# Counting of Teams in First-Order Team Logics

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

Research output: Chapter in Book/Report/Conference proceedingConference contributionScientificpeer-review

### Abstract

We study descriptive complexity of counting complexity classes in the range from $\#$P to $\#\cdot$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 FO in Tarski's semantics. Our results show that the class $\#\cdot$NP can be logically characterized by independence logic and existential second-order logic, whereas dependence logic and inclusion logic give rise to subclasses of $\#\cdot$NP and $\#$P , respectively. Our main technical result shows that the problem of counting satisfying assignments for monotone Boolean $\Sigma_1$-formulae is $\#\cdot$NP-complete as well as complete for the function class generated by dependence logic.
Original language English 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019) Peter Rossmanith, Pinar Heggernes, Joost-Pieter Katoen 15 138 Aug 2019 19:1-19:15 978-3-95977-117-7 https://doi.org/10.4230/LIPIcs.MFCS.2019.19 Published - Aug 2019 A4 Article in conference proceedings The 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019) - Aachen, GermanyDuration: 26 Aug 2019 → 31 Aug 2019https://tcs.rwth-aachen.de/mfcs2019/

### Publication series

Name Leibniz International Proceedings in Informatics (LIPIcs) Dagstuhl Publishing 138 1868-8969

### Fields of Science

• cs.LO
• cs.CC
• 111 Mathematics

## Cite this

Haak, A., Kontinen, J., Müller, F., Vollmer, H., & Yang, F. (2019). Counting of Teams in First-Order Team Logics. In P. Rossmanith, P. Heggernes, & J-P. Katoen (Eds.), 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019) (Vol. 138, pp. 19:1-19:15). (Leibniz International Proceedings in Informatics (LIPIcs); Vol. 138). https://doi.org/10.4230/LIPIcs.MFCS.2019.19