Within the model-theoretic framework for supervised learning introduced by Grohe and Tur\'an (TOCS 2004), we study the parameterized complexity of learning concepts definable in monadic second-order logic (MSO). We show that the problem of learning an MSO-definable concept from a training sequence of labeled examples is fixed-parameter tractable on graphs of bounded clique-width, and that it is hard for the parameterized complexity class para-NP on general graphs. It turns out that an important distinction to be made is between 1-dimensional and higher-dimensional concepts, where the instances of a k-dimensional concept are k-tuples of vertices of a graph. The tractability results we obtain for the 1-dimensional case are stronger and more general, and they are much easier to prove. In particular, our learning algorithm in the higher-dimensional case is only fixed-parameter tractable in the size of the graph, but not in the size of the training sequence, and we give a hardness result showing that this is optimal. By comparison, in the 1-dimensional case, we obtain an algorithm that is fixed-parameter tractable in both.

翻译：在Grohe和Turán (TOCS 2004) 引入的监督学习模型论框架下，我们研究了一元二阶逻辑（MSO）可定义概念学习的参量复杂度。我们证明：在有界团宽度的图上，从带标签样本训练序列中学习MSO可定义概念是固定参数易解的；而在一般图上，该问题对于参量复杂度类para-NP是困难的。研究发现一个重要区分在于一维概念与高维概念：k维概念的实例是图的顶点k元组。我们为一维情形获得的易解性结果更强大、更具普适性，且证明难度显著降低。特别地，高维情形下的学习算法仅在图的规模上具有固定参数易解性，但在训练序列规模上不具备，我们通过困难性结果证明这是最优的。相比之下，在一维情形中，我们获得的算法在两者上均具有固定参数易解性。