Exact reverse-mathematical strength of LTL VEL

Determine the exact reverse-mathematical strength of the Valuation Extension Lemma for Linear Temporal Logic (LTL), which lies between \(\mathrm{ACA}^{\prime}_0\) and \(\mathrm{ACA}_0\), and establish whether it is equivalent to \(\mathrm{ACA}_0\) over \(\mathrm{RCA}_0\).

Background

The paper proves VEL for LTL in ACA0′\mathrm{ACA}^{\prime}_0 and shows that it implies ACA0\mathrm{ACA}_0, but does not close the gap between these systems. Computability considerations suggest that ACA0\mathrm{ACA}_0 may be the exact strength: the unique valuation extending a computable atomic assignment is computable from the double jump and therefore cannot compute the third jump. The authors identify the need to avoid circularity in the standard automata-theoretic model-checking argument.

References

As for VEL for $\mathbf{LTL}$, its reverse-mathematical strength also lies between $\mathrm{ACA}{\prime}_0$ and $\mathrm{ACA}_0$, but its exact position remains open.

— Frame definability in second-order arithmetic  (2608.22822 - Takeda, 24 Aug 2026) in Section “Conclusion and Future Research”

Can we prove in $\mathrm{ACA}0$ that the assertion ``$\mathcal{G}{\varphi} \text{ accepts } \alpha_0$'' satisfies the conditions for an $\mathbf{LTL}$-valuation?

— Frame definability in second-order arithmetic  (2608.22822 - Takeda, 24 Aug 2026) in Question environment in Section “Conclusion and Future Research”