

What if there was another tuple {58, 103, 12/13/96} in the Reserves relation?

Sailors

Reserves

Joins (continued)

Notice how the parentheses control the scope of each quantifier’s binding.

{S | SSailors S.rating > 7 R(RReserves R.sid = S.sid

B(BBoats B.bid = R.bid B.color = ‘red’))}

Find sailors rated > 7 who’ve reserved a red boat

What does this expression compute?

sid sname rating age

22 dustin 7 45.0

31 lubber 8 55.5

58 rusty 10 35.0

sid bid day

22 101 10/10/96

58 103 11/12/96

Sailors

Reserves

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表达式的安全性

• 表达式的安全性

– 元组关系演算有可能会产生无限关系，这样的表达式是不

安全的。如{t | (t r)}，求所有不在r中的元组

• 表达式安全性判定

– 引入公式P的域概念，用dom(P)表示

dom(P) = 显式出现在P中的值 +

在P中出现的关系的元组中出现的值

– 如果出现在表达式{t | P(t)}结果中的所有值均来自dom(P)，

则称{t | P(t)}是安全的

– {t | (t r)}？

Domain Relational Calculus

Domain Relational Calculus• A nonprocedural query language equivalent

in power to the tuple relational calculus

• Each query is an expression of the form:

{ x1, x2, …, xn | P (x1, x2, …, xn)}

– x1, x2, …, xn represent domain variables

– P represents a formula similar to that of the predicate calculus

Example Queries• Find the ID, name, dept_name, salary for instructors whose

salary is greater than $80,000

– {< i, n, d, s> | < i, n, d, s> instructor s 80000}

• As in the previous query, but output only the ID attribute value

– {< i> | n, d, s < i, n, d, s> instructor s 80000}

• Find the names of all instructors whose department is in the Watson building

– {< n > | i, d, s (< i, n, d, s > instructor b, a (< d, b, a> department b = “Watson” ))}

Example Queries• Find the set of all courses taught in the Fall 2009 semester, or

in the Spring 2010 semester, or both

– {<c> | a, s, y, b, r, t ( <c, a, s, y, b, t > section s = “Fall”y = 2009 )v a, s, y, b, r, t ( <c, a, s, y, b, t >section s = “Spring”y= 2010)}

– This case can also be written as{<c> | a, s, y, b, r, t ( <c, a, s, y, b, t > section

( (s = “Fall” y = 2009 ) v (s = “Spring” y = 2010))}

• Find the set of all courses taught in the Fall 2009 semester, and in the Spring 2010 semester

– {<c> | a, s, y, b, r, t ( <c, a, s, y, b, t > section s = “Fall” y = 2009 )

a, s, y, b, r, t ( <c, a, s, y, b, t > section ] s = “Spring” y = 2010)}

Universal Quantification• Find all students who have taken all courses offered in the

Biology department

– {< i > | n, d, tc ( < i, n, d, tc > student ( ci, ti, dn, cr ( < ci, ti, dn, cr > course dn

=“Biology” si, se, y, g ( <i, ci, si, se, y, g> takes ))}

Exercises – (I)

Let the following relation schemas be given:

Let relations r(R) and s(S) be given. Give an expression in the tuple relational calculus that is equivalent to each of the following:

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Exercises – (II)

Let R = (A, B, C), and let r1 and r2 both be relations on schema R. Give an expression in the domain relational calculus that is equivalent to each of the following:

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Exercises• Practice Exercises:

– 6.2 6.8

• Exercises:

– 6.11 6.13 6.14 6.15

End of Chapter 6