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The most basic Inference Rules: Armstrong's axioms

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• Inference Rules
  • Inference rule:

    Inference rule = an assertion that we can apply to a set of functional dependencies to derive another functional dependencies

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• The basic inference rules: Armstrongs's Axioms
  • The basic inference rules used in database theory:

    Let X, Y and Z be three sets of attributes in the (same) relation R. The basic inference rules are: Reflexivity rule: If   Y ⊆ X   then:   X → Y         Augmentation rule: If   X → Y   then:   XZ → YZ         Transitivity rule: If   X → Y and Y → Z   then:   X → Z

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  • Note:

    The 3 basic inference rules are called: The Armstrong's axioms

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• Examples of the Armstrong's axioms
  • Fact:

    The Armstrong's axioms are very intuitive (and can be proved to be true)

    I will first use a example to illustrate their meaing

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  • Consider the relation:

    Employee-Department SSN fname lname DNO DName +-------------+----------+----------+----------+----------+ | 111-11-1111 | John | Smith | 5 | Research | +-------------+----------+----------+----------+----------+ | 222-22-2222 | Jane | Doe | 4 | Payroll | +-------------+----------+----------+----------+----------+ | 333-33-3333 | Pete | Pan | 5 | Research | +-------------+----------+----------+----------+----------+

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  • Examples of Armostrong axioms:

    Reflexivity rule: If   Y ⊆ X   then:   X → Y         Example: Since { fname } ⊆ { fname, lname } then: {fname, lname} → {fname} What this rule says is: If I see fname=John and lname=Smith in some tuple then: I must see the same value for fname=John in the tuple.           Comment: this is kinda obvious.... Augmentation rule: If   X → Y   then:   XZ → YZ         Example: Since {DNO} → {Dname} then: {DNO, LName} → {DName, LName} The augmentation rule says the following: It is given that DNO → DName: SSN fname lname DNO DName +-------------+----------+----------+----------+----------+ | 111-11-1111 | John | Smith | 5 | Research | +-------------+----------+----------+----------+----------+ | 222-22-2222 | Jane | Doe | 4 | Payroll | +-------------+----------+----------+----------+----------+ | 333-33-3333 | Pete | Pan | 5 | Research | +-------------+----------+----------+----------+----------+ Example:   DNO=5 → DName=Research We can verify that the following is also true: {DNO, LName} → {DName, LName} Example:   (DNO=5, LName=Smith) → (DName=Research, LName=Smith) SSN fname lname DNO DName +-------------+----------+----------+----------+----------+ | 111-11-1111 | John | Smith | 5 | Research | +-------------+----------+----------+----------+----------+ | 222-22-2222 | Jane | Doe | 4 | Payroll | +-------------+----------+----------+----------+----------+ | 333-33-3333 | Pete | Pan | 5 | Research | +-------------+----------+----------+----------+----------+ I see (DNO=5, LName=Smith) in a tuple, therefore, I must see (DName=Research, LName=Smith) also (and I do) !!! Transitivity rule: If   X → Y and Y → Z   then:   X → Z         Example: Since: {SSN} → {DNO} and {DNO} → {DName} Then also: {SSN} → {DName} The transitivity rule says the following: If SSN=111-11-1111 in a tuple, then DNO=5 in that tuple: SSN fname lname DNO DName +-------------+----------+----------+----------+----------+ | 111-11-1111 | John | Smith | 5 | Research | +-------------+----------+----------+----------+----------+ | 222-22-2222 | Jane | Doe | 4 | Payroll | +-------------+----------+----------+----------+----------+ | 333-33-3333 | Pete | Pan | 5 | Research | +-------------+----------+----------+----------+----------+ Futhermore, when DNO=5 in a tuple, then DName=Research in that tuple: SSN fname lname DNO DName +-------------+----------+----------+----------+----------+ | 111-11-1111 | John | Smith | 5 | Research | +-------------+----------+----------+----------+----------+ | 222-22-2222 | Jane | Doe | 4 | Payroll | +-------------+----------+----------+----------+----------+ | 333-33-3333 | Pete | Pan | 5 | Research | +-------------+----------+----------+----------+----------+ Then: If I see SSN=111-11-1111 in a tuple, then I must see DName=Research in the tuple: SSN fname lname DNO DName +-------------+----------+----------+----------+----------+ | 111-11-1111 | John | Smith | 5 | Research | +-------------+----------+----------+----------+----------+ | 222-22-2222 | Jane | Doe | 4 | Payroll | +-------------+----------+----------+----------+----------+ | 333-33-3333 | Pete | Pan | 5 | Research | +-------------+----------+----------+----------+----------+