arXiv2025
Extensional ESO is a fragment of existential second-order logic (ESO) that captures the following family of problems. Given a fixed ESO sentence $Ψ$ and an input structure $\mathbb A$ the task if to decide whether there is an extension $\mathbb B$ of $\mathbb A$ that satisfies the first-order part of $Ψ$, i.e., a structure $\mathbb B$ such that $R^{\mathbb A}\subseteq R^{\mathbb B}$ for every existentially quantified predicate $R$ of $Ψ$, and $R^{\mathbb A} = R^{\mathbb B}$ for every non-quantified predicate $R$ of $Ψ$. In particular, extensional ESO describes all pre-coloured finite-domain constraint satisfaction problems (CSPs). In this paper we study the computational power of extensional ESO; we ask, for which problems in NP is there a polynomial-time equivalent problem in extensional ESO?. One of our main results states that extensional ESO has the same computational power as hereditary first-order logic. We also characterize the computational power of the fragment of extensional ESO with monotone universal first-order part in terms of finitely bounded CSPs. These results suggest a rich computational power of this logic, and we conjecture that extensional ESO captures NP-intermediate problems. We further support this conjecture by showing that extensional ESO can express current candidate NP-intermediate problems such as Graph Isomorphism, and Monotone Dualization (up to polynomial-time equivalence). On the other hand, another main result proves that extensional ESO does not have the full computational power of NP: there are problems in NP that are not polynomial-time equivalent to a problem in extensional ESP (unless E=NE).