Most properties are undecidable even in $<mrow><mi\; mathvariant="sans-serif">N</mi><mi\; mathvariant="sans-serif">E</mi><mi\; mathvariant="sans-serif">x</mi><mi\; mathvariant="sans-serif">t</mi></mrow>{\mathsf{G}\mathsf{r}\mathsf{z}}_t$

We investigate decidability of properties in the lattice ${\mathsf{G}\mathsf{r}\mathsf{z}}_t$ of extensions of the Grzegorczyk tense logic ${\mathsf{G}\mathsf{r}\mathsf{z}}_t$ and the lattice ${\mathsf{S}\mathsf{4}}_t$ of reflexive and transitive tense logics, with applications to the lattice $\mathsf{b}\mathsf{i}\mathsf{I}\mathsf{P}\mathsf{C}$ of bi-superintuitionistic logics. We prove that a broad class of properties is undecidable in ${\mathsf{G}\mathsf{r}\mathsf{z}}_t$, including tabularity, Kripke completeness, the finite model property, and decidability, which also yields their undecidability in ${\mathsf{S}\mathsf{4}}_t$. We also construct infinitely many tabular extensions of ${\mathsf{G}\mathsf{r}\mathsf{z}}_t$ (and thus of ${\mathsf{S}\mathsf{4}}_t$) whose coincidence problems are undecidable, while presenting one tabular extension of ${\mathsf{G}\mathsf{r}\mathsf{z}}_t$ and infinitely many ones of ${\mathsf{S}\mathsf{4}}_t$ with a decidable coincidence problem. As a consequence, we obtain that the finite model property and tabularity are undecidable in $\mathsf{b}\mathsf{i}\mathsf{I}\mathsf{P}\mathsf{C}$, and that there are infinitely many tabular extensions of $\mathsf{b}\mathsf{i}\mathsf{I}\mathsf{P}\mathsf{C}$ whose coincidence problems are undecidable. These results clarify some similarities and differences between ${\mathsf{G}\mathsf{r}\mathsf{z}}_t$ and $\mathsf{G}\mathsf{r}\mathsf{z}$, ${\mathsf{S}\mathsf{4}}_t$ and $\mathsf{S}\mathsf{4}$, as well as $\mathsf{b}\mathsf{i}\mathsf{I}\mathsf{P}\mathsf{C}$ and $\mathsf{I}\mathsf{P}\mathsf{C}$. The proofs adapt Chagrov’s method of reducing from an undecidable problem for Minsky machines. We isolate and explicitly formulate the method of good valuations, a recurring technique underlying several proofs in the literature that use large frames, making it available for further applications.