Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

6.8 Negation as a Failure in Extended
Logic Programs
It is evident from our background in Predicate Logic that negated clauses have
a significant role in representing knowledge. Unfortunately, however, the
HORN-clause based programs do not allow negated clauses in their body. To
facilitate the users with more freedom of knowledge representation, recently,
Logic programs have been extended to include negated atomic clauses in the
body of a non-Horn clause, presented below:
p ← q, r, ¬s, ¬t
where p, q, r, s and t are atomic propositions or predicates.
The principle of negation as failure [9] states: For a given formula P, if one
cannot prove P, then it is reasonable to deduce that ¬ P is true.
For illustrating the principle consider the following extended logic program:
1. Subset (A, B) ← ¬ Non-subset (A, B).
2. Non-subset (A, B) ← Member (X | A), ¬ member (X | B).
Goal: Subset ( (2, 4, nil), (1, 2, 3, 4, nil)) →
To prove the above goal, we resolve the goal clause with clause (1) and get
← ¬ Non-subset ((2,4, nil), (1, 2, 3, 4, nil)).
Now, by negation as a failure, we want to satisfy the sub-goal
← Non-subset ((2,4, nil), (1,2,3,4, nil)).
Now, by (2) we find that
Member (X | 2 ,4,nil) , ¬ Member (X | 1,2,3,4, nil)
fails, which consequently proves that non-subset ((2,4, nil), (1,2,3,4,nil)) fails
and thus Subset ((2, 4, nil), (1, 2, 3, 4, nil)) is a valid inference.