Learning Goal-Decomposition Rules Using Exercises

Chandra Reddy, Prasad Tadepalli

Abstract: Teaching problem-solving through exercises is a widely used pedagogic technique. A human teacher selects certain problems and orders them according to their level of difficulty to form a sequence of exercises. A student starts by solving simple problems first; then, attempts harder problems applying the knowledge gained from solving the earlier problems; and then still harder problems, and so on. Machine learning of problem solving using exercises, apart from following this pedagogic tradition, offers a compromise between supervised speedup learning and unsupervised speedup learning. Supervised speedup learning, although more efficient than the latter, places the burden of providing the solutions to the training problems on the teacher-mostly a human. Unsupervised speedup learning, in contrast, expects the learner to solve the training problems, while unburdening the teacher. However, this is computationally hard for the learner for it lacks control knowledge and, hence, its only recourse is brute-force search. In exercises approach, teacher has the task of providing an exercise set-a sequence of problems ordered by difficulty. The learner has to solve the exercise problems using the bootstrapping method akin to the above-described method followed by a human student. Previously, Natarajan (1989) used exercises approach in speedup learning for learning control rules represented in the form of tree patterns. In this work, we use exercises approach for learning first-order recursive goal-decomposition rules (d-rules). A d-rule is a 3-tuple (g, c, sg) that decomposes a goal g into a sequence of subgoals sg, provided the condition c holds. The input to the system is a sequence of exercises ordered according to their difficulty levels. The difficulty levels correspond to goal-subgoal hierarchies except in the case of recursion-where the number of recursive calls is also used to set the difficulty level of a goal. This approach follows two main steps: For each exercise, (1) exercise-solver solves the exercise by searching in the space of operators and previously learned drules, and outputs the solution (the plan) and the subgoals used; and (2) first-order inductive learner forms a hypothesis d-rule using the initial state as the d-rule