# Completing the Picture: Complexity of Graded Modal Logics with Converse

## TL;DR

This paper classifies the complexity of graded modal logic with converse modalities across various frame classes, revealing increased complexity over Euclidean frames and decidability over transitive frames without graded converse modalities.

## Contribution

It completes the complexity classification for graded modal logic with converse modalities, including Euclidean and transitive frames, and confirms decidability in a specific variation.

## Key findings

- NExpTime-completeness over Euclidean frames
- Decidability over transitive frames without graded converse modalities
- Language with graded converse modalities is undecidable

## Abstract

A complete classification of the complexity of the local and global satisfiability problems for graded modal language over traditional classes of frames have already been established. By "traditional" classes of frames, we mean those characterized by any positive combination of reflexivity, seriality, symmetry, transitivity, and the Euclidean property. In this paper, we fill the gaps remaining in an analogous classification of the graded modal language with graded converse modalities. In particular, we show its NExpTime-completeness over the class of Euclidean frames, demonstrating this way that over this class the considered language is harder than the language without graded modalities or without converse modalities. We also consider its variation disallowing graded converse modalities, but still admitting basic converse modalities. Our most important result for this variation is confirming an earlier conjecture that it is decidable over transitive frames. This contrasts with the undecidability of the language with graded converse modalities.