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.

翻译：两个CNF公式若在单位子句传播（UCP）下行为相同，则称为UCP等价。若移除任意子句后所得公式与原公式不UCP等价，则该公式称为UCP不可约。根据已知结果，UCP不可约公式的规模与最小UCP等价公式的规模之比至多为$n^2$，其中$n$为变量数。本文针对对称确定Horn函数给出一个UCP不可约公式的实例，其规模比最小UCP等价公式大$\Omega(n/\ln n)$倍，因此上述比值的通用上界不可能低于该值。