The representation formalism as well as the representation language is of great importance for the success of machine learning. The representation formalism should be expressive, efficient, useful, and applicable. First-order logic needs to be restricted in order to be efficient for inductive and deductive reasoning. In the field of knowledge representation term subsumption formalisms have been developed which are efficient and expressive. In this paper, a learning algorithm, KLUSTER, is described which represents concept definitions in this formalism. KLUSTER enhances the representation language if this is necessary for the discrimination of concepts. Hence, KLUSTER is a constructive induction program. KLUSTER builds the most specific generalization and a most general discrimination in polynomial time. It embeds these concept learning problems into the overall task of learning a hierarchy of concepts.

The field of Inductive Logic Programming (ILP), which is concerned with the induction of Horn clauses from examples and background knowledge, has received increased attention over the last time. Recently, some positive results concerning the learnability of restricted logic programs have been published. In this paper we review these restrictions and prove some lower-bounds of the computational complexity of learning. In particular, we show that a learning algorithm for i2-determinate Horn clauses (with variable i) could be used to decide the PSPACE-complete problem of Finite State Automata Intersection, and that a learning algorithm for 12-nondeterminate Horn clauses could be used to decide the NP-complete problem of Boolean Clause Satisfiability (SAT). This also shows, that these Horn clauses are not PAC-learnable, unless RP = NP = PSPACE. Keywords: Inductive Logic Programming, PAC-Learning. 1 Introduction Most success within the field of Machine Learning has been achiev...

In the area of inductive learning, generalization is a main operation, and the usual definition of induction is based on logical implication. Recently there has been a rising interest in clausal representation of knowledge in machine learning. Almost all inductive learning systems that perform generalization of clauses use the relation `-subsumption instead of implication. The main reason is that there is a well-known and simple technique to compute least general generalizations under `-subsumption, but not under implication. However generalization under `-subsumption is inappropriate for learning recursive clauses, which is a crucial problem since recursion is the basic program structure of logic programs. We note that implication between clauses is undecidable, and we therefore introduce a stronger form of implication, called T-implication, which is decidable between clauses. We show that for every finite set of clauses there exists a least general generalization under T-implic...

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