Uniformity of the merge-width model-checking algorithm

Construct a single uniform first-order model-checking algorithm for bounded merge-width that avoids the current non-uniform dependence on the quantifier rank and the merge-width bound.

Background

The model-checking algorithm is non-uniform in the quantifier rank and width parameters: for each fixed pair of parameters, a separate algorithm or lookup-table construction is used. The paper conjectures that this dependence can be replaced by one uniform algorithm.

References

Moreover, while our algorithm depends non-uniformly on q and w, we conjecture that a single uniform algorithm is possible.

— Merge-width and First-Order Model Checking  (2502.18065 - Dreier et al., 25 Feb 2025) in Section 1, paragraph “Main result”