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Counting of Teams in First-Order Team Logics

Authors Anselm Haak , Juha Kontinen , Fabian Müller , Heribert Vollmer , Fan Yang

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Author Details

Anselm Haak
  • Theoretische Informatik, Leibniz Universität Hannover, Appelstraße, D-30167, Germany
Juha Kontinen
  • Department of Mathematics and Statistics, University of Helsinki, Pietari Kalmin katu 5, 00014, Finland
Fabian Müller
  • Theoretische Informatik, Leibniz Universität Hannover, Appelstraße, D-30167, Germany
Heribert Vollmer
  • Theoretische Informatik, Leibniz Universität Hannover, Appelstraße, D-30167, Germany
Fan Yang
  • Department of Mathematics and Statistics, University of Helsinki, Pietari Kalmin katu 5, 00014, Finland

Funding

Anselm Haak, Fabian Müller, Heribert Vollmer: Supported by DFG VO 630/8-1 and DAAD grant 57348395. Juha Kontinen, Fan Yang: Supported by grants 308712 and 308099 of the Academy of Finland.

Acknowledgements

We thank the anonymous referees for helpful comments.

Cite As Get BibTex

Anselm Haak, Juha Kontinen, Fabian Müller, Heribert Vollmer, and Fan Yang. Counting of Teams in First-Order Team Logics. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.MFCS.2019.19

Abstract

We study descriptive complexity of counting complexity classes in the range from #P to #*NP. A corollary of Fagin’s characterization of NP by existential second-order logic is that #P can be logically described as the class of functions counting satisfying assignments to free relation variables in first-order formulae. In this paper we extend this study to classes beyond #P and extensions of first-order logic with team semantics. These team-based logics are closely related to existential second-order logic and its fragments, hence our results also shed light on the complexity of counting for extensions of first-order logic in Tarski’s semantics. Our results show that the class #*NP can be logically characterized by independence logic and existential second-order logic, whereas dependence logic and inclusion logic give rise to subclasses of #*NP and #P, respectively. We also study the function class generated by inclusion logic and relate it to the complexity class TotP, which is a subclass of #P. Our main technical result shows that the problem of counting satisfying assignments for monotone Boolean Sigma_1-formulae is #*NP-complete with respect to Turing reductions as well as complete for the function class generated by dependence logic with respect to first-order reductions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Complexity classes
  • Theory of computation → Complexity theory and logic
Keywords
  • team-based logics
  • counting classes
  • finite model theory
  • descriptive complexity

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