Canonical Models and the Complexity of Modal Team Logic

TL;DR

This paper introduces canonical models for modal team logic (MTL), reducing satisfiability to model checking, and classifies its complexity as non-elementary, with fragments having varying complexity levels.

Contribution

It develops a canonical model construction for MTL, enabling complexity classification and demonstrating the optimality of this approach within the logic.

Findings

Satisfiability and validity of MTL are complete for the non-elementary class TOWER(poly).

Fragments with bounded modal depth are complete for levels of the elementary hierarchy.

The approach applies to both strict and lax semantics, over various frame classes.

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 non-elementary 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.