  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.
