Constructing predicate mappings for goal-dependent abstraction

Abstract

In theorem proving with abstraction, it is required for system designers to provide a useful abstraction. However, such a task is so difficult that it would be worth studying an automatic construction of abstraction. In this paper, we propose a new framework of Goal-Dependent Abstraction in which an appropriate abstraction is selected according to each goal to be proved. Towards Goal-Dependent Abstraction, we present an algorithm for constructing an appropriate abstraction for a given goal. The appropriateness is defined in terms of Upward-Property and Downward-Property. Since our abstraction is based on predicate mapping, the algorithm in fact computes predicate mappings based on which appropriate abstractions can be constructed. Given a goal, candidate predicate mappings are generated and then tested for their appropriateness for the goal. In order to find appropriate mappings efficiently, we present a property to prune useless candidate generations. The numbers of pruned candidates are evaluated in the best and worst cases. Furthermore some experimental results show that many useless candidates can be pruned with the property and the obtained appropriate predicate mappings (abstractions) fit our intuition. From the experimental results, we could expect our study in this paper to contribute to the fields of analogical reasoning and case-based reasoning as well as theorem-proving.