1. Chapter 8
Relational Calculus

2. Computational Capabilities SQL Facilities Domain Calculus
Topics in this Chapter
Tuple Calculus
Calculus vs. Algebra
Computational Capabilities
SQL Facilities
Domain Calculus
Query-By-Example
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3. Relational calculus and algebra are logically equivalent
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
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4. 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)
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5. 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>)
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6. 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> ;
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7. <bool exp>s are called well-formed formulas, WFFs, or “weffs”
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
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8. RANGEVAR SX RANGES OVER S; RANGEVAR SY RANGES OVER S;
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;
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9. RANGEVAR SU RANGES OVER ( SX WHERE SX.CITY = ‘London’ )
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
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10. 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
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11. EXISTS is the existential quantifier
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
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12. FORALL is the universal quantifier
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
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13. 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
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14. And the same can be said of the algebra
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
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15. S.STATUS < (SELECT MAX (S.STATUS) FROM S) ;
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) ;
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16. SP { S#, S#(‘S1’), P# P#(‘P1’) }
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
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17. 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
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