The Tractability Frontier of Graph-Like First-Order Query Sets

TL;DR

This paper classifies the computational complexity of model checking for graph-like first-order sentences, providing a complete tractability classification without restrictions on connectives or quantifiers.

Contribution

It introduces the first comprehensive complexity classification for graph-like first-order sentences, expanding understanding of tractability in model checking.

Findings

Complete tractability classification for graph-like first-order sentences.

Development of a new complexity-theoretic framework with novel reductions.

First classification result for a broad class of sentences without connective or quantifier restrictions.

Abstract

We study first-order model checking, by which we refer to the problem of deciding whether or not a given first-order sentence is satisfied by a given finite structure. In particular, we aim to understand on which sets of sentences this problem is tractable, in the sense of parameterized complexity theory. To this end, we define the notion of a graph-like sentence set, which definition is inspired by previous work on first-order model checking wherein the permitted connectives and quantifiers were restricted. Our main theorem is the complete tractability classification of such graph-like sentence sets, which is (to our knowledge) the first complexity classification theorem concerning a class of sentences that has no restriction on the connectives and quantifiers. To present and prove our classification, we introduce and develop a novel complexity-theoretic framework which is built on parameterized complexity and includes new notions of reduction.