Complexity of Clique-Guarded First-Order Logic with Counting

2026-06-23

AI Summary

This work investigates the computational complexity of counting-enriched clique-guarded first-order logic (cgFOC) over sparse graph classes. By integrating logical fragment modeling, VC-dimension and graph dimension analysis, and structural theory of nowhere-dense and locally bounded expansion graphs, it establishes the first unified algorithmic meta-theorem for cgFOC on classes of graphs with locally bounded expansion, enabling efficient query answering, enumeration, and PAC learning. The study further demonstrates that even a slight extension of cgFOC becomes undecidable already on trees, thereby precisely delineating the tractability frontier of this logic.

Abstract

We introduce clique-guarded first-order logic with counting (cgFOC), a fragment of the first-order logic with counting FOC [Kuske and Schweikardt, LICS 2017], and we study the complexity of this fragment. In particular, we prove computable upper bounds on the Vapnik-Chervonenkis (VC) dimension of cgFOC formulas and on the graph dimension of cgFOC counting terms on nowhere dense classes of relational structures. Furthermore, we show algorithmic metatheorems for cgFOC for query answering, enumeration, and probably approximately correct (PAC) learning for Boolean and multiclass classification problems on classes of locally bounded expansion. On the other hand, we show that a slight extension of cgFOC is already intractable on trees.

Problem

• clique-guarded logic
• first-order logic with counting
• VC dimension
• nowhere dense classes
• computational complexity

Innovation

• clique-guarded logic
• first-order logic with counting
• VC dimension
• algorithmic metatheorems
• nowhere dense classes