Canonical cover

Summary

A canonical cover for F (a set of functional dependencies on a relation scheme) is a set of dependencies such that F logically implies all dependencies in , and logically implies all dependencies in F.

The set has two important properties:

  1. No functional dependency in contains an extraneous attribute.
  2. Each left side of a functional dependency in is unique. That is, there are no two dependencies and in such that .

A canonical cover is not unique for a given set of functional dependencies, therefore one set F can have multiple covers .

Algorithm for computing a canonical cover

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  1.  
  2. Repeat:
    1. Use the union rule to replace any dependencies in   of the form   and   with  .
    2. Find a functional dependency in   with an extraneous attribute and delete it from  
  3. ... until   does not change[1]

Canonical cover example

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In the following example, Fc is the canonical cover of F.

Given the following, we can find the canonical cover: R = (A, B, C, G, H, I), F = {A→BC, B→C, A→B, AB→C}

  1. {A→BC, B→C, A→B, AB→C}
  2. {A → BC, B →C, AB → C}
  3. {A → BC, B → C}
  4. {A → B, B →C}

Fc =  {A → B, B →C}

Extraneous attributes

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An attribute is extraneous in a functional dependency if its removal from that functional dependency does not alter the closure of any attributes.[2]

Extraneous determinant attributes

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Given a set of functional dependencies   and a functional dependency   in  , the attribute   is extraneous in   if   and any of the functional dependencies in   can be implied by   using Armstrong's Axioms.[2]

Using an alternate method, given the set of functional dependencies  , and a functional dependency X → A in  , attribute Y is extraneous in X if  , and  .[3]

For example:

  • If F = {A → C, AB → C}, B is extraneous in AB → C because A → C can be inferred even after deleting B. This is true because if A functionally determines C, then AB also functionally determines C.
  • If F = {A → D, D → C, AB → C}, B is extraneous in AB → C because {A → D, D → C, AB → C} logically implies A → C.

Extraneous dependent attributes

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Given a set of functional dependencies   and a functional dependency   in  , the attribute   is extraneous in   if   and any of the functional dependencies in   can be implied by   using Armstrong's axioms.[3]

A dependent attribute of a functional dependency is extraneous if we can remove it without changing the closure of the set of determinant attributes in that functional dependency.[2]

For example:

  • If F = {A → C, AB → CD}, C is extraneous in AB → CD because AB → C can be inferred even after deleting C.
  • If F = {A → BC, B → C}, C is extraneous in A → BC because A → C can be inferred even after deleting C.

References

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  1. ^ Silberschatz, Abraham (2011). Database system concepts (PDF) (Sixth ed.). New York: McGraw-Hill. ISBN 978-0073523323. Archived from the original (PDF) on 2020-11-08.
  2. ^ a b c Elmasri, Ramez (2016). Fundamentals of database systems. Sham Navathe (Seventh ed.). Hoboken, NJ: Pearson. ISBN 978-0-13-397077-7. OCLC 913842106.
  3. ^ a b K, Saravanakumar; asamy. "How to find extraneous attribute? An example". Retrieved 2023-03-14.