R. de Wolf
http://repub.eur.nl/ppl/3317/
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RePub, Erasmus University RepositoryThe specialization problem and the completeness of unfolding
http://repub.eur.nl/pub/1434/
Mon, 01 Jan 1996 00:00:01 GMT<div>S-H. Nienhuys-Cheng</div><div>R. de Wolf</div>
We discuss the problem of specializing a definite program with respect to sets of positive and negative examples, following Bostrom and Idestam-Almquist. This problem is very relevant in the field of inductive learning. First we show that there exist sets of examples that have no correct program, i.e., no program which implies all positive and no negative examples. Hence it only makes sense to talk about specialization problems for which a solution (a correct program) exists.
To solve such problems, we first introduce UD1-specialization, based upon the transformation rule unfolding. We show UD1-specialization is incomplete - some solvable specialization problems do not have a UD1-specialization as solution - and generalize it to the stronger UD2-specialization. UD2 also turns out to be incomplete. An analysis of program specialization, using the subsumption theorem for SLD-resolution, shows the reason for this incompleteness. Based on that analysis, we then define UDS-specialization (a generalization of UD2-specialization), and prove that any specialization problem has a UDS-specialization as a solution. We also discuss the relationship between this specialization technique, and the generalization technique based on inverse resolution. Finally, we go into several more implementational matters, which outline an interesting topic for future research.Least generalizations and greatest specializations of sets of clauses
http://repub.eur.nl/pub/513/
Mon, 01 Jan 1996 00:00:01 GMT<div>S-H. Nienhuys-Cheng</div><div>R. de Wolf</div>
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.The subsumption theorem for several forms of resolution
http://repub.eur.nl/pub/514/
Mon, 01 Jan 1996 00:00:01 GMT<div>S-H. Nienhuys-Cheng</div><div>R. de Wolf</div>
The Subsumption Theorem is the following completeness result for resolution: if S is a set of clauses and C is a clause, then S logically implies C iff C is a tautology, or there exists a clause D which subsumes C, and which can be derived from S by some form of resolution. Different versions of this theorem exist, depending on the kind of resolution we use. It provides a more `direct' form of completeness than the better known refutation-completeness, which often makes the Subsumption Theorem better suited for theoretical research. In this paper we investigate for which forms of resolution the theorem holds, and for which it does not. We collect results earlier obtained by others, and contribute some results of our own. The main results of the paper are as follows. For `unconstrained' resolution, the Subsumption Theorem holds, and is equivalent to the refutation-completeness: the one can be proved from the other. The same is true for linear resolution. For input resolution, the theorem is false, even in the special case where S contains only one clause. In case of SLD-resolution for Horn clauses, the Subsumption Theorem again holds, and is equivalent to the refutation-completeness of SLD-resolution.