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 a consistent MSO-formula is fixed-parameter tractable on structures of bounded tree-width and on graphs of bounded clique-width in the 1-dimensional case, that is, if the instances are single vertices (and not tuples of vertices). This generalizes previous results on strings and on trees. Moreover, in the agnostic PAC-learning setting, we show that the result also holds in higher dimensions. Finally, via a reduction to the MSO-model-checking problem, we show that learning a consistent MSO-formula is para-NP-hard on general structures.

翻译：在 Grohe 和 Turán (TOCS 2004) 引入的监督学习模型论框架内，我们研究了学习可在一元二阶逻辑（MSO）中定义的概念的参数化复杂性。我们证明，在结构具有有界树宽以及图具有有界团宽的一维情形下（即实例为单个顶点而非顶点元组），学习一致 MSO 公式的问题是固定参数易解的。这推广了先前关于字符串和树的结果。此外，在不可知 PAC 学习设置中，我们证明了该结论在高维情形下仍然成立。最后，通过归约到 MSO 模型检验问题，我们证明在一般结构上，学习一致 MSO 公式是 para-NP 难的。