Canonical Models and the Complexity of Modal Team Logic

Abstract

We study modal team logic MTL, the team-semantical extension of modal logic ML closed under Boolean negation. Its fragments, such as modal dependence, independence, and inclusion logic, are well-understood. However, due to the unrestricted Boolean negation, the satisfiability problem of full MTL has been notoriously resistant to a complexity theoretical classification.In our approach, we introduce the notion of canonical models into the team-semantical setting. By construction of such a model, we reduce the satisfiability problem of MTL to simple model checking. Afterwards, we show that this approach is optimal in the sense that MTL-formulas can efficiently enforce canonicity.Furthermore, to capture these results in terms of complexity, we introduce a nonelementary complexity class, TOWER(poly), and prove that it contains satisfiability and validity of MTL as complete problems. We also prove that the fragments of MTL with bounded modal depth are complete for the levels of the elementary hierarchy (with polynomially many alternations). The respective hardness results hold for both strict or lax semantics of the modal operators and the splitting disjunction, and also over the class of reflexive and transitive frames.

Parameterized complexity of weighted team definability

“…For the applications, it is important to understand the complexity theoretic aspects of teambased logics. During the past ten years, the expressivity and complexity theoretic aspects of logics in first-order (also propositional Yang and Väänänen 2017, modal Hella et al 2019, temporal Gutsfeld et al 2022and probabilistic Durand et al 2018 team semantics have been studies extensively (see, e.g., Hannula et al 2018, Lück 2019, Hannula et al 2020, Durand et al 2022). The baseline for these studies is the well-known results stating that the sentences of dependence logic and independence logic are equivalent to existential second-order logic (Grädel and Väänänen 2013), while inclusion logic corresponds to positive greatest fixed point logic and thereby captures P over finite (ordered) structures (Galliani and Hella 2013).…”

A Parameterized View on the Complexity of Dependence Logic

“…Soon after the introduction of dependence logic many other interesting team-based logics and atoms were introduced such as inclusion, exclusion, and independence atoms that are intimately connected to the corresponding inclusion, exclusion, and multivalued dependencies studied in database theory [9,13]. Furthermore, the area has expanded, e.g., to propositional, modal and probabilistic variants (see a selection of works from the literature [14,15,19] and the references therein).…”