Program Extraction from Quantied Decision Trees (extended Abstract)

There are two approaches in the top-down construction of rst order decision trees that use de nite or normal clauses. The covering approach [5, 3] builds one clause at a time that covers part of positive examples, and repeats this process until no positive examples remain uncovered. The divide-and-conquer approach [1, 2] on the other hand divides the given examples into smaller subsets recursively to get ones explaining positive examples. While the divide-and-conquer approach is conceptually simple, and dividing examples by a literal is a direct extension of the propositional case, there does not seem to be much work on such approach in ILP. Quinlan [5] employed division by a literal but within the covering approach. Bostrom and Idestam-Almquist [2] adopted division by unfolding, not by a literal. Blockeel and De Raedt [1] used division by a literal but without tuple extension and their synthesized clauses have a ground head to represent a class name. In this paper, we propose a new divide-and-conquer approach to ILP to synthesize logic programs. It is based on the construction of quanti ed decision trees explained in the next section. Our approach is thought to be a logical re nement of FOIL [5] in which existentially quanti ed negative literals are used as well as existentially quanti ed positive literals. The synthesized clauses may include universally quanti ed formulas in their body but can further be compiled into de nite clauses with disequality constraints by First Order Compiler [6].