P-versus-NP-hardness classification

Classify finite-on-top-of-temporal constraint satisfaction problems according to whether they belong to P or are NP-hard.

Background

The paper classifies these CSPs with respect to definability in FPC and FP*, but not with respect to the classical P-versus-NP-hardness boundary. The authors identify this as a challenging direction because the class properly generalizes the setting of the Bulatov–Zhuk dichotomy theorem and combines quotient-clone behavior with temporal CSP algorithms. They specifically note that it is already difficult when there are two temporal copies.

References

Classify the CSPs for finite-on-top-of-temporal templates with respect to containment in P and NP-hardness. This is challenging, because it is a proper generalisation of the Bulatov-Zhuk dichotomy result; in fact, already the case $k=2$ is challenging, because it requires to deal with combinations of quotient clones that cannot be solved by Datalog with algorithms for temporal CSPs that cannot be expressed in FPC; it is in general unclear which tractable templates can arise in this setting.

— Finite-on-Top-of-Temporal CSPs in Fixed-Point Logic  (2610.18287 - Bodirsky et al., 13 Oct 2026) in Section 6, Conclusion and Open Problems, first bullet