Establish a matching lower bound for translations between pure-past and pure-future temporal logics

Determine whether the known 3EXPTIME upper bound for translating between Pure-Past Linear Temporal Logic over finite traces and its pure-future counterpart is tight by establishing a matching lower bound.

Background

The paper notes that Pure-Past Linear Temporal Logic and pure-future Linear Temporal Logic have the same expressive power over finite traces, while translations between them can require triply exponential time. It reports that the known 3EXPTIME upper bound has a matching lower bound that is not known, leaving the exact complexity of the translation problem unresolved.

References

Unfortunately, if we use /, although they have the same expressive power and the same expressive power of in expressing temporal properties on finite traces, moving from one to the other is in 3EXPTIME (matching lower bound is unknown).

— Planning for Temporally Extended Goals in Pure-Past Linear Temporal Logic: A Polynomial Reduction to Standard Planning  (2204.09960 - Giacomo et al., 2022) in Section 6, Discussion, paragraph beginning “cite{zhu2019first}”