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In [[dependency theory (database theory)|dependency theory]], a '''join dependency''' is a constraint on the set of legal relations over a database scheme. A table T is subject to a join [[dependency theory (database theory)|dependency]] if T can always be recreated by joining multiple tables each having a subset of the attributes of T. If one of the tables in the join has all the attributes of the table T, the join dependency is called trivial.
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The join dependency plays an important role in the [[Fifth normal form]], also known as ''project-join normal form'', because it can be proven that if you decompose a scheme <math>R</math> in tables <math>R_1</math> to <math>R_n</math>, the decomposition will be a lossless-join decomposition if you restrict the legal relations on <math>R</math> to a join dependency on <math>R</math> called <math>*(R_1,R_2,...R_n)</math>.
 
Another way to describe a join dependency is to say that the set of relationships in the join dependency is independent of each other.
 
Unlike in the case of [[functional dependencies]], there is no [[Soundness|sound]] and [[Completeness|complete]] axiomatization for join dependencies,<ref>S. V. Petrov. Finite axiomatization of languages for representation of system properties. ''Information Sciences'', 47:339-372, 1989.</ref> though axiomatization exist for more expressive dependency languages such as [[full typed dependencies]].<ref>Abiteboul, Hull, Vianu. Foundations of databases. Chapter 8, Bibliographic notes</ref> However, implication of join dependencies is decidable.<ref>ibid, Theorem 8.4.12</ref>
 
== Formal definition ==
 
{{quote|
Let <math>R</math> be a relation schema and let <math>R_1, R_2, ..., R_n</math> be a decomposition of <math>R</math>. <br />
The relation <math>r(R)</math> ''satisfies'' the join dependency <math>*(R_1,R_2,...R_n)</math> if <math>\bowtie_{i = 1}^n \Pi_{R_i}(r) = r</math>. <br />
A join dependency is trivial if one of the <math>R_i</math> is <math>R</math> itself.
|Silberschatz, Korth. ''Database System Concepts'', 1st Edition<ref>Silberschatz, Korth. ''Database System Concepts'', 1st Edition</ref>}}
 
2-ary join dependencies are called [[multivalued dependency]] as a historical artifact of the fact that they were studied before the general case. More specifically if ''U'' is a set of attributes and ''R'' a relation over it, then ''R'' satisfies <math>X \twoheadrightarrow Y</math> [[iff]] ''R'' satisfies <math>*(X\cup Y, X\cup(U-Y))</math>
 
== Example ==
Given a pizza-chain that models purchases in table Customer = { order-number, customer-name, pizza-name, courier}.
It is obvious that you can derive the following relations:
* customer-name depends on order-number
* pizza-name depends on order-number
* courier depends on order-number
Since the relationships are independent you can say there is a join dependency as follows: *((order-number, customer-name), (order-number, pizza-name), (order-number,courier)).
 
If each customer has his own courier however, you could have a join-dependency like this: *((order-number, customer-name), (order-number, courier), (customer-name, courier), (order-number,pizza-name)),
but *((order-number, customer-name, courier), (order-number,pizza-name)) would be valid as well. This makes it obvious that just having a join dependency is not enough to normalize a database scheme.
 
==See also==
* [[Chase (algorithm)]]
* [[Universal relation assumption]]
 
== References ==
{{Ibid|date=December 2013}}
{{Reflist}}
 
{{Database normalization}}
 
{{DEFAULTSORT:Join Dependency}}
[[Category:Database normalization]]

Revision as of 04:09, 5 March 2014

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