Capturing the polynomial hierarchy by second-order revised Krom logic
Kexu Wang, Shiguang Feng, Xishun Zhao
TL;DR
This work studies second-order revised Krom logic, $SO$-KROM$^{r}$, and an extended variant, $SO$-EKROM, to describe the polynomial hierarchy through descriptive complexity. It shows that on ordered finite structures, the existential fragment $\Sigma^1_1$-KROM$^{r}$ coincides with $\Sigma^1_1$-KROM and captures NL, while on all finite structures, alternation collapses align $\Sigma^1_k$ and $\Pi^1_k$ with $\Sigma^1_{k+1}$-KROM$^{r}$ or $\Pi^1_{k+1}$-KROM$^{r}$ depending on parity, providing an alternative PH characterization. The descriptive power of $SO$-KROM$^{r}$ collapses to its existential fragment on ordered finite structures, with data complexity residing in the polynomial hierarchy; the results connect second-order logics with PH. The extended logic $SO$-EKROM satisfies further collapses, since on ordered finite structures $SO$-EKROM $\equiv \Pi^{1}_{2}$-EKROM $\equiv \Pi^1_1$, and both $SO$-EKROM and $\Pi^{1}_{2}$-EKROM capture $\mathrm{co\text{-}NP}$, offering new tools for diagnosing the boundaries between NL, co-NP, and PH in a purely logical framework.
Abstract
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 $Σ^1_1$-KROM$^r$ equals $Σ^1_1$-KROM, and captures NL. On all finite structures, for $k\geq 1$, we show that $Σ^1_{k}$ equals $Σ^1_{k+1}$-KROM$^r$ if $k$ is even, and $Π^1_{k}$ equals $Π^1_{k+1}$-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 $Π^{1}_{2}$-EKROM and equals $Π^1_1$. Both SO-EKROM and $Π^{1}_{2}$-EKROM capture co-NP on ordered finite structures.