The Parameterized Complexity of Learning Monadic Second-Order Logic

TL;DR

This paper investigates the computational complexity of learning concepts defined by monadic second-order logic on graphs, showing fixed-parameter tractability on bounded clique-width graphs and hardness results on general graphs.

Contribution

It establishes fixed-parameter tractability results for learning MSO-definable concepts on specific graph classes and distinguishes between 1-dimensional and higher-dimensional cases.

Findings

FPT algorithm for MSO learning on graphs of bounded clique-width

Hardness results for MSO learning on general graphs

Different complexity results for 1D and higher-dimensional concepts

Abstract

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. For the higher-dimensional case, we give a learning algorithm that is 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.