Polynomial-time globally compatible strengthening of the weak CTF predicate

Determine whether the weak predicate \(\mathrm{satisfies\_ctf}\) can be strengthened to enforce global compatibility between overlapping windows while retaining polynomial-time decidability for sliding-window CNF formulas.

Background

The weak predicate $\mathrm{satisfies\_ctf}$ checks each 3-bit window independently and therefore can accept assignments whose adjacent windows disagree on their overlapping variables. The paper gives a formal counterexample demonstrating that weak predicate satisfaction does not imply the existence of a globally compatible path.

For the verified sliding-window fragment, the strong predicate supplies the needed compatibility condition, but the authors leave open whether such a strengthening can be achieved while preserving polynomial-time decidability in the relevant setting.