2 research outputs found
Least Generalizations and Greatest Specializations of Sets of Clauses
The main operations in Inductive Logic Programming (ILP) are generalization
and specialization, which only make sense in a generality order. In ILP, the
three most important generality orders are subsumption, implication and
implication relative to background knowledge. The two languages used most often
are languages of clauses and languages of only Horn clauses. This gives a total
of six different ordered languages. In this paper, we give a systematic
treatment of the existence or non-existence of least generalizations and
greatest specializations of finite sets of clauses in each of these six ordered
sets. We survey results already obtained by others and also contribute some
answers of our own. Our main new results are, firstly, the existence of a
computable least generalization under implication of every finite set of
clauses containing at least one non-tautologous function-free clause (among
other, not necessarily function-free clauses). Secondly, we show that such a
least generalization need not exist under relative implication, not even if
both the set that is to be generalized and the background knowledge are
function-free. Thirdly, we give a complete discussion of existence and
non-existence of greatest specializations in each of the six ordered languages.Comment: See http://www.jair.org/ for any accompanying file
Least Generalizations under Implication
. One of the most prominent approaches in Inductive Logic Programming is the use of least generalizations under subsumption of given clauses. However, subsumption is weaker than logical implication, and not very well suited for handling recursive clauses. Therefore an important open question in this area concerns the existence of least generalizations under implication (LGIs). Our main new result in this paper is the existence and computability of such an LGI for any finite set of clauses which contains at least one non-tautologous function-free clause. We can also define implication relative to background knowledge. In this case, least generalizations only exist in a very limited case. 1 Introduction Inductive Logic Programming (ILP) is the intersection of Logic Programming and Machine Learning. It studies methods to induce clausal theories from given sets of positive and negative examples. An inductively inferred theory should imply all of the positive, and none of the negative exam..