Existential Second-order Logic over Graphs: A Complete Complexity-theoretic Classification

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 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, Kolaitis, and Schwentick: for each possible quantifier prefix of an ESO formula, the resulting prefix class over graphs 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 et al. 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, which we show to express special constraint satisfaction problems that lie in L.