Courcelle's Theorem Without Logic
Yuval Filmus, Johann A. Makowsky

TL;DR
This paper generalizes Courcelle's Theorem by replacing the logical definability condition with a combinatorial hypothesis based on connection matrices, broadening its applicability to graph properties.
Contribution
It introduces a combinatorial framework that extends Courcelle's Theorem without relying on Monadic Second Order Logic, inspired by Lovász's work with connection matrices.
Findings
Provides a generalized version of Courcelle's Theorem
Replaces logical definability with combinatorial conditions
Broadens the scope of efficiently checkable graph properties
Abstract
Courcelle's Theorem states that on graphs of tree-width at most with a given tree-decomposition of size , graph properties definable in Monadic Second Order Logic can be checked in linear time in the size of . Inspired by L. Lov\'asz' work using connection matrices instead of logic, we give a generalized version of Courcelle's theorem which replaces the definability hypothesis by a purely combinatorial hypothesis using a generalization of connection matrices.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsHistory and Theory of Mathematics · Mathematics and Applications
