Entailment of Functional Dependencies

Tom Kelliher, CS 318

Mar. 8, 2002

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Return homework Monday. Midterm review Monday. Midterm will cover Chapters 1--7.

Assignment

From Last Time

Introduction to functional dependencies.

Outline

1. Demonstrating entailment.

2. Entailment checking algorithm.

3. Entailment checking example.

Coming Up

Midterm review.

Demonstrating Entailment

Using Armstrong's Axioms:

• .

• entails .

• and entail .

1. and entail . (Union)

2. entails and . (Decomposition)

So What?

1. Generally, when normalizing, we're interested in knowing if .

2. Armstrong's axioms are sound and complete.

   Therefore, an algorithm exists to do entailment checking.

Entailment Checking Algorithm

1. Suppose we want to see if .

2. We check this by seeing if is in the attribute closure of with respect to .
   1. This closure is defined:

      

   2. implies for each .

      By the union property then, .

   3. Note that .

      Why?

3. Attribute closure algorithm:

   closure :=
   repeat
   old := closure
   if there is an FD such that then
   closure :=
   until old = closure
   return closure

   Correctness. Use induction. (Not the entire proof.)
   1. Basis. .

      Observe . Why?

      By decomposition, .

   2. Inductive step. Assume .

      Suppose we have where . Justify each of the following steps.

      Observe .

      So .

      Therefore .

Entailment Checking Example

Suppose we have , where and contains