Lecture 6.pdf

Reasoning about FDs

R[A] - a relation F - a set of functional dependencies 𝛼 – a subset of attributes

Problems

1. compute the closure of F: F+

• the set F+ contains all the functional dependencies implied by F

• F implies a functional dependency f if f holds on every relation that satisfies F

• the following 3 rules can be repeatedly applied to compute F+ (Armstrong's Axioms): 𝛼, 𝛽, 𝛾 - subsets of attributes of 𝐴

  1. reflexivity: if 𝛽 ⊆ 𝛼, then 𝛼 → 𝛽
  2. augmentation: if 𝛼 → 𝛽, then 𝛼𝛾 → 𝛽𝛾
  3. transitivity: if 𝛼 → 𝛽 and 𝛽 → 𝛾, then 𝛼 → 𝛾

• these rules are complete (they compute the closure) and sound (no erroneous functional dependencies can be derived)

• the following rules can be derived from Armstrong's Axioms:

  𝛼, 𝛽, 𝛾, 𝛿 - subsets of attributes of �

  1. union:

     if 𝛼 → 𝛽 and 𝛼 → 𝛾, then 𝛼 → 𝛽 𝛾

     

  2. decomposition:

     if 𝛼 → 𝛽𝛾, then 𝛼 → 𝛽 and 𝛼 → 𝛾

     

  3. pseudotransitivity:

     if 𝛼 → 𝛽 and 𝛽𝛾 → 𝛿, then 𝛼𝛾 → 𝛿

     

1. compute the closure of a set of attributes under a set of functional dependencies,

   e.g. the closure of 𝛼 under F: 𝛼+

• determine the closure of 𝛼 under F, denoted as 𝛼+
• a+ the set of attributes that are functionally dependent on attributes in 𝛼 (under F)

1. compute the minimal cover for a set of dependencies