Chapter 8 Relational Calculus. Copyright © 2004 Pearson Addison-Wesley. All rights reserved.8-2...

Chapter 8

Relational Calculus

Copyright © 2004 Pearson Addison-Wesley. All rights reserved. 8-2

Topics in this Chapter

• Tuple Calculus • Calculus vs. Algebra• Computational Capabilities• SQL Facilities• Domain Calculus• Query-By-Example


Relational Calculus

• Based on predicate calculus, the relational calculus is a more natural language expression of the relational algebra

• Instead of operators used by the system to construct a result relation, the calculus offers a notation to express the result relation in terms of the source relations

• Relational calculus and algebra are logically equivalent


Relational Calculus Implementations

• Codd proposed a language called ALPHA to implement the relational calculus

• QUEL, an early competitor to SQL, was based on ALPHA

• Relational calculus came to be called tuple calculus, in distinction from domain calculus

• Domain calculus has been implemented by Query By Example (QBE)


Tuple Calculus - Syntax

<relation expression>

:= RELATION {<tuple expression commalist>}

| <relvar name>

| <relation op inv> -- “relation operator invoked”

| <with exp> -- “with expression”

| <introduced name>

| ( <relation exp>)


Tuple Calculus - Syntax

• Identical to the syntax of the algebra• Relation operator invoked now is interpreted

to mean relation definition• In addition,

<range var def> -- “range variable definition”

::= RANGEVAR <range var name>

RANGES OVER <relation exp commalist> ;


Tuple Calculus - WFFs

• <bool exp>s are called well-formed formulas, WFFs, or “weffs”

• Every reference to a range variable is either free or bound, within a context, and in particular, within a WFF


Range Variables

• RANGEVAR SX RANGES OVER S;• RANGEVAR SY RANGES OVER S;• RANGEVAR SPX RANGES OVER SP;• RANGEVAR SPY RANGES OVER SP;• RANGEVAR PX RANGES OVER P;


Range Variables

• RANGEVAR SU RANGES OVER

( SX WHERE SX.CITY = ‘London’ )

( SX WHERE EXISTS SPX

(SPX.S# = SX.S# AND

SPX.P# = P# (‘P1’) ) ) ;• In this example, SU ranges over the union of

the set of supplier tuples for suppliers located in London, and those that supply P1


Free and Bound Variables

• Every reference to a range variable is either free or bound

• A bound reference can be replaced by a reference to some other variable without changing the meaning of the expression

• A free reference is not so free


Quantifiers - EXISTS

• EXISTS is the existential quantifier• EXISTS V ( p ) -- There exists at least one

value of V that makes p true• Formally this is an iterated OR• FALSE OR p (t1) OR … OR p (tm)

-- will evaluate to false if m = 0


Quantifiers - FORALL

• FORALL is the universal quantifier• FORALL V ( p ) -- for all values of v, p is

true• Formally this is an iterated AND• TRUE AND p (t1) AND … AND p (tm)

-- will evaluate to true as long as all are true• This will evaluate to TRUE when the set is

empty


Relational Operations

• <relation op inv> in the calculus context is more a definition than an operator invocation

• <relation op inv>

::= <proto tuple> [WHERE <bool exp>]

• For example:• SX.S# WHERE SX.CITY = ‘London’

-- Get supplier numbers for suppliers in London


Calculus vs. Algebra

• Codd’s reduction algorithm reduces expressions of the calculus to algebra

• A language is said to be relationally complete if it is at least as powerful as the calculus

• And the same can be said of the algebra• QUEL, based on the calculus, can be

implemented by applying the algorithm to it, and then implementing the underlying algebra


SQL Facilities

• Because the calculus statements can be translated into algebra, they map equally well to SQL

• Example: Get supplier numbers for suppliers with status less than the current maximum status in the supplier table:

SELECT S.S# FROM S WHERE

S.STATUS < (SELECT MAX (S.STATUS)

FROM S) ;


Domain Calculus

• Its range variables range over domains (i.e. types) instead of relations

• Based on checking values in the attribute domain, a bool membership condition

• SP { S#, S#(‘S1’), P# P#(‘P1’) }

-- evaluates to true iff the shipment contains part P1 shipped by supplier S1


Query-By-Example (QBE)

• Language based on the domain calculus• Notation which is intuitive and easy to use• By making entries in blank tables, the user

specifies the conditions needed in the result• A condition box is used to allow more

complex conditions• Easy to express comparisons, uniqueness,

AND and OR, and EXISTS• Doesn’t express NOT EXISTS well, and so is

not relationally complete

Chapter 8 Relational Calculus. Copyright © 2004 Pearson Addison-Wesley. All rights reserved.8-2...

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