Counting of Teams in First-Order Team Logics

Anselm Haak, Juha Kontinen, Fabian Müller, Heribert Vollmer, Fan Yang, Michael Wagner
2019 International Symposium on Mathematical Foundations of Computer Science  
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
more » ... d-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 ⊆ #P. Our main technical result shows that the problem of counting satisfying assignments for monotone Boolean Σ1-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.
doi:10.4230/lipics.mfcs.2019.19 dblp:conf/mfcs/HaakKMVY19 fatcat:msle7mcok5cyndnisqgsnzi4ii