Attribute implications

One of the most powerful functions that can be provided by means of formal concept analysis are the attribute implications. Peter Burmeister shows amazing results in [Burm2000] based on the classification of natural numbers. Given the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 27, 64} of only 11 natural numbers and the feature set {even, odd, prime, square, cubic, non-prime , non-square, non-cubic} the following assumptions can be derived with the help of the ConImp program:

1. not-prime, not-square, not-cubic ⇒ even
   This cannot be accepted, because 15 is odd but not-prime, not-square and not-cubic
2. not-prime, not-square, not-cubic ⇒ even : acceptable
3. cubic, not-prime, not-cubic  ⇒ even, odd, prime, square, not-square
   All features appear to alert the modeler that the premise may be inconsistent. However, since the essential features of the premise negate each other, this implication is also to be accepted.
4. not-square ⇒ not-prime : acceptable
5. square, not-prime, not-square  ⇒ even, odd, prime, cubic, not-cubic :
   acceptable, because essential features of the premise negate each other
6. prime ⇒ not-square, not-cubic : acceptable
7. prime, not-prime, not-square, not-cubic ⇒ even, odd, square, cubic :
   acceptable, because essential features of the premise negate each other
8. even, odd ⇒ prime, square, cubic, not-prime, not-square, not-cubic :
   has been automatically accepted

In the continuation of the procedure it is shown that every even number could result from the sum of two prime numbers. This corresponds to the Goldbach conjecture in number theory. In summary, it can be stated that important laws and relationships can be derived from the characteristics of objects. In section BoW Attribute Implications we show a comprehensive example of attribute implications in the context of the water body ontology.