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.

@article{paperbot2204,
  title = {Capturing the polynomial hierarchy by second-order revised Krom logic},
  author = {Kexu Wang and Shiguang Feng and Xishun Zhao},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 19, Issue 3},
  year = {2023},
  doi = {10.46298/lmcs-19(3:6)2023}
}