Paraconsistent Gödel modal logic on bi-relational frames

We further develop the paraconsistent Gödel modal logic. In this paper, we consider its version endowed with Kripke semantics on $[0,1]$-valued frames with two fuzzy relations $R^{+}$ and $R^{-}$ (degrees of trust in assertions and denials) and two valuations $v_1$ and $v_2$ (support of truth and support of falsity) linked with a De Morgan negation $\mathrm{\neg }$. We demonstrate that it \emph{does not} extend Gödel modal logic and that $\mathrm{\square }$ and $\mathrm{◊}$ are not interdefinable. We also show that several important classes of frames are $\text{birelKGsquare}$ definable (in particular, crisp, mono-relational, and finitely branching). For $\text{birelKGsquare}$ over finitely branching frames, we create a sound and complete constraint tableaux calculus and a decision procedure based upon it. Using the decision procedure we show that $\text{birelKGsquare}$ satisfiability and validity are in PSPACE.