Existential Second-Order Logic Over Graphs: A Complete Complexity-Theoretic Classification

TL;DR

This paper classifies the computational complexity of various fragments of existential second-order logic over graphs, revealing that each prefix class's complexity falls into FO, L, NL, or NP, thus completing the complexity landscape.

Contribution

It provides a complete classification of the complexity of prefix classes of ESO logic over graphs, resolving previous open questions about their exact computational complexity.

Findings

Each prefix class's reduction closure is either FO, L, NL, or NP.

Containment in L is proven for certain prefixes involving existential and universal quantifiers.

Some classes express problems in L, such as specific constraint satisfaction problems.

Abstract

Descriptive complexity theory aims at inferring a problem's computational complexity from the syntactic complexity of its description. A cornerstone of this theory is Fagin's Theorem, by which a graph property is expressible in existential second-order logic (ESO logic) if, and only if, it is in NP. A natural question, from the theory's point of view, is which syntactic fragments of ESO logic also still characterize NP. Research on this question has culminated in a dichotomy result by Gottlob, Kolatis, and Schwentick: for each possible quantifier prefix of an ESO formula, the resulting prefix class either contains an NP-complete problem or is contained in P. However, the exact complexity of the prefix classes inside P remained elusive. In the present paper, we clear up the picture by showing that for each prefix class of ESO logic, its reduction closure under first-order reductions is either FO, L, NL, or NP. For undirected, self-loop-free graphs two containment results are especially challenging to prove: containment in L for the prefix $\exists R_1 \cdots \exists R_n \forall x \exists y$ and containment in FO for the prefix $\exists M \forall x \exists y$ for monadic $M$. The complex argument by Gottlob, Kolatis, and Schwentick concerning polynomial time needs to be carefully reexamined and either combined with the logspace version of Courcelle's Theorem or directly improved to first-order computations. A different challenge is posed by formulas with the prefix $\exists M \forall x\forall y$: We show that they express special constraint satisfaction problems that lie in L.