Van Lehn (1990) tells us that an effective way of teaching more complicated concepts is to build them up from simple subconcepts, as opposed to an ``all-at-once" approach. For our purposes, a disjunct can be considered one of these simple subconcepts. A lesson consists of an uninterrupted sequence of demonstrations, examples, and exercises. The length of a lesson varies. Thus, we might expect a human to better understand the concept of an armchair by presenting a series of lessons, each of which introduces a single new subconcept that builds upon the previous subconcepts. For example, a first lesson teaches the concept of a conventional chair which requires only a stable sittable surface in the correct orientation. To learn what constitutes a straightback chair, we build upon the concept of conventional chair by introducing the subconcept of back support in a second lesson. So, the second lesson broadens our notion of chairs, in general. Finally, a third lesson builds upon our understanding of a straightback chair by introducing the subconcept of arm support. By contrast, the all-at-once approach may try to explain that an armchair provides a stable sittable surface in the correct orientation with some back and arm support. Here, we are trying to teach three subconcepts at one time, and show how the three subconcepts together form the more complex concept of an armchair. Indeed, Van Lehn (1990) cites some laboratory studies which indicate that the learning task is more difficult when more than one disjunct (subconcept) is taught per lesson.
We have chosen to utilize a machine learning algorithm which has underpinnings similar to Van Lehn's one-disjunct-per-lesson assumption. In our case, concepts and subconcepts are represented by categories and subcategories. A lesson for our algorithm consists of numerous epochs of the training examples from one (sub)category. Thus, our lesson can be viewed as an uninterrupted sequence of positive examples that ``teach" the functional requirements for a single (sub)category. The length, or number of training epochs, of our lessons may vary depending on the subcategory being learned. To learn all the ranges in a category definition tree, we begin by learning the simplest concepts first. Then we learn additional more complex subconcepts by building upon the notion of the more simple concept. For example in the simplified proof tree in Figure 8, the parent category conventional chair will be learned before attempting to learn the subcategory (specialization) straightback chair. Since the subcategory straightback chair is itself a parent category, it will be learned before attempting to learn the even more complex subcategory armchair. The remainder of this subsection discusses our implementation in finer detail.
Figure 8: Simplified proof tree for an armchair object.
From an implementation standpoint, the simplest concepts are the functional properties associated with the categories that are directly linked to the root node in our category definition tree such as provides_sittable_surface and provides_stable_support for the category conventional chair. In our first lesson, we use positive examples from these ``first level" (or parent) categories to learn only those membership functions associated with these categories. Once the first level categories have been learned, their membership functions are ``frozen" and not permitted to change during subsequent lessons.
In our second lesson, only the membership functions of the ``second level" categories (i.e., the subcategories of the first level categories in the definition tree) are learned. In Figure 8, these membership functions belong to the node provides_back_support for the subcategory straightback chair. If we have learned the ``simple" functional concept associated with the parent category, the values computed for a parent category node are assumed to be reasonably accurate. For example, when the actual values in a proof tree are computed for a straightback chair training example, the actual values emanating from the parent category node conventional chair should be accurate since the concepts associated with this node have already been learned. That is, the evaluation measures for the functional properties provides_sittable_surface and provides_stable_support of a straightback chair example are assumed to be correct. This implies that the membership functions making up the functional requirement subtree (i.e., provides_back_support) are responsible for the entire error for a subcategory training example. (This explains why Equations 3 and 4 are used to propagate error through POR nodes.) Hence, the error is propagated to the modifiable leaves under a functional requirement node through a PAND subtree and learning continues as before.
The lessons continue with each parent category being learned before any of its subcategories are learned, until all subcategories have been learned. By freezing the parent category membership functions after they have been learned, we are applying to the one-subconcept-per-lesson strategy. So in Figure 8 after learning straightback chair, the membership functions for that branch are frozen and the armchair subcategory is learned by modifying the membership functions under the provides_arm_support branch of the proof tree.
OMLET begins learning by evaluating the rule base in order to determine subcategory dependencies and assigns each (sub)category in the definition tree a level in the learning hierarchy. For example, OMLET determines that the category conventional chair has no parent category and its membership functions can be learned immediately (level 1). However, the evaluation measure of the subcategory straightback chair is dependent on the parent category conventional chair. The straightback chair subcategory is assigned to learning level 2. Subcategory armchair is dependent on parent category straightback chair, and is therefore assigned to learning level 3.