# On Transitive modal many-valued logics

## TL;DR

This paper investigates the computability and expressibility of transitive modal many-valued logics based on valued Kripke frames, revealing undecidability results and comparing transitive and non-transitive cases.

## Contribution

It demonstrates the undecidability of a broad class of transitive modal many-valued logics, including those from MV and Product algebras, and analyzes their properties compared to non-transitive logics.

## Key findings

- Many transitive modal many-valued logics are undecidable.
- Transitive modal Lukasiewicz logic exhibits unique computability properties.
- Adding the Delta operation leads to undecidability of validity and local SAT.

## Abstract

This paper is focused on the study of modal logics defined from valued Kripke frames, and particularly, on computability and expressibility questions of modal logics of transitive Kripke frames evaluated over certain residuated lattices. It is shown that a large family of those logics -- including the ones arising from the standard MV and Product algebras -- yields an undecidable consequence relation. Later on, the behaviour of transitive modal Lukasiewicz logic is compared with that of its non transitive counterpart, exhibiting some particulars concerning computability and equivalence with other logics. We conclude the article by showing the undecidability of the validity and the local SAT questions over transitive models when the Delta operation is added to the logic.