Paper: On the Computational Power of Extensional ESO (at LICS 2026)

Authors: Manuel Bodirsky Santiago Guzmán-Pro

Open access: https://doi.org/10.4230/LIPIcs.LICS.2026.20

Abstract

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 A the task is to decide whether there is an extension B of A that satisfies the first-order part of Ψ, i.e., a structure B such that R^A ⊆ R^B for every existentially quantified predicate R of Ψ, and R^A = R^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).

BibTeX

  @InProceedings{BodirskyGuzmanPro-OntheComputationalP,
    author = 	 {Manuel Bodirsky and Santiago Guzmán-Pro},
    title = 	 {On the Computational Power of Extensional ESO},
    booktitle =  {Proceedings of the Forty-First Annual Symposium on Logic in Computer Science (LICS 2026)},
    year =	 {2026},
    month =	 {July}, 
    pages =      {20:1--20:26},
    location =   {Lisbon, Portugal}, 
    publisher =	 {Schloss Dagstuhl -- Leibniz-Zentrum für Informatik},
    doi =        {10.4230/LIPIcs.LICS.2026.20}
  }