The Parameterized Complexity of Learning Monadic Second-Order Logic

TL;DR

This work analyzes the parameterized complexity of learning concepts definable in monadic second-order logic ($MSO$) within the Grohe–Turán learning framework, distinguishing 1D ($k=1$) from higher-dimensional ($k>1$) concepts. On graphs of bounded clique-width, it proves that 1D $MSO$-Consistent-Learn is fixed-parameter tractable, while the problem is para-$NP$-hard on general graphs; for higher dimensions, it offers two tractability routes: a quadratic-time algorithm in the graph size $|V(G)|$ for consistent learning with a matching hardness bound, and PAC-learning results that are fixed-parameter linear on bounded clique-width classes, alongside a normal-form bound $O((m+1)^{g(c, | abla|, q, k, ar\, ext{l})}|V(G)|^2)$. The methodology relies on Courcelle-style model-checking by encoding training examples as new labels, and the authors show that MSO model-checking reduces to the 1D case, establishing para-$NP$ hardness there. Collectively, the results delineate the tractability frontier for MSO-based concept learning under bounded clique-width and clarify the effects of dimensionality and training size on algorithmic feasibility.

Abstract

Within the model-theoretic framework for supervised learning introduced by Grohe and Turán (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.

Figures (1)

• Figure 1: Graph $G$ for \ref{['example_bipartite']}. Positive examples are shown in purple, and negative examples are shown in orange.

Theorems & Definitions (5)

• Example 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5