In traditional machine teaching, a teacher wants to teach a concept to a learner, by means of a finite set of examples, the witness set. But concepts can have many equivalent representations. This redundancy strongly affects the search space, to the extent that teacher and learner may not be able to easily determine the equivalence class of each representation. In this common situation, instead of teaching concepts, we explore the idea of teaching representations. We work with several teaching schemas that exploit representation and witness size (Eager, Greedy and Optimal) and analyze the gains in teaching effectiveness for some representational languages (DNF expressions and Turing-complete P3 programs). Our theoretical and experimental results indicate that there are various types of redundancy, handled better by the Greedy schema introduced here than by the Eager schema, although both can be arbitrarily far away from the Optimal. For P3 programs we found that witness sets are usually smaller than the programs they identify, which is an illuminating justification of why machine teaching from examples makes sense at all.

翻译：在传统的机器教学中，教师希望通过有限样本集（即见证集）向学习者教授一个概念。然而，概念往往具有多种等价的表示形式。这种冗余会严重影响搜索空间，以至于教师和学习者可能难以轻易确定每种表示的等价类。在这种常见情形下，我们探索教授表示而非概念的方法。我们研究了几种利用表示和见证规模的教学生成模式（急切式、贪婪式和最优式），并分析了这些模式在某些表示语言（DNF表达式和图灵完备的P3程序）中提升教学效果的情况。我们的理论和实验结果表明，存在多种类型的冗余，本文提出的贪婪式模式比急切式模式能更好地处理这些冗余，尽管两者都可能与最优式模式相差甚远。对于P3程序，我们发现见证集通常小于它们所标识的程序，这为理解为何基于示例的机器教学具有意义提供了有力的解释。