On CNF formulas irredundant with respect to unit clause propagation

Two CNF formulas are called ucp-equivalent, if they behave in the same way with respect to the unit clause propagation (UCP). A formula is called ucp-irredundant, if removing any clause leads to a formula which is not ucp-equivalent to the original one. As a consequence of known results, the ratio of the size of a ucp-irredundant formula and the size of a smallest ucp-equivalent formula is at most $n^2$, where $n$ is the number of the variables. We demonstrate an example of a ucp-irredundant formula for a symmetric definite Horn function which is larger than a smallest ucp-equivalent formula by a factor $\Omega(n/\ln n)$ and, hence, a general upper bound on the above ratio cannot be smaller than this.

@article{savický2025_2309.01750,
  title={ On CNF formulas irredundant with respect to unit clause propagation },
  author={ Petr Savický },
  journal={arXiv preprint arXiv:2309.01750},
  year={ 2025 }
}