Fuzzy bi-Gödel modal logic and its paraconsistent relatives

We present the axiomatisation of the fuzzy bi-Gödel modal logic (formulated in the language containing $\mathrm{△}$ and treating the coimplication as a defined connective) and establish its PSpace-completeness. We also consider its paraconsistent relatives defined on fuzzy frames with two valuations $e_1$ and $e_2$ standing for the support of truth and falsity, respectively, and equipped with \emph{two fuzzy relations} $R^{+}$ and $R^{-}$ used to determine supports of truth and falsity of modal formulas. We establish embeddings of these paraconsistent logics into the fuzzy bi-Gödel modal logic and use them to prove their PSpace-completeness and obtain the characterisation of definable frames.