We study the expressive power and complexity of second-order revised Krom logic (SO-KROM$^{r}$). On ordered finite structures, we show that its existential fragment $\Sigma^1_1$-KROM$^r$ equals $\Sigma^1_1$-KROM, and captures NL. On all finite structures, for $k\geq 1$, we show that $\Sigma^1_{k}$ equals $\Sigma^1_{k+1}$-KROM$^r$ if $k$ is even, and $\Pi^1_{k}$ equals $\Pi^1_{k+1}$-KROM$^r$ if $k$ is odd. The result gives an alternative logic to capture the polynomial hierarchy. We also introduce an extended version of second-order Krom logic (SO-EKROM). On ordered finite structures, we prove that SO-EKROM collapses to $\Pi^{1}_{2}$-EKROM and equals $\Pi^1_1$. Both SO-EKROM and $\Pi^{1}_{2}$-EKROM capture co-NP on ordered finite structures.

翻译：本文研究二阶修正Krom逻辑（SO-KROM$^{r}$）的表达能力与计算复杂度。在有序有限结构上，我们证明其存在性片段$\Sigma^1_1$-KROM$^r$等价于$\Sigma^1_1$-KROM，且可刻画NL。在所有有限结构上，对于$k\geq 1$，我们证明：当$k$为偶数时，$\Sigma^1_{k}$等于$\Sigma^1_{k+1}$-KROM$^r$；当$k$为奇数时，$\Pi^1_{k}$等于$\Pi^1_{k+1}$-KROM$^r$。该结果为多项式层次结构提供了另一种逻辑刻画。我们还引入了二阶Krom逻辑的扩展版本（SO-EKROM）。在有序有限结构上，我们证明SO-EKROM可归约为$\Pi^{1}_{2}$-EKROM且等价于$\Pi^1_1$。SO-EKROM与$\Pi^{1}_{2}$-EKROM均可刻画有序有限结构上的co-NP。