Information Processing Letters
Foundations of logic programming; (2nd extended ed.)
Foundations of logic programming; (2nd extended ed.)
Existence and nonexistence of complete refinement operators
ECML-94 Proceedings of the European conference on machine learning on Machine Learning
Symbolic Logic and Mechanical Theorem Proving
Symbolic Logic and Mechanical Theorem Proving
Knowledge Acquisition and Machine Learning
Knowledge Acquisition and Machine Learning
Inductive Logic Programming: Techniques and Applications
Inductive Logic Programming: Techniques and Applications
ECML '93 Proceedings of the European Conference on Machine Learning
A completeness theorem and a computer program for finding theorems derivable from given axioms
A completeness theorem and a computer program for finding theorems derivable from given axioms
Undecidability of the Horn-clause implication problem
SFCS '92 Proceedings of the 33rd Annual Symposium on Foundations of Computer Science
Generalization of clauses under implication
Journal of Artificial Intelligence Research
Least Generalization under Relative Implication
AIMSA '00 Proceedings of the 9th International Conference on Artificial Intelligence: Methodology, Systems, and Applications
The Journal of Machine Learning Research
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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 nontautologous 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.